Armin Haupt

Nicolas Minc
email: nicolas.minc@ijm.fr

How Do Cells Sense their Own Shapes? : Mechanisms for Probing Cell Geometry and their Implications in Cellular Organization and Function

Cells come in a range of shapes that most often underlie their functions. Regulation of cell morphogenesis implicates that there are mechanisms for shape sensing which still remain poorly appreciated. Global and local cell geometrical features, such as aspect ratio, size or membrane curvature, may be probed by intracellular modules, such as the cytoskeleton, reaction-diffusion systems or molecular complexes. In multicellular tissues, cell shape emerges as an important means to transduce tissue-inherent chemical and mechanical cues into intracellular organization. One emergent paradigm is that cell shape sensing is most often based upon mechanisms of self-organization, rather than determinism. Here, we review relevant work that has elucidated some of the core principles of how cellular geometry may be conveyed into spatial information to guide processes, such as polarity, signaling, morphogenesis and division-plane positioning.

INTRODUCTION

Any biologist would acknowledge that cells are beautiful objects. The range of forms they span is not only aesthetic, but reveals the complexity and diversity of intracellular systems that serve to construct cell shape and, in most instances, reflects the function of the cell. Long polarized nerve cells make distant connections in brains and muscles; the small and flat shape of red blood cells may enhance flowing capacities needed to navigate in thin blood vessels; cuboidal epithelial cells serve as brick units for the walls of our organs; and cork-screw like bacteria can bore their way through the surface of epithelia. Besides the cell-type specific relevance of cell shape, an emergent concept is that both the regulation and functional use of cell shape, necessarily implicate geometry-sensing mechanisms. A cell that grows into a rodshape must come with designs that informs on the straightness of its long axis, or with signaling systems that detect any unwanted curvature. A symmetrically dividing cell needs to precisely locate its geometrical center and orient the division plane accounting for its global surface morphology.

According to their specific shapes, sizes and functions, cells may thus be equipped with internal systems to monitor global, local or dynamic geometrical features (Fig. 1). Those could confer robustness to cell shape, or act as means to amplify a small surface deformation, when shape changes are needed. Mechanisms that sense the straightness or curvature of the cell surface may be important to stabilize elongated cell shapes, or to detect local 3D membrane geometries, such as protrusions, or cilia [START_REF] Cannon | The Unsolved Problem of How Cells Sense Micron-Scale Curvature[END_REF]. Monitoring of the surface-to-volume ratio is thought to be important for timing cell division, or as means to locally titrate molecular reactions around cells [START_REF] Harris | Relative Rates of Surface and Volume Synthesis Set Bacterial Cell Size[END_REF][START_REF] Schmick | The interdependence of membrane shape and cellular signal processing[END_REF]. Furthermore, the global cell aspect ratio has key relevance for the positioning and orientation of the division plane and in the definition of internal polarity axes [START_REF] Minc | Establishing new sites of polarization by microtubules[END_REF][START_REF] Minc | Influence of cell geometry on division-plane positioning[END_REF] (Fig. 1).

Biologists have long recognized the relevance of feedback mechanisms that allow cells to probe their shapes and dimensions, as a basal property of all cells, even the roundest one [START_REF] Gerhart | Topobiology. An Introduction to Molecular Embryology[END_REF]Hertwig, 1884a;Hertwig, 1884b;[START_REF] Moseley | Cell division intersects with cell geometry[END_REF][START_REF] Wilson | The cell in development and heredity[END_REF].

Recent work has begun to decipher the generic designs and molecular mechanisms implicated. Dissecting specific mechanisms of geometry sensing in vivo may prove difficult, given the contributions from both chemical and mechanical cues, which also affect cell shapes in tissues. In vivo, for instance, processes such as polarity or division orientation have most often been linked to external signals, rather than as direct consequences of cell shape [START_REF] Minc | Predicting division plane position and orientation[END_REF]. However, it is becoming increasingly evident that cellular components such as the cytoskeleton and reaction-diffusion systems have self-organizing properties that can probe cellular boundary conditions [START_REF] Karsenti | Self-organization in cell biology: a brief history[END_REF]. In addition, the advent of microfabrication methods to control and manipulate shapes independently of external cues in cells ranging from microbes to mammalian cells [START_REF] Lautenschlager | Microfabricated devices for cell biology: all for one and one for all[END_REF], has served as a driving force to unequivocally demonstrate that shape has a profound impact on the spatial organization and function of most cells.

