Mercie Particulièrement

Franck Varenne

Philippe Her

Ariel Lindner

Claire Ribrault

David Tareste

Annemiek Cornelissen

Stéphane Douady

Sophie Mazan D'avoir

Monique Bioemergences

Amparo Yannick

Julien Du

Clovis Adil

Marlène, Maud Catherine Laurence

Mazyiar Julie Pierre

Samuel, Salma, Wandé Jean-Baptiste Alexandre

Jehanne, Clément Amis Des Marsouins

We propose in this thesis to characterize variability quantitatively at various scales during embryogenesis. We use a combination of mathematical models and experimental results

In the first part, we use a small cohort of digital sea urchin embryos to construct a prototypical representation of the cell lineage, which relates individual cell features with Il serait surprenant qu'une thèse sur l'aléatoire ne comporte pas une composante de hasard, celle-ci a bénéficié de nombreuses rencontres plus ou moins fortuites. Ce sont toutes ces personnes que je souhaite remercier ici. Mes remerciements vont en premier lieu à mes directeurs de thèse Giuseppe Longo et Nadine Peyriéras. Je les remercie de m'avoir soutenu tout au long de ces années de thèse. L'originalité de leurs travaux, leur ouverture d'esprit, leur énergie ont nourri ma curiosité scientifique et m'ont emmené vers des domaines nouveaux pour moi en biologie, en mathématiques, en philosophie.. Je pense en particulier à toutes les rencontres et discussions passionnantes ayant eu lieu au cours des réunions CIM organisées par Giuseppe. Je pense aussi à l'exploration permanente de Nadine sa capacité à s'enthousiasmer et sa profonde réflexion. Je les remercie de m'avoir offert autant de liberté. Je souhaite remercier plusieurs chercheurs qui ont nourri mon travail au fil des directions où mes recherches m'ont mené. Paul Bourgine pour ses intuitions, son enthousiasme et sa curiosité insatiable, mon travail sur les oursins lui doit beaucoup. Gunnar Carlsson pour sa disponibilité, sa vivacité et pour m'avoir ouvert de nombreuses portes en mathématiques. Monica Nicolau pour m'avoir permis de passer plusieurs mois à l'université de Stanford. René Doursat pour sa patience face à mes notations parfois hasardeuses! Ses idées en modélisation des systèmes vivants et sa grande clarté et ouverture d'esprit sont très appréciables.

embryo-level dynamics. This multi-level data-driven probabilistic model relies on symmetries of the embryo and known cell types, which provide a generic coarse-grained level of observation for distributions of individual cell features. The prototype is defined as the centroid of the cohort in the corresponding statistical manifold. Among several results, we show that intra-individual variability is involved in the reproducibility of the developmental process.

In the second part, we consider the mechanisms sources of variability during development and their relations to evolution. Building on experimental results showing variable phenotypic expression and incomplete penetrance in a zebrafish mutant line, we propose a clarification of the various levels of biological variability using a formal analogy with quantum mechanics mathematical framework. Surprisingly, we find a formal analogy between quantum entanglement and Mendel's idealized scheme of inheritance.

In the third part, we study biological organization and its relations to developmental paths. By adapting the tools of algebraic topology, we compute invariants of the network of cellular contacts extracted from confocal microscopy images of epithelia from different species and genetic backgrounds. In particular, we show the influence of individual histories on the spatial distribution of cells in epithelial tissues.

iii

Résumé

Nous proposons dans cette thèse de caractériser quantitativement la variabilité à différentes échelles au cours de l'embryogenèse. Pour ce faire, nous utilisons une combinaison de modèles mathématiques et de résultats expérimentaux.

Dans la première partie, nous utilisons une petite cohorte d'oursins digitaux pour construire une représentation prototypique du lignage cellulaire, reliant les caractéristiques des cellules individuelles avec les dynamiques à l'échelle de l'embryon tout entier. Ce modèle probabiliste multi-niveau et empirique repose sur les symétries des embryons et sur les identités cellulaires; cela permet d'identifier un niveau de granularité générique pour observer les distributions de caractéristiques cellulaires individuelles. Le prototype est défini comme le barycentre de la cohorte dans la variété statistique correspondante. Parmi plusieurs résultats, nous montrons que la variabilité intra-individuelle est impliquée dans la reproductibilité du développement embryonnaire.

Dans la seconde partie, nous considérons les mécanismes sources de variabilité au cours du développement et leurs relations à l'évolution. En nous appuyant sur des résultats expérimentaux montrant une pénétrance incomplète et une expressivité variable de phénotype dans une lignée mutante du poisson zèbre, nous proposons une clarification des différents niveaux de variabilité biologique reposant sur une analogie formelle avec le cadre mathématique de la mécanique quantique. Nous trouvons notamment une analogie formelle entre l'intrication quantique et le schéma Mendélien de transmission héréditaire.

Dans la troisième partie, nous étudions l'organisation biologique et ses relations aux trajectoires développementales. En adaptant les outils de la topologie algébrique, nous caractérisons des invariants du réseaux de contacts cellulaires extrait d'images de microscopie confocale d'épithéliums de différentes espèces et de différents fonds génétiques.

En particulier, nous montrons l'influence des histoires individuelles sur la distribution spatiales des cellules dans un tissu épithélial. v

General introduction

In the introduction of the Origin of species [START_REF] Darwin | On the origins of species by means of natural selection[END_REF], Charles Darwin refers to the "Mystery of mysteries" for the question of how species evolve and replace each other. Almost two centuries later, this question remains of great importance and even if some major advances have been made in the understanding of the mechanisms underlying evolution, the principle of descent with modification, which constitutes together with the principle of natural selection the grounds of Darwinian evolution, lacks a full comprehension. As a contribution to this problem, this work adresses the question of variation during embryogenesis and development.

Multicellular organisms are the result of a morphogenetic process involving multiple levels of organization, from molecules to cells to tissues to organs, interacting in a complex manner. This complexity is witnessed by causal processes that can be bottom-up, for example from the molecular level to the cellular level through gene expression, and top-down, for example from the tissue level to the cellular level through mechanical constraints. Moreover, an organism builds and maintains itself, it is self-organized. This idea of a self-organizing nature of living organisms can be traced back to Immanuel Kant's Critique of Judgment [116]. He considered organisms as entities where "every part is thought as owing its presence to the agency of all the remaining parts, and also as existing for the sake of the others and of the whole". Self-organization is at the very basis of complex systems science [START_REF] Stuart | The origins of order: Self-organization and selection in evolution[END_REF] and requires specific methodological approaches for its understanding.

The development of an organism is a dynamical phenomenon where any event occurring at a given scale at a certain time is involved in the subsequent stages of development at all level of organizations. Therefore any variation occurring in this process has consequences on other parts of the organisms at later stages. This integrated nature of the organism and its relations to variation had already been noticed by Charles Darwin who designated it as correlated variations in the fifth chapter of the Origin of species: Correlated Variations -I mean by this expression that the whole organization is 1 so tied together during its growth and development, that when slight variations in any one part occur, and are accumulated through natural selection, other parts become modified.

The question remains as to how these correlated variations occur and shape the exploration and generation of diversity.

Understanding the diversity of forms in the living begins with the question of the sources of morphological differences between individual organisms. Usually, differences between individual organisms are attributed to genetic differences and variation in environmental conditions. Recent experimental results suggest that other phenomena have to be accounted for when considering sources of diversity. One striking example is the work of Raj et al. in 2010 [START_REF] Raj | Variability in gene expression underlies incomplete penetrance[END_REF], where clonal organisms of a mutant strain of the worm Caenorhabditis Elegans are grown in homogeneous environment and result in variable phenotypes.

In this example the phenotypic variation is ascribed to stochastic gene expression. The concept of stochastic gene expression [START_REF] Michael B Elowitz | Stochastic gene expression in a single cell[END_REF] covers the processes involved in the variability of quantity of the protein expressed for a given gene. Many other mechanisms at various levels of organization are able to generate diversity; they will be described in more depth in the course of this dissertation. This raises the question of how to integrate this variety of mechanisms generating variability when considering organisms as a whole.

Recent progresses in biological imaging technology and other quantitative techniques have allowed to reconsider many biological phenomena. The observation of stochastic gene expression in single cells [START_REF] Michael B Elowitz | Stochastic gene expression in a single cell[END_REF] is a major example of the breakthroughs made possible with technological developments. In the field of developmental biology, the development of in toto and in vivo microscopy technology such as 2-photon microscopy, or single planar illumination microscopy, have opened new perspectives on the study of embryogenesis. In particular the complete digital reconstruction of the cell lineage during the first few hours of the zebrafish development has been made possible by joint innovations in imaging techniques and image processing [START_REF] Olivier | Cell lineage reconstruction of early zebrafish embryos using label-free nonlinear microscopy[END_REF]. This kind of phenomenological reconstruction provides information on developing organisms which had never been observed in a quantitative way before, leading to a reinterpretation of many of the processes. In addition to technical challenges, the very large datasets generated require new analytic methodologies to extract significant information [START_REF] Sean | Imaging in systems biology[END_REF]. In the meantime, a reinterpretation of the relations between data and models is needed, new paradigms such as data-driven models and hypothesis-driven models have emerged during the last decade [START_REF] Douglas | Here is the evidence, now what is the hypothesis? the complementary roles of inductive and hypothesis-driven science in the post-genomic era[END_REF].

These large sources of quantitative data at various scale obtained in living multicel-lular organisms shed a new light on processes occurring during development. Phenomena that have been mostly qualitatively and verbally described can now be characterized quantitatively in order to understand underlying principles [START_REF] Varenne | La reconstruction phénoménologique par simulation : vers une épaisseur du simulat[END_REF]. However, we largely lack concepts, methods and tools to use this data for deciphering biological complexity.

On a theoretical and epistemological level, biological organisms are multi-scale objects involving several levels of organization that are usually described with heterogeneous theoretical frameworks [START_REF] Buiatti | Randomness and multilevel interactions in biology[END_REF]. A central characteristic of biological objects is their historical nature, they are the result of both ontogenetic and phylogenetic trajectories, defining them as historical entities; they may not be reproducible identically. Indeed, phylogenetic trajectories are the result of an interaction and a co-constitution of organisms and their environments involving single events and small numbers [START_REF] Gould | Wonderful Life: The Burgess Shale and the Nature of History[END_REF], [START_REF] Longo | No entailing laws, but enablement in the evolution of the biosphere[END_REF], similarly ontogenetic trajectories can be considered as a sequence of symmetry breakings involving contingent events [START_REF] Longo | From physics to biology by extending criticality and symmetry breakings[END_REF], [START_REF] Bailly | Mathematics and the natural sciences: the physical singularity of life[END_REF]. This historicity and variability of biological objects is at the center of Darwinian theory of evolution and should be at the ground of our understanding of biological organization. This is an epistemological specificity of biology contrasting with the ahistoricity of most physical objects. On this aspect the following quote of the physicist Max Delbrück is particularly illuminating:

The complex accomplishment of any one living cell is part and parcel of the [fact] that any one cell represents more an historical than a physical event. These complex things do not rise every day by spontaneous generation from the nonliving matter -if they did, they would really be reproducible and timeless phenomena, comparable to the crystallization of a solution, and would belong to the subject matter of physics proper. No, any living cell carries with it the experiences of a billion years of experimentation by its ancestors. You cannot expect to explain so wise an old bird in a few simple words 1 Following this line of thought, we will consistently discuss the relevance of the mathematical concepts used and transferred from one discipline to the other by considering their epistemological justification.

Given the systemic nature of organisms and the variety of sources of variation, as well as the historicity of organisms, it is natural to ask for the characteristics of variability during embryogenesis. What makes individual singular? How variation in individuals influence variability at the population level? What are the relations between variability and probabil-CONTENTS ities? And variability and randomness? How to measure and quantify variation at several scales during development? Given the differences between individual specimen, can we define a normal prototypical development for a species?

We support the following thesis: variations at all scales during development shape the exploration of diversity of forms, specifically designed mathematical tools are required to characterize and quantify these variations.

Characterizing normality In the first part, we will consider the concept of normality and reproducibility of development. The question will be to measure how similar and how different are embryonic developments of the same species in normal conditions. To this aim, we will use a data set of five digitally reconstructed sea urchin Paracentrotus lividus embryos at the single cell resolution. The specimens of this small cohort are developing from the 32 cells stage (4 hours post fertilization) to hatching (around 500 cells, 10 hours post fertilization). Using the BioEmergences worlkflow [START_REF] Faure | An algorithmic workflow for the automated processing of 3d+time microscopy images of developing organisms and the reconstruction of their cell lineage[END_REF] their complete 3D+time cell lineage and the shape of each cell was obtained from 2-photon microscopy acquisitions.

Using this large data set, we will investigate the different levels of variability. This variability is first witnessed within an organism among the cells and underlies cell differentiation, it is the intra-individual variability. This variability is then witnessed between specimen among the cohort and is the result of individual specific histories, it is the inter-individual variability. While intra-individual variability can be well characterized by considering distributions of cell features, inter-individual variability requires to establish generic comparable features allowing to place individual specimen on the same footing without averaging out significant intra-individual variability.

Similar dynamical patterns are found at the level of the whole embryo and in each morphogenetic field. These patterns concern the evolution of the number of cells, the cell surface and volume and the number of neighbors. A preliminary linear spatio-temporal scaling is however necessary to make them fully comparable among specimen of the cohort. The value of the coefficients defining this scaling are a first step to characterize interindividual variability.

Variability in the distribution of individual cell features and symmetries of the embryos prevent to identify unique cells from one organism to the other. To compare embryos and characterize intra-individual variability, we define a generic coarse-grained level of observation based on inherent symmetries and similar fates (Mesomere, Macromeres and Large and Small Micromeres). The corresponding group of cells are clustered according to this identity and generation. They form the unite of our study of variability in the sea urchin development. Cells are considered exchangeable within these groups: the description of the distribution of cell features is not affected by permutation of the cells. Therefore we rely on the de Finetti's theorem to guarantee the use of empirical probability distribution as a good descriptor of individual cell feature distribution within a group of cells [START_REF] David | Exchangeability and related topics[END_REF], [START_REF] Yuan | Probability theory: independence, interchangeability, martingales[END_REF].

Approximated parametric probability distributions within groups of cells and approximated independency of the cell features distributions between groups of cells is the basis for a multi-level data-driven probabilistic model of the cell lineage reproducing embryolevel dynamics for each specimen from measures of individual cell features. The same structure relating individual cell features and embryo level dynamics is found in each specimen. Parameters of the probability laws estimated empirically define uniquely individual specimen. This structure serves as the basis for a prototypical model of development among the cohort representing invariant features while preserving intra-individual variability [START_REF] Rizzi | Predicting sea urchin's normal development from a small cohort of digital embryos[END_REF], [START_REF] Villoutreix | A probabilistic modeling of the cell lineage highlights interindividual variability in paracentrotus lividus early development[END_REF]. We use the framework of Information Geometry [START_REF] Amari | Methods of information geometry[END_REF], [START_REF] Nielsen | Sided and symmetrized bregman centroids. Information Theory[END_REF] to define the prototype as the Kullback-Leibler centroid in the associated statistical manifold enabling to obtain a unique set of parameters representing the cohort.

In addition to representing quantitatively every measured cellular processes, this model addresses formally and quantitatively the question of regulation in development and the concept of morphogenetic field. Moreover it can be used to characterize a notion of structural stability and irreversibility. Eventually, this multi-level data-driven probabilistic model will be employed as a basis for an hypothesis driven biomechanical model using the Mecagen modeling platform. This model enables a spatial embedding of the prototypical cell lineage, the values of biomechanical parameters are obtained by parameter exploration strategies and matching with empirical data leading to a phenotypic phase diagram [START_REF] Delile | Computational modeling and simulation of animal early embryogenesis with the mecagen platform[END_REF], [START_REF] Rizzi | Predicting sea urchin's normal development from a small cohort of digital embryos[END_REF].

Overall, this work provides a picture of development where the reproducibility of embryo level dynamics emerges from variability at the individual cell level. This picture contrasts with the traditional view of the development as a finely tuned process. Chapter 1 describes this work at the broadest level, its reading doesn't require a strong mathematical background. Chapter 2 provides a more mathematical description of the multi-level datadriven probabilistic model of the lineage and the construction of the prototype. Chapter 3 discusses some perspectives raised by the model such as a notion of irreversibility of desorganization implied by the structure found in the embryos and a formal characterization of robustness of development.

CONTENTS

Characterizing diversity

In the second part, we will study the emergence of diversity in evolution and how it is shaped by development. The question consists in how to relate variations at the individual specimen level to variation at the population level and how to characterize the influence of mutant development for diversification.

The variety of mechanisms at the origin of variation stimulate this question. In chapter 4, we try to characterize variability in biology with respect to formalization in mathematics and physics. The review of the main mechanisms sources of variation shows an heterogeneity of processes, from genetic mutations [START_REF] Salvador | Mutations of bacteria from virus sensitivity to virus resistance[END_REF] to stochastic gene expression [START_REF] Michael B Elowitz | Stochastic gene expression in a single cell[END_REF], through epigenetic changes [START_REF] Vg Allfrey | Acetylation and methylation of histones and their possible role in the regulation of rna synthesis[END_REF]. The timescales at which they operate and their various mechanisms of inheritability are obstacles for their integration, although they all contribute to the generation of diversity. We then turn to mathematics and physics to explore the characteristics of the frameworks used for the formalization of uncertainty. Probability theory offers a framework to handle events in a context of uncertainty but doesn't provide any definition of randomness itself. This framework, following Kolmogorov's axiomatisation, rely on the boolean algebra of sets to construct the set of events (σ-algebra) which is a model that may show some limitations, for example in the case of quantum mechanics. Chaos theory and ergodic theory are two frameworks in classical physics which provide two different characterizations of randomness [START_REF] Bailly | Randomness and determinism in the interplay between the continuum and the discrete[END_REF]. In both cases, it is an epistemic concept. Quantum mechanics on the other hand rely on an objective use of randomness, as an intrinsic component of the object under study. The Hilbert space structure used in quantum mechanics probabilities enables operations that were not possible in the Kolmogorovian framework such as a tensor product between space of possible corresponding to different observables [START_REF] Bengtsson | Geometry of quantum states: an introduction to quantum entanglement[END_REF]. Finally, we will argue that, given the high complexity of biological organisms and the heterogeneous nature of the mechanisms source of variability, an alternative approach consists in using models to explore the repertoire of possibles although they will always provide an incomplete description of the space of possible.

The experimental study of the squint mutant line of the zebrafish Danio rerio will then bring an example of variable phenotypic expressivity and incomplete penetrance [START_REF] Pei | Identification of common and unique modifiers of zebrafish midline bifurcation and cyclopia[END_REF].

The results of a quantitative assessement of the distribution of phenotypes in the progeny of homozygote mutants shows a discrete list of phenotypes that may be incomplete and in unpredictable proportions. This experimental study is then the basis for an analogy with the mathematical framework used in quantum mechanics, since the traditional Kolmogorovian framework shows some limitations when the description of the space of pos-sible is incomplete. The analogy rely on the possibility to use the Hilbert space vector space structure and the possibility to perform tensor products between spaces of possible. By differentiating between uncertainty at the level of the observables, at the level of possible phenotypes and the probability of obtaining these observables, we will clarify some aspects of biological variation [START_REF] Villoutreix | Biological diversity and quantum formalism: Entanglement of genotype-phenotype relations[END_REF]. In particular, this framework can handle the emergence of new observable and the emergence of new phenotypic value for a given observable. We obtain a formal analogy with entanglement in Mendel's idealized scheme of inheritance that is interpreted as a trace of biological organization.

Finally, we will explore the concept of an ontogenetic tree, which is an attempt to organize divergence patterns between developments among mutants of the same species [START_REF] Ho | Rational taxonomy and the natural system[END_REF]. This question relates to the concept of developmental constraints or canalization, and is a first step toward an understanding of how variability in development shapes the space of possible forms [START_REF] Alberch | Ontogenesis and morphological diversification[END_REF]. We show with a data set containing a large number of description of mutant developments [START_REF] Bradford | Zfin: enhancements and updates to the zebrafish model organism database[END_REF] that the zebrafish's pharyngula stage is actually the stage from which the highest number of mutants begin to diverge. However, possible biases in the data set are discussed.

Quantifying shape

In the third part, we focus on epithelial organization and on the traces left by individual histories. The relative universality of this structure allow to study variability among several species and within mutant developments. Epithelial morphogenesis results from a sequence of events involving cell proliferation, cell movement, cell death and cell extrusion, leading consequently to a complex landscape. Using a data set of epithelial images, we will consider the quantitative characterization of the network of cellular contacts [START_REF] Luis M Escudero | Epithelial organisation revealed by a network of cellular contacts[END_REF]. The network of cellular contacts gives a good estimation of epithelial organization, however traditional tools from complex networks theories show some limitations for its study since this network is highly constrainted by the underlying topology of the tissue.

A similar problem arise when trying to characterize the structure of the cosmic web. It is an historical structure shaped by random events at various scales which forms a complex landscape. Using a discrete analog of the approach developed in [START_REF] Sousbie | The persistent cosmic web and its filamentary structure-i. theory and implementation[END_REF], [START_REF] Carlsson | Topological pattern recognition for point cloud data[END_REF] we propose to compute topological invariants of the network of cellular contacts by adapting the framework of persistent homology. To this aim, we begin by considering a discrete version of the level set functions on the network using the number of neighbors as the parameter.

The number of neighbors is used as a measure of "density" in the network. Computation CONTENTS of the sub and super level sets with varying threshold for the parameter number of neighbor generates two filtrations, i.e. two sequences of nested subspaces, from a network. These sequences unfold the structure of the network. For each value of the parameter, Betti numbers can be computed on the corresponding subspaces of the filtrations. These Betti numbers roughly measure the i-th dimensional holes in the considered space and are topological invariants. The sequence of Betti numbers values obtained when exploring the range of value of the parameter enable to compute a persistence diagram which is a topological signature of the network. This signature is automatically extracted from confocal images of epithelia from Drosophila and Chick embryos. It is used to compare and classify tissues.

To make sense of these topological signatures, we introduce a model of random triangulated surface. This model has the same number of neighbors distribution as the empirical network, nodes are linked randomly to form a triangulated surface. The topological signature is computed on this model of random triangulated surface and enable to estimate the distance of empirical networks to a random spatial distribution of cells. Significant deviation from the random model is obtained for the different cases studied; non randomness of the spatial distribution is a measure of the influence of the morphogenetic process for the construction of the network of cellular contacts. It shows that events occurring at the individual cell level have an influence on the global morphology of the tissue.

The role of Myosin II, an element of the cellular cytoskeleton, is estimated by comparing the results of the method on knocked out mutants and wild type embryos. Overall we show that the level of organization of the cell cannot be uncoupled from the tissue organization.

Part I

Characterizing normality

Introduction

In this part, we investigate the question of the reproducibility of development in normal conditions. This question is crucial for developmental biology. Embryogenesis has been believed to be highly reproducible and finely tuned leading to the metaphor of the "execution of a program" encoded in the genes [START_REF] Jacob | La logique du vivant: une histoire de l'hérédité[END_REF], [START_REF] Keller | Le siècle du gène[END_REF].

In the last decade, the reproducibility of the development of the Drosophila early development has been quantified at several level of observation and has been shown to be highly reproducible in particular at early stages. At the genetic level the quantification of the morphogen concentration profiles such as Bicoid indicates a reproducibility with 10% variation, interpreted as a precise control over absolute concentrations and responses to small concentration differences. At the cellular level, the cell membrane lengthening during cellularization of the Drosophila is a highly reproducible process ( [START_REF] Lim | Kinetics of gene derepression by erk signaling[END_REF], [START_REF] Figard | The plasma membrane flattens out to fuel cell-surface growth during drosophila cellularization[END_REF], [START_REF] Julien O Dubuis | Accurate measurements of dynamics and reproducibility in small genetic networks[END_REF]) allowing to calibrate measurements very accurately. At a more macroscopic level the fly wing vein patterns are highly reproducible, the precision is in the range of a single cell width [START_REF] Abouchar | Precision and reproducibility of macroscopic developmental patterns[END_REF].

For the sea urchin, Eric Davidson proposed the idea of an "invariant cell lineage" [START_REF] Eric | Lineage-specific gene expression and the regulative capacities of the sea urchin embryo: a proposed mechanism[END_REF].

In the seminal article by Sultan and Horvitz the lineage of the nematode sea elegant Caernohabidtis elegans has been shown to be reproducible at the single cell resolution [START_REF] John | Post-embryonic cell lineages of the nematode, caenorhabditis elegans[END_REF].

However, until now, characterization of the development of the sea urchin at single cell resolution has not been performed. The work presented in this part concerns the quantitative study of sea urchin development at the single cell resolution.

Quantitative and integrative approaches in developmental biology

Quantitative approaches in developmental biology require to integrate dynamics occurring at several scales. Several steps are needed to understand the mechanisms occur-11 ring at these various scales, from data acquisition, to reconstruction, to modeling. The BioEmergences platform2 has been pioneer in this field. The approach conceptualized by Nadine Peyriéras and Paul Bourgine is summarized in the epistemological triangle as represented on figure 1. The first step consists in producing an accurate in toto and in vivo image acquisition of the embryo development. This step involve the development of efficient imaging technologies. The most popular technologies are two-photon microscopy [START_REF] Denk | Two-photon laser scanning fluorescence microscopy[END_REF], [START_REF] Helmchen | Deep tissue two-photon microscopy[END_REF] and light sheet microscopy [START_REF] Thai V Truong | Deep and fast live imaging with two-photon scanned light-sheet microscopy[END_REF], [START_REF] Pantazis | Advances in whole-embryo imaging: a quantitative transition is underway[END_REF]. The concept of two photon microscopy consists in exciting fluorophores which can be of different kinds and are distributed at relevant places in the embryo using two photons reaching the desired energy level when they converge and superpose at the focal point. This technique enables deep imaging in living tissues. The concept of the light sheet microscopy consists in using a thin sheet of light instead of light focused on one point, this technique enables quicker image acquisition but require to rotate the sample. Fluorophores consist most of the time of fluorescent proteins which are translated from RNA injected at a precocious stage. These fluorophores commonnly fuse to the membrane of the cell or to the nucleus. These techniques enable to measure developing embryo without perturbing their normal development. The data sets obtained consist in 3-dimensional images at regular consecutive time steps [START_REF] Sean | Imaging in systems biology[END_REF], therefore 4D raw data as indicated on the bottom left of the epistemological triangle on figure 1.

The second step consists in extracting biologically relevant information from the 4D raw data sets. This step is done through image processing. In particular, the position of the cell nuclei at each time step is a relevant information to reconstruct the spatiotemporal cell lineage [START_REF] Bao | Automated cell lineage tracing in caenorhabditis elegans[END_REF], [START_REF] Olivier | Cell lineage reconstruction of early zebrafish embryos using label-free nonlinear microscopy[END_REF]. In many studies the shape of individual cells carries useful information [START_REF] Rizzi | Towards 3d in silico modeling of the sea urchin embryonic development[END_REF]. From this phenomenological reconstruction, useful features can be extracted, such as the position and velocity field of cells [START_REF] Mcmahon | Dynamic analyses of drosophila gastrulation provide insights into collective cell migration[END_REF], other cinematic description of morphogenesis can be found in [START_REF] Lombardot | Description intégrative des cinématiques de la structure embryonnaire: morphogenèse précoce du poisson zébré[END_REF]. The accuracy of the phenomenological reconstruction can be assessed through visual inspection by experts using visual platform such as the MoveIt software [START_REF] Faure | An algorithmic workflow for the automated processing of 3d+time microscopy images of developing organisms and the reconstruction of their cell lineage[END_REF].

Once the phenomenological reconstruction has been established for one or several embryos, it is useful to propose theoretical hypotheses to interpret the data. These hypotheses can be extracted from the data [START_REF] Ishihara | Bayesian inference of force dynamics during morphogenesis[END_REF], [START_REF] Ishihara | Comparative study of non-invasive force and stress inference methods in tissue[END_REF], [START_REF] Bosveld | Mechanical control of morphogenesis by fat/dachsous/four-jointed planar cell polarity pathway[END_REF]. Or they can be brought from external knowledge, for example physical hypotheses or previous experimental results [START_REF] Delile | Computational modeling and simulation of animal early embryogenesis with the mecagen platform[END_REF], [START_REF] Xiong | Specified neural progenitors sort to form sharp domains after noisy shh signaling[END_REF], [START_REF] Xiong | Interplay of cell shape and division orientation promotes robust morphogenesis of developing epithelia[END_REF], [START_REF] Heisenberg | Forces in tissue morphogenesis and patterning[END_REF], [START_REF] Behrndt | Forces driving epithelial spreading in zebrafish gastrulation[END_REF], [START_REF] Forgacs | Biological physics of the developing embryo[END_REF], or information theoretic approaches such as the concept of complexity [START_REF] Geard | Developmental motifs reveal complex structure in cell lineages[END_REF], [START_REF] Ricardo Br Azevedo | The simplicity of metazoan cell lineages[END_REF].

The main question that remains is how to compare theoretical assumptions with empirical data, given the variability at all scales observed in developing embryos.

Multi-level approach for the study of the sea urchin early embryogenesis

The sea urchin Paracentrotus lividus is a model organism widely used in developmental biology. Sea urchin's embryos have many advantages that led to this status of model organism. The eggs are easily accessible, fertilization can be controlled, embryo are transparent and develop quickly, enabling their observation with microscopy techniques [START_REF] Fabrèges | Phenotypic variation and resilience in sea urchin morphogenesis[END_REF].

The use of the sea urchin for embryology can be traced back to the XIXth century, beginning in 1847 with three publications documenting fertilization, "Sur le développement des oursins" (On the development of sea urchins) by Adolphe Dufossé, "Auszug aus einem Berichte des Akameikers v. Baër, aus Triest" (Excerpt from a report by the University Graduate von Baër in Trieste), by Karl Ernst von Baër and "Observations sur le mécanisme et les phénomènes qui accompagnent la formation de l'embryon chez l'oursin comestible" (Observations on the mechanism and phenomena accompanying the formation of the embryo of the edible sea urchin), by Alphonse Derbès [START_REF] Briggs | In the beginning... animal fertilization and sea urchin development[END_REF]. These studies were followed by a famous work by Oskar Hertwig on sperm and egg pronuclear fusion in 1876. In 1891, Hans Driesch used the sea urchin to perform experiments on development, showing that a complete embryo could develop from extracted cells refuting preformation and mosaic theories.

During the XXth century the sea urchin has been the basis for major discoveries. Tim Hunt and collaborators discovered the role of cyclins in the sea urchin development, which are key regulators of the cell cycle [START_REF] Evans | Cyclin: a protein specified by maternal mrna in sea urchin eggs that is destroyed at each cleavage division[END_REF]. The sea urchin development has also been used as a basis for the comprehensive study of the gene regulatory network [START_REF] Eric H Davidson | A genomic regulatory network for development[END_REF]. The most recent development of this approach can be found on the biotapestry website 3 . [START_REF] Sf | Developmental Biology[END_REF] At the end of the cleavage period, which correspond to proliferation without cell death and without many cell movements within the blastula, the cells begin to undergo complex morphogenetic movements. Figure 3 shows a schematic drawing of this gastrulation movements which will later give rise to a free swimming pluteus larva.

The study of the sea urchin gene regulatory network (GRN) underlying development lead to the establishment of maps of interaction such as the one represented on figure 4 for Mesomere cells during the period from 6 to 17 hours. Obtaining such a map require to test each gene individually. On the scheme, each gene is represented with its name (e.g. Nodal or Lefty), and the links between the genes represent interactions such as induction or inhibition. We can observe the modularity of this network which depend only on signals from

Mesomere cells, except for some maternal inputs and the expression of the gene Wnt8 in the cells Veg2 (a subpopulation of the Macromeres lineage). The establishment of the gene regulatory network has been described in [START_REF] Eric H Davidson | A genomic regulatory network for development[END_REF] and contains more than 40 genes. The dynamics in space and time of this gene regulatory network has been studied as a boolean computational model [START_REF] Peter | Predictive computation of genomic logic processing functions in embryonic development[END_REF], it is argued in this article that the data underlying this gene regulatory network contains sufficient information to explain the complex developmental process of gene expression. The main limitation of this approach is that the spatial resolution is coarser than the individual cell level where interactions between genes take place. This limitation can be overcome with analysis at the single cell resolution such as the one presented in this part of the dissertation.

In the following we propose to investigate quantitatively the morphogenesis of the sea urchin Paracentrotus lividus from the 32-cells stage to more than 400 cells, by observing the phenomenology of individual cells, shape and proliferation dynamics.

Overview of the part

The main object of this part of the dissertation is the comparative study of a small cohort of digitally reconstructed sea urchin embryos from in toto and in vivo 2-photon microscopy imaging. Using the BioEmergences workflow, the complete reconstruction of the cell lineage and individual cell shape is obtained for 5 wild type embryos. The question which is raised by this data set is how to characterize intra-individual variability which underlies cell differentiation and inter-individual variation associated to individual developmental histories. Given the multi-scale nature of the embryonic development, the answer to this question depends on the level of observation chosen. Indeed, features at the individual cell level can be very variable from one individual to the other, whereas embryo-level dynamics appear highly reproducible.

The original approach that we develop to integrate these various levels of observation is summarized on figure 5. Based on the image acquisition and the phenomenological reconstruction, a set of features is obtained for each individual cells. These features are the length of the cell cycle, the division, the mean volume, the mean surface area, the mother/daughter volume or surface area ratio. Since we have access to the complete lineage, it is possible to follow their evolution through the genealogy. At a more macroscopic level, we can look at embryo-level dynamics such as the evolution of the number of cells through time, the evolution of the total cell volume, the evolution of the total cell surface area.

To integrate the various levels of observation, we defined an intermediate coarse-grained level of observation between individual cell features and embryo-level dynamics by clustering individual cell features into identifiable groups of cells. These groups are defined by common cell identity (Mesomeres, Macromeres, Large and Small Micromeres) and common generation (number of cycles undergone since fertilization), hence identifiable in each specimen of the cohort. This generic coarse grained level of observation is the basis unite of our comparative study. It allows to compare individual characteristics without averaging out intra-individual variability.

The link between individual cell features is characterized with a data-driven multilevel probabilistic model relying on this intermediate level of observation. Individual cell features distribution can indeed be described and approximated through parametrized probability distributions. These probability distributions are then combined using the branching structure of the cell lineage. The parameters governing the probability distributions provide a signature of each individual specimen in the cohort. They are distributed in a statistical manifold where inter-individual comparison can be performed.

Introduction

The question of finding the time and the locus for the apparition of differences between individuals has irrigated the science of embryology ( [START_REF] Jacob | La logique du vivant: une histoire de l'hérédité[END_REF]). Large genetic screens have sought to find genetic determinants of these differences ( [START_REF] Nüsslein | Mutations affecting segment number and polarity in drosophila[END_REF], [START_REF] Haffter | The identification of genes with unique and essential functions in the development of the zebrafish, danio rerio[END_REF]). The construction of the genetic regulatory network of the sea urchin should reveal the dynamics of 1. This chapter is an early version of a paper involving Barbara Rizzi, Louise Duloquin, Julien Delile, René Doursat, Paul Bourgine and Nadine Peyriéas -This study has been presented under various forms at the conferences "The developmental biology of the sea urchin" XXI and XXII, at Woods Hole, MA, USA (2012 -2014) and at the Gordon Research Conference "Stochastic physics in biology" at Ventura Beach, CA, USA (2015) 21 these determinants during development ( [START_REF] Eric H Davidson | A genomic regulatory network for development[END_REF]). However, recent results have shown that the linear relation between genetic regulatory network and phenotype can be complicated by stochastic ( [START_REF] Raj | Variability in gene expression underlies incomplete penetrance[END_REF]) effects. On the other hand, physical determination and constraints at the scale of the tissue or of the whole embryo canalize the space of possible shapes ( [START_REF] Roux | Über die Entwicklungsmechanik der Organismen[END_REF], [START_REF] He | Apical constriction drives tissue-scale hydrodynamic flow to mediate cell elongation[END_REF], [START_REF] Fleury | Clarifying tetrapod embryogenesis by a dorso-ventral analysis of the tissue flows during early stages of chicken development[END_REF]). Relating these two approaches, genetic determinants and physical constraints at the scale of the tissue, requires to understand the relations between individual cell phenomenology and transformations at the whole embryo level. Phenomenological reconstruction of live embryo development ( [START_REF] Olivier | Cell lineage reconstruction of early zebrafish embryos using label-free nonlinear microscopy[END_REF]) generates data that allow to investigate quantitatively such questions. We propose to use the full digital reconstruction of a small cohort of developing sea urchin to unfold the relations between cell, tissue and whole embryo dynamics.

The sea urchin has been studied as an animal model since over a century ([66], [START_REF] David R Mcclay | Evolutionary crossroads in developmental biology: sea urchins[END_REF]).

The study of early embryogenesis provides insight in differentiation of cells in the different layers ( [START_REF] Eric H Davidson | A genomic regulatory network for development[END_REF], [START_REF] Eric | Gene activity in early development[END_REF]). The description of the morphogenetic changes associated to these differentiation processes are lacking a quantitative description. In particular the quantitative study of the sea urchin blastula development is a good model to tackle this problem because the morphogenetic changes are undergone smoothly by the embryo allowing live imaging over a long period of cleavage.

The full digital reconstruction of live specimen reveals simultaneously quantitative features at the individual cell level and at the scale of the whole embryo. The development is orchestrated by changes in size, shape, number, position and gene expression of cells. The question remains of how these quantities are related to each other and in which way do the macroscopic dynamics of the development emerge from the micro characteristics of the cells. Are the individual cell features aggregated together in a unique precise way that would create the precise patterns that we observe, suggesting a precise developmental mechanism regulating the development, or the global dynamics emerge from loose relations between the cells, favoring robust emerging process.

A small cohort of digital sea urchin throughout their cleavage period

The quantitative comparison of the cell lineage and cell behaviors was achieved through the full reconstruction of digital specimens from 2-photon microscopy imaging of live embryos. Nuclear and membrane staining obtained by 1-cell stage injection ( [START_REF] Faure | An algorithmic workflow for the automated processing of 3d+time microscopy images of developing organisms and the reconstruction of their cell lineage[END_REF]) of syn- and the segmentation of cell shapes (Figure 1.1 -3rd line). The visualization interface Mov-IT ( [START_REF] Olivier | Cell lineage reconstruction of early zebrafish embryos using label-free nonlinear microscopy[END_REF]) was used to validate and correct the cell tracking and manually label cells according to their distribution in known populations ( [START_REF] Lynne | Animal-vegetal axis patterning mechanisms in the early sea urchin embryo[END_REF]).

Feature Extraction and Measuring

To define normal sea urchin's development we used 5 developing specimens imaged in similar experimental condition with the same set up. Acquisition lasted from 2 to 8 consecutive hours with a time resolution of 2 to 4 min, beginning 4 to 5 hours post fertilization (32 cells -Figure 1 To compare quantitatively cell features among specimens of the cohort, it is necessary to find generic coarse-grained levels of description because symmetries of the embryo prevent to identify individual cells (Chapter 2). Relying on exchangeability of the cell fate at this period for cells belonging to the same population (Mesomeres, Macromeres, Large micromeres, Small micromeres), cells were clustered in groups of cells sharing identity and generation. These groups of cells forms the basic unit for the comparison and modeling in this study of the sea urchin development.

Emergence of embryo-level dynamics from individual cell features

To relate the macroscopic dynamics with the individual cell features, we propose a data-driven multi-level probabilistic model of the cell lineage which rely on its branching nature. After each cell cycle, a cell divide into two. The features of the cells are chosen randomly using the corresponding probability distributions defined for each groups of cells independently of any genealogical relationships (Chapter 2 -section 2.5). The macroscopic dynamics observed in each embryo and within each morphogenetic fields are ac- The variation in the total cellular volume and surface area result from variable mean characteristics with invariant cell dynamics (Chapter 2). These results suggest that the high reproducibility of embryo-level dynamics emerges from individually loosely regulated cell features within a branching structure and population level characteristics.

Each specimen of the cohort is represented by a set of parameters sufficient to reproduce embryo-level dynamics. The cohort is represented as a set of points in the associated statistical manifold and the prototypical representation of the cohort is defined as the centroid of the specimens in this parameter space (Chapter 2 and [START_REF] Amari | Methods of information geometry[END_REF], [START_REF] Nielsen | Sided and symmetrized bregman centroids. Information Theory[END_REF]). Prototypical statistics for the groups of cells are defined using this methodology (Figure 1

Spatial modeling

To understand the relations between the individual cell features and the shape of the embryo, the prototypical model of the cell lineage is embedded in space with a biomechanical model using the MecaGen modeling platform (Figure 1.3 A - [START_REF] Delile | Computational modeling and simulation of animal early embryogenesis with the mecagen platform[END_REF], [START_REF] Delile | Mecagen: A cell-based computational model of embryogenesis coupling mechanical behavior and gene regulation[END_REF]). Each cell is represented by a single cylindrical particle oriented along the apico-basal axis of the epithelium (Figure 2.18). As cells are extremely small and sticky (low Reynolds number, [START_REF] Purcell | Life at low Reynolds number[END_REF]), inertia is neglected in favor of viscosity forces and the cell displacement is caused by their immediate mechanical interactions. Thus, the relation between displacement and the net force applied on cells by their neighbors is ruled by an overdamped equation of motion. At the bastula stage, the increasingly epithelial nature of the cells induces a decomposition of the force exerted between two neighbors into a set of tangential and normal components: the attraction-repulsion force maintains the integrity of the cell volume and controls the stiffness and the adhesion of interaction in the tangential direction, and the planarity conservation force maintains the planarity of the monolayered sea urchin epithelium (Figure 1.3 B,C). In between a pair of neighbor cells, the attraction coefficient varies depending whether the pair belongs to the same subpopulation (homotypic) or not Parameter space exploration determines the parameter sets which govern realistic spatial enfolding of the sea urchin embryo development. Validation uses the sphericity of the global embryo shape, the maintenance of the monolayered epithelium and the similarity of the inter-subpopulation border shapes with those observed in the digital embryos (Chapter 2 section 2.7). The best fitting domain is obtained for low heterotypic adhesion, confirming that clear-cut border can be obtained without the need of biasing the division orientation (Figure 1.3 Db:Mean). Moreover, when the attraction-force intensity is larger than the planarity force intensity, the embryo epithelium agglomerate into 3D aggregate. This transition runs through different phenotypic states: from the highly spherical and planar embryos to the collapsed magma of cells, some starfish shaped embryos may appear in the parameter region where planarity is obtained even in the absence of sphericity (Figure 1.3 -D).

Discussion

In summary, we demonstrate that, despite interindividual variability during cleavage period, it is possible to uncover underlying invariant structures driving macroscopic dynamics and morphological changes. Their reproducibility emerge from loosely regulated cell features along the branching cell lineage. In particular we show that the desynchronization of cell division increases continuously during cleavage, and not as successive period of synchronicity, metasynchronicity and asynchronicity as previously stated ( [START_REF] Sf | Developmental Biology[END_REF]).

And if a pseudo gradient from vegetal to animal has been suspected ( [START_REF] Duncan | The echinoid mitotic gradient: effect of cell size on the micromere cleavage cycle[END_REF]), our study suggests that it results from the characteristics of cell life length variability and does not necessarily need a material support.

By finding a relevant generic coarse-grained level of observation to compare specimen we overcome the problem of intraindividual variability and symmetries that arise from multiscale observations. Modeling of the cell lineage and its 3D biomechanical embedding assure the sufficiency of the characteristics defined at the level of groups of cells to describe accurately the development of one specimen of the cohort, thus suggesting a regulation at the level of populations of cells and not at the level of individual cells. This period of development may not need to require a fine tuned genetic regulation. This sparse description of the development for each specimen lead to the modeling of a prototypical embryonic development which is the centroid of the cohort in the space of models. This prototypical representation defines the normal development with a generality level guaranteed by the number of specimen in the cohort. This framework may form a basis to bridge the gap between experimental biology and theoretical biology.

Chapter 2

Variability in the sea urchin development:

A multi-level data driven probabilistic model Abstract This chapter describes a data-driven multi-level probabilistic underlying the comparative study of a small cohort of digital sea urchin embryos presented in chapter 1.

To relate individual cell features with embryo-level dynamics, it is necessary to define an intermediate generic coarse-grained level of observation. This level enables to characterize

probability laws that are the basis of a comprehensive probabilistic model. Corresponding parameters are distributed in a statistical manifold where each embryo is identified by a small set of points. A prototype is obtained by computing the Kullback-Leibler centroid among specimen of the cohort. This prototype serves as a basis for a spatial embedding through biomechanical modeling with the MecaGen platform. 1

Introduction

The development of the sea urchin blastula from 32 to 540 cells happens through cell proliferation with no cell death. Early embryonic territories and cell morphology allow to categorize cells according to known cell types, namely Mesomeres, Macromeres, Large Micromeres, and Small Micromeres [START_REF] E H Davidson, R A Cameron | Specification of cell fate in the sea urchin embryo: summary and some proposed mechanisms[END_REF] as represented on figure 2.2. Symmetry by rotation along the animal vegetal axis as well as cellular variability prevent to identify individual cells from one specimen to the other.

Using digital reconstructions of in toto and in vivo developing sea urchin embryo, we propose a data-driven multi-level probabilistic model that relates individual cell features with embryo-level dynamics. These embryo level dynamics are the number of cells, the cell volume and the cell surface area. The high reproducibility of these dynamics is observed after performing a linear spatio-temporal rescaling.

We propose to model the cell lineage tree as a binary branching process where the division probability of a cell depends on its age. Each cell of the lineage lives for a random time before giving rise to two new cells. The same process applies to these two new cells. This approach can be compared to a binary Bellman-Harris process, yet identical probability among generations will not be assumed [START_REF] Kimmel | Branching Processes in Biology[END_REF]. Once the cell genealogy has been established, the dynamics of the cellular volume and surface is investigated along the cell lineage. An intermediate level of description is defined by clustering cells by common cell type and cell cycle, allowing interindividual comparison. Life lengths and division times in these groups of cells are found to be well described with gaussian probability distributions, mean volume and surface area and mother/daughter ratio for the volume and the surface area are found to be well described with log-normal distributions. The same multilevel probabilistic model relates distribution of individual cell features in these groups of cells with embryo-level in each specimen of the cohort. The geometry of the space of probability distributions allow to assess the accuracy of our model which is found to be high compared to inter-individual variability. Moreover, this model provides a description of the propagation of intra-individual variability in the cell lineage. Finally, based on this multi-level probabilistic model, we define a unique prototypical representation of this period of development by aggregating individual specimen statistics as Kullback-Leibler centroids in the relevant morphospace. This prototypical representation is used to generate artificial cell lineages which will be the basis of a biomechanical model. Parameters of the biomechanical model are fitted to the empirical data by comparing the simulations with the reconstruction.

Image acquisition and digital reconstruction

This section describes the protocol to obtain images of developing sea urchin embryos, and the image processing algorithms used to reconstruct the cell lineage and the cell shape from these images. This section describes Louise Duloquin's work for the image acquisi-tion and Barbara Rizzi's work for the image processing [START_REF] Faure | An algorithmic workflow for the automated processing of 3d+time microscopy images of developing organisms and the reconstruction of their cell lineage[END_REF].

Image acquisition

To obtain microscopy images of developing sea urchin embryo, it was first necessary to inject oocytes from Paracentrotus lividus with 150 µg/ml H2B-mcherry and 150 µg/ml eGFP-ras synthetic mRNAs [START_REF] Andrew P Mcmahon | Introduction of cloned dna into sea urchin egg cytoplasm: replication and persistence during embryogenesis[END_REF]. Embryos were either maintained between slide and coverslip covered with protamine or embedded in 0.25% low-melting-point agarose sea water at the center of a 3 cm Petri dish. Image acquisition was performed with 2-photon microscopy with simultaneous excitation at two different wavelengths (1030 nm and 980 nm). Acquisition of 5 developing specimen is obtained while maintaining similar experimental conditions. They lasted from 2 to 8 hours with a time resolution of 2 to 4 min, beginning 4 to 5 hours post fertilization.

Image processing

Nucleus center detection The detection of the position of cell nuclei relies on the intensity of the signal emitted by the fluorescent histone fusion protein H2B-mcherry. A difference of gaussian algorithm is applied on the images, resulting in a band pass filter sharpening cell nuclei signal, which enables center detection by maxima identification on the image. Value of the parameters were manually chosen with the MoveIt software superimposing raw data with the detected cell positions.

Cell tracking

The cell tracking consists in the complete spatio-temporal genealogy of the cells, it is obtained by linking cell nuclei from one time frame to the other when they represent the same cell, or two daughters after mitosis. The tracking is obtained in five steps. The first one consists in linking nearest nuclei when this relation is reciprocal (it is not necessarily the case when there is a mitosis). The second step accounts for mitoses by connecting nuclei without predecessor with the nearest nuclei in the previous frame. The third step refine the tracking obtained by assuming that the cells have small displacement and hence don't change of neighborhood quickly. An energy function is minimized by simulated annealing. The fourth step consists in considering cells that do not live long as false positive nuclear centers. The fifth step consists in visual inspection of the lineage obtained with the MoveIt software which superimposes raw data with the reconstructed lineage.

Cell segmentation

The cell segmentation consists in the reconstruction of individual cell shapes. The fluorescent signal associated to the cell membrane is obtained thanks to the GFP-Ras fluorescent protein. The cell membrane shape are reconstructed using a generalized version of subjective surfaces method [START_REF] Zanella | Cells segmentation from 3-d confocal images of early zebrafish embryogenesis[END_REF]. This method enable to reconstruct missing boundaries, relevant in the context of the images presenting a low signal-to-noise ratio and incomplete membrane contours. This method is performed on the images that are first smoothed with a geodesic mean curvature flow filter [START_REF] Krivá | 3d early embryogenesis image filtering by nonlinear partial differential equations[END_REF]. Numerical implementation of both filtering and segmentation algorithms has been performed on finite difference schemes [START_REF] Osher | Fronts propagating with curvature-dependent speed: algorithms based on hamilton-jacobi formulations[END_REF][START_REF] Sarti | Subjective surfaces: A method for completing missing boundaries[END_REF]. Further details about the employed methods can be found in [START_REF] Krivá | 3d early embryogenesis image filtering by nonlinear partial differential equations[END_REF] and [START_REF] Zanella | Cells segmentation from 3-d confocal images of early zebrafish embryogenesis[END_REF].

Using these image acquisition and image processing techniques, we obtained a small cohort of 5 digital embryos with completely validated reconstructed cell lineages and reconstructed individual cell shape.

Multi-level measures and rescaling

The digital reconstruction of a small cohort of developing embryos was analyzed at different levels of observation, from individual cell features to embryo-level dynamics via relevant coarse-grained cell groups. We observed similar patterns of evolution in the measured quantities, such as cell volumes and numbers of neighbors, across all specimens of the cohort. Interindividual variability was captured by a linear temporal rescaling of the cell number charts, and a spatial rescaling of the geometric charts. Spatial symmetries in the embryos defined groups of exchangeable cells that constitute the basis of the probabilistic model.

Individual cell features

The digital reconstruction of entire sea urchin specimens allowed us to extract a variety of features at the level of individual cells as presented. For any cell i , we defined the following quantities (Fig. 2.1):

-its life length: x i , corresponding to the time between two consecutive divisions -a mitosis time: m i , denoting the moment at which the cell i divides into two daughter cells (and ceases to exist as such) -its current volume: v i (t ), with the average volume over its life length (using discrete

v i = 1 x i m i m i -x i v i (t )d t ≃ 1 x i m i -1 t =m i -x i v i (t ) (2.1)
-its current surface area: s i (t ), with the average surface area over its life length:

s i ≃ 1 x i m i -1 t =m i -x i s i (t ) (2.2)
-a generation number: n i , denoting the number of past divisions on cell i 's lineage branch, which includes itself and all its ancestors since fertilization -a cell type: k i , taking one of four possible values: 1 for mesomeres (Mes), 2 for macromeres (Mac), 3 for large micromeres (LMic), and 4 for small micromeres (SMic) [START_REF] E H Davidson, R A Cameron | Specification of cell fate in the sea urchin embryo: summary and some proposed mechanisms[END_REF] (Fig. 2.2)

-its current degree: d i (t ), equal to the number of neighbors of cell i at time t , assuming that the spatial distribution of cells can be represented by an undirected graph of cellular contacts, in which cells correspond to nodes and cell-cell interactions to edges [START_REF] Matthew C Gibson | The emergence of geometric order in proliferating metazoan epithelia[END_REF][START_REF] Luis M Escudero | Epithelial organisation revealed by a network of cellular contacts[END_REF] (Fig. 2.

3).

Each cell i also has a mother identified by j (a shortcut notation for the function j (i ) mapping any cell index to its mother's index), which is used in the definition of two additional features:

-a daughter/mother average volume ratio: a i = v i /v j -a daughter/mother average surface ratio: b i = s i /s j and three relationships:

-the mitosis time of a cell is equal to the sum of its life length and the mitosis time of its mother:

m i = m j + x i
-the generation is incremented by one at each division: n i = n j + 1 -starting from the 32-cell stage, each cell type is conserved across divisions throughout the period considered here (development of the blastula):

k i = k j .

Morphogenetic fields and global state variables

The cell lineage, denoted by L , is the set of all cells that the embryo contained during a given period of time. From there, various subsets were defined:

-the current embryo, the set of all cells alive at time t :

L (t ) = {i ∈ L | m i -x i ≤ t < m i }
morphogenetic fields, the sets of cells of a given type k across all generations:

L k = {i ∈ L | k i = k}
current fields, the morphogenetic fields at time t :

L k (t ) = {i ∈ L (t ) | k i = k}.
At the level of the entire embryo, and at each time step t , the global measures were:

-the number of cells, cardinality of the current embryo:

N (t ) = |L (t )|
-the total cellular volume, sum of all individual cell volumes:

W (t ) = i ∈L (t ) v i (t )
-the total cellular surface area, sum of individual cell surface areas:

Z (t ) = i ∈L (t ) s i (t )
-the total number of contacts, sum of the local degrees:

C (t ) = i ∈L (t ) d i (t )
These quantities were also calculated inside each current morphogenetic field k, replacing L (t ) by L k (t ) in the above definitions and using the notations N k (t ), W k (t ), Z k (t ), and

C k (t ).

Normalized embryo dynamics

The number of cells, total cellular volume, and total cellular surface area were measured empirically in each embryo of the cohort E , indexed by e ∈ [START_REF] Abouchar | Precision and reproducibility of macroscopic developmental patterns[END_REF][START_REF] Alberch | Ontogenesis and morphological diversification[END_REF] and denoted N e (t ),

N k e (t ), and so on. The evolution over time of these variables was plotted concurrently (Fig. 2.4A,D,G). We observed that the patterns of evolution across the various specimens were similar but did not overlap: some were globally faster, some slower; some were larger, other smaller. To filter out this variability and facilitate interindividual comparison, we applied temporal and spatial rescaling functions.

Temporal rescaling

For each observed cell number N e (t ), we considered its inverse function t e (N ) equal to the time at which embryo e contained a given number N of cells (symmetric of Fig. 2.4A). Since N e (t ) is a monotonically increasing function (no cell death during the period considered), so is t e (N ). However, because of the relative synchrony among cell cycles, N e can remain constant for several time steps (plateaus in Fig. 2.4A). Therefore, to obtain a single-valued function, we defined t e (N ) = min{t | N e (t ) = N } over the size interval of e, [N e,min , N e,max ] (if N was not part of the observed discrete values, t e (N ) was interpolated). Then, to minimize the discrepancy between the observed time of an embryo, t e (N ), and the mean time of the five specimens, 〈t 〉(N ), an affine transform was performed on the time axis using a cost function:

F e (α, β) = N e,

Intermediate cell groups

Symmetries in the embryo, such as the rotational symmetry around the animal-vegetal (AV) axis, prevent the identification and matching of individual cells from one specimen to another. Unique identification of cells based on their morphological characteristics cannot be done without ambiguity either. To overcome these issues, a generic coarse-grained level of observation was needed. We chose here to define new subsets of L (adding to the list of Section 2.3.1):

generational cell groups, or simply "cell groups", the subsets of generation-n cells inside morphogenetic fields k:

L n,k = {i ∈ L | n i = n & k i = k}.
Note that L n,k is not the same as L k (t ): the latter is a snapshot of morphogenetic field k at time t and therefore may contain a mix of generations if some cells have divided more frequently than others. Conversely, cells in the former group may have to be taken at different times. These sets offer two useful viewpoints on the embryo, one focusing on its global state, the other on cell statistics. In any case, the greater n and t , the more distant these two sets are likely to be. Since each cell gives rise to two daughter cells at each cycle and there is no cell death, the number of cells of a given type continued doubling, i.e. |L n,k e | = 2 n-6 |L 6,k e | for n ≥ 6 during the period of interest. The six cell features considered here are the life length x i , mitosis time m i , average cell volume v i , average cell surface area s i , and the daughter/mother ratios of average volume a i and surface area b i . The dispersion of these features observed in each group L n,k e of the empirical data is represented by distributions, modeled by sequences of random variables. For example, (x 1 , x 2 , ..., x m ) denotes the sequence of the values that the first feature, life length, takes in the m cells of a given group L n,k e . Each x i is the realization of a random variable X i pertaining to cell i . Cells being indistinguishable within the same group, however, their order in the sequence is irrelevant. This property is referred to as "exchangeability" of the sequence of random variables and is formalized as follows: for any permutation π of the indices [1, m], the joint probability distribution of these variables is the same:

P (X π(1) , X π(2) , ..., X π(m) ) = P (X 1 , X 2 , ..., X m ).
(2.5)

This property leads to de Finetti's theorem, which can be summarized by saying that an infinite sequence of exchangeable random variables is a "mixture" of independent and identically distributed (i.i.d.) sequences [START_REF] David | Exchangeability and related topics[END_REF][START_REF] Yuan | Probability theory: independence, interchangeability, martingales[END_REF]. As for a finite exchangeable sequence, it can also be approximated by a mixture of i.i.d. sequences [START_REF] Diaconis | Finite exchangeable sequences[END_REF]. One consequence of this theorem is that the following empirical measure P m,X g constitutes a summary statistic for the probability density P of the X sequence:

P m,X g (I ) = 1 m m i =1 δ X i (I ), with δ x (I ) = 1 if x ∈ I and δ x (I ) = 0 if x ∉ I , (2.6)
where I is an interval of values or union of intervals (i.e. an element of the "σ-algebra" on R) and δ x is the Dirac measure. If m could increase to ∞, P m,X g would converge asymptotically to the underlying theoretical probability distribution P of the sequence of exchangeable random variables.

The motivation of the probabilistic model presented here and in the next section was to provide an idealized representation of the dynamics of multicellular development relating the distributions observed in each cell group with the global dynamics of the cell lineage. To simplify notations, let us define an embryo-specific cell group index g = (e, n, k) so that L n,k e can be equivalently written L g . With this, our goal was the following:

-link the distribution in each group L g of life lengths: X g ∼ {x i | i ∈ L g } and the distribution of mitosis times: M g ∼ {m i | i ∈ L g } to the evolution of the total number of cells in the embryo N e (t )

-link the distribution in each group L g of average volumes V g and the distribution of daughter/mother volume ratios A g to the evolution of the total cellular volume

W e (t )
-link the distribution in each group L g of average surface areas S g and the distribution of daughter/mother surface ratios B g to the evolution of the total cellular surface area Z e (t ).

Observation and approximation of multi-level statistics

From the empirical distributions of individual cell features, we derived parametrized representations approximating their probability distributions in each cell group. Combining cell features along the cell lineage required considering simple models of inheritance.

As explained below, we calculated correlations based on an assumption of linearity in in-tercellular dependencies and, obtaining low values, concluded that the individual features of a cell were largely independent from its mother's or sister's features. This independence was then taken as a founding hypothesis of our model.

Estimation of cell feature distributions in cell groups

To interpret the empirical feature distributions, capture their evolution and predict the developmental dynamics, it was best to describe them in terms of parametric models. First, however, because of the finite and relatively short duration of the period of observation, some groups L g were incompletely represented in the dataset. Therefore, a distribution of cell features in a group was considered to be significant only if the number of observed cells constituted at least 95% of the full cardinality |L g | calculated above.

Moreover, certain features, such as the volume v i and surface area s i , required the observation of their evolution during whole cell cycles. Incomplete cell cycles were taken into account only if the mean period of observation of individual cell dynamics in a group was at least 20 min. Based on these criteria, we retained 252 distributions of individual cell features across all cell groups L g (corresponding to a rough average of 4 × 3 significant distribution-generation pairs in each one of the 5 × 4 embryo-cell type fields).

Graphical assessment of these histograms led us to categorize them into two major types: normal distributions (i.e. Gaussian curves) for life lengths x i and mitosis times m i ; and log-normal distributions for average cell volumes v i , average cell surface areas s i , daughter/mother volume ratios a i and surface area ratios b i (where a random variable X is said to be log-normally distributed if the random variable Y = log X is normally distributed).

We performed a chi-squared goodness-of-fit test on each distribution to validate our assessment. This test evaluates the proximity of the empirical frequency with the theoretical one, i.e. whether the random variables are i.i.d. under a normal distribution or a log-normal distribution. To this aim, we first calculated the empirical mean and standard deviation of each random variable in each cell group L g using a classical maximum likelihood estimation [START_REF] Casella | Statistical inference[END_REF]. For example, in the case of life lengths X g these two quantities are the expected distribution was given by P m, X g (I h ), where X g is the theoretical normal distribution calculated from ( µ X g , σ X g ). The chi-squared statistic computes the discrepancy between both distributions as follows:

µ X g ≈ µ X g = 1 N g i ∈L g x i (2.7) σ X g ≈ σ X g = 1 N g -1 i ∈L g (x i -µ X g )
χ 2 g = l h=1 (P m,X g (I h ) -P m, X g (I h )) 2 P m, X g (I h ) . ( 2.9)
This statistic was assumed to follow a chi-squared distribution with κ = l -3 degrees of freedom [START_REF] John | Handbook of biological statistics[END_REF] (the number of frequencies, l , reduced by the number of parameters of the fitted distribution, µ and σ, and the constraint h P m,X g (I h ) = 1). This gave the p-value, which is the probability of observing a test statistic at least as extreme, i.e. 1 minus the cumulative distribution function of χ 2 g for κ degrees of freedom. Finally, our Gaussian-fit model was deemed adequate for a given feature in a group L g if its p-value was greater than 0.05 (Fig. 2

.6).

Note that the chi-squared test of goodness-of-fit produces inaccurate results if the size of the sample is too small-i.e., by convention, less than 5 elements in a bin [START_REF] John | Handbook of biological statistics[END_REF]. Reducing the number of frequencies l increases the number of elements in each bin, but also decreases the number of degrees of freedom κ, which must remain greater than 3 for the test to be applicable here. Based on these constraints, 128 distributions out of 252 had to be excluded from the evaluation.

For most of the remaining 124 distributions, the validity of our assumptions about their type (normal or log-normal) could be retained. Only 20 of them had a p-value under 0.05, among which 12 were under 0.01 (Fig. 2 Ideally, the worst-fit cases could be used to subdivide cell groups L g into smaller classes (e.g., oral and aboral cells) reflecting the multimodal aspect of certain distributions, or to adopt different parametric models (e.g., exponential instead of Gaussian curves).

Given the small size of the cohort, however, and the non-reproducibility of these outlier distributions in various specimens, it was difficult to rely on such deviations to define new cell types or new models. Moreover, biological relevance of the inferred cell clusters would have to be validated by correlation with genetic expression in the form of a developmental "atlas" [START_REF] Castro-González | A digital framework to build, visualize and analyze a gene expression atlas with cellular resolution in zebrafish early embryogenesis[END_REF], which is out of the scope of this study. This is why we preferred keeping the simplicity of normal and log-normal approximations as they captured the essential characteristics of statistical distributions (their mean and variance) even in marginal cases. For now, we left out any potential additional information for the sake of understandability and interpretability of the whole dataset. The description of distribution shapes can be refined in future work by using a larger cohort of specimens.

Cell volume and surface area dynamics

The volume and surface area of a cell are not constant during its life length because of various contractions and dilations. To highlight the fluctuations of the cellular geometry around its average, we define two other cell quantities:

-the normalized volume:

ω i (u) = v i (t )/v i -the normalized surface: φ i (u) = s i (t )/s i
where t = m j + ux i (m j is the mother's mitosis time) and u ∈ [0, 1] represents the cell's normalized age, which can be equivalently defined by t = m i -(1-u)x i . Then, the microdynamics of cell geometry can be described by taking the mean of these quantities in each distributions of cell features. The six variables considered here were X g , M g , logV g , log A g , log S g and log B g across the five embryos e, four cell types k, and all generations n within the period during which the cell feature was observable. group L g (Fig. 2.9,2.10):

ω g (u) = 1 N g i ∈L g ω i (u) and φ g (u) = 1 N g i ∈L g φ i (u), (2.10)
making the assumption that the functions ω g and φ g are essentially deterministic, i.e. the variations in volume and surface normalized in time and amplitude are the same for all cells i ∈ L g (hence the notations ω g , φ g instead of µ Ω g , µ Φ g ). Their particularity compared to the other six empirical means calculated previously (equation 2.7) is their dependency on time (normalized to align the signals).

Independence along the lineage

Having defined parametric representations ( µ, σ) for each cell feature distribution within each cell group L g , we wanted to investigate the relationships among these variables.

Since cells are genealogically related, it was natural to hypothesize some degree of cor-

SS res = N i =1 (y i -( λ + µx i )) 2 = N i =1 ǫ 2 i , (2.11)
where ǫ i denotes the residual error terms such that y i = λ + µx i + ǫ i and is assumed to be normally distributed around zero. Then, to characterize the accuracy of this linear estimation, we used the coefficient of determination, R 2 , which measures the percentage of variation of one variable explained linearly by the other [START_REF] Rao | Linear statistical inference and its applications[END_REF][START_REF] Casella | Statistical inference[END_REF]. If we define the total sum of squares by

SS tot = N i =1 (y i -y) 2 with y = 1 N N i =1 y i , (2.12)
where y is the empirical mean of Y , then the percentage of variation that remains unexplained linearly is given by the ratio SS res /SS tot , and the coefficient of determination is equal to its complement:

R 2 = 1 - SS res SS tot . (2.13)
For a "simple" linear regression (i.e. one with a single explanatory variable), it can be shown that the coefficient of determination is equal to the square of the estimated Pearson's coefficient of correlation:

R 2 = ρ 2 X ,Y , where ρ X ,Y = cov(X , Y ) σ X σ Y = µ X Y -µ X µ Y σ X σ Y . (2.14)
The greater R 2 , the more X and Y tend to be linearly dependent, with a binary decision threshold generally set at 0.6. Note that R 2 does not measure nonlinear dependencies, thus can remain low even if X and Y are strongly related through quadratic or logarithmic curves, for example.

Here, this coefficient was applied to six pairs of features calculated among the 252 distributions deemed sufficiently represented in the dataset (Section 2.4.1, Fig. 2.5). In the following, for a group index g = (e, n, k) we use the shorthand notation g ′ = (e, n -1, k) to refer to the previous-generation group in the same embryo.

Daugther's life length vs. mother's mitosis time As above, we assumed a simple model of linear dependency between the life length x i of a cell i ∈ L g and the mitosis time m j of its mother j ∈ L g ′ :

x i = λ + µm j . The coefficients of determination R 2 = ρ 2 X g ,M g ′ were computed between the distributions X g ∼ {x i | i ∈ L g } and M g ′ ∼ {m j | j ∈ L g ′ } at several
generations n. For this calculation, we used the 23 groups L g of Fig. 2.5A in which all cell cycles were completely known, i.e. had observed start and end mitosis times, and their corresponding 23 groups L g ′ . In the end, the values of R 2 that we obtained fell in the range [3e-3, 0.51], which indicated only weak or inexistant linear dependencies.

Sisters' life lengths For each pair of sister cells

i 1 , i 2 ∈ L g , ordered such that x i 1 > x i 2 ,
we also assumed x i 1 = λ + µx i 2 , where the longer life length was included in a set X g ,1 and the shorter one in another set X g ,2 . The coefficient of determination R 2 = ρ 2 X g ,1 ,X g ,2 was then computed in the same 23 groups and the values obtained belonged to the interval [0.095, 0.77], revealing some linear dependency among sisters' life lengths within certain groups. More precisely, R 2 coefficients scored above 0.6 in the following seven groups g = (e, n, k): (4, 7, Mac), (5, 7, Mac), [START_REF] Adcock | The ring of algebraic functions on persistence bar codes[END_REF][START_REF] David | Exchangeability and related topics[END_REF]Mac), [START_REF] Aegerter-Wilmsen | Exploring the effects of mechanical feedback on epithelial topology[END_REF][START_REF] David | Exchangeability and related topics[END_REF]Mac), (1, 7, Mes), (3, 7, Mes) and (3, 9, Mes). In the other 16 groups, there was no clear linear dependency.

Sisters' volume ratios

Concerning the volume and surface area, several models of dependency can be explored. Usually, it is assumed that the volume of a mother is conserved in its two daughters through mitosis, meaning that the average volumes should verify:

v i 1 + v i 2 = v j . Dividing by v j , this is equivalent to a linear relationship a i 1 = 1 -a i 2
between the average daugther/mother volume ratios of the sister cells. To verify if there is such a dependency, we computed the coefficient of determination R 2 = ρ 2 A g ,1 ,A g ,2 as above between pairs of sister cells i 1 , i 2 ∈ L g ordered such that a i 1 > a i 2 . Over the 38 cell groups L g of Fig. 2.5C in which all volumes could be confidently measured, the coefficients of determination belonged to [1.4e-3, 0.88], with 34 values below 0.61 and the other four values in [0.75, 0.88] corresponding to cell groups (3, 7, SMic), (3, 7, Mac), (4, 8, Mac) and

(1, 7, Mes). However, the λ and µ parameters for these groups were found in [-0.45, 0.12] and [0.87, 3.02] respectively, i.e. far from 1 and -1. Altogether, these results largely invalidated the hypothesis of volume conservation across mitosis.

Daughter's volume ratio vs. mother's volume It was also legitimate to ask whether the average daughter/mother volume ratio a i of a cell i ∈ L g and the mean volume v j of its mother j ∈ L g ′ were correlated. However, the coefficients of determination R 2 computed in the same 38 groups as above ranged in [1.9e-4, 0.43], indicating no linear relationship. Sisters' surface ratios; Daughter's surface ratio vs. mother's surface Finally, the same tests were performed for surface area features, yielding similar negative results. All relationships are summarized in Table 2.1 and the histogram of all the coefficients of determination evaluated is shown on Fig. 2.11.

Overall, given the weak values of R 2 obtained for the above six relationships of individual features in the different cell groups (R 2 was greater than 0.6 in only 20 cases out of 198 investigated pairs of distributions), we adopted the viewpoint that the various random variables were independent between and within cell groups. Although some of these correlation values may be ascribed to underlying deterministic factors responsible for cell-tocell variability [START_REF] Sandler | Lineage correlations of single cell division time as a probe of cell-cycle dynamics[END_REF], our goal in the present work is to find parameters that can summarize relationships between random variables, hence the poor statistical significance of the linear regression test led us to a global assumption of independence. The following section examines the relations between probability laws in the different cell groups and how they are combined in the cell lineage.

Multi-level probabilistic model

In this section we use the parametrized description of the distributions of individual cell features obtained for each cell group in the preceding section and their combination in the cell lineage. The individual cell features considered are the life length, the mitosis time, the volume and the surface area. The resulting probabilistic model predicts accurately the final distribution of individual cell features comparatively to inter-individual dif-

daughter i ∈ L g mother j ∈ L g ′ sister i 1 ∈ L g sister i 2 ∈ L g R 2 min R 2 max #groups g such that R 2 > 0.6
x i m j 3e-3 0.51 0 of 23

x i 1 > x i 2 0.

Proliferation dynamics

To model the dynamics and evolution of the cell lineage, the relations between the moment of division and the life length of each cell in the cell lineage have to be established.

X n,k is the random variable describing the cell cycle length (time between two consecutive mitoses) within the group L n,k , and M n,k is the random variable describing the mitosis time in L n,k . Based on these notations, we propose the following model, derived from the analysis of the data:

M n,k = X n,k + M n-1,k (2.15) M n,k ∼ N (µ n,k M , σ n,k M ) (2.16) X n,k ∼ N (µ n,k X , σ n,k X ) (2.17) P (X n,k | M n-1,k ) = P (X n,k ) (2.18)
where N represents a normal distribution of mean µ and variance σ 2 [START_REF] Casella | Statistical inference[END_REF]. Equation 2.18

shows that X n,k and M n-1,k are assumed to be independent from each other, P represents the probability law governing X n,k .

This model is instantiated independently in each branch of the cell lineage with the following relation among mitosis times:

m i = m j + x i (2.19)
where i ∈ L n,k and j ∈ L n-1,k are the indices of a cell and its mother, m i is a realization of M n,k and represents the mitosis time of cell i , x i is a realization of X n,k and represents the life length of cell i , m j is a realization of M n-1,k and represents the mitosis time of cell j .

We can relate the parameters governing the different distributions:

M n,k ∼ N µ n,k M , σ n,k M (2.20) µ n,k M = n r =n 0,k +1 µ r,k X + µ n 0,k ,k M (2.21) (σ n,k M ) 2 = n r =n 0,k +1 (σ r,k X ) 2 + (σ n 0,k ,k M ) 2 (2.22)
where n 0,k indicates the initial cycle at which the cells of type k are found. The variability of mitosis times, represented by the variance of the Gaussian distribution is the sum of the variability of cycle lengths in each cell group along the lineage tree. Similarly, the mean mitosis time is the sum of mean cell cycle lengths over the preceding cell cycles in the lineage.