We will here focus on key questions underlying the mechanisms of shape-sensing, and on the functional interplays between shape and internal organization: How do molecules and/or molecular assemblies probe cellular shapes? How do they use such information to control cell behavior and morphogenesis? Is cell-shape sensing a conserved trait in evolution needed for fitness? We will introduce the various geometrical parameters that may be sensed in different cells and discuss mechanisms, which link shape-sensing to the spatial control of processes, such as polarity, signaling, and division positioning. Although cell size is another important geometrical feature, we will not cover it here and refer the readers to recent reviews on this topic [START_REF] Amodeo | Cell-Size Control[END_REF][START_REF] Levy | Biological Scaling Problems and Solutions in Amphibians[END_REF].

CELL SHAPE SENSING AND POLARITY

Microtubules as cell shape sensors for cell polarity

Cell polarity underlies the ability of cells to define subcellular domains of activity at their surface. This allows cells to grow or migrate directionally, specify a front and a rear, or position internal organelles, such as the nucleus, the centrosome or the Golgi relative to each other [START_REF] Etienne-Manneville | Cdc42--the centre of polarity[END_REF]. Polarization domains can spontaneously assemble as a result of positive feedbacks between interacting proteins and cytoskeleton elements that amplify local fluctuations into mature polar domains [START_REF] Irazoqui | Scaffold-mediated symmetry breaking by Cdc42p[END_REF][START_REF] Wedlich-Soldner | Spontaneous cell polarization through actomyosin-based delivery of the Cdc42 GTPase[END_REF]. Extrinsic or intrinsic chemical, mechanical or electrical cues, can then orient or stabilize those domains along specific directions [START_REF] Casamayor | Bud-site selection and cell polarity in budding yeast[END_REF][START_REF] Etienne-Manneville | Cdc42--the centre of polarity[END_REF][START_REF] Haupt | Electrochemical regulation of budding yeast polarity[END_REF][START_REF] Ladoux | Front-Rear Polarization by Mechanical Cues: From Single Cells to Tissues[END_REF].

As polarity ultimately directs processes such as cell growth or migration, it serves as a key input to define cell shapes over time. This has been extensively studied in the model fission yeast. These cells grow into a rod by targeting polarity and growth factors, such as the Rho-GTPase Cdc42, actin and cell wall remodeling enzymes to cell tips (see [START_REF] Chang | Shaping fission yeast with microtubules[END_REF] for a detailed review). Key to the tip-localization of polarity zones are dynamic microtubule (MT) bundles, which emanate from the nucleus and grow near parallel to the long cell axis to deposit MT-plus-end-associated landmark proteins at cell tips, which then recruit downstream elements. Mutants with defective MT organization have abnormal bent or branched shapes, and many mis-shaped mutants have defective MT organization, which has long suggested the existence of interplays between shape, MT organization and polarity [START_REF] Verde | Fission yeast cell morphogenesis: identification of new genes and analysis of their role during the cell cycle[END_REF].