Proof. Let's first establish a recurrence relation between the random variables:

M n,k = X n,k + M n-1,k = X n,k + X n-1,k + M n-2,k = X n,k + X n-1,k + ... + X n 0,k +1,k + M n 0,k ,k
The observation window of the developing embryo in our data is finite. The observation begins at the 32 cells stage at the earliest, i.e. with cells found at the 6th cell cycle in each of the four subpopulations. The observation ends at the 408 cell stage at the latest, i.e. with cells found at the 10th cell cycle for the Mesomeres and the Macromeres, at the 9th cell cycle for the Large Micromeres and at the 7th cell cycle for the Small Micromeres. Since the recurrence cannot be traced back to the origin, we have to provide a boundary condition for the initial distribution of mitosis time, M n 0,k ,k .

The first distribution of mitosis time M n 0,k ,k is a Gaussian distribution:

M n 0,k ,k ∼ N (µ n 0,k ,k M , σ n 0,k ,k M ) (2.23)
where n 0,k indicates the initial cycle at which the cells of type k are found.

The probability distribution of mitosis time, M n,k , is obtained recursively from the parameters of the cycle length distribution and the fact that the density of two independent random variables is the convolution of their densities. Since all the random variables are independent, we obtain the following relations between the probability densities:

f M n,k = f X n,k * f M n-1,k (2.24) f M n,k = f X n,k * f X n-1,k * f M n-2,k (2.25) f M n,k = f X n,k * f X n-1,k * ... * f X n 0,k +1,k * f M n 0,k ,k (2.26)
where * denotes the convolution product and f RV denotes the probability density of a random variable RV .

In the case of Gaussian distributions, the convolution of two distributions is another Gaussian whose mean and variance are the sums of their respective components. [START_REF] Bromiley | Products and convolutions of gaussian distributions[END_REF].

The probability density of a Gaussian distribution is f

(x | µ, σ) = 1 2πσ 2 . exp(- (x-µ) 2
2.σ 2 ). We obtain for two Gaussian distributions with parameters (µ 1 , σ 1 and µ 2 , σ 2 ):

f (x | µ 1 , σ 1 ) * f (x | µ 2 , σ 2 ) = f (x | µ 1 + µ 2 , σ 2 1 + σ 2 2 ) (2.27) Each distribution of the cell cycle lengths are Gaussian distributions, ∀n, k, f X n,k = f (x | µ n,k X , σ n,k X ) and f M n 0,k ,k = f (x | µ n 0,k ,k M , σ n 0,k ,k M
). Consequently, combining equation 2.26 with equation 2.27, we obtain:

f M n,k (x) = f (x | n r =n 0,k +1 µ r,k X + µ n 0,k ,k M , n r =n 0,k +1 (σ r,k X ) 2 + (σ n 0,k ,k M ) 2 ) (2.28)
This result guarantees that the knowledge of the distributions of cell cycle lengths in each cell group is sufficient to characterize the distributions of mitosis time for each of them.

Cell volume dynamics

The cell volume varies throughout cell proliferation. The variation of volume is decomposed into a stochastic and a deterministic component. The stochastic component concerns the evolution of the mean volume of the cells along the cell lineage and the deterministic component serves at modulating the cell volume around its mean by contraction and expansion throughout the cell cycle.

To model the first component, we use a random variable V n,k describing the mean cell volume in the group L n,k , and another, A n,k , describing the daughter/mother mean volume ratio in the same group. We propose the following model that relate the mean volume and the daughter/mother ratio:

V n,k = A n,k V n-1,k (2.29) V n,k ∼ l nN (µ n,k V , σ n,k V ) (2.30) A n,k ∼ l nN (µ n,k A , σ n,k A ) (2.

31)

P (A n,k | V n-1,k ) = P (A n,k ) (2.32)
where l nN represents the log-normal distribution with parameters µ and σ, A n,k being log-normally distributed is equivalent to log(A n,k ) being normally distributed with mean µ and variance σ 2 [START_REF] Limpert | Log-normal distributions across the sciences: Keys and clues[END_REF][START_REF] Casella | Statistical inference[END_REF]. Equation 2.32 shows that the random variables A n,k and V n-1,k are assumed to be independent, P is the probability law governing A n,k .

This model is instantiated in each branch of the cell lineage with the following relation,

for a cell i ∈ L n,k with mother j ∈ L n-1,k : v i = a i v j (2.33)
where a i represents the ratio of a cell mean volume and its mother mean volume, it is a realization of the random variable A n,k . v j and v i are realizations of V n,k and V n-1,k and represent the mean volume of cell i and j respectively.

The volume of a cell is not constantly equal to its mean value during the cell cycle.

The deterministic function ω n,k to which we refer as the volume micro dynamic model the variation of volume which occurs between two consecutive mitoses. This function is defined for each cell group L n,k and represents the expansion and contraction that a cell volume undergoes during its cell cycle. Its domain is the interval [0, 1] which is the percentage of elapsed cell cycle (the function is equal to 0 outside this interval) and its range is the interval [0.8, 1.3] and it is a normalized function:

1 0 ω n,k = 1. Fig. 2.9 shows the estimated shape of ω n,k for the different cell groups.

Using this function ω n,k and keeping the same notations as in the equation 2.33, the

volume v i is obtained at each time step t for a cell i ∈ L n,k with mother j ∈ L n-1,k : v i (t ) = v i ω n,k (u i (t )) (2.34) = a i v j ω n,k (u i (t )) (2.35)
where the function u transports the time t to values in the interval [0,1] which is the domain of ω n,k and describe the percentage of elapsed life length. u is defined for cell i as

u i (t ) = t -m j m i -m j if m j ≤ t ≤ m i and u i (t ) = 0 otherwise.
The parameters of the various probability laws can be related in the following way:

V n,k ∼ l nN µ n,k V , σ n,k V (2.36) µ n,k V = n r =n 0,k +1 µ r,k A + µ n 0,k ,k V (2.37) (σ n,k V ) 2 = n r =n 0,k +1 (σ r,k A ) 2 + (σ n 0,k ,k V ) 2 (2.38)
where n 0,k indicates the initial cycle at which the cells of type k are found. As for the mitosis times, the variability of the volume (σ n,k V ) 2 is the sum of the variabilities in the daughter/mother coefficients, (σ r,k A ) 2 , along the lineage tree with the initial variability (σ n 0,k ,k

V ) 2 .
Proof. Using the initial distribution and the parameters of the daughter/mother mean volume ratio µ n,k A , σ n,k A at each subsequent cell cycle, the mean volume distribution is calculated with a recurrence relation and the property that the product of two independent random variables log-normally distributed is log-normally distributed.

We have the recurrence:

V n,k = A n,k V n-1,k (2.39) = A n,k A n-1,k V n-2,k (2.40) = A n,k A n-1,k ...A n 0,k +1,k V n 0,k ,k (2.41)
we can take the logarithm:

log(V n,k ) = log(A n,k ) + log(V n-1,k ) (2.42) = log(A n,k ) + log(A n-1,k ) + log(V n-2,k ) (2.43) = log(A n,k ) + log(A n-1,k ) + ... + log(A n 0,k +1,k ) + log(V n 0,k ,k ) (2.44)
The last equation can be re-written:

log(V n,k ) = n r =n 0,k +1 log(A r,k ) + log(V n 0,k ,k ) (2.45) where ∀n, k, log(A n,k ) ∼ N (µ n,k A , σ n,k A ) and ∀n, k, log(V n,k ) ∼ N (µ n,k V , σ n,k V ).
All the random variables are independent. As already stated in equation 2.27, the sum of two independent random variables governed by a Gaussian distribution is a Gaussian

Mean volume

Mean surface area

V n,k ∼ l nN (µ n,k V , σ n,k V ) ←→ S n,k ∼ l nN (µ n,k S , σ n,k S ) Mother/

daughter volume ratio

Mother/daughter surface ratio

A n,k ∼ l nN (µ n,k A , σ n,k A ) ←→ B n,k ∼ l nN (µ n,k B , σ n,k B ) Volume micro dynamic Surface area micro dynamic ω n,k ←→ φ n,k
Table 2.2: Identification between variables describing cell volume and cell surface area in any cell group L n,k distribution whose parameters are the sum of the parameters of the initial distributions.

Therefore we obtain:

log(V n,k ) ∼ N (µ n,k V , σ n,k V ) (2.46) i .e. V n,k ∼ l nN (µ n,k V , σ n,k V ) (2.47) µ n,k V = n r =n 0,k +1 µ r,k A + (µ n 0,k ,k V ) (2.48) (σ n,k V ) 2 = n r =n 0,k +1 (σ r,k A ) 2 + (σ n 0,k ,k V ) 2 (2.49)

Cell surface area dynamics

Cell surface areas and cell volumes are modeled in the same way, identifications between the quantities involved in their model are summarized in the table 2.2:

Model summary

The description of the model is summarized in tables 3.1 and 3.2. Table 3.1 describes the relations between the random variables at the level of the cell groups and table 3.2 describe how the model is instantiated in each branch of the cell lineage tree.

A cell lineage is computed with this model by beginning with the 32 cells stage (16 Mes, 8 Mac, 4 LMic, and 4 SMic), using the fact that each cell divide into two and the relations described in table 3.2 to generate the microscopic features. Relying on the cell lineage we can compute macroscopic dynamics in each morphogenetic field, namely the number of cells N k (t ), the cellular volume W k (t ), and the cellular surface area Z k (t ), as described in section 2.3.1. These macroscopic features are themselves random variables. However, we will not give an analytical expression of their probability distribution since they are difficultly tractable and will be simulated, leading to a numerical representation of their properties.

Parameters are estimated from digital embryos as described in section 2.4.1. We recall that it requires a first spatial and temporal rescaling over the cohort to overcome a first level of inter individual variability (section 2.3.1).

Cell group

L n,k Cardinality |L n,k | = 2 * |L n-1,k | = 2 n-n 0,k |L n 0,k ,k | Division time M n,k = X n,k + M n-1,k = n r =n 0,k +1 X r,k + M n 0,k M n,k ∼ N (µ n,k M , σ n,k M ) X n,k ∼ N (µ n,k X , σ n,k X ) P (X n,k | M n-1,k ) = P (X n,k ) µ n,k M = n r =n 0,k +1 µ r,k X + µ n 0,k ,k M (σ n,k M ) 2 = n r =n 0,k +1 (σ r,k X ) 2 + (σ n 0,k ,k M ) 2 Mean volume V n,k = A n,k V n-1,k = n r =n 0,k +1 A r,k V n 0,k ,k V n,k ∼ l nN (µ n,k V , σ n,k V ) A n,k ∼ l nN (µ n,k A , σ n,k A ) P (A n,k | V n,k ) = P (A n,k ) µ n,k V = n r =n 0,k +1 µ r,k A + µ n 0,k ,k V (σ n,k V ) 2 = n r =n 0,k +1 (σ r,k A ) 2 + (σ n 0,k ,k V ) 2
Mean surface area

S n,k = B n,k S n-1,k = n r =n 0,k +1 B r,k S n 0,k ,k S n,k ∼ l nN (µ n,k S , σ n,k S ) B n,k ∼ l nN (µ n,k B , σ n,k B ) P (B n,k | S n,k ) = P (B n,k ) µ n,k S = n r =n 0,k +1 µ r,k B + µ n 0,k ,k S (σ n,k S ) 2 = n r =n 0,k +1 (σ r,k B ) 2 + (σ n 0,k ,k S ) 2
Table 2.3: Summary of the relations between the different random variables in the modelthe definitions of the different notations can be found in the previous sections 2.5, 2.5, 2.5

Cells j ∈ L n-1,k and i ∈ L n,k
i is one of the two daughters of j Division time

m i = x i + m j x i is a realization of X n,k m j is a realization of M n-1,k Volume v i (t ) = a i .v j .ω n,k (u i (t )) a i is a realization of A n,k v j is a realization of V n-1,k ω n,k is the deterministic volume micro dynamic u i (t ) is the percentage of elapsed life length of cell i Surface area s i (t ) = b i .s j .φ n,k (u i (t )) b i is a realization of B n,k s j is a realization of S n-1,k
φ n,k is the deterministic surface area micro dynamic u i (t ) is the percentage of elapsed life length of cell i Table 2.4: Instantiation of the model independently in any branch of the cell lineage. The stochastic components are highlighted in bold-the definitions of the different notations can be found in the previous sections 2.5, 2.5, 2.5

Model evaluation

To assess the significance of our model, we propose to measure the distance between the empirical distribution (M n,k , V n,k , S n,k ) with the distribution predicted by the model.

For a cell cycle n, the parameters µ n,k and σ n,k of the distribution of individual cell features can be predicted from the values of the parameters governing the distributions of previous cell cycles with the following relationships.

For the moment of division:

M n,k ∼ N µ n,k M , σ n,k M (2.50) µ n,k M = n r =n 0,k +1 µ r,k X + µ n 0,k ,k M (2.51) (σ n,k M ) 2 = n r =n 0,k +1 (σ r,k X ) 2 + (σ n 0,k ,k M ) 2 (2.52)
for the volume:

V n,k ∼ l nN µ n,k V , σ n,k V (2.53) µ n,k V = n r =n 0,k +1 µ r,k A + µ n 0,k ,k V (2.54) (σ n,k V ) 2 = n r =n 0,k +1 (σ r,k A ) 2 + (σ n 0,k ,k V ) 2 (2.55)
and for the surface area:

S n,k ∼ l nN µ n,k S , σ n,k S (2.56) µ n,k S = n r =n 0,k +1 µ r,k B + µ n 0,k ,k S (2.57) (σ n,k S ) 2 = n r =n 0,k +1 (σ r,k B ) 2 + (σ n 0,k ,k S ) 2 (2.58)
Since we measure both the statistics of M n,k , V n,k , S n,k and X n,k , A n,k , B n,k for all cell groups L n,k . We can compare the empirical statistics of M n,k , V n,k , S n,k and the values of the parameters predicted by the model.

A good measure of distance between probability distributions is the symmetrized Kullback-Leibler divergence [START_REF] Amari | Methods of information geometry[END_REF][START_REF] Rao | Information and accuracy attainable in the estimation of statistical parameters[END_REF][START_REF] Nielsen | Sided and symmetrized bregman centroids. Information Theory[END_REF]. The Kullback-Leibler divergence between two probabil-ity distributions, having p and q as probability densities, is defined as:

K L(p||q) = x p(x) log( p(x) q(x) ).d x (2.59)
The Kullback-Leibler divergence between two Gaussian distributions p(x|µ p , σ 2 p ) and p(x|µ q , σ 2 q ) is:

K L(p(x|µ p , σ 2 p )||p(x|µ q , σ 2 q )) = 1 2 .(2. log σ q σ p + σ 2 p σ 2 q + (µ q -µ p ) 2 σ 2 q -1) (2.60)
and can be symmetrized into:

D(p(x|µ p , σ 2 p ), p(x|µ q , σ 2 q )) = 1 2 .K L(p(x|µ p , σ 2 p )||p(x|µ q , σ 2 q )) + 1 2 .K L(p(x|µ q , σ 2 q )||p(x|µ p , σ 2 p )) (2.61) = 1 4 .( σ 2 q σ 2 p + σ 2 p σ 2 q + (µ q -µ p ) 2 ( 1
σ 2 p + 1 σ 2 q ) -2) (2.62)
For each specimen of the cohort we obtain the comparison between the empirical mea-

sures of coefficients M n fin,k ,k , V n fin,k ,k , S n fin,k ,k ,
where n fin,k represents the last cell cycle observed in the window of observation (we write n fin when the cell type is obvious). We can estimate the distance between the model and the data. In the following, the exponent emp stands for measured statistics and the exponent mod corresponds to the parameters predicted by the model.

For the mitosis times:

D(M emp n,k , M mod n,k ) = 1 4 .   (σ n,k M ,mod ) 2 (σ n,k M ,emp ) 2 + (σ n,k M ,emp ) 2 (σ n,k M ,mod ) 2 + (µ n,k M ,mod -µ n,k M ,emp ) 2 ( 1 (σ n,k M ,emp ) 2 + 1 (σ n,k M ,mod ) 2 ) -2   (2.63)
where

µ n,k M ,mod = n r =n 0,k +1 µ r,k X ,emp +µ n 0,k ,k M ,emp and (σ n,k M ,mod ) 2 = n r =n 0,k +1 (σ r,k X ,emp ) 2 +(σ n 0,k ,k M ,emp ) 2 .
For the volume:

D(V emp n,k ,V mod n,k ) = 1 4 .   (σ n,k V,mod ) 2 (σ n,k V,emp ) 2 + (σ n,k V,emp ) 2 (σ n,k V,mod ) 2 + (µ n,k V,mod -µ n,k V,emp ) 2 ( 1 (σ n,k V,emp ) 2 + 1 (σ n,k V,mod ) 2 ) -2   (2.64)
where

µ n,k V,mod = n r =n 0,k +1 µ r,k A,emp + µ n 0,k ,k V,emp and (σ n,k V,mod ) 2 = n r =n 0,k +1 (σ r,k A,emp ) 2 + (σ n 0,k ,k V,emp ) 2 .
And for the surface area:

D(S emp n,k , S mod n,k ) = 1 4 .   (σ n,k S,mod ) 2 (σ n,k S,emp ) 2 + (σ n,k S,emp ) 2 (σ n,k S,mod ) 2 + (µ n,k S,mod -µ n,k S,emp ) 2 ( 1 (σ n,k S,emp ) 2 + 1 (σ n,k S,mod ) 2 ) -2   (2.65) where µ n,k S,mod = n r =n 0,k +1 µ r,k B,emp + µ n 0,k ,k S,emp and (σ n,k S,mod ) 2 = n r =n 0,k +1 (σ r,k B,emp ) 2 + (σ n 0,k ,k S,emp ) 2 .
The measure of distance in itself is not easily interpretable because we don't have a benchmark for it. To assess relevant orders of magnitude, we compute the same distance between the measures obtained in the various embryo of the cohort. For a specimen e of the cohort, the distance between the model and the empirical distribution is computed for the last observable cell cycle n fin,k . This quantity is normalized with the averaged inter individual difference. We evaluate the normalized difference between the model and the data for an embryo e with the function ev defined, for the mitosis times, as:

ev(M , k) = D(M emp n fin ,k,e , M mod n fin ,k,e ) 1 |E |-1 l ∈E ,l =e D(M emp n fin ,k,e , M emp n fin ,k,l ) (2.66)
for the volumes:

ev(V , k) = D(V emp n fin ,k,e ,V mod n fin ,k,e ) 1 |E |-1 l ∈E ,l =e D(V emp n fin ,k,e ,V emp n fin ,k,l ) (2.67)
and for the surface areas:

ev(S, k) = D(S emp n fin ,k,e , S mod n fin ,k,e ) 1 |E |-1 l ∈E ,l =e D(S emp n fin ,k,e , S emp n fin ,k,l ) (2.68)
These measure of differences between the model and the data are computed for each morphogenetic field k, in each embryo such that n fin,k > n 0,k . They are computed for division times, volumes and surface areas. The results are grouped in the histogram on Fig. 2.12. 36 predicted sets of parameters among 48 (75%) show a distance significantly smaller than the reference distance (ev < 0.5), 10 predicted sets of parameters are in the same order of magnitude than the reference distance (0.5 ≤ ev < 1.5), and 2 predicted sets of parameters differ significantly from the empirical data; they correspond to the statistics of volume and surface in the embryo 5 for the Mesomere (k = 1). As a conclusion, except for these two "outlying" distributions, the model predicts accurately the dynamics of the Figure 2.12: Histogram of the score of the model for all morphogenetic field k such that n fin,k > n 0,k in each embryo for division times, volumes and surface areas, using the evaluation functions defined in equations 2.66, 2.67, 2.68 sea urchin development.

Simulation of artificial cell lineages with volume and surface area dynamics

300 realizations of the cell lineage were simulated in order to generate significant model output. Matlab random number generator was used to simulate the realizations of random variables. Macroscopic features were computed and compared to the empirical value obtained for each specimen, as represented on figures 2.13 and 2.14.

Each embryo e of the cohort E has a specific window of observation. This window determines the observed of cell groups L n,k e . n 0,k denotes the initial cell cycle where cells of type k are observable in a significant number and n f i n,k the last cell cycle where cells of type k are observable in a significant number (we write n 0 and n f i n when the cell type k is obvious), as defined in the section about parameter estimation. The algorithm used to generate the cell lineage is a straightforward implementation of the model described in the section 2.5.

The finite range of observation of the cell lineage prevents to have access to the complete cell cycle for the first and last generation, thus the volume (ω n,k ) and surface area (φ n,k ) micro dynamics cannot be estimated for these cycles. The shaded area on the right of the plot corresponding to the cell volume and the cell surface area represents the incomplete cell cycles on figures 2.13 and 2.14.

Prototype

The probabilistic model provides an invariant structure relating individual cell features with embryo level dynamics by using an intermediate coarse-grained level (cell groups clustered by common cell type and generation). The same structure appears in each specimen of the cohort. Sufficient statistics are identified for each cell group, providing a "signature" identifying each specimen of the cohort. To get a unique representation of the sea These aggregated measures associated to the identified invariant structure are meant to provide a prototypical representation of the cohort. The aggregation procedure is based on the geometrical concept of centroid of a set of points which generalizes the notion of average in higher dimensional spaces. The mathematical framework of information geometry is relevant here to deal with spaces of probability distributions needing to be endowed with a relevant notion of distance [START_REF] Amari | Methods of information geometry[END_REF][START_REF] Rao | Information and accuracy attainable in the estimation of statistical parameters[END_REF]. Bregman divergences generalize euclidean distance and unify it to the statistical Kullback-Leibler divergence. In the following we use the Kullback-Leibler divergence as the relevant notion of distance to compare probability distribution.

Kullback-Leibler centroid

We give in this section the main formulas needed to compute the centroid of the cohort of sea urchin. More refined mathematical developments can be found in [START_REF] Nielsen | Sided and symmetrized bregman centroids. Information Theory[END_REF].

The Kullback-Leibler centroid is defined as follow:

c = arg min c∈S 1 n . n i =1 K L(p i ||c) + K L(c||p i ) 2 (2.69)
Therefore combining equation 2.60 and 2.71, we obtain:

c = arg min c∈S 1 n 1 4 n i =1 (2. log σ c σ i + σ 2 i σ 2 c + (µ c -µ i ) 2 σ 2 c -1) + (2. log σ i σ c + σ 2 c σ 2 i + (µ i -µ c ) 2 σ 2 i -1)
which simplifies into

c = arg min c∈S 1 n 1 4 n i =1 σ 2 i σ 2 c + σ 2 c σ 2 i + (µ i -µ c ) 2 .( 1
σ 2 c + 1
σ 2 i ) -2 (2.70)

Prototype as a centroid in the relevant morphospace

The probabilistic model provides an invariant structure relating individual cell features with embryo level dynamics by using an intermediate coarse-grained level (cell groups clustered by common cell type and generation). The same structure appears in each specimen of the cohort. Sufficient statistics are identified for each cell group, providing a "signature" identifying each specimen of the cohort. To get a unique representation of the sea urchin blastula normal development, measures of each specimen have to be aggregated.

These aggregated measures associated to the identified invariant structure are meant to provide a prototypical representation of the cohort. The aggregation procedure is based on the geometrical concept of centroid of a set of points which generalizes the notion of average in higher dimensional spaces. The mathematical framework of information geometry is relevant here to deal with spaces of probability distributions needing to be endowed with a relevant notion of distance [START_REF] Amari | Methods of information geometry[END_REF][START_REF] Rao | Information and accuracy attainable in the estimation of statistical parameters[END_REF]. Bregman divergences generalize euclidean distance and unify it to the statistical Kullback-Leibler divergence. In the following we use the Kullback-Leibler divergence as the relevant notion of distance to compare probability distribution.

Kullback-Leibler centroid

We give in this section the main formulas needed to compute the centroid of the cohort of sea urchin. More refined mathematical developments can be found in [START_REF] Nielsen | Sided and symmetrized bregman centroids. Information Theory[END_REF].

The Kullback-Leibler centroid is defined as follow:

c = arg min c∈S 1 n . n i =1 K L(p i ||c) + K L(c||p i ) 2 (2.71)
Therefore combining equation 2.60 and 2.71, we obtain:

c = arg min c∈S 1 n 1 4 n i =1 (2. log σ c σ i + σ 2 i σ 2 c + (µ c -µ i ) 2 σ 2 c -1)+(2. log σ i σ c + σ 2 c σ 2 i + (µ i -µ c ) 2 σ 2 i -1) (2.72) which simplifies into c = arg min c∈S 1 n 1 4 n i =1 σ 2 i σ 2 c + σ 2 c σ 2 i + (µ i -µ c ) 2 .( 1
σ 2 c + 1
σ 2 i ) -2 (2.73)

Prototype as a centroid in the relevant morphospace

The prototypical representation of the sea urchin blastula normal development is modeled with the same structure as for each individual specimen. This structure is summarized in section 2.5 and relates individual cell features with global dynamics through statistics at the level of groups of cell. Using this multi-level probabilistic model, the prototype is defined as a centroid in the relevant morphospace (the idea is schematized on Fig. 2.15). Because the statistics at the level of cell groups are assumed to be either normal or log-normally distributed, the relevant morphospace consists in the statistical manifold S defined for each random variable by the coordinates system (µ, σ 2 ). Finally, the prototypical volume and surface area microdynamics ω(t ) and φ(t ) which are deterministic functions are obtained for each cell group L g by a simple average over the cohort. They are represented in bold on the figures 2.9 and 2.10.

(µ p , σ p ) = arg min (µ,σ)∈R 2 + 1 n 1 4 e∈E σ 2 e σ 2 + σ 2 σ 2 e + (µ e -µ) 2 .( 1
σ 2 + 1 σ 2 e ) -2 (

Simulation

300 realizations of prototypical cell lineages are produced and the macroscopic features are compared with the empirical values as shown on Fig. 2.17.

Artificial cell lineages decorated with cellular volume and cellular surface can be provided for the prototypical model in the same way as for each embryo. We used as input

coefficients: µ X ,p , σ X ,p , µ A,p , σ A,p , µ ω , µ B,p , σ B,p , µ φ , n 0 = 6 (the initial stage is the 32 cell stage) and µ n 0 ,k M ,p , σ n 0 ,k M ,p , µ n 0 ,k V,p , σ n 0 ,k V,p , µ n 0 ,k S,p , σ n 0 ,k S,p .
The simulation ends at the 408 cell stage with n f i n,1 = n f i n,2 = 10 for the Mesomeres and the Macromeres, n f i n,3 = 9 for the Large Micromeres and n f i n,4 = 7 for the Small Micromeres. Section 2.5.1 defines the notation and the procedure of estimation of their value.

Biomechanical model description

The prototypical model of the cell lineage, with cell volume and cell surface area dynamics, is used as a basis for a biomechanical modeling of the sea urchin early embryogenesis. This model is obtained with MecaGen modeling platform corresponding to Julien Delile's work under the supervision of René Doursat and Nadine Peyriéras ( [START_REF] Delile | Computational modeling and simulation of animal early embryogenesis with the mecagen platform[END_REF], [START_REF] Delile | Mecagen: A cell-based computational model of embryogenesis coupling mechanical behavior and gene regulation[END_REF]). This model describes an organism as a set of interacting particles, representing the cells. Cells are small and "sticky", which is a main property of their physical behaviors.

The consequence is the disappearance of inertial forces [START_REF] Purcell | Life at low Reynolds number[END_REF] and low value of Reynolds number. In this situation, applied forces produce a velocity and not an acceleration. The displacement is proportional to the instantaneous force and inversely proportional to a damping coefficient λ. For a multicellular system, the motion of each cell i with an interacting neighborhood N i of cells j is governed by the equation: where F i j is the interaction force exerted by cell j over cell i .

λ i v i = j ∈N i F i j (2.75)
Each particle correspond to a cell, each of them having a certain shape. This shape is modeled as a cylinder as shown on figure 2.18. Cell shape is defined by a lateral radius R, a half-height R ⊥ and a normal vector U . The cylinder axis U is oriented orthogonally to the epithelial surface. Throughout this study, we assume that the depth of the tissue remain constant and identical for each cell as the embryo develops itself, thus R ⊥ i = R ⊥ 0 for each cell. And therefore, the relation between the tangential radius of each cell i and its current volume V i (t ) reads:

R i (t ) = V i (t ) 2πR ⊥ 0 (2.76)
The motion of each cell is determined by its interactions with its neighborhood. The establishment of this neighborhood is performed in two steps, first a metrical assessment of the neighborhood and then a topological criteria to filter results of the first step. This two-step procedure, which is described in ( [START_REF] Delile | Computational modeling and simulation of animal early embryogenesis with the mecagen platform[END_REF], [START_REF] Delile | Mecagen: A cell-based computational model of embryogenesis coupling mechanical behavior and gene regulation[END_REF]) enable to compute for each cell i its interacting neighborhood N i .

Once the topological list of neighbors is determined, the surface orientation of the monolayered sea urchin can be expressed at each cell location by averaging the outward normal vectors of the n triangles formed by n topological neighbors (Fig 2 .19).

The interaction force F i j exerted by cell j over cell i leads the swarm of cells toward an -an attraction-repulsion interaction force F ∥ i j , which maintains the integrity of the cell's volume and controls both the stiffness and the adhesion of the interaction in the direction of the surface plane; this force can be modulated via the stiffness and adhesion coefficients, and -a planarity conservation interaction force F ⊥ i j , which maintains the planarity of a monolayered epithelium; this force can be modulated via a planar rigidity coefficient.

Moreover, the damping coefficient λ i , which plays here a role somewhat equivalent to the mass m i in Newton's Second Law, is proportional to the surface of the cell:

λ i = λ 0 S i with S i = 2πR i (R i + 2R ⊥ i )
as the cell shape is considered cylindrical. The equation of motion used throughout the study will be:

v i = 1 λ 0 S i j ∈N i F ∥ i j + F ⊥ i j (2.77)
The attraction-repulsion interaction force F ∥ i j is a spring-like force derived from an elastic potential. Its expression is given by the following system of equations

F ∥ i j =            -w rep (r i j -r max i j ) 2 (r i j -r eq i j )
. u i j if r i j < r eq i j -w adh (r i j -r max i j ) 2 (r i j -r eq i j ). u i j if r i j ≥ r eq i j and r i j < r max i j

0 if r i j ≥ r max i j (2.78)
where w rep and w adh are a repulsion and an adhesion coefficient, r eq i j is an equilibrium distance between two cells corresponding to the densest packing in the 2D plane, r max The planarity conservation interaction force ensures that the spatial configuration of the swarm of cells remain planar during the simulation. Between two neighboring cells, i and j , this planar rigidity coefficient is proportional to the planar rigidity coefficient k r i g , and to the dot product between the neighborhood vector ( X j -X j ) ( X j and X j are the positions of i and j ) and the normalized sum of both neighbor normal vectors n i j =

U i + U j U i + U j
. The force is oriented in the direction of n i j and the force complete expression reads: The equilibrium distance is r eq = r eq i j = c eq (R i + R j with R i = 0.1 and R j = 0.15. The solid line is the attraction/repulsion potential F ∥ i j with w adh = w rep = 500. The dashed line is a force derived from the Morse potential, its equation reads: 2Dke -k(r -r eq ) -2Dke -2k(r -r eq ) with D = 0.265 and k = 10. The dot-dashed line is force derived from the Lennard-Jones potential, its equation reads:

-24ǫ σ (2( r σ ) -13 -( r σ ) -7
) with ǫ = 0.117 and σ = 2 -1/6 r eq . The force parameters were selected in order that the equilibrium distance and the maximum value of the three curves match approximately on the plot. Figure by Julien Delile

F ⊥ i j = k rig ( X j -X i ). n i j n i j
The notations are described on figure 2.21. Similarly, cell volumes are computed with the prototypical probabilistic model. As mentionned in section before, the apico-basal radius of a daughter cell remains equal to the one of its mother

R ⊥ m = R ⊥ d1 = R ⊥ d2 = R ⊥ 0 .
The daughter cell lateral radii are deduced from their respective volume.

Comparison to experimental data

This section describes the metric used to compare the simulation of the biomechanical model embedding the prototypical probabilistic model in space. The metric enables to compare the simulation to the empirical data-set with varying values of the free parameters such as the adhesion coefficient w adh . This approach validates the hypotheses underlying the model. Three metrics based on three different aspects of the topology of the embryo are designed (D n for the number of neighbors, D c for the number of neighbors variation, and D b for the inter-population borders). In addition, we also calculate four objective functions characterizing some properties of simulated embryos only (C s evaluates the cohesion of the tissue, S s and N s the sphericity of the simulated embryo shapes, and P s the planarity of the epithelium). The evaluation of these comparisons is performed in section 2.7.4.

In the following, we will associate the letter s with the spatial simulations and e with the digital reconstructions.

Metrics

Comparison protocol To compare a spatially simulated embryo with the cohort, we used the three topological features: 1) the distribution of the number of neighbors 2) the distribution of the rate of neighborhood change and 3) the distribution of the neighbors shared at the border.

Topological features

The topology of an embryo is well described by the network of cellcell contacts which can be used as an estimator of the shape of the embryo. At time t, the vertices of the network are the nuclei of the cells, denoted by L (t ) (with the same notation as in section 2.3.1). The edges of the network are the junctions between cells, denoted by

E (t ).
The neighborhood of a given cell i at time t is

N t i = j ∈ L (t ) : (i , j ) ∈ E t .
The number of neighbors (degree) d t i of this cell can be calculated with the cardinality of its neighborhood:

d t i = |N t i |
The network is evolving through time. To get an idea of the dynamics that it undergoes, we computed the rate of change of the neighborhood per unit of time, c t i . We compared the neighborhood at time t and t + d t and made the sum of the number of lost cells and gained cells. More formally it can be described as:

c t i = 1 d t (N t i ∆N t -d t i ) = 1 d t (N t i ∪ N t -d t i ) \ (N t i ∩ N t -d t i )
were ∆ represents the symmetrical difference between two sets.

The embryos are composed of different subpopulations k that have already been presented above, in section 2.3.1. The topological network of cellular contacts allows to define a notion of boundary between them, it will be the set of cells of a given type that share a contact with the cells of another type. Formally, the border of the cell of type k 1 with the cell of type k 2 is:

B t k 1 →k 2 = i ∈ L k 1 (t ) : ∃ j ∈ N t i , such that j ∈ L k 2 (t )
. With this definition of the border, the number of contacts of a given cell i (type k 1 ) shared at the border

B t k 1 →k 2 can be written: b t i (k 1 → k 2 ) = | j ∈ N t i : type( j ) = k 2 |

Distributions of topological features in the groups of cells

We would like to compare embryos based on these topological features. We have shown that it is impossible to compare developing embryos at the level of the individual cell because of an inherent variability in the cellular behaviors. We chose to compare these features at the level of the subpopulations of cells to be coherent with the models whose parameters are defined at this resolution. We therefore focused our attention on the distribution of topological features in the groups of cells.

At a given time t, for a subpopulation k, we can define the distribution of the number of neighbors using its normalized histogram h d ,k (t ). Since the number of neighbors of a cell is a discrete quantity, we defined naturally the bins as being the number of neighbors.

Namely for a bin n ∈ N, we compute the frequencies of occurrence as:

p t d ,k (n) = 1 |L k (t )| .| i ∈ L k (t ) : d i (t ) = n | and the complete histogram is the vector, h d ,k (t ) = (p t d ,k (1), p t d ,k (2), ..., p t d ,k (N ))
, N being the maximum number of neighbors observed. The histogram is normalized: Σ N k=1 p t d ,k (k) = 1, it can be viewed as a discrete probability density.

In the same manner we computed a normalized histogram for the rate of change of the neighborhood per unit of time h t c,g . To link the bins with interpretable biological quantity, we considered as a base unit one change in the neighborhood in 4 minutes (equal to rate of change of 0.25 per minute). The histogram h c,k (t ) is defined on the set X = ([0, 0.25], ]0.25, 0.5], ..., ]C -0.25,C ]), with C being the maximum value observed for the rate of change in the neighborhood. The frequencies of occurrence for a bin x ∈ X can be calculated with the following formula:

p t c,k (n) = 1 |L k (t )| * 0.25 .| i ∈ L k (t ) : c t i ∈ x | the complete histogram being the vector h c,k (t ) = (p t c,g (0), p t c,g (1), ..., p t c,g (C )), with Σ C k=0 p t c,g (k) * 0.25 = 1.
The histogram can be considered as a discretized continuous probability density.

For the boundaries, we considered all the possible cases of contact between cells. There are four cell populations, SMic, LMic, Mac and Mes, which gives us 12 boundaries (considered asymmetrically): (SMic → LMic, SMic → Mac, ..., Mac → Mes, Mes → Mac). For each of them (symbolized as g → l ), we computed the normalized histogram of the shared number of contacts at the border B t g →l , h t b,g →l . As for the number of neighbors, natural bins are formed by the integers. The frequency of occurrence of a bin n ∈ N is :

p t b,g →l (n) = 1 |B t g →l | .| i ∈ B t g →l : b t i ,g →l = n |
the complete histogram is the vector h b,g →l (t ) = (p t b,g →l (1), p t b,g →l (2), ..., p t b,g →l (S)), with S with the maximum number of neighbors shared at a border. The histogram is normalized Σ S k=1 p t b,g →l (k) = 1 and can be considered as a density distribution.

Measures of similarity

To measure the similarity of two histograms we chose to compute the Matusita distance which is related to the cosine of the angle between the two histograms and varies between 0 (identical) to 1 (opposite). First we define the Bhattacharyya coefficient (Bh) between two histograms p and q having the same set of bins X.