Studies that manipulated fission yeast shape by physically bending cells in microfabricated devices have been instrumental in deciphering feedbacks between shape and polarity [START_REF] Minc | Establishing new sites of polarization by microtubules[END_REF][START_REF] Terenna | Physical mechanisms redirecting cell polarity and cell shape in fission yeast[END_REF]. Subsequent to cell bending, MT organization dramatically changes, with most MTs not reaching the cell extremities anymore, but instead touching the cell sides, yielding the formation of ectopic polarization sites (Fig. 2A). In a reciprocal manner, round mutants with defective MT organization forced into a straight shape, recovered near wild-type MT orientation and polarization at cell tips [START_REF] Terenna | Physical mechanisms redirecting cell polarity and cell shape in fission yeast[END_REF]. Thus, MTs through their dynamic properties and straightness, serve as robust shape-sensors to promote polarization along a linear geometrical axis. This feedback between shape, MT orientation and polarity, may confer robustness to polarized growth and ensure the maintenance of rod-shape morphogenesis through generations of dividing cells [START_REF] Drake | Model of fission yeast cell shape driven by membranebound growth factors and the cytoskeleton[END_REF]). Yet, polarizing spores or spheroplasts can readily elongate a first straight rod-shape even in the absence of MTs and landmarks; suggesting the existence of other systems promoting directed growth [START_REF] Bonazzi | Symmetry Breaking in Spore Germination Relies on an Interplay between Polar Cap Stability and Spore Wall Mechanics[END_REF][START_REF] Kelly | De novo growth zone formation from fission yeast spheroplasts[END_REF].

Work on epithelial cells in fly embryonic tissues has suggested similar influence of cell shape on MT organization (Fig. 2B) [START_REF] Gomez | Microtubule organization is determined by the shape of epithelial cells[END_REF], with plausible implications on the definition of planar polarity axis [START_REF] Butler | Planar cell polarity in development and disease[END_REF]. Alignment of MTs along the long axis has also been documented in adherent cells, and proposed to serve as a mechanism to limit the length of spreading cells [START_REF] Levina | Cytoskeletal control of fibroblast length: experiments with linear strips of substrate[END_REF][START_REF] Picone | A polarised population of dynamic microtubules mediates homeostatic length control in animal cells[END_REF].

This effect may also contribute to guide the polarization of the nuclear-centrosome axis, to enhance persistent directional migration of cells which migrate along their long axis, although patterns of adhesion could serve as dominant spatial cues when cell geometry is mostly isotropic (Fig. 2B andC) [START_REF] Jiang | Directing cell migration with asymmetric micropatterns[END_REF][START_REF] Mcwhorter | Modulation of macrophage phenotype by cell shape[END_REF]Thery et al., 2006;[START_REF] Vignaud | Reprogramming cell shape with laser nano-patterning[END_REF]. Thus MT-based geometry-sensing may have broad relevance to polarity establishment and maintenance of eukaryotic cells.

Subcellular micron-scale curvature sensing

The rod-shape morphogenesis is also a common feature of many bacteria, including E. coli [START_REF] Chang | How and why cells grow as rods[END_REF]. In those cells, insertion of cell wall material occurs along the sides of the cylinder, not at cell poles, a process mediated by the actin homolog MreB [START_REF] Furchtgott | Mechanisms for maintaining cell shape in rod-shaped Gram-negative bacteria[END_REF]. A recent study found that MreB polymers tend to preferentially bind to regions of positive curvatures along the cylinder [START_REF] Ursell | Rod-like bacterial shape is maintained by feedback between cell curvature and cytoskeletal localization[END_REF]. This observation supported a model, in which curvature detection and consequent local promotion of cell wall insertion and growth by MreB allowed cells to actively straighten their shape (Fig. 2D). However, a recent paper indicates that this feedback may not be sufficient to account for cell straightening upon larger deformation [START_REF] Wong | Mechanical strain sensing implicated in cell shape recovery in Escherichia coli[END_REF]. Thus, bacterial cells, which lack long-range cytoskeletal systems, may ensure rod-shape maintenance through local curvature sensing, likely encoded in the structural properties of MreB polymers [START_REF] Colavin | Effects of polymerization and nucleotide identity on the conformational dynamics of the bacterial actin homolog MreB[END_REF].