Bh(p, q) = Σ x∈X p(x) q(x) = cos(θ)
where θ is the angle between the vectors ( p(1), p(2), ..., p(N )) and ( q(1), q(2), ..., q(N )), equal to 1 when the distributions are identical. This coefficient is defined for two continuous density distribution as Bh(p, q) = p(x) q(x)d x. The Matusita distance is defined as Mat(p, q) = ( p(x) -q(x)) 2 d x, the two quantities are related by the relationship:

Mat(p, q) = 2 * (1 -Bh(p, q)). In the following, we will use the discretized version of this distance defined for discrete histograms (p(1), p(2), ..., p(N )), (q(1), q(2), ..., q(N )):

d (p, q) = 1 2 Σ x∈X (( p(k) -q(k)) 2 * db)
where db is the size of the bin x. This distance is equal to zero if the two distributions are identical and equal to one when they are opposite, it is applicable to any kind of distribution and even if any bin is empty (which is a reason of preferring this measure to the chi-squared statistics).

Histogram resampling to calculate similarities

As stated previously the first step of the comparison between the embryo and the spatial simulation consists in comparing the distribution of the features at different time steps. An issue is that the digital embryos have all their own time sampling, ranging from 2min per time step to 5min. To overcome this problem, we decided to use a common time sampling for all the embryos and for the spatial simulation. We used a time interval ranging from 240 min post fertilization to 702 min, with a sample every 3 min. We interpolated the value of the histograms used for the comparison at each time point of this time interval. d d ,k,e ,d c,k,e ,d b,g →l ,e ; end average over all the subpopulations: d n,e ,d c,e ,d b,e ; end average over the cohort:

D n = d n , D c = d c , D b = d b ;
Algorithm 1: Comparison protocol for the three topological metrics With this procedure, we obtain a measure of similarity for each of the three topological feature that was used to define a protocol of comparison as shown on the Algorithm 1.

Objective functions

In addition to the three metrics introduced above, we measure intrinsic features of the simulated embryos via objective functions. These functions are defined in the following for a given time step. They are subsequently averaged over the entire time interval.

Tissue cohesion objective function C s

The tissue cohesion objective function, which represents the spacing between neighbor cells. Neighbor cells i , j tend to maintain their effective distance r i j close to their equilibrium distance r eq i j but they may detach from each other depending on their biomechanical properties. The definition of C s reads:

C s = 1 |E | i ∈L j ∈N i r i j -r eq i j r eq i j
where E is the of edges in the network of cellular contacts, L the set of cells and for a cell i , N i is its neighborhood.

Sphericity objective function S s and N s An important criteria to evaluate the model is the shape of the simulated embryos. We designed two functions, S s and N s for that purpose. S s is the ratio between standard deviation and the average of the set of distances between the cell positions X i and the embryo center X . The rationale is that this function has a low value for a small standard deviation and a large average. Its value is null in the case of a spherical embryo.

S s = σ µ where µ = 1 |L | i ∈L X i -X and σ = 1 |L | i ∈L ( X i -X -µ) 2
The second sphericity objective function N s is similar to S s and its rationale is that in the case of a spherical embryo, the cell axes U i are equal to their normalized relative

position vector X i ,center = X i -X X i -X
. The set of dot products between U i and X i ,center should have a minimal standard deviation and an average value close to 1. The equation of the function N s reads:

N s = σ µ where µ = 1 |L | i ∈L 1 + U i • X i ,center 2 and σ = 1
|L | i ∈L ( 1 + U i • X i ,center 2 -µ) 2
Planarity objective function P s For each cell, the planarity objective function P s aims at characterizing the unicity of the epithelium layer by measuring the ratio of cells belonging to the surrounding apico-basal neighborhood. We split the set of neighbors N i into two complementing sets, one covering the lateral domain N ∥ i and the other the apical domain N ⊥ i . They are defined as the following:

N ∥ i = j ∈ N i : X j -X i X j -X i • U i < η N ⊥ i = j ∈ N i : X j -X i X j -X i • U i ≥ η
A threshold value η controls the repartition between the two sets. Here, we set η = cos( π 4 ). The planarity objective function is written:

P s = 1 |L | i ∈L |N ⊥ i | |N i |
In the case of a mono-layered epithelium P s value is 0 and its value increases as the cells agglomerate in a 3D tissue.

Initial State

For the spatial simulation, an initial spatial state of the cells was needed in order to begin the simulation at the 32 cells stage. We choose to use the measured initial state of one of the five embryos of the cohort. This solution, although not optimal, was more satisfying than an artificial 32 cells stage. We wanted to avoid a spatial averaging that would have been without biological meaning.

Validation -Parameter space

The purpose of this exploration study is to determine the parameter sets which are responsible for realistic spatial enfolding of the sea urchin embryo development. The criteria of validation are the sphericity of the global embryo shape, the maintenance of the monolayered epithelium and the similarity with the empirical degree, rate of change of the neighborhood and the inter-subpopulation border shapes. This parameter space of the simulation is 4D, comprising the planar rigidity coefficient k rig , the gabriel criteria coefficient α gab and two specific adhesion coefficients controlling the attractive part of the relaxation force F ∥ i j : w adh,e between pairs of heterotypic cells and w adh,o between pairs of homotypic cells. The other free parameters of the model are set at constant values (Table 2.5). The timestep ∆t between two simulated time step is 6 seconds.

Note concerning the force amplitude coefficients w adh,e , w adh,o , w rep and k rig : these coefficients are all divided by the damping coefficent λ 0 in the master equation of motion.

Simulations are strictly equivalent if the ratio force amplitude coefficient over damping remains constant. The data-driven multi-level probabilistic model obtained in the previous chapter can be used to discuss several aspects of developmental processes. Among these aspects, we will explore the ideas of robustness, developmental irreversibility, chreodes and epigenetic landscape. The main goal of this chapter is to identify some interesting perspectives.

We will begin by characterizing this idea of developmental irreversibility that arise from the structure of the model.

Let's first recall the main characteristics of the multi-level probabilistic model obtained in the previous section operating at the level of the group of cells L n,k and their combination through the lineage :

85

Cell group L n,k Cardinality |L n,k | = 2 * |L n-1,k | = 2 n-n 0,k |L n 0,k ,k | Division time M n,k = X n,k + M n-1,k = n r =n 0,k +1 X r,k + M n 0,k M n,k ∼ N (µ n,k M , σ n,k M ) X n,k ∼ N (µ n,k X , σ n,k X ) P (X n,k | M n-1,k ) = P (X n,k ) µ n,k M = n r =n 0,k +1 µ r,k X + µ n 0,k ,k M (σ n,k M ) 2 = n r =n 0,k +1 (σ r,k X ) 2 + (σ n 0,k ,k M ) 2 Mean volume V n,k = A n,k V n-1,k = n r =n 0,k +1 A r,k V n 0,k ,k V n,k ∼ l nN (µ n,k V , σ n,k V ) A n,k ∼ l nN (µ n,k A , σ n,k A ) P (A n,k | V n,k ) = P (A n,k ) µ n,k V = n r =n 0,k +1 µ r,k A + µ n 0,k ,k V (σ n,k V ) 2 = n r =n 0,k +1 (σ r,k A ) 2 + (σ n 0,k ,k V ) 2
Mean surface area

S n,k = B n,k S n-1,k = n r =n 0,k +1 B r,k S n 0,k ,k S n,k ∼ l nN (µ n,k S , σ n,k S ) B n,k ∼ l nN (µ n,k B , σ n,k B ) P (B n,k | S n,k ) = P (B n,k ) µ n,k S = n r =n 0,k +1 µ r,k B + µ n 0,k ,k S (σ n,k S ) 2 = n r =n 0,k +1 (σ r,k B ) 2 + (σ n 0,k ,k S ) 2
Table 3.1: Summary of the relations between the different random variables in the modelthe definitions of the different notations can be found in the previous sections 2.5, 2.5, 2.5

Cells j ∈ L n-1,k and i ∈ L n,k
i is one of the two daughters of j Division time

m i = x i + m j x i is a realization of X n,k m j is a realization of M n-1,k Volume v i (t ) = a i .v j .ω n,k (u i (t )) a i is a realization of A n,k v j is a realization of V n-1,k
ω n,k is the deterministic volume micro dynamic u i (t ) is the percentage of elapsed life length of cell i Surface area

s i (t ) = b i .s j .φ n,k (u i (t )) b i is a realization of B n,k s j is a realization of S n-1,k
φ n,k is the deterministic surface area micro dynamic Proof. Let's denote S the set of gaussian distribution and * the convolution product.

u i (t )
(S , * ) is a monoid if the operation * is stable, associate and has an identity element.

stability For two gaussian distributions with parameters ((µ 1 , σ 1 ), (µ 2 , σ 2 )), we have

f (x|µ 1 , σ 1 ) * f (x|µ 2 , σ 2 ) = f (x|µ 1 + µ 2 , σ 2 1 + σ 2 2 )
. The convolution product of two gaussian distributions is a gaussian distribution.

∀(x, y) ∈ S 2 , x * y ∈ S commutativity For two gaussian distributions with parameters ((µ 1 , σ 1 ), (µ 2 , σ 2 )), we have

f (x|µ 1 , σ 1 ) * f (x|µ 2 , σ 2 ) = f (x|µ 1 + µ 2 , σ 2 1 + σ 2 2 ) = f (x|µ 2 + µ 1 , σ 2 2 + σ 2 1 ) = f (x|µ 2 , σ 2 ) * f (x|µ 1 , σ 1 )
Therefore, ∀x, y ∈ S 2 , x * y = y * x.

associativity For three gaussian distributions with parameters ((µ 1 , σ 1 ), (µ 2 , σ 2 ), (µ 3 , σ 3 )),

we have:

( f (x|µ 1 , σ 1 ) * f (x|µ 2 , σ 2 )) * f (x|µ 3 , σ 3 ) = f (x|µ 1 + µ 2 , σ 2 1 + σ 2 2 ) * f (x|µ 3 , σ 3 ) = f (x|(µ 1 + µ 2 ) + µ 3 , (σ 2 1 + σ 2 2 ) + σ 2 3 ) = f (x|µ 1 + (µ 2 + µ 3 ), σ 2 1 + (σ 2 2 + σ 2 3 )) = f (x|µ 1 , σ 1 ) * f (x|µ 2 + µ 3 , σ 2 2 + σ 2 3 ) = f (x|µ 1 , σ 1 ) * ( f (x|µ 2 , σ 2 ) * f (x|µ 3 , σ 3 ))
The convolution product is an associative operation on the set of gaussian distribution S .

identity element An element e ∈ S is an identity element if ∀x ∈ S , e * x = x * e = x. * is commutative, thus we need to find e with parameters (µ e , σ e ) ∈ R 2 such that ∀x ∈ S with parameters (µ,

σ) ∈ R 2 , f (x|µ e , σ e ) * f (x|µ, σ) = f (x|µ e + µ, σ 2 e + σ 2 ) = f (x|µ, σ).
The solution is a gaussian distribution with parameters µ e = 0, σ e = 0. This element is the Dirac delta function and is denoted by δ.

Given the properties of the convolution product on the set of gaussian distributions, (S , * ) is said to form a monoid.

A corollary of this result is that the tendency towards more desynchronization is irreversible. Indeed, the successive convolutions of gaussian distributions necessary lead to a gaussian distribution with a greater variance. This operation is not invertible (except for gaussian distribution with null variance).

In biological terms, it means that the rounds of division are less and less synchronized throughout development. Similarly the dispersion of the volume and surface area increases with time, even if their mean values are decreasing. This irreversibility seems to underlie the process of cell differentiation. As written in Davidson et al. [START_REF] Eric H Davidson | A genomic regulatory network for development[END_REF] the fundamental feature of developmental transcriptional systems in higher (bilaterian) animals is that it always moves inexorably forward, never reversing direction Here, it is the dispersion of the morphological features that goes inexorably toward being more spread.

However, this result rely on the fact that the random variables defined in each groups of cells are supposed independent of each other. It is possible to conceive mechanisms that would enable cells to control dispersion while proliferating. Regulation of the cell cycle and other morphological features are required to avoid going inexorably toward more disorganization.

Formalization as a dynamical system

This underlying algebraic structure as a monoid allows to describe the development as a dynamical system. To define this system (following the definition on wikipedia) it is necessary to have a tuple (T, M , ψ) where T is a monoid, M is a set and ψ is a function:

ψ : U ⊂ T × M → M
and ψ has the properties of a flow:

-I (x) = {t ∈ T : (t , x) ∈ U } -ψ(0, x) = x -ψ(t , ψ(s, x)) = ψ(t + s, x)
The structure of monoid here is given by the set of gaussian distribution together with the convolution product and the set of M correspond to the set of groups of cells embedded in the manifold defined by the coordinate system (µ, σ), which is the phase space of the system. The function ψ is the evolution function, and an orbit/trajectory of the system is the set ψ(t , x) : t ∈ I (x) and correspond to the development of one specimen in the cohort.

However, the function ψ is unknown and can only be obtained empirically for now, i.e. interpolated from the sequence of points for each developmental trajectories. With a dynamical system formalism and a phase space, we can define a notion of structural stability and a notion of vector field in the phase space. The vector field in the phase space can be interpolated between the measured points and from that we can expect to compute Lyapounov experiments or topological entropy that gives a measure of the complexity of development. The structural stability could be investigated by characterizing singularity points of the vector field and the periodic trajectories. The fundamental solutions of this equation can be fitted to the empirical gaussian distributions obtained in the sea urchin embryo. The use of this equation on our system could be interpreted as the random diffusion of the cells in a morphological space with a drift generated by cell differentiation.

Fokker-Planck Equation

Fluctuation theory and robustness

It could be interesting to characterize the robustness of the model by studying its behavior when submitted to small fluctutations. We know that after three cell cycles, the probability density of a random variable f 123 such as the division time is the result of the convolution of probability densities of random variables such as the life length ( f 1 , f 2 , f 3 ).

We have the equality

f 123 = f 1 * f 2 * f 3 1. "
= ∆ f 1 * f 2 * f 3 + f 1 * ∆ f 2 * f 3 + f 1 * f 2 * ∆ f 3 + 2.(∆ f 1 * ∆ f 2 * f 3 + ∆ f 1 * f 2 * ∆ f 3 + f 1 * ∆ f 2 * ∆ f 3 ) + ∆ f 1 * ∆ f 2 * ∆ f 3 can be simplified into ∆ f 123 = ∆ f 1 * f 2 * f 3 + f 1 * ∆ f 2 * f 3 + f 1 * f 2 * ∆ f 3 by considering only small ∆.
However, to define these fluctuations, we would be interested to keep the same metric that we have used to compare embryos and to define the prototype as a centroid. These equalities would require to combine Kullback Leibler divergence (relative entropy) with a convolution.

On this point, we can quote Oliver Johnson ([114] p.33):

Although we would like to prove results concerning the behaviour of relative entropy on convolution, it proves difficult to do so directly, [...] Specifically the logarithm term in the definition of entropy behaves in a way that is hard to control directly on convolution. Therefore, even if the metric that we have used to define the prototypical probabilistic model is highly efficient to compute a prototype in the statistical manifold it is less suited for a perturbation theory in developmental biology.

Parameters evolution

Waddington's epigenetic landscape

The multi-level probabilistic model obtained for each specimen of the cohort of sea urchin can be fruitfully brought closer to the old idea of an epigenetic landscape during development. Indeed, the highest unknown that stays after having formalized this model is the following question "Why does the probability laws have such parameters values?".

The life length increases at each generation, the mean volume is divided, but not exactly in half.

One possibility is that the value of these parameters is the result of dynamics of the underlying genetic network that governs these characteristics and which may imply a certain level of variability as well. In that case, the concept an epigenetic landscape is a relevant illustration of this idea, see figures 3.1 and 3.2. In our model, at each cell cycles, the valleys widen as the result of the increase in variability (the standard deviations are added). This increase in variability may well result in differentiation of cells into subpopulations.

The idea of the cells being pushed forward into the valleys is also relevant since the model present an additive behavior with no regulation.

Kupiec's ontophylogenesis

Finally this work can be brought closer to Kupiec theory of ontophylogenesis. Jean-Jacques Kupiec claimed that the processes occurring during ontogenesis, i.e. during development, could be considered as probabilistic ( [START_REF] Kupiec | Ni Dieu ni gène, pour une autre théorie de l'hérédité[END_REF], [START_REF] Kupiec | L'origine des individus[END_REF]). The reproducibility of the development coming from a principle analog to natural selection within the organism. The results of our study don't disprove this approach since individual cell features can indeed be modeled with probability distribution and the parameters values of these probability distributions may well arise from the interaction between the cells and their environment.

Overall, this probabilistic perspective on development sheds new light on the robustness of development. The high reproducibility of development seems to emerge from highly variable individual cell features. This perspective is at the opposite of the idea of the execution of a finely tuned developmental program. Therefore, we may need a new theoretical framework to understand development. Figure 3.2: Epigenetic landscape II. "The complex system of interactions underlying the epigenetic landscape. The pegs in the ground represent genes, the strings leading from the chemical tendencies that the genes produce. The modeling of the epigenetic landscape, which slopes down from above one's head towards the distance, is controlled by the pull of these numerous guy-ropes which are ultimately anchored to the genes." (Waddington CW. The Strategy of the Genes. London, Allen and Unwin (1957), [START_REF] Hal | The strategy of the genes[END_REF])

Conclusion

To conclude on this part we have shown the first example of a complete integrative approach for the study of multi-scale dynamics during the sea urchin development. This This data-driven prototypic representation of the cell lineage was then used as a basis for a biomechanical modeling using the MecaGen platform. This model allowed to infer the value of adherence parameters by systematic parameter exploration and fit to empirical data. Because of its data-driven nature, the prototype was able to free the biomechanical model of a certain number of parameters governing the proliferation rates or the shape of cells. The remaining free parameters were associated to the hypotheses underlying the biomechanical model.

Introduction

We propose to study in this part of the dissertation the theoretical status of biological variability and the relations of variability during development with darwinian evolution as well as its relation to randomness.

In the first chapter, we introduce the question of the relations between randomness, variability and diversity in biology. We review the mechanisms at the origin of variation, how uncertainty is modeled in mathematics and physics and how hypothesis based model can help to explore the range of possible at the price of approximations on other levels of organization.

In the second chapter of this part, we present the experimental results obtained when studying variable phenotypic expression and incomplete penetrance in the squint c z35 zebrafish mutant line. The results of this experiment show that the list of possible phenotypes is discrete, these phenotype are obtained in unpredictable proportions and the variations in the list of possible phenotypes may be incomplete even if the parents are homozygote mutants. These experimental results suggest a need for a clarification of the different levels of variability in biology, this is the subject of the following chapter.

In the third chapter of this part we study the relations between variability during development and evolution. Using a formal analogy with quantum mechanics, we propose a clarification of the various levels of variability; probability of obtaining a given phenotype, uncertainty on the set of possible phenotypes for a given observable and unpredictability at the level of the observable itself. Uncertainty on the set of observable is a much stronger form of uncertainty than a probability of obtaining a given phenotype, it is specific to the historical nature of biological objects and prevent to consider the space of possible a priori. Surprisingly, we find a formal analogy between quantum entanglement and Mendel's idealized scheme of inheritance which we relate to biological organization.

In the fourth chapter of this part, we consider the relationships between variations in individual development and observed diversity of phenotypes. Given the path-dependency 99 and historical nature of development we propose to gather individual developments, normal and pathological, into an ontogenetic tree. This ontogenetic tree structure enables to define a developmental proximity between phenotypes revealing the influence of developmental stages. By considering a large number of empirical descriptions of zebrafish developments we show that the pharyngula stage has the highest number of diverging developmental paths, suggesting an empirical basis for its status of phylotypic stage. However, the data set used may include possible biases.

Chapter 4

Sources of biological diversity and randomness

Abstract In this chapter, we review the various concepts of variability developed in biology, from gene mutation to stochastic gene expression and epigenetic effects. We then turn to mathematical and physical theories of randomness and models of uncertainty. Probability theory provides a framework to handle unpredictable events, but doesn't provide a definition for randomness. This theory is extended to capture specific aspects of quantum mechanics. Characterizations of randomness are provided for chaotic systems and in ergodic theory. To overcome the complexity of biological organisms, one possibility consists in using modeling approaches describing some mechanisms and explore regions of the space of possible. However, the multi-scale nature of dynamics occurring in organisms make those models necessarily incomplete. We propose to use multi-scale prototypical model to ground modeling approaches.

This chapter is an extended and translated version of [START_REF] Villoutreix | Vers une modélisation multi-échelle de la variabilité biologique[END_REF]. The idea of this chapter is to review the various mechanisms that can be sources of variability in biology, from gene mutation, to epigenetic and stochastic effects. Once their main characteristics have been exhibited, we then turn to mathematics and physics to see how unpredictable phenomena have been formalized. However, we show that the different frameworks are heterogeneous. We then explore the possibility to use models in biology to describe space of possible associated to some mechanisms. While reducing the generality of the statements, these models may integrate several aspects of biological variability. However, they are necessarily incomplete regarding the multi-scale and historical nature of organisms. These obstacles may be overcome by the use of multi-scale prototype of biological organ-101 isms.

Sources of variability in biology

In this section we review the main mechanisms sources of variability. We begin with the most established results before considering more recent ones on stochasticity in gene expression.

Gene mutations

Permanent alteration of the DNA sequence constitutes the most famous source of variation in biology. Some mutations correspond to a punctual alteration of the nucleotide sequence: they can lead to changes in the encoded protein by the mutated gene, or to a complete absence of its expression. Other mutations correspond to a more significant modification of the DNA, as a chromosome gain for example (long chain of nucleotides).

In any cases, if the mutations are not silent, they alter the considered organism. For example, with the zebrafish Danio rerio, if the gene encoding the Oep protein, receptor of the Nodal signaling pathway, the development fails: fishes become cyclops [START_REF] Pei | Identification of common and unique modifiers of zebrafish midline bifurcation and cyclopia[END_REF].

It is usually assumed that these mutations happen spontaneously, independently of their fitness in a given environment. The question of the influence of the environment on the variation of organism's traits is associated to a debate between Jean-Baptiste de Lamarck and Charles Darwin theories on evolution. For the first one, whose theory is the "transformisme Lamarckien", variations in the environment induce variations in organisms and these variations are oriented, with the internal efforts of the organism, toward the goal of being adaptive. For example, the fact that food is more abundant in height would have led giraffes to increase the length of their neck and their front legs while trying to reach it 1 . For Darwin, and in contrast to Lamarck, variations occur independently of their fitness in the environment. Living beings are continuously varying. Less adapted individuals will be disadvantaged with respect to others and thus reproducing in smaller 1. « Relativement aux habitudes, il est curieux d'en observer le produit dans la forme particulière et la taille de la girafe (camelo-pardalis) : on sait que cet animal, le plus grand des mammifères, habite l'intérieur de l'Afrique, et qu'il vit dans des lieux où la terre, presque toujours aride et sans herbage, l'oblige de brouter le feuillage des arbres, et de s'efforcer continuellement d'y atteindre. Il est résulté de cette habitude, soutenue, depuis longtemps, dans tous les individus de sa race, que ses jambes de devant sont devenues plus longues que celles de derrière, et que son col s'est tellement allongé, que la girafe, sans se dresser sur les jambes de derrière, élève sa tête et atteint à six mètres de hauteur (près de vingt pieds). » [START_REF] Lamarck | Philosophie Zoologique[END_REF], p. 256.

proportions. A clear separation between variability and fitness can be found in Darwin's theory. ing that this acquisition occurs spontaneously, and thus independently of their fitness, their non Lamarckian nature was validated. Therefore, this result contributed to give to genetic mutation the status of main source of diversity for the living [START_REF] Morange | Histoire de la biologie moléculaire[END_REF], [START_REF] Merlin | Le "hasard évolutionnaire" de toute mutation génétique, ou la vision consensuelle de la synthèse moderne[END_REF].

It has been shown since, that the mutation rate of some bacteria could be increased in response to variations in the environment [START_REF] Bjedov | Stress-induced mutagenesis in bacteria[END_REF]. Some mechanisms have a direct action on the rate at which the genome varies in time. This is the case of the SOS system in Escherichia coli, or as a response to the stress induced by changes in the temperature of the environment or changes in the amount of nutrient. These examples show that internal variations depend on environmental variations. The environment has a direct influence on the mutation rate. Mutations are however not directed towards a better fitness: this phenomenon does not correspond to a movement back toward Lamarckism. The environment can induce a more or less important capacity to vary, without reconsidering the idea that these variations are independent of their fitness.

Genetic mutations are an important source of variation in biology. They form the basis of many mathematical evolutionary theories, particularly in population genetics [START_REF] Lynch | The origins of genome complexity[END_REF]. In these theoretical frameworks, mutations are supposed to be spontaneous, independent of each other and independent of their fitness. However, these assumptions carry at least two presuppositions that can be discussed. The first one corresponds to the assumption of a direct (linear) relation between a gene and a phenotypic trait; the second one corresponds to the hypothesis of an independent variation between genes or phenotypic traits within an organism and with respect to the environment, this is an assumption on the modularity of variation 2 . The organization of living beings, the coupling between several levels of organization, the fact that the whole and the parts form a complex network of relations where retroactions are numerous, and where relations with the environment modify the internal space, are many reasons to suppose on one side that the linear relation between a gene or a set of genes and a phenotypic trait is not necessarily unequivocal, and on the other side that variation cannot be split up among the different parts of an organism. 3 2. Some models take into account interactions between genes in the formation of phenotypes with the concept of epistasis, complicating the picture [START_REF] Patrick | Epistasis-the essential role of gene interactions in the structure and evolution of genetic systems[END_REF]. Conversely, some models take into account the effect of mutations on several phenotypes [START_REF] Frank | One hundred years of pleiotropy: a retrospective[END_REF] 3. Darwin's definition of correlated variations is very meaningful here; "Correlated Variations -I mean by this expression that the whole organization is so tied together during its growth and development, that when slight variations in any one part occur, and are accumulated through natural selection, other parts become modified." [START_REF] Darwin | On the origins of species by means of natural selection[END_REF] -chapter 5

Epigenetic and stochasticity

In parallel to the study of genetic mutations, works have been conducted on other mechanisms sources of variation. It is the case for the phenomena grouped together under the term "epigenetic", although associated concepts have evolved with time. More recently study of stochasticity in single cell reveal new mechanisms. We will see here what are the differences that these studies have contributed to bring to the established conceptions of biological variations.

It is difficult to give a unique definition of what is epigenetic because the phenomena described with this word are numerous and its definition has evolved with scientific discoveries. Historically, this term has been introduced by Conrad Waddington in the 1940s to name all the mechanisms involved in the process of expressing the genotype into the phenotype. As from the 1960s, following the discovery of the molecular mechanisms of the lac operon by Jacob and Monod [START_REF] Jacob | Genetic regulatory mechanisms in the synthesis of proteins[END_REF], the role previously attributed to epigenetic is captured by genetic through gene regulatory networks and the metaphor of the genetic program. To transform a static code stored in the DNA into a phenotype, it wasn't necessary to invoke an external mechanism anymore; genes could act directly on each other during the execution of the genetic program [START_REF] Monod | Le hasard et la nécessité[END_REF]. Starting from there, epigenetic has become the study of the modulation of the genetic expression by way of chromatin modification (DNA and its protein skeleton). One of the main mechanism is histone methylation. An history of the concept of epigenetic can be found in [START_REF] Jablonka | The changing concept of epigenetics[END_REF]. The first study on histone methylation is [START_REF] Vg Allfrey | Acetylation and methylation of histones and their possible role in the regulation of rna synthesis[END_REF]. Histone methylation is a process leading to a change in chromatin spatial organization. Opening or closing of chromatin will allow or not transcription of certain genes.

These variations in the access to DNA don't constitute a definite modification of the transcription mechanism, but still have a certain stability in time. One of the main characteristic of the epigenetic modifications of chromatin is their transmission across generation, through mitoses and in certain conditions through meiosis, i.e. within an organism or during sexual reproduction. This research area is very active and has been joined by the study of various mechanisms that share an effect on the modulation of genetic expression. Those mechanisms are for example the inactivation of the X chromosome in mammalian development [START_REF] Okamoto | Epigenetic dynamics of imprinted x inactivation during early mouse development[END_REF], the process by which one of the two copy of the X chromosome is inactivated in female XX mammals during early embryogenesis; or the role of the maternal proteome in the gamete, i.e. the set of all the proteins already present in the female gamete before fertilization; or study on prion, a protein whose spatial folding is transmissible to other prion protein [START_REF] Michael P Mckinley | A protease-resistant protein is a structural component of the scrapie prion[END_REF].

Epigenetic modification modify the gene expression level whereas DNA mutations alter the structure of the proteins and hence the qualitative effect resulting from the gene expression. It can however be noticed that quantitative modification can lead to important qualitative effect. For example, during embryonic development, a change in the timing of expression period of a gene can imply an important effect on the final phenotype of an organism 4 [START_REF] Moss | Heterochronic genes and the nature of developmental time[END_REF]. These effects are called developmental heterochronies and the observation of the developmental sequence at every level of organization is a widely used tool to compare embryogenesis among species. Conceptually, epigenetic modifications are close to DNA mutations, although there is a slight difference with respect to heredity, since the first ones can be reversed, or more precisely are less stable in time, whereas the second ones are considered as permanent [START_REF] Oliver | Timescales of genetic and epigenetic inheritance[END_REF].

Besides, recent results have demonstrated the existence of stochastic phenomena in gene expression. In particular, the work of Michael Elowitz's team has shown with timelapse visualization of the expression of two fluorescent probes with two different colors associated to the same regulatory sequence, that the level of protein expression is variable in time within a cell and occur in a stochastic manner as shown on figure 4.2. Using these two different fluorescent probes, we can differentiate between the component of variability associated to the expression of each coding region depending on the same regulatory sequence, it is the so-called intrinsic noise. The common component of variability is associated to environmental variation, the so-called extrinsic noise [START_REF] Michael B Elowitz | Stochastic gene expression in a single cell[END_REF]. This stochastic gene expression is called noise because it can be associated to random fluctuations but doesn't change qualitatively gene expression. One of the reason for the success of this approach comes from the mathematics used to describe this experiment. The variability observed can be characterized with the Fano Factor η 2 t ot defined as the ratio of the variance of the measure over the mean:

η 2 t ot ≡ σ 2
x 〈x〉 2 . η t ot is the noise in gene expression. It can be shown that in the dual reporter experiment, the noise can be decomposed into an intrinsic part associated to the decorrelated variations of the two genes within a cell, i.e. an intrinsic source of noise within the process, η 2 i nt , and into an extrinsic part associated to the correlated variations of the two genes within a cell, i.e. a common external influence on the process, η 2 ext . The relation reads η 2 t ot = η 2 i nt +η 2 ext ( [START_REF] Peter S Swain | Intrinsic and extrinsic contributions to stochasticity in gene expression[END_REF], [START_REF] Michael B Elowitz | Stochastic gene expression in a single cell[END_REF], [START_REF] Paulsson | Summing up the noise in gene networks[END_REF]). These relations assume a static environment and have to be changed to account for a fluctuating environment [START_REF] Hilfinger | Separating intrinsic from extrinsic fluctuations in dynamic biological systems[END_REF].

4. These temporal variations in the sequence of events during development can also take place at other, more macroscopic, levels. It is the case of the maxillary bone between various groups of salamanders which is morphologically highly different because of variations in the growth periods and morphogenesis periods. These results are presented in [START_REF] Alberch | Size and shape in ontogeny and phylogeny[END_REF] This the case of the 1957 article "Enzyme induction as a all-or-none phenomenon" ( [START_REF] Novick | Enzyme induction as an all-or-none phenomenon[END_REF]). However, they weren't able to measure gene expression in time to prove a form of stochasticity in the process.

Another work reports an experiment combining stochasticity with inheritance. The article shows that cells can inherit a stochastic switch. A population of yeast Saccharomyces cerevisiae change their phenotype in a stochastic way between two semi stable epigenetic states. This stochastic switch is correlated among related individual cells, indicating an inheritance of epigenetic determinants of stochasticity [START_REF] Benjamin B Kaufmann | Heritable stochastic switching revealed by single-cell genealogy[END_REF].

It can be shown more generally that phenotypic variability in a clonal population of cells can reflect noise in the transcriptome underlying the choice of fate between several metastable states of the gene regulatory network [START_REF] Hannah H Chang | Transcriptome-wide noise controls lineage choice in mammalian progenitor cells[END_REF], [START_REF] Losick | Stochasticity and cell fate[END_REF].

When considering multicellular organisms, this noise may have dramatic effects on development. This is the case of a mutant line of the nematode Caenorhabditis elegans whose capacity to form intestinal cell precursor is highly variable for genetically identical individual in the same environmental conditions. This variability can be explained by the stochasticity of the gene expression and the amplification of small variation by a bimodal gene regulatory network [START_REF] Raj | Variability in gene expression underlies incomplete penetrance[END_REF]. A bimodal gene regulatory network is a network that has two stable states. The bimodal gene regulatory network is very sensitive to fluctuations in gene expression. The small number of molecules and the possibility of degrading RNA molecules enable these small variations. Amplified by highly non-linear responses of some gene regulatory network, these small variations, considered as noise, can have dramatic consequences on development and thus on organisms phenotypic traits and fitness 5 .

In addition to these stochastic phenomena during gene expression, we can consider the stochastic events occurring during cell division. These events contribute to differentiate cells from each other. The random distribution of the proteome during mitoses is the most studied phenomenon [START_REF] Huh | Random partitioning of molecules at cell division[END_REF]. In the same manner than for stochastic gene expression, if the number of molecules is small and the gene regulatory network is non linear, the effects associated to variations related to cell division can be highly amplified and have consequences on the whole organism.

The set of epigenetic phenomena, the stochastic effects during gene expression or during cell division constitute an important source of variability for the living, different from classical genetic mutations. Their status is also important to understand evolution, even if they are less permanent than mutations [START_REF] Oliver | Timescales of genetic and epigenetic inheritance[END_REF]. In particular these variations will have a higher effect if they occur early in development for multicellular organisms.

It is possible to consider with Jean-Jacques Kupiec that organisms are built around this intrinsic variation and a principle of natural selection acting within the organism. This is an extension of darwinian evolution to the formation of the individual in an "ontophylogenetic" process [START_REF] Kupiec | L'origine des individus[END_REF], [START_REF] Kupiec | Ni Dieu ni gène, pour une autre théorie de l'hérédité[END_REF]. It seems necessary to take into account the specific development of every organism as an essential component of biological variation in order to understand the exploration of possible forms during evolution. The specific development of each organism will be the sequence of singular events which has come along the constitution of an organism [START_REF] Longo | From physics to biology by extending criticality and symmetry breakings[END_REF].

By looking at the mechanisms sources of variability, we notice two important aspects, on one side these mechanisms involve the systemic structure of organisms, in the sense that several processes coming from different mechanisms are mixed in an organism, and on the other side they are involved in their ontogenesis, meaning that the sequence of events occurring during development will be decisive for the constitution of an organism.

The processes described are heterogeneous. Can they all be assembled and qualified as random? How should this randomness modeled? Can it be associated to forms that have 5. For a review on epigenetic phenomena during development, see [START_REF] Huang | Non-genetic heterogeneity of cells in development: more than just noise[END_REF] already been defined in physics? And more generally, what are the impacts of these aspects of variability on models of morphogenesis ?