Eukaryotic cells may also possess micron-scale curvature sensing systems. Septins are conserved oligomeric protein complexes that can assemble into non-polar filaments and bind to the cell cortex. They are important to many processes including cell shape, protein scaffolding and disease [START_REF] Bridges | Septin Form and Function at the Cell Cortex[END_REF][START_REF] Cannon | The Unsolved Problem of How Cells Sense Micron-Scale Curvature[END_REF]. Yeast and mammalian septins have been recently demonstrated to have intrinsic affinities to bind surfaces with a defined positive curvature at the micron scale. Even though individual oligomers exhibit selective curvature binding, they require polymerisation to bind curved membranes within cells [START_REF] Bridges | Micron-scale plasma membrane curvature is recognized by the septin cytoskeleton[END_REF]. In cells, micron-scale positive curvature can be found on edges between spheres and protrusions (a topology often referred to as 'saddle', see Box 1), such as at the bases of dendritic spines, axons and cilia, in the yeast bud neck or branches of fungal hyphae, all structures where septins have been found to localize before (Fig. 2E) [START_REF] Bridges | Septin Form and Function at the Cell Cortex[END_REF]. The function of septins at some of these sites has been linked to a function as diffusion barriers or scaffolds (Hu et al., 2010;[START_REF] Merlini | The mother-bud neck as a signaling platform for the coordination between spindle position and cytokinesis in budding yeast[END_REF]. Thus, the inherent anisotropic curvature of saddle structures might promote the formation of specific higher-order septin organization, which is important for their biological functions.

This curvature-dependent septin recruitment might be used as a cellular readout. In budding yeast, the formation of a bud is monitored through the morphogenesis checkpoint that delays mitosis onset until bud emergence [START_REF] Kang | How do cells know what shape they are?[END_REF]. The presumptive bud site is labelled by a septin ring, which might rearrange into higher-order structures that are aligned along the saddle that is formed upon bud emergence. This alignment could trigger the recruitment of Elm1, a kinase which influences mitotic entry (Fig. 2F) [START_REF] Kang | Sensing a bud in the yeast morphogenesis checkpoint: a role for Elm1[END_REF].

Therefore, septin-mediated curvature sensing could also serve cells to monitor shape changes and time them with cell cycle progression.

Another prime example of local curvature sensing was recently proposed in the context of tip branching regulation during endothelial cell migration. Branching and general curvature formation in these cells was shown to be limited by acto-myosin cortex contraction [START_REF] Elliott | Myosin II controls cellular branching morphogenesis and migration in three dimensions by minimizing cell-surface curvature[END_REF][START_REF] Fischer | Local cortical tension by myosin II guides 3D endothelial cell branching[END_REF]. Myosin II was found to preferentially bind to flat membrane regions of low local curvature, with a binding efficiency that negatively correlates with absolute curvature. This generates a positive feedback loop, in which Myosin II may promote its own binding by actively reducing the curvature of bent membrane regions. In the context of branching morphogenesis, this feedback allows Myosin II to stabilize branches that form at small angles to pre-existing protrusions, while suppressing those that arise perpendicular to existing ones, thus polarizing branching along the migration direction (Fig. 2G). Curvaturesensing of Myosin was there again proposed to be encoded in the elongated morphology of filaments, which could favor their binding to actin on flat membranes [START_REF] Elliott | Myosin II controls cellular branching morphogenesis and migration in three dimensions by minimizing cell-surface curvature[END_REF].

Curvature dependence of acto-myosin forces has also been proposed to be important in the context of cytokinesis to promote faithful centripetal ring closure [START_REF] Dorn | A theoretical model of cytokinesis implicates feedback between membrane curvature and cytoskeletal organization in asymmetric cytokinetic furrowing[END_REF][START_REF] Zhou | The contractile ring coordinates curvature-dependent septum assembly during fission yeast cytokinesis[END_REF].

Evidence that cells may have developed ways to probe their local curvature on the micron scale through means other than direct polymer recognition of curved surfaces, has recently been obtained in fission yeast. In this system, the width of functional polarity domains scales with the local radius of curvature in rounded spores and rod-shaped cells [START_REF] Bonazzi | Actin-Based Transport Adapts Polarity Domain Size to Local Cellular Curvature[END_REF]. This scaling was shown to rely on actin-dependent vesicle transport and fusion that may dilute and spread polar domains in a dose-dependent manner with the local curvature.