Randomness and its formalisms in mathematics and physics

Probability theory

The main problem when we try to model situations involving random events is the problem of prediction. We would like to determine in advance the set of all possible results for a given experiment and the way these possible results can occur, the frequency at which they appear for example. In mathematics, the idea of randomness first appeared with the concept of probability. The probability concept is rooted in the work of Blaise Pascal and was also developed in the "Logique de Port-Royal" 6 in the middle of the 17th century. Pascal's wager is one of the first example of a mathematical reasoning in a context of uncertainty. In this wager, he computes the gain of believing in god, given a probability that god exists, in order to justify to believe in god even if its existence cannot be proved (the whole wage can be found in Pensées part III, §233). Mathematization of this concept has taken some time before being considered as a true branch of mathematics. By enunciating a small set of axioms in 1933, Kolmogorov contributed to set probability theory on a firm ground [START_REF] Kolmogorov | Grundbegriffe der Wahrscheilichkeitsrechnung[END_REF]. See [START_REF] Shafer | The sources of kolmogorov's" grundbegriffe[END_REF] for an historical perspective of the mathematical context surrounding the enunciation of the axioms, for an historical account of the emergence of probability see [START_REF] Hacking | L'émergence de la probabilité[END_REF]. Axioms of probability theory give constraints on the way to compute probabilities and hence on the tools enabling to measure randomness. They however avoid to define randomness as such. Probabilities always have to be interpreted. Probabilities are sometimes considered as deriving from imperfect knowledge of the experimenter relative to his object of study; they are considered as subjective. Probabilities are some other time considered as an objective property of the object; they are considered as objective. Additional epistemological interpretations of probabilities can be found between these two extreme positions, see for example [START_REF] Gillies | Philosophical theories of probability[END_REF] for a philosophical review of the various meanings of probability. Probabilities don't tell us anything about randomness in itself since the mathematical theory is compatible with these various interpretations. Kolmogorov's six axioms of probability are enunciated in the following way ([124],

6. The Port-Royal Logic is a very influential book, first published anonymously in 1662 in Paris, it is acknowledged to Antoine Arnauld and Pierre Nicole. It has been a reference in language theory and logic until the 19th century [START_REF] Shafer | The sources of kolmogorov's" grundbegriffe[END_REF]). Let E denote a set, the set of elementary events, and F denote a set of subsets of E , which are the random events:

1. F is a field of sets (it's an algebra over E , i.e. it is closed under the intersection and union of pairs of sets and under complements of individual sets) 2. F contains E 3. To each set A from F is assigned a nonnegative real number P (A). This number P (A) is called the probability of the event A 4. The probability of the E is P (E ) = 1 5. If A and B are disjoint, then P (A ∪ B ) = P (A) + P (B )

6. If A 1 ⊇ A 2 ... is a decreasing sequence of events from F with ∩ ∞ n=1 A n = , then lim n→∞ P (A n ) = 0
The sixth axiom, also called axiom of continuity, is equivalent to countable additivity of P when the first five axioms are given. It means that, for any infinite sequence of disjoints sets A 1 , A 2 , ... in F , P satisfies the equality:

P ( ∞ i =1 A i ) = ∞ i =1 P (A i )
Concerning this sixth axiom, it is interesting to read its interpretation by Kolmogorov (quotation translated in [START_REF] Shafer | The sources of kolmogorov's" grundbegriffe[END_REF], [START_REF] Kolmogorov | Grundbegriffe der Wahrscheilichkeitsrechnung[END_REF]):

Since the new axiom is essential only for infinite fields of probability, it is hardly possible to explain its empirical meaning ... In describing any actual observable random process, we can obtain only finite fields of probability. Infinite fields of probability occur only as idealized models of real random processes. This understood, we limit ourselves arbitrarily to model that satisfy Axiom VI. So far this limitation has been found expedient in the most diverse investigations

We understand here that this asymptotic axiom which is needed for the probability calculus has no empirical counterpart. It is an infinity, useful to construct the space of possible of probability theory. It may be interesting to compare it to other forms of infinity used to construct spaces of possible in mathematics and physics [START_REF] Longo | Mathematical infinity "in prospettiva" and spaces of possibilities. Visible[END_REF].

Another way formalization of probability theory states that (Ω, F , P ) is a measure space with P (Ω) = 1. Formalizing probability theory as a special case of measure theory is the major contribution made by Kolmogorov. Ω is the sample space, F is the space of events and P is the probability measure. F is a σ-algebra; i.e. closed under complement, union of countably many sets and intersection of countably many sets. This is the model of events at play. These rules are those of boolean logic on sets. This logic will constrain the way phenomena can be represented through these probabilities. In particular, it is worth noting that the closure by complement require to have a space of possibles a priori that gives the complementary of any events. Moreover, the closure by union of countably many sets requires to assume a modularity of the modeled events, that is to say that any two events can be associated in any order without having any influence on each other. For more detail and interpretations, see [START_REF] Harthong | Probabilités & statistiques. De l'intuition aux applications[END_REF].

To illustrate these notions, we can describe the example of dice throwing. The set of elementary equally likely events correspond to each face of a dice Ω = {1, 2, 3, 4, 5, 6}.

If we consider the experiment consisting in throwing two times consecutively the dice, then we may want to compute the probability that the dice have the same value. Let's denote this event by A in F . All 36 possible sequences of two throws of dice correspond to {{1, 1}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 1}, {2, 2}...}, they are all the possible variants that are a priori possible as outcomes of the experimental conditions. All 6 sequences of identical consecutive throws corresponding to {{1, 1}, {2, 2}, {3, 3}, {4, 4}, {5, 5}, {6, 6}} are included in the set A, therefore, in each of these 6 cases, event A has taken place. If all the sequences of two consecutive throws are considered equally likely, then the probability of A is P (A) = 6 36 = 1 6 . Studying the axioms shows clearly that probability theory is a framework to handle random events, but doesn't give any definition of randomness itself. The specific characteristics of the space of possible, of the probability measure, of the space of events, all depend on the modeled situation.

Randomness in algorithmic theories

We have to turn our attention to algorithmic theories of information to find attempts at defining randomness in itself. In the 1960s, the problem of characterizing an infinite random sequence of symbols was open. Kolmogorov proposed a concept of incompressibility for finite sequences of symbols; incompressible sequences are those which can not be produced by a shorter program that the size of the sequence itself using a universal Turing Machine. In 1966, the swedish mathematician Martin Löf proposed a definition of infinite random sequence of symbols by considering all the sequences that possess all conceivable statistical properties of randomness using algorithmic theory to formally define the notion of a test of randomness [START_REF] Martin-Löf | The definition of random sequences[END_REF]. It is ultimately this definition, which is considered the best definition of randomness in mathematics. It has been shown that under certain conditions given by Chaitin, infinite random sequences are exactly those which have all finite incompressible initial segments. This definition of randomness is widely accepted as the fundamental characterization of the notion of a random sequence. It should be noted however, that this definition is based on sets of infinite sequences and statistical tests which are not computable, leaving therefore a large gap between this definition and empirical results.

For example, the infinite binary sequence 0101010101010101... is not random since it can be compressed as a repetition of 01. On the hand, we can consider a Chaitin Omega number which is defined as the halting probability of a universal self-delimiting Turing machine. It is a computable enumerable and (algorithmically) random number. The first 64 bits of a Chaitin Omega have been computed by Calude, Dinneen and Shu in [START_REF] Cristian S Calude | Computing a glimpse of randomness[END_REF]: 0000001000000100000110001000011010001111110010111011101000010000. Some results bridge the gap between these abstract characterizations of randomness and physical systems in asymptotic cases [START_REF] Galatolo | Effective symbolic dynamics, random points, statistical behavior, complexity and entropy[END_REF]. We will now focus on physical systems presenting unpredictable behaviors.

Randomness in dynamical systems and ergodic theory

Randomness in dynamical systems The idea of randomness in physics cannot be approached without considering mathematical determinism as it appears in the study of dynamical systems and chaos theory which is its unpredictable counterpart. To understand randomness in dynamical systems, we have to go back to the XIXth century. Pierre Simon Laplace published A Philosophical Essay on Probabilities within which he developed his views on determinism. In his view, the world is ruled by a set of causes and effects relationships and only an incomplete knowledge about the state of the world at a given moment could be an obstacle to a perfect knowledge of these relationships and hence to a prediction of its futures states. This perspective is clearly expressed in the following famous sentence where he develops the idea of a demon with an infinite knowledge about the world:

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes. This view is mathematically supported by the Cauchy-Lipschitz theorem, (also known as the Picard-Lindelöf theorem). For a differential equation

∂y(t ) ∂t = f (t , y(t ))
with initial condition y(t 0 ) = y 0 , if the function f is regular enough (lipschitz continuous)

with respect to y and continuous in t, then for some ǫ > 0, there is a unique solution on the interval [t 0 -ǫ, t 0 + ǫ]. The proof for this theorem involve the use of a fixed point result.

However the works of Henri Poincaré in the late nineteenth century on the stability of the solar system and the three-body problem have weakened this position and laid the foundation to what would become chaos theory. In a seminal article Poincaré shows that in some dynamical system, ruled by a set of non linear differential equation, a small uncertainty on the initial conditions can lead to very large fluctuations on futur behavior [START_REF] Poincaré | Sur le problème des trois corps et les équations de la dynamique[END_REF]. This contradicts Laplace's view since for the latter, an approximate knowledge on the initial conditions should result in an approximation of the same magnitude for the knowledge on the evolution of the system. This means that perturbations could not have an important effect on trajectories. Laplace, like his contemporaries, was confident in linearization methods of differential equations [START_REF] Marinucci | Tra ordine e caos: metodi e linguaggi tra fisica, matematica e filosofia[END_REF]. Poincaré's result is therefore very important because it shows that even if the phenomena we observe are very well defined, it may be impossible to predict the evolution of the system. Perturbations that may seem irrelevant can actually impact the evolution of the system highly significantly. On this point, it is interesting to quote Poincaré from his book Science et méthode, 1908 [START_REF] Poincaré | Science and Method[END_REF]:

A very small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say that that effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But, even if it were the case that the natural laws had no longer any for us, we could still only know the initial situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all that we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may hap-pen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible and we have the fortuitous phenomenon.

Unpredictability resulting from sensitivity to initial conditions can be viewed as randomness. Even if we knew perfectly the laws of evolution of a system, our inability to know perfectly its state leads to its unpredictability. It is an epistemic kind of randomness steming from the interstice between the theory and the physical object. For this reason the status of measure becomes dominating, it determines our access to the objects and thus our ability to know the object [START_REF] Bailly | Randomness and determinism in the interplay between the continuum and the discrete[END_REF].

A chaotic behavior can be quantified with Lyapunov exponents. They express mathematically the sensitivity to initial conditions and the rate at which two close solutions of dynamical system diverge. Given two trajectories in a phase space, their initial separation is given by |δZ 0 |. The Lyapounov exponent λ is defined as

|δZ (t )| ≈ e λt |δZ 0 | where |δZ (t )|
is the difference between the two solutions at time t .

Ergodic theory When considering gas, another sort of problem for the prediction of the system's evolution appears. Following the work of Ludwig Boltzmann, gaz are composed of a high number of atoms and molecules, each of them having its own dynamic. Therefore this high number of elements makes it impossible to follow each particle individually. It is thus assumed that each individual particle has a random motion and collide with other particles in a stochastic way. In order to derive the macroscopic behavior from the behavior of each individual particle, one of the necessary condition was the ergodic hypothesis [START_REF] Frigg | The ergodic hierarchy[END_REF]. This assumption is required to prove the equipartition of energy in the kinetic theory of gases. This assumption says that the average of a process over time of a system and the average over the statistical ensemble are the same [START_REF] George | Proof of the ergodic theorem[END_REF]. It means that each element of the system goes everywhere in the space of possible in a uniform fashion. This statement of uniformity of individual behavior is a form of randomness and involve an asymptotic statement which require the use of infinity. This randomness results from the too high number of elements in interaction.

These two forms of randomness found in dynamical systems theory and in ergodic theory are somehow opposed, they correspond respectively to a too tiny cause or too many causes. However, they both relate to an epistemic notions. They are defined in the frame of classical physics where trajectories of individual particles are assumed. Quantum physics has a radically different approach of the problem of randomness.

Randomness in quantum mechanics -Quantum mechanics as a generalized probability theory

Quantum physics studies phenomena at the scale of atoms. The formalization of quantum mechanics, which happened in the beginning of the XXth century, is an epistemological revolution. The concept of trajectory itself has been changed compared to classical physics. Schrödinger equation which describes the evolution of a quantum system, no longer governs the path of a body but the trajectory of a probability amplitude (a normalized vector state in a Hilbert space). When a quantum system is measured, the result of the measure is related to the kind of measure applied and the results are predicted in probability. A quantum state is described as a linear combination of possible measure results.

Each of the coefficients of this combination gives the probability to obtain the associated possible result according to the Born rule.

It is worth noting that the classical probability structure relying on Kolmogorov's axioms (as presented above) has been replaced by a structure using a normalized state vector in a Hilbert space. The σ-algebra has been replaced by an orthoalgebra within which various experimental contexts, sometimes incompatible, can be combined. It becomes however difficult to use the word object, and it is more common to refer to a process inseparable of the experimental conditions which are also the very conditions of its existence [START_REF] Bitbol | La mécanique quantique comme théorie des probabilités généralisées[END_REF]. The traditional configuration where an observer observe a system from the outside is blurred, this situation is part of the formalism. In this field, the theory is build around a probability amplitude. The concept of randomness is at the center, although it seems to be more related to our relations with the object of the study rather than to the properties of the object as such. In any case, the form of randomness used in quantum mechanics can be considered as objective, since the theory cannot overcome it. The violation of Bell inequalities and their experimental measurement proves that no theory with local hidden variables is compatible with the quantum mechanics framework. This theory has a high predictive power. There is however no definition for the randomness in itself or characterization for the mechanisms sources of randomness.

After reviewing these model of randomness in physics and mathematics, we understand that this notion covers very rich and varied fields. Formalisms used can be very different and are not necessarily compatible. To summarize, the main differences rely on the choice of the space of possibles to describe the results of an experiment, the capacity to separate the object from the context of the experiment, the experimental reproducibility and finally the asymptotic character of the studied system. Non predictability is the underlying concept, the inability to predict uniquely the result of an experiment. Probabilities provide a calculus framework on the results of an experiment with uncertain results. The choice of the rules used in the computations depend on the model of the experiment and thus on the concept of randomness at play.

As presented in the first part of this chapter, the sources of variability in biology are numerous and of various nature. We need a notion of variability, of non predictability and thus of randomness that could place these effects on the same footing. The various time scales at play, the diversity of mechanisms at play, the systemic organization of organisms coupling these different phenomena, makes it difficult to reduce biological variability to one or the other form of randomness met in physics. It is for example possible to have a combination of chaotic and quantum effects [START_REF] Buiatti | Randomness and multilevel interactions in biology[END_REF]. The question remains as to how these couplings between all these phenomena are an obstacle to models of biological variability.

Is it possible to decompose in a meaningful way these phenomena or to state the adequate simplifying hypotheses in order to obtain a model of variability? Can we isolate some morphogenetic processes and model them independently to capture one or the other aspect of variability?

Variability and models in biology 4.3.1 Models and simulation as tools for exploring some dynamics of the living

Here and in the following, we will consider that a simulation is the computation of a specific realization of a model, whereas the model itself is a set of assumptions that aims to describe a target phenomenon -with a lower degree of generality than a theory.

One of the first work describing at the same time a model and a simulation of a morphogenetic process in the living is due to Alan Turing. In his 1952 article « The chemical basis of morphogenesis » [START_REF] Mathison | The chemical basis of morphogenesis[END_REF], he explores the possibilities of a mathematical model of morphogenesis. He proposes a system of differential equations which describe the spatial distribution of chemical elements in an idealized biological tissue through a reaction diffusion mechanism. This non-linear system of equation generates stationary spatial patterns from an initially homogeneous situations (symmetrical with respect to the op-eration of interchanging the cells [START_REF] Mathison | The chemical basis of morphogenesis[END_REF] p.42) and with an initial local perturbation. Although it is possible to solve this problem analytically and thus to exhibit explicitly some solutions with simplifying hypotheses, Alan Turing recommend in his article the use of computer simulations 7 . These simulations solve numerically the system of differential equations. They exhibit some valid solutions and enable us thus to construct a representation of some solutions. They allow avoiding simplifying assumptions, opening the way to more complicated cases than those presented in the article, for example by integrating mechanical constraints between cells in addition to chemical reactions. This article shows how a mathematical model which is "a simplification and an idealization" sheds light on some mechanisms of the living by describing fundamental characteristics. This model of reaction-diffusion is still at the center of numerous contemporary research [START_REF] James | Mathematical biology i: An introduction[END_REF]. The use of computer simulation enables to describe explicitly solution of models by sacrificing mathematical generality in favor of the intelligibility of specific solutions. Computer simulation is a tool for exploring mathematical models.

This practice carries with it a set of constraints that should be taken care of for the study of biological variation. Indeed, all computer simulations rely on a Turing machine.

A Turing machine is a discrete state machine, based on a Laplacian paradigm of identical iteration. As shown by Giuseppe Longo, the perfect iteration that underlies all computer sciences introduces an inevitable difference between a simulation and some dynamics occuring in the physico-chemical continuum like chaotic systems [START_REF] Longo | Laplace, turing et la géométrie impossible du "jeu de l'imitation" aléas, déterminisme et programmes dans le test de turing[END_REF]. Turing's 1952 work on morphogenesis, subsequent to his invention of the so-called Turing machine, is a work on the non-linear dynamics defined in the mathematical continuum. These dynamics are sensitive to initial conditions, he uses the words "catastrophic growth" ( [START_REF] Mathison | The chemical basis of morphogenesis[END_REF] p. 64). Turing needed the mathematical continuum to make assumptions on morphogenetic mechanisms in the living. This mathematical continuum is then approximated to be simulated on a computer. More precisely, if one considers a chaotic system, the study of its behavior is limited by the ability to reproduce the initial conditions whose access is limited through a necessarily finite measure. Simulation of this kind of system would be identically reproducible if the same parameters are set as inputs. These discussions on the profoundly reproducible status of computer simulation highlight the difference in the epistemological status between the model which can be developed in the mathematical continuum and the simulation which happen in the discrete. They should however not hide the ex-7. "It might possible be possible, however, to treat a few particular cases in detail with the aid of a digital computer. This method has the advantage that it is not so necessary to make simplifying assumptions as it is when doing a more theoretical type of analysis [START_REF] Mathison | The chemical basis of morphogenesis[END_REF] p.72 ploratory power of computer simulation.

Many examples of computer simulations allow to convince oneself of the relevance of some mechanisms in the morphogenesis of the living. Refinements of Turing's reactiondiffusion system describe well the formation of some patterns in the development of the zebrafish [START_REF] Nakamasu | Interactions between zebrafish pigment cells responsible for the generation of turing patterns[END_REF], [START_REF] Kondo | Reaction-diffusion model as a framework for understanding biological pattern formation[END_REF]. Cellular Potts model is another approach which tries to describe relations between cells in terms of adhesion within a tissue by using method from statistical physics [START_REF] Graner | Simulation of biological cell sorting using a two-dimensional extended potts model[END_REF]. By generalizing the Ising model, Potts model describes some cellular displacements. The Ising model is one the simplest model of statistical physics presenting a phase transition. Moreover, the deformation of a 2 dimensional tissue in a 3 dimensional structure has been modeled with a vertex model during Drosophila morphogenesis, the tension differences between cells is used to faithfully reproduce empirical measures [START_REF] Osterfield | Three-dimensional epithelial morphogenesis in the developing Drosophila egg[END_REF].

At a more molecular scale, the gene regulatory network in the development of the sea urchin has been modeled with dynamical boolean network [START_REF] Peter | Predictive computation of genomic logic processing functions in embryonic development[END_REF] (it is a network where the nodes can take values from a discrete set of values depending on their interactions). This model enable to reproduce the main steps of the dynamic of the gene regulatory network as it is observed. To take into account epigenetic effects, it is possible to introduce probabilistic interactions between genes [START_REF] Shmulevich | Probabilistic boolean networks: a rule-based uncertainty model for gene regulatory networks[END_REF]. Finally, recent attempts try to model the coupling between gene regulatory network and mechanical interactions between cells during early zebrafish embryogenesis, explaining some phases of development 8 .

The question then is to disturbe the mechanisms reproducing empirical phenomena in order to test the predictive power of the model. Given the high complexity of organism, it is to be expected that the mechanisms observed and modeled have a narrow range of applicability and that the mechanisms change highly when considering adjacent situation. These different models aiming at describing embryogenesis offer sets of reasonable assumptions for some specific phenomena at a given level of organization at a given time of

Living organisms are organized objects involving different levels of organization with heterogeneous dynamics

These various mechanisms are not sufficient to describe the complexity of an organism if they don't allow to integrate the various levels of organization. The experimental results presented in the first section of this chapter show how the coupling between different levels of organization can have an effect on biological variability. It should hence be taken into account in models of variability. Most of the models in developmental biology assume a separability between the phenomena occurring at various scales concerning measurement and modeling. For example Eric Davidson's work consists in modeling the gene regulatory network in the sea urchin embryo 9 from experimental results. This model is simulated with a boolean network. It may show some limitation as it cuts the embryo in parts of tissues that are coarser than the level of the cell and in time period of several hours, much longer than a cell cycle. The problem is that genetic interactions occur within a cell where genetic expression takes place. The simplifying hypothesis which consists in abstracting the level of organization of the unique cell may lead to wrong interpretations of the dynamics of gene expression. On the opposite, one can consider work coupling the dynamics of genetic expression at the cellular level with physical processes at play at the tissue level in the Drosophila during mesoderm invagination [START_REF] Wang | Differential positioning of adherens junctions is associated with initiation of epithelial folding[END_REF]. The problem which may arise here is the role of previous developmental steps in the phenomenon. For example early events in development may have a later effect on the tissue organization. The simplifying assumption here consists in focusing on a short time period without considering earlier steps in study of causal mechanisms. These two approaches may show some limitations in explaining embryogenesis while considering organisms as a whole.

It is possible to consider an alternative approach in some multi-scale approach such as the phenomenological reconstruction of a complete organism [START_REF] Olivier | Cell lineage reconstruction of early zebrafish embryos using label-free nonlinear microscopy[END_REF] as shown in the first part of this dissertation. This work consists in reconstructing in an automated way the dynamics occurring at different scales. From 3D+time image acquisition, raw data are processed in order to automatically detect the position of the nuclei of each cells in time as well as the shape of the cellular membrane. From these data, it is possible to obtain the spatio-temporal cell lineage. This approach enable the access to the dynamics of each individual cells, as well as the dynamics of the tissue and the whole embryo. The obtained data set is a digital embryo. This reconstruction of a process involving contingent events 9. http://sugp.caltech.edu/endomes/ at various levels enable to construct a duplicate of the empirical space [START_REF] Varenne | La reconstruction phénoménologique par simulation : vers une épaisseur du simulat[END_REF] were the phenomena with heterogeneous dynamics are combined, and can be studied as a whole. The process of automated reconstruction introduces some assumptions on the nature of the observed phenomena it is the "thickness" of the digital object. By offering an access to the living, which is digital, quantitative, multi-scale, taking into account the temporal dimension of the embryonic development, this phenomenology reduces the distance between the simulations and the empirical results. It reduces this distance in two ways. First, because it allows to confront quantitatively the simulation and experimental data, which is rarely the case in developmental biology. Secondly, because the underlying assumptions required for the automatic processing of the data are assumptions on the observed objects, and are thus close to a model. The aim differs however from an explicative model since the idea is to reproduce accurately the empirical data and not to test the existence of mechanisms allowing to explain them.

Thanks to this kind of approaches, the question of variability can be asked in a multiscale and temporal manner. In particular, the phenomenology of variability at various levels of organization can be studied. Variability can be transmitted from one level of organization to the other. Similarly, its transmission through time within the embryo is an interesting question, for example along the cell lineage. By studying the phenomenological reconstruction of five sea urchin embryos during the first hours of development, we could quantify this variability at different scales, from the individual scale to the groups of cells sharing the same type and generation to the whole individual. (see chapter 1 and 2 of this dissertation). We show that the inter individual variation is high for the proliferation dynamics and cellular volume if only the cellular level is considered, although it decreases when considering a coarse-grained level, cells clustered by cell type and generation. Similarly, we show that the variation observed at a given generation is only weakly transmitted from generation to generation at the level of groups of cells. These groups of cells are defined with morphological criteria identifiable for each specimen and seem to be a relavant level of observation to describe embryonic development. Using these two results, the dynamics at the level of the group of cells can be averaged over the cohort into a prototypical representation of development. This prototypic description takes into account intrinsic variability of the groups of cells through the use of a probability distribution and allow to describe the behavior at the scale of the whole embryo by combining the behavior of each of the groups. This is a multi-scale quantitative prototypic representation of embryogenesis. By shedding light on the relations between different levels of organization and by identifying invariant features among individuals, this kind of object seems to be a good candidate to be at the basis of a multi-scale model of morphogenesis with a certain level of generality and leading thus to a theoretical level of explanation.

The prototypical reconstruction of the sea urchin is a very simple case, on a model organism which is expected to be highly reproducible. The question of integrating the variability stemming from the different levels of organization is open in most of the cases.

Conclusion

The two main obstacles for modeling possible forms in the living are on one side the systems aspect, which couples various levels of organization and heterogeneous dynamics within a single organism and on the other side the temporal aspect, the fact that all the steps of the ontogenetic trajectory constituted of variable phenomenon at all scales has an impact on the organism. Biological objects are the result of an evolutive history, phylogenetic and of an individual trajectory, constituted of singular events, epigenetic and stochastic. We have shown that the sources of this variability are multiple, precluding a priori to reduce these phenomena to one or the other form of randomness already studied in physics. Moreover, probabilities only provide a framework for calculus to describe phenomena and don't say anything on the phenomena themselves. They are useful once the model of variability has been established. Biological objects are organized, multi-scale, the result of a complexe morphogenetic process. Thus, models focusing on one level of organization and including too strong hypotheses on other levels of organization may not have a sufficient generalizing power in other experimental frameworks and may neglect important phenomena. Phenomenological reconstruction such as presented in the first part of this dissertation allow to access multi-scale dynamics with heterogeneous dynamics. Using these multi-scale dynamics, variability can be measured at all levels of organization. It is possible to identify the relevant levels of organization to describe a model of variability. The multi-scale prototypical representation of the embryonic development of the sea urchin embryo is an example showing that it is possible to study variability in an integrative way, overcoming some limitations described in this article. Models relying on this kind of approach could acquire sufficient generality to reduce the distance between empirical and theoretical biology.

Chapter 5

Variable phenotypic expressivity and incomplete penetrance of the zebrafish mutant line squint cz35

Abstract This chapter studies a phenomenon of incomplete penetrance in the squint zebrafish mutant line. By measuring the distribution of phenotypes (interocular distance) in progeny of identified pairs of homozygote parents, it is shown that the list of possible phenotypes is discrete with intermediate phenotypes and in unpredictable proportions. The statistical distributions put aside factors related to genetic backgrounds or environment.

These complex inheritance patterns suggest a variety of possible relationships between early molecular events and late phenotypes.

Introduction

This chapter explores a phenomenon of incomplete penetrance and variable phenotype expression in a zebrafish mutant line. The experimental results and their interpretation presented here also serve as a basis for the next chapter.

The notion of incomplete penetrance describes the situation when individuals carrying identical mutants alleles show either mutant or wild-type phenotypes as it has been observed in the nematode Caenorhabditis elegans [START_REF] Horvitz | Isolation and genetic characterization of celllineage mutants of the nematode caenorhabditis elegans[END_REF] by Horvitz and Sulston in 1980, and more recently in 2010 by Raj et al. [START_REF] Raj | Variability in gene expression underlies incomplete penetrance[END_REF] . The introduction of this concept can be traced back to Timoféeff-Ressovsky [START_REF] Nw Timoféeff-Ressovsky | Über den einfluss des genotypus auf das phänotypen auftreten eines einzelnes gens[END_REF] and Romaschoff [START_REF] Dd Romaschoff | Über die variabilität in der manifestierung eines erblichen merkmales (abdomen abnormalis) bei drosophila funebris f[END_REF] who noticed it independently in 1925. The notion of variable (phenotypic) expressivity describes the idea that 123 individuals with identical mutant alleles have variable phenotypes, not necessarily wild type, although it has been found associated to incomplete penetrance [START_REF] Horvitz | Isolation and genetic characterization of celllineage mutants of the nematode caenorhabditis elegans[END_REF]. Incomplete penetrance and variable expressivity can sometimes be explained by variation in environmental conditions [START_REF] Jane | Phenotypic plasticity and the origins of diversity[END_REF], variation in the genetic background [START_REF] Carlborg | Epistasis: too often neglected in complex trait studies?[END_REF] but also by stochastic effects in gene expression [START_REF] Raj | Variability in gene expression underlies incomplete penetrance[END_REF]. In normal conditions this variability in gene expression may be buffered to ensure proper development. We explore here quantitatively a zebrafish mutant line presenting incomplete penetrance and variable phenotypic expressivity. After having described the phenomenology of the phenotypic variation, we will point out methodological problems raised when trying to explain this observed variability with underlying molecular variability. The zebrafish is a vertebrate model organism appreciated for its high manipulability in laboratories, its rapid development and its transparency during early embryogenesis. It is also known for its regenerative abilities. The main stages of its development are shown on figure 5.1. After fertilization, the egg proliferate on the yolk sack before undergoing complex morphogenetic movements such as epiboly, and convergence extension.

Materials and methods

Mutant line squint c z35 -10 pairs identified by PCR The squint c z35 mutation consists in an insert of 1.9 kb in the squint gene which is a ligand of the Nodal signaling pathway leading to a truncated expression of the squint gene. To perform our experiment we identified 10 pairs of squint mutant sq t c z35 by fin-clipping male and female squint adults and genotyping by PCR with specific primers [START_REF] Feldman | Zebrafish organizer development and germ-layer formation require nodal-related signals[END_REF]: reverse primers detecting the presence (5'-ATATAAAATCAGTACAACCGCCCG-3') or absence (5'-GCCAGCTGCTCGCATTTTATTCC-3') of the insertion were used with a forward primer (5'-GAGCTTTATTTCAATAACTGCGTG-3') present in wild-type and sqt c z35 alleles. See figure 5.3 A.

Fixation at 30 hpf Controlled fertilization was performed by using aquariums where pairs of fishes were maintained isolated. The night before the egg laying a transparent wall separates male and female, in the morning the wall is removed and freshly layed eggs are controlled every 15 minutes. Fertilized eggs were then stored in an incubator at 28 • C. Environmental conditions were similar during the 8 weeks of the experiment. After 30 hours post fertilization (30hpf ), embryos were fixed with paraformaldehyde (PFA) to stop the development and maintain the morphology comparable within the assemblies of embryos.

Photographs and measures -Fiji

Once the embryos were fixed, they could be photographed to provide a quantitative measure of their morphology. They were dechorionated to be better manipulated and have a better resolution. They were positioned with the head in the good orientation, by using rails in agarose. Photographs were taken with a binocular microscope in bright light. Measures of interocular distance have been manually extracted with FIJI software as shown on figure 5.4. A total of 928 measures is obtained.

Results

Protocol -Number of embryos in each egg clutch

The experimental protocol is summarized on the figure 5.4. It consisted in controlling the spawning of identified pairs (male and female breeding couple) of homozygote squint mutants during 8 consecutive weeks.

Pictures of each embryo of the progeny were then taken at 30 hours post fertilization. They were used to measure the interocular distance. The number of embryo in the progeny of each pairs was very variable through time although the conditions were maintained identical from one embryo to the other and through time. Table 5.1 shows the distribution of the number of embryos through time for the different pairs considered. We see that pair 1 and pair 5 generated embryos on the most regular basis, they were breeding 6 times among the 8 weeks of the experiment. Pair

Squint penetrance

The measures performed on the embryos enable to compute the number of cyclops during each week of the experiment, they are reported in the following table 5.2 and correspond to the penetrance of the squint phenotype:

The values of penetrance shown in the table 5.2 are very variable among the different couples (from 0% to 75%) and throughout the experiment for an identified pair, for example for couple 5 from 3.3% to 32.4%. In contrast to the results obtained in [START_REF] Pei | Identification of common and unique modifiers of zebrafish midline bifurcation and cyclopia[END_REF], these results suggest that the variations in the penetrance rate of cyclopia in Sqt-/-embryos may be determined by other factors than the genetic background.

Discussion

Overall, this experiment on the distribution of phenotypes in the progeny of homozygote squint mutant shows variable phenotypic expression in unpredictable proportions and with a possible incomplete list of accessible phenotypes. This last point is underlied by the fact that the list of intermediate phenotypes is not the same among the various couples. Therefore it can be supposed that other couples of mutants present other intermediate phenotypes.

The observed variation in the phenotypes could result from a cryptic variation of βcatenin, involved in the nodal signalling pathway, revealed by the absence of squint. This maternal protein has a redundant role compared to the protein squint [START_REF] Schneider | βcatenin translocation into nuclei demarcates the dorsalizing centers in frog and fish embryos[END_REF]. To find correlation, and possible causal links, between early phenotypes such as the spatial distribution of β-catenin and late phenotypes such as the interocular distance requires to measure the spatial distribution of β-catenin. This measure should lead to various phenotypic classes. The question that will arise then is how to relate these statistical distributions? Figure 5.7 shows the hypothetical causal links between early molecular events, early phenotypes, and late phenotypes. The case (a) corresponds to a direct link between an early molecular event and a late phenotype, for example the absence of oep lead to cyclopia with complete penetrance. The case (b) corresponds to situations when the symmetry breaking hasn't occurred yet, for example in a bimodal gene network two metastable states are possibles. The case (c) corresponds to convergence during development, when for example two embryos with different phenotypes converge to same phenotype. The case (d) corresponds to the case when an early phenotype is observed but doesn't have a late counter part, because this causal link has been observed only once and only at the molecular level for example. The case (e) is the same idea as (d) but when only the late phenotype has been observed. Finally it can be assumed that some causal links may oc-cur but have not been observed yet. surement (d,e) and possible new causal relationships are important methodological obstacles for statistical inference. What does these methodological obstacles imply for biology? Can it be assumed that this situation is the norm in biology instead of a special case of inheritance and determination? Thus, the choice of the space of possibles is a central problem, it is the central problem of the next chapter.

Introduction

The two principles by which Darwin proposed to explain the diversity of biological forms are descent with modification and natural selection [START_REF] Darwin | On the origins of species by means of natural selection[END_REF]. The sources of variation underlying descent with modification are multiple, from combinatorial genetic lottery, to mutations of the DNA, to epigenetic effects such as DNA methylation or histone modification, to stochastic effects during gene expression, [START_REF] Huang | The molecular and mathematical basis of waddington's epigenetic landscape: A framework for post-darwinian biology[END_REF], [START_REF] Raj | Variability in gene expression underlies incomplete penetrance[END_REF]. However, the theoretical status of variability is still unclear.

Following a line of thoughts initiated by Longo, Bailly and Montévil [START_REF] Bailly | Mathematics and the natural sciences: the physical singularity of life[END_REF], [START_REF] Longo | Perspectives on Organisms: Biological Time, Symmetries and Singularities[END_REF], [START_REF] Longo | From physics to biology by extending criticality and symmetry breakings[END_REF] individual organisms can be considered as resulting from a specific onto-and phylogenetic 135 trajectory involving contingent events. They are thus historical entities. This perspective has strong implications for the theoretical status of biological objects. In particular, the space of possible is not completely predictable, radically new observables are likely to emerge in the course of evolution, for example a new organ or a new function for an already existing organ [START_REF] Longo | No entailing laws, but enablement in the evolution of the biosphere[END_REF].

In this article, we propose to analyze this unpredictability in the light of the quantum mechanics formalism in order to characterize the various levels of uncertainty met in biology. Previous approaches have brought closer biology and quantum mechanics.

Schrödinger in his book "What is life" [START_REF] Schrödinger | What is life[END_REF], Rosen in a paper discussed quantum information as an approach for describing genetics [START_REF] Rosen | A quantum-theoretic approach to genetic problems[END_REF]. More recently, Khrennikov and colleagues have extended the use of quantum mechanics formalism to biological situations [START_REF] Basieva | Quantumlike interference effect in gene expression: glucose-lactose destructive interference[END_REF], with a fine analysis of epigenetic evolution [START_REF] Asano | A model of epigenetic evolution based on theory of open quantum systems[END_REF]. The mathematical structure of quantum mechanics contains several epistemological characteristics of quantum mechanics, such as the dependency of the empirical results to the experimental context, and more particularly the measure instruments [START_REF] Bitbol | Théorie quantique et sciences humaines[END_REF], [START_REF] Bitbol | Mécanique quantique: une introduction philosophique[END_REF]. Other disciplines sharing similar epistemological constraints may benefit from the use of the quantum formalism. The emerging field of quantum interaction provides examples of successful applications in cognitive sciences, decision theory, computer sciences.

Variability in biology, emergence of new phenotypes

Any biological inquiry begins by studying a particular observable. An observable corresponds to a measurable characteristic that can provide knowledge about an organism.

It can be a phenotypic trait, a genotype, a function. Drawing a direct analogy with quantum mechanics, a biological observable will be represented as an hermitian operator in a hilbert space. The eigenvectors of the operator being the possible values of the observable.

For example, the observable can correspond to the "wrinkled" or "smooth" character of a pea as in Mendel's experiments. The observable can also be the presence or not of eyes, such as in the species of fish Astyanax mexicanus which comes in at least two forms, with or without eyes (for the blind cave form) [START_REF] William | Cavefish as a model system in evolutionary developmental biology[END_REF].

Following the classical framework of quantum mechanics [START_REF] Cohen-Tannoudji | Quantum Mechanics[END_REF], in a population where only two possible phenotypes coexist: phenotype 1 and 2 can be symbolized by the two vectors |p 1 〉 and |p 2 〉, the observable as A. A population of similar organisms in homogeneous environment is described by a normalized state vector ψ = a 1 . |p 1 〉 + a 2 . |p 2 〉 with parameters a 1 and a 2 in R such that a 2 1 + a 2 2 = 1. Following the Born rule, the a i coefficient corresponds to a probability amplitude and a 2 i is the probability to measure the value |p i 〉 of the observable A. In practice this probability is obtained empirically from the observed frequency f i . It captures a first level of uncertainty about the result of biological experiments. The use of a probability amplitude leads to an additional degree of freedom in the choice of the sign: a i = -+ f i . We notice here that we restrict our use of probability amplitude to real value and not complex values as in quantum mechanics to avoid unnecessary degrees of freedom. The choice of the observable depends on the observer, its possible values and their empirical frequencies depend on the experimental system, this is the main difference between A, |p i 〉 and a i .

For example, in Mendel's experiment, the observable A corresponds to the outer appearance of peas, its possible values are "wrinkled" (symbolized by the vector |w〉) or "smooth" (symbolized by the vector |s〉) and their probability f i can be measured empirically, leading to the probability amplitude a i = -+ f i and the descriptor of a population of peas as a normalized state vector ψ = a 1 |w〉 + a 2 |s〉. Similarly, the fishes of the species Astyanax mexicanus can be characterized with an observable A detecting the presence of the eyes, its possible values are with eyes (|eyes〉) or eyeless (|eyeless〉) and as in the previous example, the distribution of phenotypes in a population of similar organisms can be described with the normalized state vector ψ = a 1 . |eyes〉+a 2 . |eyeless〉. If a finer approach is necessary, the observable can correspond to size of the eye instead of a binary detection of presence.

To summarize, we can formalize the distribution of phenotypes in a population of similar organisms in the following way:

-an observable A which is associated to a measurement procedure -a set of possible phenotypes, |p i 〉 i , corresponding to the possible values of the observable actually observed, they form the basis of eigenvectors of A -a set of probability amplitude, {a i } i , associated to the empirical frequency f i = a 2 i of each phenotype |p i 〉. To reduce the number of degree of freedom, we limit them to real values.

-a state vector ψ = i a i |p i 〉 describing the distribution of phenotypes in the observed population By increasing the size of the population of similar organisms under study it is possible to obtain new unexpected result for the observable, because the exploration of diversity is still at work (even in homogeneous environment). This is where historicity of individual organisms plays a major role. Indeed, an ontogenetic trajectory is a sequence of highly contingent events, for example stochastic gene expression, asymmetric cell division, epigenetic effects. These contingent events are constitutive of the final phenotype of an organism and thus play a role in the diversity observed. The list of possible phenotypes may not be completely predictable because of yet unobserved contingent event.

If a new unexpected phenotype is observed, it is necessary to update the space of possible. Say, we know that the space of possible H is well described with the list of possible phenotypes (|p 1 〉 , |p 2 〉 , ..., |p n 〉) for observable A. The new unexpected phenotype will be denoted |p n+1 〉. Then the new space of possible H new will be defined by the basis

(|p 1 〉 , |p 2 〉 , ..., |p n 〉) × (|p n+1 〉) = (|p 1 〉 , |p 2 〉 , ..., |p n 〉 , |p n+1 〉)
and the corresponding observable A will add |p n+1 〉 to the set of its eigenvectors. While not changing fundamentally the observable, this approach allows to capture a second level of uncertainty on the set of possible values of an observable. The dimension of the space of possible related to the observable is increased in a linear way. The fact that the space of possible is defined as a vector space allows to increment its set of possible values with a cartesian product. The use of a normalized state vector allows to avoid renormalization techniques when considering previous experimental results. Using this approach shows that the prediction of the possible results of a biological experiment is related to the knowledge of the observer and the context of previously observed phenotypes. We presuppose the unability to predict completely the list of possible phenotypes for a given observable.

It is possible, moreover, that during the course of evolution, a radically new observable emerges which doesn't fit in the space of possible of a previously defined observable, or, stated otherwise, that cannot be described with the same act of measuring. This is Goldsmith's hypothesis of hopeful monster. This level of uncertainty is captured by defining a new observable B . It is a much more stronger form of uncertainty than a probability since it requires to define a new space of possibles.

There is therefore a strong difference between an unpredictability in terms of probability of having a given phenotype, at the level of the set possible phenotypes for a given observable and the unpredictability of the observable itself. To illustrate these ideas, we present unpublished experimental data.

Squint experiment -an incomplete list of phenotypes

The squint c z35 mutant line of the zebrafish presents a phenomenon of incomplete penetrance and variable phenotypic expressivity ( [START_REF] Pei | Environmental and genetic modifiers of squint penetrance during zebrafish embryogenesis[END_REF] and chapter 5 of this dissertation).

Crossing homozygote mutants lead to progeny with various phenotypes, from complete cyclopia to two well formed eyes in unpredictable proportions. To quantify the variation in phenotypes we conceived a protocol characterizing embryos in the most reproducible way. Figure 5.4 presents the design of the experiment, 10 couples of identified homozygote mutant (Sqt -/-) were crossed during 8 consecutive weeks and progeny grown in controlled conditions. Each week, pictures of each embryo of the progeny were taken at 30 hours post fertilization. These pictures were used to measure an interocular distance. The number of embryo in the progeny of each pair was very variable through time even if the conditions were kept identical from one pair of embryo to the other. Table 5.1

shows the number of embryos through time for the different pairs considered. Notably, we can see that couple 2 and 10 were unable to produce embryo. Couple 6, 7, 8, 9 produced very few embryos during the course of the experiment (a total of 4, 10, 17, 8 respectively).

Couple 5 provided embryos on a regular basis.

Histograms corresponding to distributions of measures during 5 weeks are shown on figure 6.2 for couple 5 (pictures of week 2 were unfortunately unusable, reducing the number of samples). The first observation is that the proportion of cyclops is variable through time (embryo with null interocular distance) from 2% to 32% of the progeny. The second observation is that the distribution of phenotypes doesn't form a continuum between cyclops and two eyed phenotypes. We then observe that embryos with non-zero interocular distance can be classified according to a reduced set of phenotypic classes. In the course of the experiment, 3 phenotypic classes are identified, associated to the colors orange, green and blue on figure 6.2. These classes are associated to specific morphologies as shown on the photographs presented on the figure 6.2, and to specific values of inte-new phenotype orange would have emerged leading to increment the space of possible with the new phenotype: H new = H ×ket orange = (ket cyclops, ket green, ket blue, ket orange).

This example shows that it is possible to observe the emergence of new phenotypes for a given observable, here the interocular distance. It is not impossible that by continuing this experiment several more weeks, other new unexpected phenotypes would emerge.

The structure of the space of possible as a hilbert space is necessary to perform a cartesian product with the new direction defined by a new phenotype. The use of state vector allow to define probabilities in a space whose dimensions can be incremented while keeping its normalization. Figure 6.2: Distributions of interocular distance (id) among progeny for the pair number 5 during 5 weeks. The proportion of cyclops varies from 3.3% to 32% (id = 0µm). Mutants with wild type phenotypes are observed (blue id ≥ 140µm). Two intermediate phenotypes can be observed (orange 0 < id < 80µm and green 80µm ≤ id < 140µm). This example, is a good example of an experiment where, for a given observable, the list of possible phenotypes is not completely predictable and where the probabilities for each possible phenotypes seem unpredictable. Finding causal links for the constitution of the various phenotypes could allow to predict with certainty the value of the observable. However, these causal links can only be obtained by correlating the values of several observables.

Correlating several observables

Given populations of similar organisms, we may be interested in correlating various observables A, B . For example correlating phenotypes with genotypes or phenotypes with various environments. Most of the time, causal links are difficult to identify. Organisms involve multiple levels of organization where the causal relations can be top-down and bottom-up, for example for the molecular level to the cellular level through gene expression, or from the tissue level to the cellular level through mechanical constraints. In the work of Raj et al. [START_REF] Raj | Variability in gene expression underlies incomplete penetrance[END_REF], the phenotype under study is the intestinal cell fate during embryogenesis of the mutant nematode Caenorhabditis elegans, and it is correlated to stochastic gene expression in an underlying bimodal gene regulatory network. If considered jointly the space of possible becomes the tensor product of the two individual spaces of possible H A ⊗ H B since all possible combinations of observables values can be expected. The result of an experiment correlating two observables can be modeled as a state vector |ψ〉 in the product space. It can be written as a combination of the elements of the basis of

If considered independently, two observables gives two state vectors |ψ〉

H A ⊗ H B . |ψ〉 = i , j c i , j . |a i 〉 ⊗ |b j 〉, (i , j ) ∈ {1, ..., n} × {1, ..., m} (6.1)
The size of the new space of possible is the product of the size of the two spaces associated to each observables. The presence of non-linear causal chains, possible bifurcations along ontogenetic paths, complex interactions between levels of observations, implies that the structure of the tensor space may be non trivial.

Mendel's model of inheritance: a formal analog of entanglement

In this section, we use an idealization of Mendel's experimental results on transmission of hereditary character to show the necessity of a tensor product for the space of possible of two correlated observables. Mendel's experiment consisted in constructing pure lines of peas, where crossing two individuals with a given phenotypic traits led to individual with the same phenotypic traits [150]. These pure lines were then hybridized, giving rise to a

However this equality leads to the following system of equations .4 shows the non separability of state vector on the spaces corresponding each observable's subspaces. This is a formal analog of quantum entanglement [START_REF] Einstein | Can quantum-mechanical description of physical reality be considered complete?[END_REF]. This phenomenon stems from biological organization, the fact that multiple levels of organization are related to each other. Indeed, mixes of |P d om 〉 and |P r ec 〉 do not exist at the same time in a single organism.

                                           a.e = 0.

Discussion

We have shown in this article that biological variation can be handled to a certain extend with the mathematical framework of quantum mechanics. The distinction has been made between the unpredictability of the list of possible phenotypes corresponding to a well defined observable and unpredictability of possible observables. Variations in the choice of phenotypes is described with a probabilistic framework. This quantum probabilistic framework uses the structure of the hilbert space to be easily updated when new possible phenotypes are observed with a simple cartesian product between the actual space of possible and the direction defined by the new phenotype, a non trivial operation in Kolmogorov's simplex framework. The possibility to perform tensor product between spaces of possibles of several observables enables to consider complex causal relationships in a sound way.

To summarize, three levels of unpredictability are defined. The first one corresponds to the incompleteness of the set of observables, which is closely related to historicity of evolution and to the knowledge of the observer. The second one is related to the possible value of the observable, it involves the historicity of each developmental path and can be described with classical and quantum probabilities, as exemplified with the squint experiment. The third one is related to the complex correlations that relates various observables and is formalized with a tensor product, it is related to the organization of biological systems as shown with Mendel's idealized scheme of inheritance.

With no assumption on the mechanisms underlying biological variation, this approach throws light on the theoretical status of randomness in biology. The strongest form of randomness and its specificity with respect to physics is the unpredictability at the level of the set of observables [START_REF] Cs Calude | Classical, quantum and biological randomness as relative incomputability[END_REF], [START_REF] Buiatti | Randomness and multilevel interactions in biology[END_REF], [START_REF] Heams | Randomness in biology[END_REF]. It involves the observer as situated in space and time because the definition of the set observables is indexed on the knowledge of the observer, relying on retrospective analysis of the measure and an impossibility to predict future observable.

Chapter 7 Evolution and development: toward an ontogenetic tree

Abstract This chapter investigates the relations between individual developments, normal and pathological, and the space of possible forms. The ontogenetic tree structure is proposed to organize observations on mutant developmental paths. It serve as a basis to define a developmental distance between developments. This developmental distance can be compared to a phenotypic distance to quantify the path dependency of a phenotype. Finally, the measure of the rate of diversification in the zebrafish development supports the description of the pharyngula stage as a phylotypic stage.

Introduction

In this chapter, we propose some perspectives on the relationships between development and evolution. The interaction between development and evolution is a major question of evolutionary developmental biology ( [START_REF] Jay | Ontogeny and phylogeny[END_REF], [START_REF] Alberch | Size and shape in ontogeny and phylogeny[END_REF], [START_REF] Alberch | Ontogenesis and morphological diversification[END_REF], [START_REF] Smith | Developmental constraints and evolution: a perspective from the mountain lake conference on development and evolution[END_REF], [START_REF] Webster | The origin of species: a structuralist approach[END_REF]). Originally, Darwin's theory of evolution uses as a fundamental principle the notion of descent with modification but without characterizing the underlying mechanisms sources of variation [START_REF] Darwin | On the origins of species by means of natural selection[END_REF].

On the other hand Mendel's quantitative work on inheritance proposed a model explaining the distribution of phenotypes in controlled population through the transmission of inheritance factors [150].

Later during the XXth century, the proponents of the "Modern Synthesis" provided a synthesis between these two theories [START_REF] Huxley | Evolution. the modern synthesis. Evolution. The Modern Synthesis[END_REF]. Associated to this theory comes the idea that genotype and phenotype should be clearly separated, and that the random variations oc-147 cur at the level of the genotype. These approaches of population genetics lead to systems of equations such as P = G + E , a phenotypic trait P is the sum of genetic G and environmental E effects and thus Var(P ) = Var(G) + Var(E ) + 2.Cov(G, E ) where Var(P ) represents the variance of a phenotypic trait P , Var(G) represents the genetic variance, Var(E ) the variance in environmental conditions and Cov(G, E ) the covariance between variations in the genotype and the environmental conditions [START_REF] Kempthorne | An introduction to genetic statistics[END_REF]. As explicitly represented in this equation, this theory completely abstracts out the concept of development, and considers a direct mapping between the genotype and the phenotype, with selection acting at the level of the phenotype and variations occurring at the level of the genotype. However, these kinds of models have often been criticized for reducing organisms to their genes in a so-called "beanbag genetics" [START_REF] Mayr | Animal species and evolution. Animal species and their evolution[END_REF].

Taking the development into account as a phenomenon extended in time and occurring at different scales when considering evolutionary theories requires to introduce other concepts. The first problem consists in being able to compare developing organisms leading to phenotypes that are not necessarily comparable, for example between different species. Before the advent of developmental genetics, the concept of heterochrony during development has been proposed as a good operational tool to compare development in close or distant species ( [START_REF] Jay | Ontogeny and phylogeny[END_REF], [START_REF] Alberch | Size and shape in ontogeny and phylogeny[END_REF]). Heterochrony, which has first been introduced by Ernst

Haeckel, characterizes the changes in the timing of events during development. After having identified homologous parts in various organisms, it is possible to classify the transformation between developments in several classes, Acceleration, Neoteny, Hypermorphosis, Progenesis, Post displacement and Pre displacement depending of the changes in the control parameters which are the growth rate, the shape changes and the time of onset and offset of development of a given shape. These concepts have been successfully used to compare and understand the differences in the morphology of three species of salamanders.

Embryonic developments among animals of the vertebrate phylum has been shown to go through a very similar stage after presenting great variations in the early stages. This is the so-called phylotypic stage, leading to a developmental hourglass model. First proposed by Von Baer in 1828 [START_REF] Baer | Über Entwicklungsgeschichte der Thiere. Beobachtung und Reflexion.-Königsberg[END_REF], this model is considered as showing developmental constraints in development. The phylotypic stage has been supported by recent molecular evidence [START_REF] Kalinka | Gene expression divergence recapitulates the developmental hourglass model[END_REF], [START_REF] Irie | Comparative transcriptome analysis reveals vertebrate phylotypic period during organogenesis[END_REF].

Many morphological approaches have been put aside for many years after the development of developmental genetics in the 1980s. In particular, homeotic genes have been shown to be involved in the establishment of the body plan in many organisms [START_REF] Nüsslein | Mutations affecting segment number and polarity in drosophila[END_REF] , for example Hox genes in the Drosophila [START_REF] Regulski | Homeo box genes of the antennapedia and bithorax complexes of drosophila[END_REF]. After these discoveries, the new field of evolutionary developmental biology was concerned with explaining the evolution of these genes which regulate development in many organisms.

Opposite to this last perspective is the "physicalist" approach developed by Vincent Fleury who explains the body plan only with physical constraints on the tissue generated by the early embryo, considered as a "tissue flow" [START_REF] Fleury | Clarifying tetrapod embryogenesis, a physicist's point of view[END_REF]. This approach provides a very interesting perspective in terms of constraints undergone by the tissue which is a physical material. These physical constraints are relatively universal because they apply to any tissue having the same boundary conditions. However this approach only characterizes one level of organization, the tissular one, and neglect the behavior of individual cells.

The key concepts in the approaches bringing together development and evolution are the notions of canalization, plasticity, developmental stability, developmental constraints, evolvability and how development shapes evolution and how evolution constrain development [START_REF] Debat | Mapping phenotypes: canalization, plasticity and developmental stability[END_REF], [START_REF] Brigandt | From developmental constraint to evolvability: how concepts figure in explanation and disciplinary identity[END_REF], [START_REF] Peter | Evolution and development of shape: integrating quantitative approaches[END_REF]. Cryptic developmental variation can also be considered as an explanation for buffered phenotypic variability [START_REF] Duveau | Role of pleiotropy in the evolution of a cryptic developmental variation in caenorhabditis elegans[END_REF].

In the following, we would like to consider the developmental paths and the distribution of symmetry breaking as a way to characterize diversification. Following suggestions made in the frame of the so called dynamical structuralism, we propose here to link a historical perspective of morphogenesis with a comparison of resulting phenotypes using the zebrafish as a model organism [START_REF] Webster | The origin of species: a structuralist approach[END_REF]. First we will explore a concept proposed to correlate phenotype and genotype while including development in the Drosophila embryogenesis, under the name ontogenetic tree. We propose then to use this concept to link a "developmental proximity" with a "phenotypic proximity" in order to highlight structural constraints in developing zebrafish.

Reconstructing the ontogenetic tree of the Danio rerio embryogenesis 7.2.1 The concept of an ontogenetic tree

In a series of papers, Ho and Saunders develop the idea of an ontogenetic tree as a basis for a rational approach of taxonomy [START_REF] Ho | Rational taxonomy and the natural system[END_REF], [START_REF] Ho | An exercise in rational taxonomy[END_REF]. They refer explicitly to the Romantische Naturphilosophie in Germany after the seminal work of Goethe where he introduced the notion of morphology and the theory of transformation from an Urpflanze in 1790 [START_REF] Wolfgang Von Goethe | Versuch die Metamorphose der Pflanzen zu erklären[END_REF]. This relates also to the debate between Cuvier and Geoffroy Saint-Hilaire in 1830 on the existence of a unique body plan for all animals [START_REF] Piveteau | Le débat entre cuvier et geoffroy saint-hilaire sur l'unité de plan et de composition[END_REF], which later led to comparative anatomy. Of course, these conceptions have to be considered in their historical context and should not be transposed as such for contemporary science. On the other hand, in Darwin's evolutionary theory, published a few years later, the theory of forms and transformation is not central anymore, the main driving process comes from natural selection and thus adaptation. Morphologies diverge from each other among the genus, taxa, species, as the result of random variations. Although very interesting, these historical considerations belong to history of science and epistemology and should be discussed elsewhere. In their articles, Ho and Saunders refer to a set of experimental results resulting from genetic perturbations of the Drosophila embryos. These perturbations affect the segmentation process. What they argue is that these perturbations lead to phenotypic changes that cannot be completely arbitrary and necessarily depend on the developmental path.

Therefore, the final phenotypes are constrained by the intermediate stages through which the embryos are passing during development. The genetic perturbations generate successive bifurcations from the normal development. All of these developments are gathered together within an ontogenetic tree. Each bifurcation will generate a new branch in the tree. Figure 7.1 shows an illustration of the concept of ontogenetic tree for the Drosophila melanogaster in [START_REF] Ho | Rational taxonomy and the natural system[END_REF] based on data published in [START_REF] Ho | Ether-induced segmentation disturbances in drosophila melanogaster[END_REF].

The approach and the structure of an ontogenetic tree are very appealing concepts, although the underlying assumptions for the process of segmentation may not be relevant anymore in regard of contemporary results. The structure of a tree can be formalized mathematically. The distance between resulting phenotype can be characterized by considering the length of the common developmental path, which constrain highly the subsequent development.

Formalization of the tree

Developmental processes are usually described with successive developmental stages corresponding to the setting up of the different parts of the organisms. Because any variation in the development will affect all the subsequent stages of the process, we can describe hierarchically the possible paths that stem from each mutational event. The pooling of different development paths that arise from different mutational profiles results in a hierarchical tree where each bifurcation corresponds to a symmetry breaking. The symmetry breakings that we consider here consist, for any part of the organism, in 1) change in timing, 2) abnormal development, or 3) absence of development. We will call this tree an ontogenetic tree in reference to previous work done on the drosophila segmentation [START_REF] Ho | Rational taxonomy and the natural system[END_REF] presented in the previous subsection. Thus, we can compute a "developmental distance", two organisms will be developmentally close if they share long branches, i.e. if they share a long common history.

As a toy example, let's consider the following set of developmental sequences and symmetry breaking presented in table 7.1. Suppose that the development of the wild type (WT) is composed of four developmental stages {S1, S2, S3, S4}. Then, the developmental sequence associated to the wild type is {0, 0, 0, 0}. A symmetry breaking in a mutant development will be denoted by a 1 in the developmental sequence at the corresponding developmental stage. For example the mutant M1 undergoes a symmetry breaking compared to the wild type at the third stage, therefore its developmental sequence is denoted as {0, 0, 1, 0}. If several mutant developments are considered, they can be ordered with a

Observing the phylotypic stage

To explore these ideas further, we used the Zebrafish Model Organism Database (zfin.org) that gather current works on the zebrafish [START_REF] Bradford | Zfin: enhancements and updates to the zebrafish model organism database[END_REF]. This database relies on a temporal ontology that describes the main developmental stages and a morphological ontology that describes the main parts of the organisms. 7317 mutations have been registered, with the description of their development using this ontology. We transformed each development description in a binary sequence that summarizes the normal stages and the symmetry breakings as defined in the previous section. After having ordered them using a lexicographical order we were able to map them on the same tree as exemplified on figure 1.

The first observation that can be made is that the distribution of developmental paths is spread over most of the developmental stages. At first sight, density of branches among the various developmental stages seems uniformly distributed. Moreover, we have plotted in different colors the mutant developmental paths affecting the eyes (yellow) and the developmental paths described without indication of an effect on the eye. The two colors seem to be well distributed along the developmental stages, indicating that there are no preferential stages for effects on the eyes.

Are the mutant developmental paths distributed uniformly along the ontogenetic tree?

When and where do the higher number of development paths arise? One way to compute answers to these questions is by considering for each developmental stage the number of developmental path which undergo a first symmetry breaking at this stage, i.e. diverging from the wild type development. For example, with the toy data set presented on the table 7.1, there is one developmental path diverging from the wild type at stage S1 (M4), two at stage S2 (M2, M3) and one at stage S3 (M1).

Figure 7.3 shows the result of the computation of the number of diverging developmental paths for each stage of the zebrafish embryogenesis. This number can be interpreted as a diversity potential. We can see that except for the early embryogenesis the diversity potential increases with the succession of the developmental periods until the pharyngula period before decreasing with the hatching period. The pharyngula period corresponds to the period of organogenesis. This result is coherent with the fact that this period has been considered as the phylotypic stage, the main axis are already established and the variation can therefore take place on less critical parts of the organism.

Discussion and conclusion

The main flaws of this study reside in the nature of the dataset which can be biased by 1) the chosen structures and phenotypes for the ontology used to describe the development (epistemic flaw) 2) the description biases related to the teams making the experiments (epistemic flaw) 3) the mutagenesis methodology (sampling methodology) .

It could be interesting to provide a null model of tree, to show that the diversity potential is really significantly higher for the pharyngula period compared to randomly distributed changes along the developmental paths.

Overall, we see that the concept of an ontogenetic tree can be useful to compare a large number of developmental sequences. The divergence pattern between developmental sequences can explain some of the morphological similarity between species, since divergence is more likely after critical stages of development having been gone through.

Therefore the early stages of embryogenesis canalize the possible forms of final phenotype. Variation seems to proceed mostly on already established organization. Variation

Conclusion

The main idea unifying the studies presented in this part of the dissertation is to understand the relation between variations in individual development and generation of diversity. The multi-scale nature of organisms, i.e. their organization and the path dependency of development, i.e. their historicity, require to give to variability a specific role.

The review of the main mechanisms at the origin of diversity in Chapter 4 points out that variability can emerge at multiple scales in an organism, from gene mutation to stochastic effects and epigenetic events. The heterogeneous nature of these mechanisms make them difficult to integrate in a common framework. However they are relevant to understand diversity since they all have an effect on resulting phenotypes upon which natural selection act.

Turning to mathematics and physics doesn't simplify the picture. Indeed, the formalisms developed rely on different hypotheses, for example in chaotic systems theory and ergodic theory, the characterizations of randomness are different. The mathematics of probability theory provide a framework enabling calculus on uncertain events without defining any notion of randomness. The assumptions underlying the axioms, however, involve some modeling hypotheses on the described situation. These assumptions may not be always the most suited as witnessed by the probability theory developed in quantum mechanics, generalizing Kolmogorovian probability theory which is the most common framework used in classical situations.

Given the heterogeneous nature of mechanisms at the origin of diversity, and the heterogenous nature of mathematical frameworks used to model situation of uncertainty, it is argued that, as an alternative to a theoretical integration of sources of diversity, it is possible to use hypothesis based modeling and simulations to explore the range of possibles allowed by a set of identified mechanisms. However, the simplifying assumptions necessary to establish such models reduce the generality of their results.

After these general considerations on variability in the living, we turn our attention, in 157

Chapter 5, to a specific experimental system which shows surprisingly high levels of variability. Crossing homozygote zebrafish mutants lead to a phenomenon of variable phenotypic expressivity and incomplete penetrance, in the squint mutant line. The mutation affect the Nodal signaling pathway which is involved in the establishment of the main body axes of the zebrafish. A quantitative description of the distribution of phenotypes (interocular distance) in the progeny of identified homozygote mutant parents leads to the following properties of variability: the discrete list of possible phenotypes is explored in unpredictable proportions, this list is possibly incomplete. Trying to find determinants to this variability involve complex causal relationships.

Theoretical implications of the results of the squint experiment are investigated in Chapter 6. In this chapter, we propose an analogy with quantum mechanics to clarify the different levels of variability encountered in biology. Indeed, variability leads, for a given observable, such as the interocular distance in the squint experiment, to the observation of phenotypes in variable proportions. These proportions can be described in terms of probability. However, this list of possible phenotypes may be extended with the observation of an emerging new phenotype. In that case, the space of possible associated to the observable needs to updated, this can be performed with a cartesian product between the previous space of possible and the new direction defined by the new phenotype. This operation requires a vector space structure for the set of probability distributions. Finally, in the course of evolution a radically new phenotype can emerge, requiring new observables for its description. In that case, the new observables define new spaces of possibles.

This level of unpredictability is much higher than the one described by probability theory in a pre-given set of possible phenotypes. It is specific of the historical nature of biological objects. After having clarified these levels of uncertainty formally with a mathematical framework analog to the framework developed in quantum mechanics, it is possible to describe a way to infer causal links or correlation between observable values. Using Mendel's scheme of inheritance as an example, we show that the dominance-recessivity relationships imply a phenomenon formally analog to quantum entanglement. This effect can be explained by the fact, that given biological organization, two phenotypes cannot coexist in the same organism.

The path dependency of developmental processes implies that the relations between genotype and phenotype are not linear. The squint experiment shows that the relations can be highly complex. However, it is possible to try to measure and quantify the influence of path dependency on resulting phenotypes. This is the object of Chapter 7. In this chap-ter, we propose a formal structure, an "ontogenetic tree", to gather mutant developments and characterize their diverging patterns in relation to each other. Using this tree structure, we can compute a developmental distance between phenotypes by measuring the length of their common development. This developmental distance has to be compared to a phenotypic distance in order to highlight the effect of the different developmental stages on the resulting phenotype. As an example we use a data set of numerous descriptions of zebrafish mutant developments. By computing the ontogenetic tree, we show that the pharyngula period of development generates the highest number of diverging developmental paths, canalizing therefore subsequent degrees of freedom for development.

These results should however be manipulated carefully since multiple biases can exist in the data set.

Altogether, we show that variability at all scales has an effect on diversity. The integration of this variability in a unique framework is an open and difficult problem which is not reducible to already existing mathematical framework because of the organization and historicity of biological objects. It is a central problem of biology that should be considered in any study.

Part III

Quantifying biological shapes

Introduction

In this part, we consider the question of shape. We propose to study epithelial organization in developing embryos. Epithelial tissues constitute one of the simplest structure found in development. The question is to find invariants in this structure among several species and genetic backgrounds and way to quantify variability in a generic manner on this relatively universal structure.

One approach for the study of epithelial organization considers the network of cellular contacts from static images. This mathematical structure enables to characterize the spatial distribution of cells. Previous studies have used the tools of complex networks studies, however these approaches underestimate the role of underlying topological constraints of the tissue which reduce the power of these measures. Developmental histories leading to epithelial organization prevent to find simple symmetrical structures, epithelia are indeed the result of cell proliferation, cell motility and cell extrusion producing complex structures.

Departing from an analogy with the structure of the cosmic web which is highly historical in nature, we propose to define topological invariants based on the tools of persistent homology. Using this approach we were able to compare and classify a wide range of tissues and get back to species and genetic background classification.

To assess the significance of this approach, we introduce a model of random triangulated surface. This model constrains local characteristics: the degree distribution in the network is set. The choice of neighbors in the network is free and random, the only condition being that the resulting network can be embed in a surface. This approach uncovers the higher-order spatial constraints in the distribution of the cells and shows the existence of patterns involving groups of cells in significant proportions. These constraints can be interpreted as the mark of individual developmental histories. Relying on a data set consisting in confocal images of epithelia from Drosophila and Chick embryos, we perform a comparative study in Chapter 8.

163

Chapter 9 proposes some perspectives on a dynamical characterization of the network of cellular contacts. In particular we argue that time should not be considered as a similar dimension as space dimensions since its evolution changes the nature of the network. Finally we propose a measure that combine the study of genealogical relations between cells and their spatial organization at the same time. This measure should be able to quantify the unfolding of shape in space and time. this process results in planar graphs with appropriate degree distribution. We explore the topological characteristics of these surfaces and compare them to a set of empirical data.

Differences between the null model and the data provide insights for the understanding of underlying biological processes and equip us with a notion of "level of randomness". This notion can be used to evaluate the contribution of genetic factors such as the presence of Myosin II. 1

Introduction

Embryonic morphogenesis is a complex process involving regulations at all scales. Epithelia are simple tissular structures found in a wide range of embryo. An epithelium is a

1. The study presented in this chapter has greatly benefited from the supervision of Gunnar Carlsson during an extended stay at Stanford University Mathematics Department (Feb.-Jul. 2014). This work has been presented at the conference "Algebraic Topology -Methods, Computation and Science 6" at Vancouver, Canada (05/26-30/2014) 165 monolayer of densely packed cells. The spatial distribution of cells forms a complex structure shaped by cell proliferation, cell motility and cell extrusion throughout development [START_REF] Guillot | Mechanics of epithelial tissue homeostasis and morphogenesis[END_REF].

Resulting spatial structure and individual cell shapes within the tissue depend on this sequence of events. Cell shapes in 2D can be modeled as polygons. This model allow to relate cell proliferation to distribution of cell shapes in epithelia [START_REF] Matthew C Gibson | The emergence of geometric order in proliferating metazoan epithelia[END_REF]. A generalization of this approach considers the whole network of cellular contacts [START_REF] Luis M Escudero | Epithelial organisation revealed by a network of cellular contacts[END_REF]. The network of cellular contacts is an abstract representation of the epithelial organization. Each cell is a node and the contacts between cells are represented as edges. Similar patterns of cellular contacts has been studied in crack patterns [START_REF] Bohn | Hierarchical crack pattern as formed by successive domain divisions[END_REF], [START_REF] Bohn | Four sided domains in hierarchical space dividing patterns[END_REF], or in foam dynamics [START_REF] Kern | Two-dimensional viscous froth model for foam dynamics[END_REF].

The authors of [START_REF] Luis M Escudero | Epithelial organisation revealed by a network of cellular contacts[END_REF] import the tools of complex networks for the study of the network of cellular contacts. This approach is pursued in a subsequent study [START_REF] Sánchez-Gutiérrez | Topological progression in proliferating epithelia is driven by a unique variation in polygon distribution[END_REF]. We take in this chapter a different approach to study the network. Using invariants defined in algebraic topology, we propose to quantify high-order structures in the network of cellular contacts.

We develop an algorithm of discrete pattern recognition to describe structure involving small groups of cells in the network of cellular contacts. This approach gives us access to intrinsic and tissue-level characteristics of the network of cellular contacts.

Global characterization of epithelial tissues

Network of cellular connectivity

The structure of an epithelium can be described with the network of connectivity between the cells: two cells are connected if their membranes are in contact. This network can be formalized as a couple (V, E ) where V = {v 1 , v 2 , ...v n } is the set of vertices in the network corresponding to each cell in the tissue and E = v i , v j ∈ V 2 is the set of unoriented edges corresponding to each cellular contact in the tissue [START_REF] Albert | Statistical mechanics of complex networks[END_REF]. Figure 8.1 shows a representation of this network in a sample of tissue. This network is obtained by segmenting the cells in the image. Although it is represented in space, it is a purely combinatorial structure. In figure 8.1, the nodes positions correspond to mass center of the associated cells. It differs from a voronoi tessellation of the space based on the cell centers because regions defined by cell membranes may not be convex.

In a network, the degree of a node corresponds to the number of edges coming to this node, it corresponds to the number of neighbors of a cell. The frequency distribution of n-sided cells in a tissue as described in [START_REF] Matthew C Gibson | The emergence of geometric order in proliferating metazoan epithelia[END_REF] is exactly the degree distribution in the network of cellular connectivity. Figure 8.2 shows this distribution in different type of epithelia.

We can see that the shape of this distribution varies greatly from one type of tissue to the other, although they are all centered on 6-sided cells. paper [START_REF] Luis M Escudero | Epithelial organisation revealed by a network of cellular contacts[END_REF]. Authors of this paper use three measures to characterize this network: the degree, the clustering coefficient and the average degree of neighbors. The degree is the number of neighbors of a cell. The clustering coefficient is a measure of local packing which is defined for a cell as the ratio between the number of observed interconnections in its neighborhood and the number of interconnections in a complete graph having the same number of nodes (i.e. where all the nodes are interconnected). The average degree of neighbors is the averaged degree over the neighbors of the considered node.

Complex networks approach shows some limitations

The degree, the clustering coefficient and the average degree of neighbors are measures of network developed in the context of complex systems approaches [START_REF] Da F Costa | Characterization of complex networks: A survey of measurements[END_REF]. If complex network approaches are relevant for the study of sociological networks, they may show some limitations for the study of epithelia which are highly constrained by the underlying topology. Network of cellular contacts in epithelia can be assimilated to a triangulated planar graph as a first approximation. A planar graph has the property to be embeddable in a plane without any edge crossing. A triangulated planar graph has for dual network, a network where each node has degree 3. It is easy to show that in a planar triangulation, the clustering coefficient of a node having k neighbors is equal to 2 k-1 . Therefore, the measure of the clustering coefficient doesn't provide more information than the number of neighbors. The average degree of neighbors is a measure the connectivity of neighbors and gives information about wider context in the network. The accuracy of this last measure is nevertheless dependent on the dispersion of the degree in the network, and may not provide better information than the degree distribution itself for networks with low degree variation. These classical measures of complex networks have been introduced in the context of communications or sociological networks where the underlying space has a complex topology [START_REF] Albert | Statistical mechanics of complex networks[END_REF]. They have been very successful to describe Internet topology or networks of actors. The limitations presented here derive from the global topology of epithelia. It is thus necessary to introduce new measure that would be able to provide global information about the network of cellular contacts, and which are not reducible to the number of neighbors.

Alternative approaches can be found in the study of topological properties of networks.

There is a large mathematical literature that describe the topology of network ( [START_REF] Gross | Topological graph theory[END_REF]), yet the traditional characteristics like the euler characteristics or the genus may not be accurate to distinguish between different tissues since most of the time epithelia are planar graphs (or close to it). They are contractible, that means that they have the homotopy type of a point. Some differences may appear due to transient processes such as rosette formation that induces 4-sided or 5-sided junctions and appear transiently during the process of cell intercalation [START_REF] Blankenship | Multicellular rosette formation links planar cell polarity to tissue morphogenesis[END_REF], [START_REF] Trichas | Multi-cellular rosettes in the mouse visceral endoderm facilitate the ordered migration of anterior visceral endoderm cells[END_REF]. They are however described as unstable features of the networks and may not form the basis of stable signature of a tissue topology. The following will present more sophisticated tools aimed at finding robust signatures of the topology of the networks and unfolding the richness of their organization.

Persistent homology

A useful way to approach the topology of a complex landscape is to use the framework of persistent homology ( [START_REF] Carlsson | Topological pattern recognition for point cloud data[END_REF], [START_REF] Carlsson | Topology and data[END_REF], [START_REF] Edelsbrunner | Computational topology: an introduction[END_REF]). This method has been applied successfully in cosmology to characterize the filamentary structure of the cosmic web ( [START_REF] Sousbie | The persistent cosmic web and its filamentary structure-i. theory and implementation[END_REF]) among many examples. The idea is to define a sequence of nested subspaces of the investigated space indexed by a parameter and to observe how the topology of these spaces change with the parameter. These changes are then summarized by a barcode or a persistence diagram, which gives a topological signature of the space. This method refine the description of the space by providing a multi scale representation, unfolding its organization. Moreover, stability results show that this description is robust [START_REF] Chazal | Gromov-hausdorff stable signatures for shapes using persistence[END_REF]. can be interpreted as robust features features of the space whereas small bars can be induced by local small variations. Implementation of the algorithm is described in [START_REF] Zomorodian | Computing persistent homology[END_REF]. We used the javaplex library [START_REF] Adams | Javaplex tutorial[END_REF] to compute a barcode for each epithelium. Barcodes computed over the filtrations Sub * and Sup * defined above offer a multi scale representation of the epithelial topology. They carry useful information that can be used to investigate the global organization of the network of cellular connectivity. These barcodes provide a qualitative information that can be compared from one sample to the other because they represent similar features in the networks. Their differences reflect changes in the organization. To make these comparisons possible on a broad range of tissue samples it is however necessary to introduce quantitative features associated to each barcodes.