According to this model, local geometry constrains could limit the volume that cortical actin cable networks can probe, yielding different cargo flux to dilute the cap (Fig. 2H). As polar domains built around conserved small GTPases such as Cdc42 usually scale to the size of cells ranging from small yeast to large eggs [START_REF] Bonazzi | Actin-Based Transport Adapts Polarity Domain Size to Local Cellular Curvature[END_REF][START_REF] Jost | Probing Yeast Polarity with Acute, Reversible, Optogenetic Inhibition of Protein Function[END_REF], it will be interesting to test if this curvature-sensing mechanism is based on similar actindependent transport processes in other cell types. Thus, the ability of actin or actinhomologues and their interactors to probe local micron-scale curvatures could serve as an important conserved shape-sensing design in morphogenesis.

Local surface to volume ratio as a cue for polarization and signaling

One interesting implication of the existence of flat, curved or narrow regions inside cells is that they may affect the local surface-to-volume ratio. The global surface-to-volume ratio, set by cell size, has been well documented to influence division timing or the scaling of internal organelles to cell size, but local effects are still poorly appreciated [START_REF] Harris | Relative Rates of Surface and Volume Synthesis Set Bacterial Cell Size[END_REF][START_REF] Wilbur | Mitotic spindle scaling during Xenopus development by kif2a and importin alpha[END_REF]. The impact of the local surface-to-volume ratio on local cell biochemistry was first explored theoretically using the example of a membrane bound kinase, which could phosphorylate a cytoplasmic substrate [START_REF] Meyers | Potential for control of signaling pathways via cell size and shape[END_REF]. Competition between the kinase reaction rate and substrate diffusion results in a gradient of phosphorylation from the membrane to the cell center. A cell with a protruding flat front would in that context be able to set a different cytoplasmic biochemical activity, simply as a result of the increased local surface-to-volume ratio. This could in principle serve as a 'ruler' to measure cell shape and size (Fig. 2I) and could even recognize small membrane deformations. Yet, one important limitation of this model is that the lateral diffusion of the kinase in the membrane will rapidly disperse this gradient. As a consequence, this effect could only serve as a transient shape sensor, unless the kinase is continuously maintained in a polarized subcellular localization through reinforcements by cytoskeletal elements or other external cues [START_REF] Schmick | The interdependence of membrane shape and cellular signal processing[END_REF]. A recent study explored this mechanism in a systematic manner for reactions that occur within the membrane using an enzyme that diffuses in the cytoplasm or in the extracellular space, and proposed that a transient localized increase in reaction products could be achieved theoretically and experimentally, purely as a result of cell deformation, which even influenced signaling globally [START_REF] Rangamani | Decoding information in cell shape[END_REF].

Pioneering studies using micropatterned islands have long demonstrated that cell geometry could influence such important processes as differentiation, growth and death [START_REF] Chen | Geometric control of cell life and death[END_REF][START_REF] Kilian | Geometric cues for directing the differentiation of mesenchymal stem cells[END_REF][START_REF] Mcwhorter | Modulation of macrophage phenotype by cell shape[END_REF]. However, in this context, geometry was thought to indirectly affect the mechanical state of cells, which then influenced its fate [START_REF] Chen | Geometric control of cell life and death[END_REF]. Thus, while the above-mentioned concept that cell shape could directly modulate signaling reactions await further generalization, it supports a provocative view that, at least in certain cell types, geometry per se could serve as a hub to mediate key cellular decision during tissue development and homeostasis [START_REF] Rangamani | Decoding information in cell shape[END_REF][START_REF] Ron | Cell shape information is transduced through tensionindependent mechanisms[END_REF][START_REF] Schmick | The interdependence of membrane shape and cellular signal processing[END_REF].

CELL SHAPE AND DIVISION POSITIONING

Cell shape and division in evolution

Cell shape also has important influence on the positioning and orientation of the cell division plane, a process with profound implications for cell size control, stem cells and tissue morphogenesis [START_REF] Minc | Predicting division plane position and orientation[END_REF]. While some cells exhibit marked asymmetric divisions, most cells tend to divide in a symmetric manner by bisecting the mother cell into two halves of near exact same volume. Symmetric division is inherently linked to cell geometry, as here cells need to locate their geometrical center. In addition, apart from rare exceptions [START_REF] Leisch | Growth in width and FtsZ ring longitudinal positioning in a gammaproteobacterial symbiont[END_REF], most elongated prokaryotic and eukaryotic cells tend to bisect their longest axis during symmetric division. This conserved behavior is fascinating in terms of evolution because mechanisms that position cytokinesis or septation vary largely between kingdoms [START_REF] Oliferenko | Positioning cytokinesis[END_REF]. In animal cells, cytokinesis is positioned from spatial cues that are provided by the mitotic spindle towards anaphase. In most fungi, cytokinesis and septum ingression are pre-specified from the position of the nucleus and the spindle usually aligns orthogonal to the septum. In plants, the preprophase band also forms around the nucleus and serve as a landmark for spindle orientation and cell plate growth. In bacteria and other prokaryotes, septum positional information arises from protein gradients in a manner mostly independent of DNA or any cytoskeletal structure (reviewed in [START_REF] Oliferenko | Positioning cytokinesis[END_REF].

These considerations raise the question of what kind of evolutionary pressure could have driven cells, which span a large range of size, shape, and environment, to divide along their long axis. One possibility is that this geometrical design could provide the largest cytoplasmic space for DNA segregation [START_REF] Cadart | Exploring the function of cell shape and size during mitosis[END_REF][START_REF] Lancaster | Mitotic rounding alters cell geometry to ensure efficient bipolar spindle formation[END_REF]. Another proposed idea is that this could promote the stability of ring position during ingression, at least in some cell types [START_REF] Mishra | Cylindrical cellular geometry ensures fidelity of division site placement in fission yeast[END_REF]. We also foresee that such a setting may minimize the possible error transfer between the detection of a geometrical center and the volume repartition between daughter cells. Finally, cell division along the long axis could just reflect inherent geometrical sensing properties of division-positioning machineries, which naturally tend to self-organize along the longest shape axis [START_REF] Karsenti | Self-organization in cell biology: a brief history[END_REF]. Recent evo-devo studies suggest that variations in asymmetric division plane position are scaled with cell size, thereby ensuring a stable volume ratio between daughter cells [START_REF] Farhadifar | Scaling, selection, and evolutionary dynamics of the mitotic spindle[END_REF]. Similar evodevo approaches assaying the variations in symmetric division position and orientation as a function of cell shape in closely related species should help address those important questions.

Microtubule forces as geometrical rulers to target the cell center

Historically, the relevance of cell shape for division positioning was first appreciated in large dividing animal embryos, leading to the formulations of early empirical "geometrical" rules for cell divisions (Hertwig, 1884a;[START_REF] Hertwig | Ueber den Werth der ersten Furchungszellen fuer die Organbildung des Embryo[END_REF], see [START_REF] Minc | Predicting division plane position and orientation[END_REF] for review. Indeed, most invertebrate and vertebrate zygotes cleave exactly in the cell center. This is challenging, because eggs are very large, up to a millimeter in amphibian species, and because fertilization, which brings centrosomes attached to the sperm pro-nucleus into the egg, is dramatically asymmetric. The questions thus arises of how can the nucleus and centrosome move to the exact geometric center of those large cells [START_REF] Wuhr | How does a millimeter-sized cell find its center?[END_REF]? Apart from rare exceptions (see hereafter), nuclear centration in animal cells is mediated by MT-mediated forces, with actin being mostly dispensable. MTs nucleate around the centrosome and grow to form an aster, which then exerts forces to move the nucleus and centrosome to the cell center [START_REF] Reinsch | Mechanisms of nuclear positioning[END_REF][START_REF] Wuhr | How does a millimeter-sized cell find its center?[END_REF].