Quantitative comparison by computing features on top of barcodes

In order to compute quantitative features based on the barcodes which are the objects summarizing the evolution of the topological features of the spaces along the filtration, we used a methodology proposed in [START_REF] Adcock | The ring of algebraic functions on persistence bar codes[END_REF]. In this article the authors suggest to use the coordinates of the bars of a bacode as the domain of functions that will return quantitative value characterizing the barcodes. These functions have to satisfy a certain number of criteria that makes them sufficiently generic over different types of barcodes. These criteria Given the apparent separability of the clusters, it is expected that more sophisticated statistical analysis and machine learning tools would capture precisely the value of the features obtained for each type of tissue. These observations prove that images of epithelia can be compared and classified using persistent homology. Moreover, we will see below that computing persistent homology enables to characterize some aspects of the organization that are missing when considering the degree distribution alone.

Summary

The automatic pipeline that has been developed in this section is described by the following steps (see table 8.

Random surfaces with arbitrary degree distribution to model tissue topology

In the previous section we provided a method to compare samples of tissues taking into account their global organization. However, to gain a biological understanding of this organization, the differences observed in the features values need to be interpreted. We

propose in this section a model of random graphs that tries to make sense of the observed differences. The question that will support these investigations is the question of how local relations between cells affect the global organization.

Use of a null model

One way of answering the question of how local interactions between cells influence the global organization of the tissue is to use a null model that will match some properties of the network while keeping other properties free. Since the question is to understand how relations between cells affect the global topology of the network, the properties that will be set are the local properties of cells, thus the degree. The choice of the type of connections between vertices will then affect the global organization of the network. Comparing an empirical network with its random counterpart having the same degree distribution will highlight the role of the connections between nodes on topological features.

Topological hypotheses are necessary

The simplest model of random graph with arbitrary degree distribution is the Newman Strogatz model [START_REF] Mark Ej Newman | Random graphs with arbitrary degree distributions and their applications[END_REF]. In this model each node has a degree randomly drawn from the degree distribution. The nodes are then randomly connected to each other. With this construction process, the degree distribution is prescribed in the network and nothing is assumed about the relations between the nodes.

We generated random networks with empirical degree distribution for each sample of tissue. The previous section described a way to quantify global topology of any combinatorial network, thus random networks can be studied in the same way. This difference mostly rely on the barcodes obtain for Betti 1 (cycles). The barcodes corresponding to Betti 0, describing the simply connected components, are very similar in the two situations. First order topology is captured by the degree distribution. Thus, the barcode of Betti 0 seem to be in that case another representation for the degree distribution which is independent of the way nodes are related to each other. Higher order structures are much more different as highlighted by the differences in Betti 1. High order structures arise from the interconnection between nodes.

The topology corresponding to higher order structures seems to have a very specific organization in the actual sample of tissue compared to the random network. Connecting nodes randomly induces a complex underlying topology which is very far from the highly constrained network observed in biological tissues, which can be embeddable in the plane. This property canalizes these higher order structures. Our approach using persistent homology is proved to be relevant here, since the difference between the random network and the empirical data wouldn't have been noticeable based only on a comparison of the degree distribution.

Randomly gluing polygons

One way of describing epithelial topology, already proposed in [START_REF] Matthew C Gibson | The emergence of geometric order in proliferating metazoan epithelia[END_REF], is to approximate cells with polygons and consider epithelia as tessellation of the plane. Using this idea, a random epithelium can be modeled as a random tessellation that would maintain and control the distribution of polygons. Random orientable surfaces can be obtained by randomly gluing triangles [START_REF] Pippenger | Topological characteristics of random triangulated surfaces[END_REF]. The resulting triangulation forms a planar graph embeddable in the sphere. However these kinds of model don't control the degree distribution.

We generalize this model by randomly gluing polygons (they inherit the characteristics of the random triangulated surfaces since the dual graph is a random triangulation), while keeping the orientation.

The steps of construction of the random surface obtained by gluing polygons are described on the figure 8.11. The idea is to begin first with a randomly chosen polygon, then to fill its empty sides with randomly drawn polygons (while preserving the orientation and the degree distribution), and to reiterate with its neighbors. Whenever possible empty sides are connected with already existing empty sides (while preserving the orientation).

The orientability of the whole surface is guaranteed by constraining the orientation when adding any new polygon to the construction.

This model of random network imposes in addition to a prescribed degree distribution that the dual graph has degree three. It is the easiest way to construct a graph that is embeddable in the plane.

Any degree distribution can be provided as an input, we show several examples on figure 8.12. to the graph to be embeddable in the plane, confirming that this is a major constraint on this type of networks.

Topological characteristics of random surfaces

Before going into a systematic comparison of the data and the model, the stability and robustness of the model needs to be assessed. To evaluate the stability of the model, it is relevant to study its dependency to input parameters. The parameters are the number

Comparison of the null model and the data for each of the features

For each sample we computed the corresponding random network by using the empirical degree distribution and the same number of nodes. 5 realizations of the random model have been simulated for each sample of tissue.

For each of the features

We found that the data and the model have similar features values for the features 1,2 and that they vary more for features 3 and 4. These features have been defined in the first Distance between the null model and the data -"randomness level"

For each epithelium a degree distribution is computed using the network of cellular contacts, 5 random triangulated surfaces are generated according to this degree distribution. The 16 features described in section 8.2.5 are computed for the random surfaces. Distribution of random surfaces and the empirical samples are compared in the 16-dimensional space with a measure of proximity d .

For principal component i of the PCA defined in section 8.2.5

d i = e i . µ emp i -µ r and i σ emp i (8.8)
where e i is the eigenvalue associated to principal component i , µ

Discussion and Conclusion

The use of persistent homology introduced in this article provides a new tool to quantify patterns in networks of cellular contacts. A set of features characterizing the topology of the network at several scales can be obtained automatically from raw data using exclusively topological information. Epithelia of different types have reproducible features which can be used to classify and quantify them. Although those features are less optimal than the degree distribution alone to discriminate different tissue types, they are useful for characterizing patterns at the level of groups of cells. These patterns at the level of groups of cells (neighborhood) are inadequately described with traditional features used in complex networks analysis because they are highly constrainted by the (quasi)-triangulated nature of the network of cellular contacts. Using a null model of random triangulated surface with arbitrary degree distribution we quantify the presence of high-order patterns in the network. The significance of their presence varies between the various epithelial types and can be related to the specifities of the underlying biological processes.

Using this methodology, we quantified the contribution of Myosin II and cellular cytoskeleton to the formation of spatial patterns at the level of groups of cells. Indeed, mutant specimens present a higher proximity to null model and thus less significant presence of ordered pattern than the wild type. However, we see that chicken neuroepithelium (cNT) is even closer to the null model. The role of Myosin II is therefore not completely determinant for the presence of spatial patterns and should be combined to other processes to explain the structure of epithelia. The different nature of epithelia, squamous, tubular, may also be relevant to understand the specificities of the spatial distribution of cells. Finally the sequence of events, such as cell proliferation, cell motility, and cell extrusion, may be the most significant contributions to the morphogenesis of epithelia. To test and quantify the contribution of the various biological processes, it seems necessary to systematically perform perturbation experiments leading to a quantification of ordered patterns in the resulting networks of cellular contacts.

In order to increase the applicability of the proposed method, the approach could be generalized to characterize patterns and organization in 3D structures within epithelia without limits a priori. Dynamical evolution of epithelia may be a more difficult problem since the number and structure of the simplicial complex associated to the network of cellular contacts may vary from one time step to the other with cell motility and cell pro-

Introduction

In the previous chapter, we have proposed an approach characterizing the network of cellular contacts from static images. This approach was able to uncover specific higherorder constraints in the spatial distribution of cells that we ascribe to developmental histories. However, to confirm these assumptions and describe morphogenesis, it would be necessary to characterize dynamically the evolution of these networks. We propose in this chapter some perspectives in this direction.

A previous study which has gained much attention characterizes the dynamic nature of epithelial organization by means of cell division [START_REF] Matthew C Gibson | The emergence of geometric order in proliferating metazoan epithelia[END_REF], [START_REF] William | Cell topology, geometry, and morphogenesis in proliferating epithelia[END_REF]. A simple model describes the evolution of the cellular contacts as shown on figure 9.1. This model leads to the establishment of a markov chain whose evolution converges toward a distribution of polygons (degree distribution) found in various epithelia (Drosophila, Hydra and Xenopus).

However the distribution of polygons found in [START_REF] Matthew C Gibson | The emergence of geometric order in proliferating metazoan epithelia[END_REF] is not as universal as expected and patterns, or cell displacements to explain shifts from this theoretical distribution [START_REF] Sebastian A Sandersius | Correlating cell behavior with tissue topology in embryonic epithelia[END_REF],

[4], [START_REF] Ankit | Modeling and inferring cleavage patterns in proliferating epithelia[END_REF], [START_REF] Zallen | Planar polarity and tissue morphogenesis[END_REF], [START_REF] William T Gibson | Control of the mitotic cleavage plane by local epithelial topology[END_REF], [START_REF] Xiong | Interplay of cell shape and division orientation promotes robust morphogenesis of developing epithelia[END_REF], [START_REF] Sahlin | A modeling study on how cell division affects properties of epithelial tissues under isotropic growth[END_REF].

The original model proposed by Gibson et al. [START_REF] Matthew C Gibson | The emergence of geometric order in proliferating metazoan epithelia[END_REF] rely on the geometric properties of the cells in an epithelium. It postulates that the cells don't move and that the orientation of division is chosen randomly. These two hypotheses seem realistic and form the basis of the model, they can be modulated depending on the modeling context, for example if the polarity of the cell division is highly constrained or if the cells are highly mobile. In addition to this two hypotheses, the model use a mean field assumption. Under this assumption, it is postulated that the cells divide synchronously and that each cell gain one new contact from a neighboring dividing cell in average. This last assumption seem to be the most difficult, in particular in development, because epithelial organization is far from an equilibrium or asymptotic state. As shown in the previous chapter, patterns involving groups of cells are present in significant proportions in epithelia and the underlying topology constrains the network. Neighborhoods cannot be considered uniformly with a mean field assumption, they are biased by these patterns.

As shown in the previous chapter, when using a measure of organization taking into account higher order spatial structures, we find that the developmental sequence of events has had an influence on the resulting shape of the tissue. We will present in the next sections a measure considering at the same time the spatial organization and the history of the tissue after a short review of already existing measures.

Time evolving networks 9.2.1 Time evolution of static measurements

The most straightforward approach consists in observing the dynamical evolution of measures defined on static networks. For example, we can look at the average degree in an evolving network. The average and standard deviation of the degree distribution obtained from the network of cellular contacts in one of the digitally reconstructed sea urchin presented in Chapter 1 of this dissertation is shown on figure 9.2. This measure shows a tendency towards 6 neighbors in average and an increasing standard deviation.

Although very useful, this kind of approach can only describe the system at a macroscopic level since the detail of individual cell trajectories is not taken into account. Relating this measure with individual cell dynamics require a model, such as [START_REF] Matthew C Gibson | The emergence of geometric order in proliferating metazoan epithelia[END_REF]. A barcode can be represented as a persistence diagram where the coordinate axis are birth and death of these bars. A generalization of the barcode for time varying system is the concept of Vineyard as represented on figure 9.3 adapted from [START_REF] Munch | Applications of Persistent Homology to Time Varying Systems[END_REF], [START_REF] Cohen-Steiner | Vines and vineyards by updating persistence in linear time[END_REF]. The points in the persistence diagram are extended to "spaghetti" or "vine" according to a temporal dimension. This construction has been successfully used for the study of evolving set of sensors.

The main limitation of this approach is that vineyards require a continuity of the set in time, meaning that mitoses or cell extrusion should be excluded in our case. Moreover, this object is not completely stable and computable. The persistence diagram is updated at each time step -Image adapted from [START_REF] Munch | Applications of Persistent Homology to Time Varying Systems[END_REF] We see that extension of measure of static networks in time is appealing but not completely efficient to solve our problem.

Looking at spatiotemporal networks

Another line of thoughts consists in considering time evolution as another parameter of the space under consideration.

In algebraic topology, at least two possibilities have been developed. The first one is multidimensional persistence. This is the generalization of persistent homology to two or n-parameters [START_REF] Carlsson | The theory of multidimensional persistence[END_REF]. The main problem that appears with this approach is the fact that it has been shown that there is no simple summary like the barcode for 1-dimensional persistence [START_REF] Zomorodian | Computing persistent homology[END_REF]. Moreover, as for the vineyards, the multi-dimensional approach assume an identical underlying topological space in each direction. Yet, in our case, cells are proliferating, thus changing the nature of the underlying topological space.

The second possibility stems from a collaboration between the mathematician Edelsbrunner and the biologist Heisenberg [START_REF] Edelsbrunner | The medusa of spatial sorting: topological construction[END_REF]. They propose the concept of the Medusa to characterize the phenomenon of cell sorting during development, with an approach considering space and time as two similar dimensions. Cell sorting is a phenomenon where two types of cells that are initial mixed segregate into two spatially distinct populations.

This mathematical construction seem to be relevant for a problem where the parts are moving in time. However, they don't take the proliferation dynamics into consideration.

Using genealogy as a parameter -historical dependency of shape

As already suggested in [START_REF] Petit | Methods in clonal analysis and applications[END_REF] and in the previous chapter, the combination of cell proliferation and cell displacement produces the specific organization of the network of cellular contact. A good measure of this process should integrate these two phenomena. A solution can be found by considering the relatedness between cells as the criteria to construct the sequence of nested spaces in the filtration.

Given a network (V, E ) of cellular contacts, we can define relations among cells. Let's define a function R that returns the degree of relatedness of two cells. If two cells (i , j ) ∈ V 2 are sisters, i.e. they share the same mother cell, we will write R(i , j ) = 1. If the two cells (i , j ) ∈ V 2 are cousins, i.e. they share the same great mother cell, we will write R(i , j ) = 2. We can notice that R(i , j ) = 1 ⇒ R(i , j ) = 2, but the opposite is not necessarily true.

Similarly, if two cells (i , j ) ∈ V 2 share the same great great mother, we will write R(i , j ) = 3, with R(i , j ) = 1 ⇒ R(i , j ) = 2 ⇒ R(i , j ) = 3. R is defined in the same way for higher degree of relatedness. If no ancestor is known for a couple of cells (i , j ) ∈ V 2 , we write R(i , j ) = 0. This filtration contains for each value of the parameter k, all the vertices of the network.

The edges are included in the filtration with a condition on relatedness. The higher degree of relatedness is considered, the more complete is the filtration. By taking the clique complexes Fil * (k) on this filtration for each value of k as described in Chapter 8 section 8.2.3, we obtain the filtration Fil * on which we can apply persistent homology. In that case, persistent homology algorithm will measure the mixing of cells through displacements and proliferation.

For example if cells only divide with no displacement in a tissue, then the algorithm will detect a decreasing number of connected components and no holes. Whereas, if the cells are highly mixed, the number of connected components will decrease less rapidly and the number of holes will increase.

This measure is a first strong step toward solving the problem of the quantification of the intertwining between cell proliferation and cell displacement in an evolving network of cellular contact.

Conclusion

The perspectives presented in this chapter show that the problem of understanding the spatial organization of epithelia in the light of cell displacement and cell proliferation is interesting and require to be studied. The simple extension of static characterization of networks in a dynamical framework is not that simple. Several problems may arise from the use of another dimension when considering topological characterization. The proliferation undergone by the cells raises numerous problems as the topological nature of the tissue is changed in time. It is a difficult problem to match two simplicial complexes, it is much harder when the number of elements vary in time.

However, we present in the last section an appealing perspective that would allow to combine cell lineage and cell spatial organization. This measure which integrate cell proliferation and cell movements should help to understand the unfolding of tissue shape during morphogenesis.

Conclusion

The main goal of this part was to uncover the relations between spatial organization of tissue and development in epithelia, which is one of the most simple tissular structure.

This approach which tries to characterize invariant features among embryo of different species and between mutant and normal development is intended to reveal some universal pattern in the shape of embryo. Shape in a developing organism is dependent on the specific sequence of events occurring during its developmental trajectory.

The first problem raised by such aims is the question of finding a generic way of describing tissues. The network of cellular contacts is a good candidate. It is composed by nodes corresponding to the cells and by edges corresponding to their contacts. It is a purely combinatorial structure and is thus not affected by questions of size. Moreover, it allows to consider the level of individual cells in relation to the whole tissue.

Descriptive tools developed in previous studies investigating this question show some limitations because they neglect the high constraints of the underlying tissue topology. We proposed to reveal underlying topological invariants using the mathematical framework of persistent homology. This mathematical framework has been proved to be relevant in the study of the structure of the cosmic web which is a historical structure in the sense that its establishment involve random events at multiple scales and cannot be reduced to a small number of parameters.

To apply the mathematical framework of persistent homology on images of epithelia, we defined for each sample of tissue two filtrations using the number of neighbors as a proxy for density in the tissue. These filtrations, which are nested sequences of simplicial complexes, are the basis for the establishment of a multi-scale topological signature of the tissue. This signature can then be used to compare and classify tissue between samples from Drosophila and Chick embryos. Signature of common tissue types are clustered, suggesting the possibility of an automatic classification of embryo images.

The significance of these measures is assessed by comparing the topological signa-201 ture of a null model of random surface with arbitrary degree distribution. This model constrain local features, degree distribution, while letting the connection between nodes unconstrained. The construction process ensure that the resulting network can be embeddable in the plane, a realistic topological constraint. By comparing empirical samples with their random counterparts, randoms network with similar degree distribution, we show that the different types of tissues present different "degrees of randomness" in their spatial organization. Low "degree of randomness" reflects the significant presence of patterns involving groups of cells. These differences between tissues reflect the type of epithelium, squamous or tubular, and the presence of molecular determinant of shape such as Myosin II. We conjecture that these patterns of groups of cells are the traces of the sequence of developmental events.

Measures combining spatial organization and developmental history of a tissue are discussed in Chapter 9. It is shown that the dynamical study of measures defined on static networks doesn't seem sufficiently informative to understand the effect of individual cell histories without the use of models. Looking at the evolving network as a spatio-temporal structure is an appealing direction. However, the proliferative nature of the cells change the nature of the topological space under investigation in time, leading to difficult mathematical formalization. Finally, we propose a way to combine genealogical information with spatial organization by adapting the persistent homology approach defined in Chapter 8. This approach which uses the cell lineage should reveal the interaction between cell displacement and cell proliferation as the origin of tissular organization.

Altogether, we see that, even for a simple structure as an epithelium, the multi-scale nature of biological objects raises challenges that require the design of specific mathematical approaches. These mathematical approaches should tackle the question of organization and developmental history of organisms.

General conclusion

The approach developed throughout this work is a combination of theoretical, mathematical and experimental considerations concerning the measure and interpretation of variability in animal embryogenesis. This very general subject has led to a certain number of more specific studies, the main contributions are the following: The historicity of the developmental process is witnessed by the accumulation of variability and consequently the increase of the spread of distributions of individual cell features through time. This process underlies cell differentiation (chapter 3). These interesting relations between individual cell features and embryo-level dynamics shed new light on the sources of reproducibility. Where a finely-tuned process at the cellular level would have been expected, we find an averaged process over variable cell features.

⋆
The variety of mechanisms sources of variability in individual cells and of diversity in population of organisms is reviewed in chapter 4. This multiplicity of phenomena is the main obstacle for a proper characterization of biological variability. Their heterogeneity prevent their multi-scale integration. Indeed, we show that several concepts exist in mathematics and physics for the formalization of the idea of randomness which underlies frameworks modeling situations of uncertainty. These frameworks are not compatible as such. Two common properties are the need for an a priori space of possible and the use of an infinity to define this space of possible, or fluctuations in the case of chaotic systems.

To avoid having to integrate these various forms of randomness, an alternative can be to use relevant simplifying assumptions on other levels of organization or on past history.

The price being a reduced generality of the results.

An experiment presenting complex patterns of variability is presented in chapter 5.

This experiment involves the squint mutant line of the zebrafish. The results of this experiment show in the progeny of homozygote couples of mutants a phenomenon of variable phenotypic expressivity and incomplete penetrance; from complete cyclopia to two well formed eyes. We obtain embryos with phenotypes from a list of phenotypes in unpredictable proportions, without being able to ensure the completeness of the list. These results suggest complex scheme of inheritance of maternal factors and/or complex causal processes during development, supported by the various possible sources of variability described in chapter 4.

The variability observed and measured in chapter 4 motivates the reflections developed in chapter 5. These reflections rely on an analogy with the quantum mechanics mathematical framework as an attempt to account for the various levels of variability observed in the living. The interest of this analogy is to be able to consider sets of probability distributions as vector spaces. The problem consists in clarifying the difference between variable phenotypes in a fixed list (which can be described with classical probability), new emerging phenotypes to be added to the list of already known phenotypes (which is best described as a cartesian product) and the emergence of a new observable which require a new space of possible. These emergences of new phenotypes or new observables are derived from the historical nature of onto-and phylogenetic trajectories. In addition, this framework can account for complex relations between observables, with a tensor product coupling the space of possible corresponding to each of these observables. This property is used to account for Mendelian scheme of inheritance and is shown to be formally analog to quantum entanglement. We interpret it as a consequence of the organization of biological organisms preventing the coexistence of two phenotypes in the same individual.

To study the relations between variations in individual developments and patterns of diversification, we consider in chapter 6 the concept of an ontogenetic tree which enables to define a notion of developmental proximity. This approach is intended to compare a phenotypical similarity with the developmental one in order to quantify the influence of path dependency. Several assumptions are made concerning the canalization on possible phenotypes as a result of the constraints imposed by the already undergone developmental stages before diverging, this approach is underlied by an empirical data set describing numerous mutant zebrafish developments.

This second part on diversity and diversification involves many different situations.

The organized and historical aspects of development drive and canalize diversification, while being at the same time the result of this evolutionary history. And if biological variability cannot be reduced to randomness as defined in physics and mathematics, this dependence to an evolutionary history may give it its specific status. Biological variability implies a stronger form of randomness, at the level of the observable and thus of the space of possible itself.

Finally in chapter 7 and 8 we turned to a more specific problem, the characterization of epithelial organization. Although epithelia are simple cellular structures compared to complete organisms, their presence in many species during embryonic development gives to this problem a high level of generality. Epithelial organization is formalized as a network of cellular contacts whose underlying topological invariants are found to enable comparison and classification over a wide range of specimens. The framework of persistent homology is used to define these topological invariants. By comparing these tissue shapes to a null random model, we found that they contain pattern involving small groups of cells in significant number. We interpret these patterns as resulting from the sequence of events involved in the morphogenesis of these tissues. To characterize the relation between ep-

Figure 1 :

 1 Figure 1: BioEmergences epistemological triangle showing the various steps required from data acquisition, to phenomenological reconstruction, to modeling -Image courtesy of the BioEmergences platform see footnote 2

Figure 2 :

 2 Figure 2: Diagram representing the first steps of the sea urchin cleavage patterns, from[START_REF] Sf | Developmental Biology[END_REF]

Figure 3 :

 3 Figure 3: Normal sea urchin development, following the fate of the cellular layers of the blastula. (A) Fate map of the zygote. (B) Late blastula with ciliary tuft and flattened vegetal plate. (C) Blastula with primary mesenchyme. (D) Gastrula with secondary mesenchyme. (E) Prism-stage larva. (F) Pluteus larva. Fates of the zygote cytoplasm can be followed through the color pattern. (Courtesy of D. McClay.) -Image and caption from[START_REF] Sf | Developmental Biology[END_REF]

Figure 4 :

 4 Figure 4: Diagram representing the gene regulatory network of the Mesomeres during 6 to 17 hours post fertilization in the sea urchin Strongylocentrotus purpuratus from Eric Davidson's website at Caltech http://sugp.caltech.edu/endomes/ as of January 21, 2015

Figure 5 :

 5 Figure 5: Scheme representing the proposed approach to quantify inter individual variability in the sea urchin development and produce a prototypic model.

Figure 1 . 1 :

 11 Figure 1.1: (a) Reconstruction of digital specimens from 3D+time in toto imaging. First line -Volume rendering of raw images from 2-photon laser scanning microscopy, H2B-mCherry and farnesylated eGFP staining. Second line -Nuclei detection and cell tracking. Coloured dots represent the cell positions, cell trajectories displayed as streamlines over the next 5 consecutive time steps. Third line -Surface rendering of segmented cell membranes. Color code: Mesomeres (blue), Macromeres (red), Large (pink) and Small (purple) Micromeres. (b) Temporal sequences covered by the imaging of the five different specimens analyzed in this study.

  .1 (d)). Cell lineage combined with cell segmentation provides the life length and division time of cells, as well as the volume and surface area (Chapter 2section 2.3.1). Mesomeres, macromeres and large and small micromeres cell populations were marked at the 32-cell stage. Morphological changes at the embryo-level are witnessed by the evolution of the number of cells, the cell volume and the cell surface area (Chapter 2 -Figure 2.4 A, D, G). These dynamics have similar patterns in each specimen of the cohort. However spatio temporal rescaling overcome a first level of interindividual variability, making these dynamics comparable from one embryo to the other, in whole embryo and in each morphogenetic field (Chapter 2 -Figure 2.4 B, E, H). It consists in an affine transformation of the time dependency (two parameters -Chapter 2 -Figure 2.4 C) and a linear transformation of the spatial dependency (one parameter -Chapter 2 -Figure 2.4 F).

Figure 1 . 2 :

 12 Figure 1.2: Probabilistic modeling. Each individual of the cohort is associated to one color. The black lines with grey intervals correspond to 300 realizations of the model, mean (black) and standard deviation (grey). The first column (A,D,G) shows the value of the parameters, cell cycle lenth, cell volume, cell surface area for the prototypical representation of the cohort. The second column (B, E, H) corresponds to the model for one individual of the cohort. The third column (C, F, I) corresponds to the prototypical model over the cohort. The first line (B,C) shows the evolution of the number of cells, relying on the distribution of life lengths in the different subpopulations (A). The second line (E,F) shows the evolution of the cellular volume, relying on the distribution of the volume (D). The third line (H,I) shows the evolution of the cellular surface area, relying on the distribution of the surface area in the different subpopulations (G).

  .2 A, D, G). Intraindividual variability is represented by the prototypical standard deviation computed for each cell feature. A representation of the normal development of the sea urchin during cleavage is obtained by simulating prototypical embryo-level dynamics from the probabilistic model (Figure 1.2 C, F, I).

Figure 1 . 3 :

 13 Figure 1.3: A. Each dot represents a cell center, the edges relating them are calculated from the two steps spatial neighborhood algorithm, via metric and topological criteria. The red axes represent the cell axis U i oriented along the apico-basal direction. Membrane surfaces are calculated a posteriori for rendering purpose. Color indicates indentity similar to figure 1.1. B. Intensity of the attraction-repulsion force F ∥ exerted between two neighbor cells as a function of their relative distance. This force displays a similar shape to interatomic potential derived forces like Morse or Lennard-Jones (Chapter 2 -Figure 2.6). The force well is modulated by varying the appropriate attraction coefficient w adh . C. The cell axis U i (red arrow) is calculated by averaging the 6 surrounding triangle outward normal vectors (light red arrows). D. Phenotypic phase diagram, axis correspond to (w adh,e , w adh,o ) with k rig = 10000 and α gab = 1.0. Four distinct phenotype domain are obtained when exploring the parameter space.Figure by Julien Delile

  Figure 1.3: A. Each dot represents a cell center, the edges relating them are calculated from the two steps spatial neighborhood algorithm, via metric and topological criteria. The red axes represent the cell axis U i oriented along the apico-basal direction. Membrane surfaces are calculated a posteriori for rendering purpose. Color indicates indentity similar to figure 1.1. B. Intensity of the attraction-repulsion force F ∥ exerted between two neighbor cells as a function of their relative distance. This force displays a similar shape to interatomic potential derived forces like Morse or Lennard-Jones (Chapter 2 -Figure 2.6). The force well is modulated by varying the appropriate attraction coefficient w adh . C. The cell axis U i (red arrow) is calculated by averaging the 6 surrounding triangle outward normal vectors (light red arrows). D. Phenotypic phase diagram, axis correspond to (w adh,e , w adh,o ) with k rig = 10000 and α gab = 1.0. Four distinct phenotype domain are obtained when exploring the parameter space.Figure by Julien Delile

Figure 2 . 1 :

 21 Figure 2.1: Sample of the cell lineage indicating how life lengths are calculated.

Figure 2 . 2 :

 22 Figure 2.2: The four cell populations identifiable at the 32-cell stage: 16 mesomeres, 8 macromeres, 4 large micromeres, and 4 small micromeres.

Figure 2 . 3 :

 23 Figure 2.3: Example of a cell with seven neighbors (visualized with the Mov-IT software).

Figure 2 . 4 :

 24 Figure 2.4: Embryo-level dynamics before and after temporal and spatial rescaling. Starting from the 32-cell stage, the global measures of five specimens e = 1, ..., 5 are plotted over their respective imaging periods. A) Number of cells N e (t ) in each embryo. Original periods in minutes post-fertilization (mpf ): 225-396 mpf (blue), 345-507 mpf (green), 260-588 mpf (orange), 300-575 mpf (cyan), and 260-638 mpf (pink). B) Temporally rescaled number of cells in each embryo, N e (α e t + β e ) (top curves), and each morphogenetic field, N k e (α e t + β e ) (four groups of lower curves, from top to bottom: Mes, Mac, LMic, SMic). The rescaled periods are: 240-398 mpf (blue), 357-539 mpf (green), 246-702 mpf (orange), 291-619 mpf (cyan), and 265-531 mpf (pink). C) Affine transform parameters (α e , β e ) used in the previous top curves. D) Total cellular volumes W e (t ). E) Temporally and spatially rescaled total cellular volumes, (γ e ) 3 W e (α e t + β e ) (top), and field cellular volumes (γ e ) 3 W k e (α e t +β e ) (same order as B), using the same parameters (α e , β e ) as before. F) Coefficients γ e used in the previous top curves. G) Total cellular surface areas Z e (t ). H) Temporally and spatially rescaled total cellular surface areas, (γ e ) 2 Z e (α e t +β e ) (top), and field cellular surface areas (γ e ) 2 Z k e (α e t +β e ) (same order as B), using the same parameters (α e , β e ) and γ e as before. I) Temporally rescaled total numbers of contacts C e (α e t + β e ) and field numbers of contacts C k e (α e t + β e ).

  At this intermediate scale, it became possible to identify and map cell groups L n,k e across embryos. At the lower level, cells belonging to the same group were expected to display similar individual features. At the sixth generation, each embryo e contained a total of 32 cells composed of 16 Mes, 8 Mac, 4 LMic, and 4 SMic cells, which can be written: |L 6,1 e | = 16, |L 6,2 e | = 8, |L 6,3 e | = |L 6,4 e | = 4 for all e ∈ [1, 5].

Figure 2 . 5 :

 25 Figure 2.5: Mean and standard deviation bars representing the normal and log-normal approximations of the cell feature distributions in various cell groups L g . For each feature (one per frame), the cell groups g = (e, n, k) cover all four cell types k (except life length), one or several generations n ∈ [6, 10] per type, and most of the five embryos e per generation-type combination (using the same colors as Fig. 2.4).

Figure 2 . 6 :

 26 Figure 2.6: The 12 cell feature distributions (two per variable type) with the highest pvalue, i.e. the best cases of fit by normal or log-normal curves with respect to the chisquared test. Left to right, top to bottom: the two X g and two M g distributions closest to normal curves; the two V g , two S g , two A g and two B g distributions closest to log-normal curves. Titles indicate the embryo e, cell type k, generation n and p-value.

Figure 2 . 7 :

 27 Figure 2.7: The 12 worst-fit cases (p-value < 0.01) of cell feature distributions, where the assumption of normal or log-normal curves could not be retained with respect to the chisquared goodness-of-fit test. They fall into four n, k categories: 8, Mac; 8, Mes; 9, Mac; and 9, Mes. Most of them concern X and M .

Figure 2 . 8 :

 28 Figure 2.8: Histogram of all p-values obtained by chi-squared goodness-of-fit test on 124distributions of cell features. The six variables considered here were X g , M g , logV g , log A g , log S g and log B g across the five embryos e, four cell types k, and all generations n within the period during which the cell feature was observable.

Figure 2 . 9 :

 29 Figure 2.9: Estimated cell volume microdynamics ω g (u) in various groups L g of types k = 1, 2, 3 and generations n = 6, 7, 8, 9. Thin lines correspond to individual embryos e of the cohort E (one or several per group). Thick lines represent cohort averages 〈 ω g (u)〉 e∈E .

Figure 2 . 10 :

 210 Figure 2.10: Estimated cell surface microdynamics φ g (u) in various groups L g of types k = 1, 2, 3 and generations n = 6, 7, 8, 9. Thin lines correspond to individual embryos e of the cohort E (one or several per group). Thick lines represent cohort averages 〈 φ g (u)〉 e∈E .

Figure 2 . 11 :

 211 Figure 2.11: Histogram of the coefficient of determination R 2 = ρ 2 X ,Y for the various pairs of distributions X and Y displayed in Table 2.1. Only 20 values rank above 0.6.

Figure 2 . 13 :

 213 Figure 2.13: Probabilistic modeling for each embryo of the cohort E A) D) G) in color number of cells in embryo 1,2,3 -in black and grey mean and standard deviation of 300 realizations of the model B) E) H) in color cellular volume in embryo 1,2,3 -in black and grey mean and standard deviation of 300 realizations of the model C) F) I) in color cellular surface in embryo 1,2,3 -in black and grey mean and standard deviation of 300 realizations of the model -The shaded area on the right of B, C, H and I refers to incomplete cell cycles

Figure 2 .

 2 Figure 2.14: Fig. 2.13 continued J) M) in color number of cells in embryo 4,5 -in black and grey mean and standard deviation of 300 realizations of the model K) N) in color cellular volume in embryo 4,5 -in black and grey mean and standard deviation of 300 realizations of the model L) O) in color cellular surface in embryo 4,5 -in black and grey mean and standard deviation of 300 realizations of the model -The shaded area on the right of K, L, N and O refers to incomplete cell cycles

Figure 2 . 15 :

 215 Figure 2.15: Schematic representation of the concept of prototype in the relevant morphospace -the morphospace considered here is the statistical manifold S and the prototype is the centroid of the cohort computed as a minimization problem described in equation 2.74

2 . 74 )

 274 To actually compute the coefficients used in the simulations, this minimization problem is solved numerically with the estimated empirical parameters for each cell group L g . The results of the computation are shown on Fig.2.16. The distributions of life length X and mitosis time M within cell groups L g are assumed to be normal distributions, the prototypical coefficients are computed directly from the equation 2.74. The distributions of volume V , surface area S, daughter/mother mean volume ratio A and mother/daughter mean surface area ratio B are assumed to be log-normal distributions, thus the prototypical coefficients are computed from the logarithm of these random variables which are normally distributed.

Figure 2 . 16 :

 216 Figure 2.16: Prototypical coefficients for each group of cell L n,k A) Mean and standard deviation of the life lengths B) Mean and standard deviation of the division times C) Mean and standard deviation of the log of the volume D) Mean and standard deviation of the log of the surface area E) Mean and standard deviation of the log of the daughter/mother volume ratio F) Mean and standard deviation of the log of the daughter/mother mean surface ratio

Figure 2 . 18 :

 218 Figure 2.18: Cells are represented by cylindrical particle. Each cell is defined by a lateral radius R, a half-height R ⊥ and a normal vector U . -Figure by Julien Delile

Figure 2 . 19 :

 219 Figure 2.19: Cell axis determination procedure. The cell axis U i (red arrow) is calculated by averaging the 6 surrounding triangle outward normal vectors (light red arrows).In order to define the surrounding triangles, each neighbor position is first projected on the plane orthogonal to U i (here, only p 1 the projection of the neighbor a 1 is displayed). Then neighbors are sorted according to the angle α j formed by the vector (i , p j ) and an arbitrary vector belonging to the orthogonal plane (the light red and dashed line). Figure by Julien Delile

  under which cells can adhere, finally r i j corresponds to the distance between the two cells. The magnitude of this force is plotted on figure 2.20 (a) with varying values of w adh . Alternative possible models are shown on figure 2.20 (b).

Figure 2 .

 2 Figure 2.20: (a)Plot of the attraction-repulsion force F ∥ modulated with the variable values of w adh (b) Comparison of the attraction-repulsion force used here with alternative classical attraction-repulsion forces. The equilibrium distance is r eq = r

Figure 2 . 21 :

 221 Figure 2.21: Schematic of the planar rigidity force between neighbor epithelial cells.Here, cells i , j and k are represented in 2D, with their respective normal axes U i (yellow arrow), U j (red arrow) and U k (blue arrow). The planar rigidity forces exerted between neighbor cells i and j , and between i and k are aligned colinearly with the shared normal vectors n i j (orange vector), and n i k respectively. Forces are displayed by the thicker arrows. The intensity of the force is lesser between i and k because the relative position vector u i k and n i k are nearly orthogonal. As a result of these forces, neighbor cells are always attracted back to the lateral domain of each cell and thus maintain the planarity of the tissue. Figure by Julien Delile

Abstract

  In this chapter we explore perspectives raised by the data-driven multi-level probabilistic model developed in the previous chapter. We begin by characterizing the algebraic structure underlying the relations between random variables, leading to identyifing a Monoid structure. This structure can form the basis of a dynamical system representation of the model enabling to derive some properties such as stability of the dynamics or law of evolution. We also use this model to study its relation to previous theoretical proposition in developmental biology.