While much debate is still ongoing on the exact nature and regulation of MT-mediated forces, one common well-accepted aspect of shape sensing and aster centration is that MT-exerted forces depend on average on the length of astral MTs. The combination of an aster-like geometry with the dynamic growth and shrinkage properties of MTs, and their lengthdependent forces provides a simple yet extremely robust design for the cell to be able to probe the cellular volume and center the centrosome (Fig. 3A) [START_REF] Bornens | The centrosome in cells and organisms[END_REF][START_REF] Holy | Assembly and positioning of microtubule asters in microfabricated chambers[END_REF][START_REF] Kimura | Computer simulations and image processing reveal lengthdependent pulling force as the primary mechanism for C. elegans male pronuclear migration[END_REF][START_REF] Tanimoto | Shape-motion relationships of centering microtubule asters[END_REF][START_REF] Wuhr | A model for cleavage plane determination in early amphibian and fish embryos[END_REF]. Length-scaling may arise from compressive pushing forces owing to MT-plus-end polymerization against the cell surface. Because of dynamic instabilities, more MTs will push on the centrosome as it is being brought closer to the cortex. Additionally, pushing may be limited by buckling in a length-dependent manner, so that longer MTs will buckle more than shorter ones, and these will thus exert comparatively more pushing forces on the centrosome (Fig. 3A) [START_REF] Howard | Elastic and damping forces generated by confined arrays of dynamic microtubules[END_REF][START_REF] Howard | Physical Limits on the Precision of Mitotic Spindle Positioning by Microtubule Pushing forces: Mechanics of mitotic spindle positioning[END_REF]. Indeed, such a mechanism has been demonstrated to promote nuclear centering in small yeast cells [START_REF] Daga | Asymmetric microtubule pushing forces in nuclear centering[END_REF][START_REF] Tolic-Norrelykke | Nuclear and division-plane positioning revealed by optical micromanipulation[END_REF][START_REF] Tran | A mechanism for nuclear positioning in fission yeast based on microtubule pushing[END_REF] and proposed to contribute to spindle centration in Caenorhabditis elegans [START_REF] Garzon-Coral | A force-generating machinery maintains the spindle at the cell center during mitosis[END_REF][START_REF] Howard | Physical Limits on the Precision of Mitotic Spindle Positioning by Microtubule Pushing forces: Mechanics of mitotic spindle positioning[END_REF]. In vitro work and simulations suggest, however, that pushing could yield to unstable situations that tend to decenter asters if, for instance, MTs are allowed to pivot around centrosomes [START_REF] Letort | Centrosome centering and decentering by microtubule network rearrangement[END_REF][START_REF] Pinot | Effects of confinement on the self-organization of microtubules and motors[END_REF]. Whether those situations are relevant to cells remains to be tested.

In larger cells, such as eggs and early blastomeres of marine invertebrates, amphibians and fish, mounting evidence suggests that MT pushing is only poorly efficient in promoting aster centration (Hamaguchi and Hiramoto, 1986;[START_REF] Tanimoto | Shape-motion relationships of centering microtubule asters[END_REF][START_REF] Wuhr | A model for cleavage plane determination in early amphibian and fish embryos[END_REF].

Length-scaling in these cells is thought to be driven from dynein motors that directly pull on MTs from sites in the cytoplasm [START_REF] Barbosa | Dynactin binding to tyrosinated microtubules promotes centrosome centration in C. elegans by enhancing dynein-mediated organelle transport[END_REF]Hamaguchi et al., 1986;[START_REF] Kimura | Intracellular organelles mediate cytoplasmic pulling force for centrosome centration in the Caenorhabditis elegans early embryo[END_REF][START_REF] Longoria | Cargo transport by cytoplasmic dynein can center embryonic centrosomes[END_REF]. The mechanical coupling of dyneins to the cytoplasm is thought to be mediated by the friction of cargos or endo-membranes that are moved by dyneins to the aster center. Accumulation of dynein-cargo complexes onto MTs in a length-dependent manner naturally results in MT-pulling forces that scale with the length of MTs; but even if those complexes are diluted in the aster, the pulling forces are still predicted to scale to the length of MTs (although in a non-linear manner) (Fig. 3A). It is still unclear though whether there are types of endo-membrane or cargo that are more suitable for MT forces in bulk cytoplasm, and how these cytoplasmic anchors may be recycled back to the surface to be able to continuously sustain this effect.