Table 3 . 2 : 5 3. 1 The probabilistic model implies a monoid structure 3 . 1 . 1

 3251311 is the percentage of elapsed life length of cell i Instantiation of the model independently in any branch of the cell lineage. The stochastic components are highlighted in bold-the definitions of the different notations can be found in the previous sections 2.5, 2.5, 2.Monoid Structure It is interesting to look at the algebraic structure underlying the successive addition, or multiplication, of independent variables, either for the relation:M n,k = X n,k + M n-1,k or for the relations V n,k = A n,k .V n-1,k and S n,k = B n,k .S n-1,k, which are equivalently described as addition of independent variables as lnV n,k = ln A n,k + lnV n-1,k and ln S n,k = ln B n,k + ln S n-1,k . The basic property of a sum of independent random variables is that their probability laws are convolved. This property is particularly useful in the case of gaussian distributions because the convolution of two gaussian distributions is a gaussian distribution. Since all the probability considered here are gaussian distributiion (for the ln version of the volume and surface area), we obtain a monoid structure for the random variables equipped with the addition.Sum of independent gaussian random variableIt can be shown that the set of gaussian distributions with the convolution product forms a Monoid. The neutral element is N (0, 0), the convolution of a gaussian distribution gives a gaussian distribution (stability). If gaussian distribution has a strictly positive standard deviation, then it cannot be inversed. The evolution of the distribution of mitosis can only be toward more desynchronization and thus disorganization. The development can be seen as the monoid action on the group of cells. In terms of symmetry, it can be said that the symmetry identified with the exchangeability principle are conserved by the monoid action.

  One possibility for the analytic characterization of the function ψ is to characterize the evolution of probability distribution through time by way of an equation. One candidate equation could be the Fokker-Planck equation, indeed Brownian motion of a particle is described by a stochastic differential equation d X t = µd t +σdW t , where the X t are particle positions in R n . µ is the drift velocity, σ is an n × n matrix and dW t represents an n-dimensional normal Wiener process. The Fokker-Planck equation (also called forward Kolmogorov equation) describes the temporal evolution of the probability density p(X t ): ∂p ∂t = -∇.(µ.p) + ∇.(D∇p), where D = 1 2 σσ T If µ and D are constant, the Fokker-Planck equation reduces to a drift-diffusion equation that can be solved analytically. The fundamental solutions are Gaussian distributions which drift and widen with time. 1

Figure 3 . 1 :

 31 Figure 3.1: Epigenetic landscape I. "The path followed by the ball, as if rolls towards the spectator, corresponds to the developmental history of a particular part of the egg." (Waddington CW. The Strategy of the Genes. London, Allen and Unwin (1957), [215])

  work aimed at characterizing intra-and inter-individual variability at the level of the individual cell. By studying the dynamics occurring in a cohort of 5 digital embryos, we showed that an intermediate level of observation was required between reproducible embryolevel dynamics and variable individual cell features. This coarse-grained level of observation served as a basis for the establishment of a data-driven multi-level probabilistic model. This model was found to have an invariant structure among individuals of the cohort. Taking advantage of the branching structure of the cell lineage, the value of parameters governing probability in the model were sufficient to identify uniquely specimen and reproduce embryo-level dynamics. The invariance of the structure relating individual cell features and embryo-level dynamics enabled to define a prototypical representation of the cell lineage associated to individual cell features.

Finally, this

  data-driven multi-level probabilistic model sheds new light and opens some perspective on theoretical conception of the development. Indeed, the traditional view of development as a finely tuned process is weakened by the observed variability at the individual cell level. The good adequacy of the empirical data with the probabilistic description enables to suggest some theoretical hypothesis, such as the development could rely on this variability at the individual cell level. This could be interpreted as an[START_REF] Heams | Randomness in biology[END_REF]

Figure 4 . 1 :

 41 Figure 4.1: Illustration of Luria Delbrück fluctuation test. Schematic trees represent cell proliferation in an environment containing a virus. Red colonies survive the virus attack. Mutations induced by the presence of virus, case (A), are present in similar proportions in every tree. Spontaneous mutations enabling survival to virus attack, case (B), appear randomly in the trees resulting in highly fluctuating size of colonies -Adapted from wikipedia

Figure 4 . 2 :

 42 Figure 4.2: This figure is adapted from Elowitz et al. 2002 paper on stochastic gene expression [65]. A and B show the basic concept of the experiment: two genes having the same regulatory sequence express two different flurorescent protein. In the case A, both genes have the same expression pattern, in the case B, the two genes have differing gene expression patterns. Figure C shows the result of an experimental measurement for this experiment in a strain of Escherichia coli (RP22). The distribution of colors from green to red proves the presence of stochastic gene expression in individual cells

  embryogenesis. They allow to describe and explore the possible results of these morphogenetic processes. They give an idea of what may be the repertoire of possible forms determined by the mechanisms. Yet, the question of knowing what is the scope of these models is still open. To what extend can we draw general conclusions from the study of these mechanisms considered separately? What is the legitimacy of the assumptions made in the models? Are these approaches accounting for the aspects of variability described previously? 8. Thèse Julien Delile : « From Cell Behavior to Tissue Deformation: Computational Modeling and Simulation of Early Animal Embryogenesis », under the supervision of Nadine Peyriéras and René Doursat

Figure 5 . 1 :

 51 Figure 5.1: Main stages of Zebrafish development. The names of the stages and the corresponding developmental times are indicated below the figures. For the first stages, cells are represented in blue and the yolk sac in yellow. After 12h of development the distinction between the yolk sac and the embryo is clear. Adapted from[START_REF] Kimmel | Branching Processes in Biology[END_REF]

  At 30 hours post fertilization (hpf) the embryo shows an almost final morphology, in particular its head is well formed, and its two eyes are separated by the forebrain. One of the main signaling pathway involved in this morphogenesis is called Nodal. Defects in Nodal related genes such as squint, oep or cyclop lead to dramatic phenotypic defects in the midline such as cyclopia and in the establishment of the embryonic axes [169] [44]. Similar defects are found in human embryos; defects of the midline division of the developing forebrain into cerebral hemisphere with concomitant facial midline defects are known as holoprosencephaly (HPE) [173]. Examples of the consequences of these developmental defects in humans and mice are shown on figure 5.2.

Figure 5 . 2 :

 52 Figure 5.2: Spectrum of craniofacial phenotypes in humans (a-e) and Twsg1-/-mice (fj). (a) Single central incisor; (b) microcephaly, midface hypoplasia with bilateral cleft lip and palate; (c) cyclopia with proboscis above the fused eye; and (d) hypotelorism and a single nostril. (e) Agnathia with downward displacement of the ears and microstomia. (f) Wild type; (g) severe anterior truncation; (h) cyclopia with proboscis; (i) single nostril with agnathia; and (j) agnathia.Figure and caption from[START_REF] Petryk | Holoprosencephaly: signaling interactions between the brain and the face, the environment and the genes, and the phenotypic variability in animal models and humans[END_REF]

Figure 5 . 3 :

 53 Figure 5.3: A. Figure showing the insert of 1.9kb in squint gene (mutant squint c z35 ) [71] B.Incidence of cyclopia in crossing of identified couple of mutants, from[START_REF] Pei | Identification of common and unique modifiers of zebrafish midline bifurcation and cyclopia[END_REF] C. Wild type phenotype in a homozygote squint c z35 mutant D. Cyclopia phenotype in a homozygote squint c z35 mutant E. Proportion of cyclopia in random crossings of homozygous mutants rule out simple models of inheritance, results from[START_REF] Pei | Identification of common and unique modifiers of zebrafish midline bifurcation and cyclopia[END_REF]

Figure 5 . 4 :

 54 Figure 5.4: Protocol. Every week during 8 weeks, all embryo of the progeny of identified couples of homozygote squint mutants were photographed at 30 hpf in controlled fertilized condition.

3 and pair 4

 4 generated embryos half of the time of the experiment, i.e. 4 times. Pair 7 and pair 8 generated embryos two times with a low number of embryos (max 16 embryos). Pair 6 and pair 9 generated embryos only once and it was less than 8 embryos. Pair 2 and pair 10 were unable to reproduce during the experiment.

Couple 5

 5 We first consider a single pair throughout the experiment. The pair number 5 has the highest number of embryo (283) during the 8 weeks of the experiment providing therefore the most significant results. The distribution of interocular distance are reported on figure 5.5, five histograms are shown corresponding to weeks 1, 3, 4, 6, 8. The experiment performed during week 2 could not be exploited for measurement of the interocular

Figure 5 . 5 :

 55 Figure 5.5: Proportion of Cyclops and intermediate phenotypes for the pair number 5. Histograms correspond to week 1, 3, 4, 6, 8 of the experiment and show the distribution of interocular distance (id). The proportion of Cyclops varies from 3.3% to 32.4%. Mutants with wild type phenotypes are observed (blue id ≥ 140µm). Two intermediate phenotypes can be observed (orange 0 < id < 80µm and green 80µm ≤ id < 140µm).

Figure 5 . 6 :

 56 Figure 5.6: Each histogram represents the distribution of every egg clutches merged. Results are shown for each 8 pairs having been able to have a progeny (pairs: 1, 3, 4, 5, 6, 7, 8, 9).

Figure 5 . 7 :

 57 Figure 5.7: Possible causal chains between late phenotypes (eg. interocular distance) and early phenotypes (eg. spatial distribution of β-catenin). Each experiment trying to link early phenotypes with late ones should deal with these causal pathways a) simplest causal chain where an early phenotype can be linked uniquely to the late phenotype b) symmetry breakings occur lately in development c) two early different phenotypes are buffered into one late phenotype d) the late phenotype counterpart of an early phenotype has not been observed e) the early counterpart of a late phenotype has not been observed f) early and late phenotypes have not been observed but are possible

Figure 6 . 1 :

 61 Figure 6.1: Protocol. Every week during 8 weeks, progeny of homozygote squint mutants (Sqt -/-) couples were systematically photographed at 30 hours post fertilization (hpf ) in controlled fertilized condition to measure interocular distance

  A and |ψ〉 B , each of them being a distribution of phenotypes in a population of similar organisms in homogeneous environment. They are linear combination of their eigenvectors. These eigenvectors define two spaces of possible, H A and H B having for basis (|a 1 〉 , |a 2 〉 , ..., |a n 〉) and(|b 1 〉 , |b 2 〉 , ..., |b m 〉) respectively.

Figure 6 . 3 :

 63 Figure 6.3: Scheme representing Mendel's model of inheritance in the first generation from the hybrids. On the left, the distribution of inheritance factors in the population with their relative frequencies (genotype). On the right, the distribution of phenotypes with their relative frequencies.

  first generation of hybrids showing only one of the two phenotypes. The first generation from hybrid was then presenting the two ancestors phenotypes in proportion 3 to 1. To understand these distributions of phenotypes, Mendel proposed a model of inheritance involving hereditary factors and a phenomenon of recessivity and dominance. This model is summarized on figure 6.3 for the first generation from the hybrids. The first observable is the genotype, with the hereditary factors A and a. In the first generation of hybrids possible genotypes are: {|A A〉 , |a A〉 , |Aa〉 , |aa〉}. The second observable is the phenotype ("wrinkled" or "smooth"), the set possible phenotypes is {|P d om 〉 , |P r ec 〉}, dominant and recessive phenotype respectively. The state vector describing the distribution of phenotypes |ψ〉 can be written, according to the idealized experimental results: |ψ〉 = 0.25. |A A〉⊗|P d om 〉+ 0.25. |Aa〉⊗|P d om 〉+ 0.25. |a A〉⊗|P d om 〉+ 0.25. |aa〉⊗|P r ec 〉 (6.2) If the two spaces of possible were separable, we should be able to write the state vector as |ψ〉 = (a. |A A〉 + b. |a A〉 + c. |Aa〉 + d . |aa〉) ⊗ e. |P d om 〉 + f . |P r ec 〉

25 a

 25 . f = 0 b.e = 0.25 b. f = 0 c.e = 0.25 c. f = 0 d .e = 0 d . f = 0.25 which has no solution. Therefore we have |ψ〉 = (a. |A A〉 + b. |a A〉 + c. |Aa〉 + d . |aa〉) ⊗ e. |P d om 〉 + f . |P r ec 〉 (6.4) ∀(a, b, c, d , e, f ) ∈ R Equation 6

Figure 7 . 1 :

 71 Figure 7.1: The ontogenetic tree of the observed morphologies in Drosophila with hypothetical intermediate phenotypes (dotted outline) based on developmental rules. Figure adapted from [101]

Figure 7 . 2 :Figure 7 . 3 :

 7273 Figure 7.2: Reconstruction of the zebrafish ontogenetic tree using zfin.org database. Each leaf of this tree corresponds to a set of mutant phenotypes that share a similar history, each node corresponds to a developmental stage, each branching corresponds to a symmetry breaking. Nodes are colored in yellow if they correspond to phenotypes having a default on the eye and in red if not

  cannot be considered as leading to any arbitrary possible phenotype. This view relates to von Baer conception of diversification by successive specialization. Understanding the development in different species, could explain and help classify the morphological differences observed in later stages.

Abstract

  Epithelial tissues are simple cellular structures found in developing embryo in different species. The organization of these tissues has been studied using local properties of the network of cellular contacts. However, accounting for global and spatial properties requires extending these approaches. Using persistent homology on the network of cellular contacts reveals global topological characteristics. To assess the significance of these characteristics, we provide a model of random triangulated surfaces with arbitrary degree distribution. These oriented surfaces are obtained by randomly gluing oriented polygons;

Figure 8 . 1 :Figure 8 . 2 :

 8182 Figure 8.1: Sample of epithelium from the Drosophila Wing Prepupa where the cells membranes are represented in black and the network of cellular connectivity is visualized in blue. The vertices of the network are placed on the mass centers of the cells, although the network of cellular connectivity is a purely combinatorial object

Figure 8 . 3 :

 83 Figure 8.3: Schematic representation of the network of cellular contacts. k blue edges connect a node to its neighbors. Red edges connect neighbors to each other. The clustering coefficient is defined for a cell as the ratio between the number of observed interconnections in its direct neighborhood and the number of interconnections in a complete graph having the same number of nodes (i.e. where all the nodes are interconnected). A) Most common case, clustering coefficient C = 2/(k -1) B) Case with a 3-sided cell, C = 2.(k + 1)/(k.(k -1)) C) Case with a 4-sided junction, C = 2.(k + 1)/(k.(k -1))

Figure 8 . 4 :

 84 Figure 8.4: Example of a Clique complex on top of a network. Nodes are in red, edges in black, triangles in light blue when three nodes are connected, tetrahedron in deep blue when four nodes are connected -From wikipedia

Figure 8 . 5 :

 85 Figure 8.5: Sequence of simplicial complexes Sub * (k) k for the same sample of tissue represented in figure 8.1 (from Drosophila Wing Prepupa), with k = 5, k = 6, k = 7 and k = 8. Cell membranes are represented in black and simplicial complexes in blue. Nodes are positioned on the mass center of cell surface area, although simplicial complexes are a purely combinatorial structure.

Figure 8 . 6 :

 86 Figure 8.6: Example of a torus, its Betti numbers have for values β 0 = 1, β 0 = 2, β 0 = 1 -Source wikimedia

Figure 8 . 7 :

 87 Figure 8.7: Barcodes for Sub * for the sample of tissue represented in figures 8.1 and 8.5. Each individual blue line correspond to a generator of the i-dimensional homology. Dimension 0 shows the number of connected component. Dimension 1 shows the number of holes. In each graph abscissa correspond to the value of the parameter in the filtration, here the degree of a node in the network

Figure 8 . 8 :

 88 Figure 8.8: The 52 samples of epithelium are well separated using the main characteristics of the degree distribution. Each point corresponds to one sample of tissue and is plotted in the space defined by the mean number of neighbors versus the standard deviation. Each color corresponds to one type of epithelium -Magenta: Drosophila Wing Prepupa (dWP) -Green: Drosophila Mutant Wing Prepupa (dMWP) -Red: Chick Neuroepithelium (cNT) -Blue: Chick Ectoderm (cEE)

Figure 8 . 9 :

 89 Figure 8.9: Principal component analysis performed on the quantitative features, defined in section 8.2.4, associated to each sample of tissue -Magenta: Drosophila Wing Prepupa -Green: Drosophila Mutant Wing Prepupa -Red: Chick Neuroepithelium -Blue: Chick Ectoderm

↓

  filtration on the sub and super level sets of the number of neighbors ↓

Figure 8 . 10 shows

 810 the comparison of the barcodes between an actual sample of tissue (on the left) and the random network having the same degree distribution (on the right). The comparison of these two figures shows a clear difference in the barcodes of these two networks.

Figure 8 . 10 :

 810 Figure 8.10: Comparison of the barcodes for an actual sample of tissue (a) and its random counterpart (b) using the Newman Strogatz model of random networks

Figure 8 .

 8 Figure 8.13 shows the barcode for a sample of tissues and its random counter part obtained by randomly gluing polygons. Barcodes for Betti 0 are similar, which is analogous to what has been obtained in the case of the Newman Strogatz random network. This similarity is explained by the fact that the number of simply connected components is mostly dependent on the shape of the degree distribution. Barcodes for Betti 1 are very similar between the data and the model, this is a clue toward the fact that this model is able to capture the features present in the global organization of the tissue. These results show that the random model captures well the global organization of the network by imposing

Figure 8 . 11 :Figure 8 . 12 :Figure 8 . 13 :

 811812813 Figure 8.11: Steps of construction of the random triangulated surface with appropriate degree distribution. Every polygons are drawn from the same degree distribution. The order of gluing is indicated by the number on the polygons.

Figure 8 . 14 :Figure 8 . 15 :

 814815 Figure 8.14: Scaling of the topological features with the model's parameters and independence of the topological features' values with respect to the number of polygons in the model. Degree distribution were chosen as normal distribution with mean (µ) taking values in the set (5,6,7,8) and a standard deviation taking values in the set (0.4,0.8,1.2), the number of polygons varied from 100 to 100 000. Points are colored according to standard deviation value. Graphs (a) correspond to the evolution of feat 1/σ 2 . Feat 1 has been defined according to equation 8.4. Left graph correspond to Sub * and right graph to Sup * . Graphs (b) correspond to the evolution of feat 2/σ 3 . Feat 2 has been defined according to equation 8.5. Left graph correspond to Sub * and right graph to Sup * .

Figure 8 .

 8 Figure 8.17 shows the results. Chick Neuroepithelium, in red, is the closest of the four epithelial types examined here to the null model. It means that in this kind of epithelial organization do not present patterns at the scales of the groups of cells, as much as the other types of epithelia do. The Drosophila Wing Prepupa is the most distant from the null model, it means that this tissue has the most constrained organization. Drosophila Mutant Wing Prepupa is as expected closer to the null model than the wild type. Indeed, Myosin II which is an element of the cytoskeleton is involved in epithelial organization.However, surprisingly, this distance is higher than for the Chick Neuroepithelium. That means that other factors than Myosin II contribute to epithelial organization.

Figure 8 . 16 :Figure 8 . 17 :

 816817 Figure 8.16: Computation of the features for the data and the model for sub-(a) and super-(b) level sets. The results for the empirical data are plotted with the name superimposed and a rounded label. The results for the null model are plotted with squared label. Magenta: Drosophila Wing Prepupa -Green: Drosophila Mutant Wing Prepupa -Red: Chick Neuroepithelium -Blue: Chick Ectoderm

Abstract

  liferation. Finally developing the null model into a more analytical object could lead to the characterization of statistical distribution of spatial networks, and for example the computation of p-value for the ordered nature of the organization of the network of cellular contacts. This chapter proposes some perspectives for the characterization, quantification and comparison of developing tissues as evolving networks. It is shown that time should not be considered as simple dimension since the nature of the network is changed through its flow. An adaptation of the persistent algorithm is proposed using genealogy and relatedness as a parameter. Its use should highlight the intertwining between cell proliferation and cell displacement.

Figure 9 . 1 :

 91 Figure 9.1: Empirical evidence underlying the establishment of the model of cell division in an epithelial tissue in [80] "d, Dilation of the junctional lattice permits rounding of a seven-sided mitotic cell during stages corresponding to prophase-metaphase. Owing to compression and stretching of the pseudocoloured neighbours, no cell-neighbour exchanges occur (n = 18 dilating cells). Units of t are in minutes. e, During stages corresponding to anaphase through cytokinesis, local topology (connectivity between cells) remains unchanged; the mitotic cell approaches abscission surrounded by the same cohort of seven neighbours (n = 23 cytokinetic cells). f, Two-cell clones marked by heritable expression of GFP (green) imaged at the level of the septate junctions stained with anti-Dlg (red). g, In approximately 94% of cell divisions, cytokinesis resolves with formation of a new cell interface, resulting in the type I conformation of mitotic siblings. h, Summary diagram of topology changes during cell division."

Figure 9 . 2 :

 92 Figure 9.2: Evolution of the mean number of neighbors (red) and standard deviation (grey) through time in the sea urchin development in one of the 5 specimens studied in chapter 1

Figure 9 . 3 :

 93 Figure 9.3: Vineyard as mathematical characterization of an evolving topological space.The persistence diagram is updated at each time step -Image adapted from[START_REF] Munch | Applications of Persistent Homology to Time Varying Systems[END_REF]

An illustration of this function is shown on figure 9 . 4 .

 94 In a similar way as in Chapter 8, we can use the sub level sets of this function R to define a filtration Fil = {Fil(k)} k∈N on the network of cellular contacts:Fil(k) = ({v i ∈ V }, {e = {v i , v j } ∈ E : R(v i , v j ) ≤ k}) ∀k ∈ N (9.1)

Figure 9 . 4 :

 94 Figure 9.4: Visualization of the function R that returns the degree of relatedness of two cells. A cell lineage is represented in black. When considering cells at the bottom of the tree, any couple of cells in the branches below the red circles have relatedness indicated by the value of R

  In Part I, we considered the question of the reproducibility of development. The study of intra-and inter-individual variability in a small cohort of digitally reconstructed sea urchins has led to the establishment of multi-level data-driven probabilistic model relating variable cell features with reproducible embryo-level dynamics. This model forms the basis for a prototypical representation of the sea urchin development as the centroid of the cohort, based on empirical parameters estimation. Surprisingly, variability at the individual cell level plays a significant role in the establishment of reproducible embryo-level dynamics, weakening the traditional view of the development as a finely-tuned process at the individual cell level. ⋆ In Part II, we considered the question of the diversity of phenotypes in the living, or how variations in individual developments affect the space of possible. The main contribution consists in using a formal analogy with the mathematical framework of quantum mechanics to clarify the various levels of randomness in biology. This approach enables to account for the high variability observed in the zebrafish squint mutant line and classical Mendelian scheme of inheritance. The latter is shown to be formally analog to quantum entanglement. We also discuss the relations between variability in biology and randomness in physical and mathematical theories. ⋆ In Part III, we study epithelial structure using the network of cellular contacts. By using topological invariants and a generalized model of random network, we show the presence of patterns involving groups of cells. We assume that these patterns result from the sequence of events involved in the morphogenesis of tissues. We propose a measure to characterize spatial organization which incorporates lineage 203 relationships between cells. ********** This work is part of a larger movement introducing quantitative approaches and computation in biology. The wealth of data generated by new imaging technologies and ever increasing computational power has opened the way for new perspectives on the understanding of biological problems such as embryonic development or evolution. The quantitative turn undergone by biology in the last decade demands a reinterpretation of many biological concepts. These challenges are associated to developments and extensions of mathematical approaches which are required to handle and make sense of these, sometimes massive, data sets. However, these mathematical developments cannot be performed by themselves, they require an underlying theoretical and epistemological basis to keep them interpretable. These observations form the general framework within which this work has taken place.The main concepts around which this thesis has evolved are the notions of organization and of historicity of organisms. These notions are closely related to variability, the central theme of this thesis. Indeed, organization is the result of an evolution, whose main ingredient is variation, on the other hand, historicity stems from this intrinsic variability which is ubiquitous in biology. The concept of organization expresses itself in the multiscale nature of organisms involving dynamics that can be heterogeneous in nature. The concept of historicity expresses itself in development by the path-dependency of the process, meaning that the specific sequence of events has an effect on the final phenotype.Historicity in evolution is associated to the contingency of events resulting from variability at all scales in biological organisms. In this context, we have tried in the course of this dissertation to characterize mathematically some notions of variability at different scales, thus in relation to the concept of organization, and along the different stages of development in animal embryogenesis, thus in relation to historicity of development.More precisely, in the comparative study of a small cohort of digitally reconstructed sea urchin embryos developed in chapters 1 and 2 we are confronted with these two notions of organization and historicity. Embryo-level developmental dynamics present similar patterns from one specimen to the other, whereas individual cell features are variable and spatial symmetries of the embryo prevent to compare individual cell between specimen of the cohort. This multi-scale nature of the development is captured by defining groups of cells as an intermediate level of observation which takes advantage of these symmetries.

while coefficients γ e are shown in Fig. 2.4F. Finally, we used the same coefficients squared for the rescaled cellular surface areas, (γ e ) 2 Z e (α e t + β e ) and

  Spatial rescalingBased on the same temporal rescaling, a linear transform was also performed along the spatial dimensions to minimize the discrepancy between the temporally rescaled cellular volume of an embryo, W e (α e t +β e ) and its mean value over all specimens:

		max N =N e,min	αt e (N ) + β -〈t 〉(N )	2 with 〈t 〉(N ) =	1 |E N | e∈E N	t e (N ),	(2.3)
	where E N is the sublist of embryos e such that N ∈ [N e,min , N e,max ] (including by inter-polation). For each embryo, the parameters adopted for the transform satisfied (α e , β e ) = arg min F e (α, β). Finally, taking the inverse again, we obtained five rescaled total numbers
	N e (α e t +β e ) and four groups of five rescaled field numbers N k e (α e t +β e ), which are plotted in Fig. 2.4B. The five parameter pairs (α e , β e ) are shown in Fig. 2.4C.
	G e (γ) =	(t e,max -β e )/α e t =(t e,min -β e )/α e	γ 3 W e (α e t + β e ) -〈W 〉(t )	2 with 〈W 〉(t ) =	1 |E t | e∈E t	W e (α e t + β e ), (2.4)
	where E t is the sublist of embryos e such that (α e t + β e ) ∈ [t e,min , t e,max ] (including by interpolation). For each embryo, γ e = arg minG e (γ) was then defined as the optimal co-efficient. The original volumes W e (t ) are plotted in Fig. 2.4D. The five rescaled total vol-
	umes (γ e ) 3 W e (α e t + β e ) and four groups of five rescaled field volumes (γ e ) 3 W k e (α e t + β e ) each are plotted in Fig. 2.4E, (γ e ) 2 Z k e (α e t + β e ) (Fig. 2.4H; the original curves Z e (t ) are shown in Fig. 2.4G).
	Neighborhoods The total numbers of contacts in each embryo and each morphogenetic
	field did not require an extra spatial transform to highlight interindividual similarities. Af-
	ter carrying over the temporal rescaling parameters found above, the curves C e (α e t + β e ) and C k e (α e t + β e ) already presented a high degree of overlap among embryos e ∈ [1, 5] (Fig. 2.4I). This is consistent with the fact that neighborhood relationships are topological
	features, thus independent from metric deformations.

  = |L g |. The same formulas were applied to variables M g , logV g , log S g , log A g , or log B g , for each one of the 252 available distributions, yielding parameters ( µ M g , σ M g ), ..., ( µ B g , σ B g ). the distribution of life lengths {x 1 , x 2 , ..., x m } over the m cells of group L g was categorized into a fixed number l of discrete bins, ( x1 , x2 , ..., xl ), where x1 = min{x i } andxh+1 -xh = ∆ x = (max{x i } -min{x i })/l for all h ∈ [1, l -1]. In this histogram, the observed distribution corresponds to the number of values falling in each bin and was given by the empirical measure P m,X g (I h ) (equation 2.6), where I h = [ xh , xh +∆ x) and I l is closed, while

	1/2
	2	(2.8)
	where N g All 252 (µ, σ) pairs are plotted in Fig. 2.5.
	Then, based on the empirical mean and standard deviation, our categorization hy-
	pothesis was challenged in each distribution by calculating a p-value corresponding to
	the probability to find a good fit with a normal or log-normal curve. Taking again X g as
	an example,

Table 2 .

 2

1: Summary of the linear regression test for six pairs of features across 198 distributions in 61 groups.

Table 2 .

 2 5: Range and cardinalities of the parameters explored in this study. Results of the exploration of the parameter space are shown on figure 1.3 D.

		Min. Max. Cardinality
	w adh,e	10	1000	20
	w adh,o	10	1000	20
	w rep	100	100	1
	k rig	6000 16000	11
	α gab	0.9	1.3	0.02
	λ 0	3000	3000	1
	c max	2	2	1

  We would like to relate small deviation on the density distribution ∆ f 1 , ∆ f 2 , ∆ f 3 with deviation on the final probability density ∆ f 123 . The result should indicate how the perturbations at different stages of development affect the final result. It is likely that earlier fluctuations are more amplified than late ones.More formally, we would like to show a relation such that ∆ f 123

		Brownian	Motion	in	2D	and	the	Fokker-Planck	Equation"	from	the	Wol-
	fram	Demonstrations		Project	-	http://demonstrations.wolfram.com/
	BrownianMotionIn2DAndTheFokkerPlanckEquation/

Table 8 .

 8 1: Steps of the automatic pipeline enabling comparison and classification of epithelial tissues

http://www.bioemergences.eu

http://www.biotapestry.org/

The establishment of the multi-level data-driven probabilistic model has highly benefited from advices by Paul Bourgine whom we warmly acknowledge

Remerciements

extension of the evolutionary process within the organism. Although empirical results are also compatible with a more traditional epigenetic interpretation such as the one developed by Waddington by means of an epigenetic landscape. To sum up the content of this part, there is an intrinsic variability at the individual cell level underlying cell differentiation and this variability is involved in the reproducibility of embryo-level dynamics. ematical approaches to be grounded on stable settings. It will hopefully enable to link deeply development and evolution, while giving major roles to the notions of organism and of historicity of living objects.

Placing variation at the foundation of his theory of evolution is one of the major paradigm shifts operated by Charles Darwin. Our results suggest that we may need the same kind of revolution for developmental biology.

Part II

Characterizing diversity [START_REF] Bohn | Hierarchical crack pattern as formed by successive domain divisions[END_REF]

.1: Number of fertilized eggs during the 8 week of the experiment for each couple of identified homozygote squint mutant rocular distance i d which are identified manually. Cyclops corresponds to id = 0, orange phenotypes to 0 < id < 80µm, green phenotypes to 80µm ≤ id < 140µm and blue phenotypes to id ≥ 140µm We obtain the following empirical frequencies and state vectors:

-Week 1 -cyclops : 17,5%, orange : 2,5%, green : 5%, blue : 75% ψ 1 = 0, 42.ket cyclops + 0, 16.ket orange + 0, 2.ket green + 0, 87.ket blue -Week 3 -cyclops : 22,414%, orange : 1,7241 %, green : 8,6207%, blue : 67,241% ψ 3 = 0, 47.ket cyclops + 0, 13.ket orange + 0, 29.ket green + 0, 82.ket blue -Week 4 -cyclops : 32,432 %, orange : 0 %, green : 2,7027%, blue : 64,865% ψ 4 = 0.57.ket cyclops + 0.ket orange + 0.16.ket green + 0.80.ket blue -Week 6 -cyclops : 31,034%, orange : 0%, green : 2,2989%, blue : 66,667% ψ 6 = 0, 56.ket cyclops + 0.ket orange + 0, 15.ket green + 0, 82.ket blue -Week 8 -cyclops : 3,2787 %, orange : 1,6393 %, green : 4,918%, blue : 90,164% ψ 8 = 0, 18.ket cyclops + 0, 13.ket orange + 0, 22.ket green + 0, 95.ket blue During the first week of observation, the 4 phenotypic classes were already observed.

However, if the measures had begun at week 4, the phenotype orange wouldn't have been observed until week 8. The space of possible phenotypes would then have been the Hilbert space of dimension 3, H = (ket cyclops, ket green, ket blue) for week 4, 5, 6, 7. At week 8 the lexicographical order as shown in the table 7.1.

There is a first symmetry breaking at stage S1, separating the developments of (WT, M1, M2, M3) on one side and (M4) on the other side. Then, a second symmetry breaking is observed at stage S2 separating the developments of (WT, M1) on side and (M2, M3) on the other side. In the branch gathering (WT, M1), a symmetry breaking is observed at stage S3 separating the two developmental paths. In the branch gathering (M2, M3) the two developmental paths diverge at the stage S4.

If we define a developmental proximity d as the length of the common developmental path, we obtain the following inequalities:

The values of these developmental proximities can then be compared to a phenotypic proximity between the resulting embryos, e.g. based on the morphological similarity or on the functional similarity.

In the case of the network of cellular connectivity, we can use the number of neighbors as a proxy for the density of the space. For each network (V, E ), two filtrations are defined.

A filtration is a nested sequence of spaces. The filtrations considered here are indexed with a parameter k (natural integer) corresponding to values of degree. They are defined as a discrete analog of the sub-and super level sets functions. We first define two sequences of nested networks, Sub = {Sub(k)} k∈N and Sup = Sup(k) k∈N :

Sub(k) and Sup(k) are subnetworks of (V, E ) such that each node and each edge connect nodes having at most (at least respectively) k neighbors in (V, E ). We have therefore The idea of the persistent homology algorithm is to describe the topology of the spaces contained in the filtrations Sub * and Sup * for each value of the parameter k. Persistent features will correspond to intrinsic features of the object, whereas features associated to noise will be unstable and won't last. The topological features considered are the Betti numbers, which measure the i-dimensional "holes" of a space. For example, for a torus, as represented on figure 8.6, the number of 0-dimensional holes (β 0 ) is 1, it is the number of connected components. The number of 1-dimensional holes (β 1 ), i.e. "circles", is 2, the one that encircles the void in the middle of the torus and the one that encircles the torus.

The number of 2-dimensional holes (β 2 ), i.e. "voids", is 1, the one inside the torus.

Computing these topological invariants for each subspaces of the filtrations results in a multi-scale object called a barcode that summarizes the evolution of these features are sufficient to define functions in terms of tropical polynomials that will be used in the following.

For each Betti number, corresponding barcodes can be written as sets of couples (x i , y i ) i , with (y i > x i ), each of them representing a bar. Only bars of finite length will be considered. We denote by n the number of bars. We define the functions that are relevant to our analysis in the following way:

Each of these functions returns a unique real value associated to one of the Betti num-

Classification of tissues

Using the data set provided with the article [START_REF] Luis M Escudero | Epithelial organisation revealed by a network of cellular contacts[END_REF], we used these vectors of features as a basis for a statistical analysis over different type of epithelia. The data set consists of sample of tissues extracted from live embryo and imaged with confocal microscopy. 9 to 15 samples have been imaged in each of the different following situations: drosophila Wing ithelial shape and the developmental sequence we propose a methodology based on the cell lineage and cell network of cellular contacts. Overall, we show in this last part that organization and historicity are intertwined in morphogenesis.

Perspectives raised by this dissertation are hopefully numerous. We have made some suggestions in the course of the manuscript, particularly in chapters 3 and 9. However, here are some perspectives

The data-driven multi-level prototypical probabilistic model obtained in chapter 2 can serve as a basis for a dynamical system description of development, a law of evolution needs to be found to relate the value of the parameters. The approach is general enough to be expected to be found applicable in the development of other species where there is relative synchrony of cell cycles using the same mathematical framework. Moreover, the approach consisting in defining a relevant coarse-grained level of observation preventing averaging out significant information and being generic enough to enable comparison between specimens is likely to be transferable in other developing systems. As an extension of the model it could be interesting to combine the analysis of the cell lineage with a spatial analysis of the distribution of cells.

The probabilistic approach developed in the first part raises numerous questions on the reproducibility and robustness of development. An interesting line of research could be to perform experimental perturbations of the system to see if the mathematical structure uncovered stays the same. For example by modulating the length of the cell cycles.

In the second part, we have considered the diversity of the living. The main perspective here would be to be able to integrate the variability at all scales in a theory of organisms.

Such a theory would enable to describe more formally this notion of biological randomness at the level of the space of possible. This would lead to better understanding of the concept of prediction in biology

In the third part, we have considered epithelial organization, and its relations to cell proliferation and displacement. It could be interesting to perform perturbations on developing systems, e.g. on cell proliferation or cell motility, to quantify their influence on the resulting shape. Another perspective consists in generalizing the approach to more complex tissues, for example during epithelial folding to describe the transition from two to three-dimension, by taking into account the curvature for example. This approach can also be generalized to three-dimensional situations.

Overall, we anticipate that the field of embryology will greatly benefit from an understanding of variation at every scales. This understanding requires quantitative and math-