MT-based centering systems have remarkable abilities in tracking the geometrical confines of a cell. For instance, sperm asters, which nucleate on the side of the egg at fertilization, can precisely target the center of large eggs in tens of minutes without any prior positional information. This is because they can continuously monitor the local cell geometry through MT growth, length-limitation by the cortex and length-dependent forces. This was clearly demonstrated by manipulations of egg shape and sites of sperm entry, which all led to astercentering paths with sharp turning points that reflected the local cell geometry explored by the aster during centration (Fig. 3B) [START_REF] Tanimoto | Shape-motion relationships of centering microtubule asters[END_REF][START_REF] Tanimoto | Quantitative approaches for the study of microtubule aster motion in large eggs[END_REF]. Another property of length-dependent MT forces, is that they act as elastic springs with respect to the geometry of the cell. In recent work, magnetic beads attached to centrosomes in C. elegans metaphase spindles allowed to apply external calibrated forces to asters [START_REF] Garzon-Coral | A force-generating machinery maintains the spindle at the cell center during mitosis[END_REF]. Centrosomes moved away from the cell center to stop at a position that depended on the applied force and rapidly repositioned to the center after force cessation, much like a spring (Fig. 3C). The stiffness value of this spring is of key biological significance. A small stiffness would permit aster fluctuations, rotations or displacement owing to other cellular cues (e.g. an asymmetric cortical domain), whereas a high stiffness would tend to "freeze" the aster at the cell center [START_REF] Howard | Physical Limits on the Precision of Mitotic Spindle Positioning by Microtubule Pushing forces: Mechanics of mitotic spindle positioning[END_REF]. How stiffness and overall mechanical properties of centering machineries vary with shape, size and among different cell types, and how they relate to aster dynamics and division phenotypes is an exciting future research direction.

Finally, it is worth noting that some cells may promote nuclear or centrosome centration relative to cell shape using systems that may not solely rely on MT forces. Actomyosin contraction might produce forces that could directly influence MT dynamics, aster motion and position [START_REF] Burakov | Centrosome positioning in interphase cells[END_REF][START_REF] Zhu | Finding the cell center by a balance of dynein and myosin pulling and microtubule pushing: a computational study[END_REF]. Remarkably, mouse oocytes, which lack centrosomes and asters, are capable of moving nuclei to the center of large eggs, albeit in a much slower manner than in embryos discussed above [START_REF] Almonacid | Active diffusion positions the nucleus in mouse oocytes[END_REF]. This is thought to be mediated by a gradient of actin-driven cytoplasmic agitation, with more agitation close to the cell cortex. As a result of this gradient, a nucleus positioned off-center will experience more "kicks" on the cortex-facing side, which will move it to the center of the cell. Thus in some cases, actin may contribute to geometry sensing to move large structures and organelles to the center of cells.

Orienting division with cell shape

After the nucleus has reached the center, the centrosome duplicates and defines an orientation axis for the mitotic spindles. As observed almost 150 years ago, this axis most often correlates with the long axis of the cell, but how cells probe their geometry to orient division accordingly remained mysterious. Recent studies of the cleavage patterns of early embryos suggested that length-dependent MT forces could robustly function in orientating the division axis with respect to cell shape anisotropies [START_REF] Minc | Influence of cell geometry on division-plane positioning[END_REF][START_REF] Wuhr | A model for cleavage plane determination in early amphibian and fish embryos[END_REF]. In essence, the distribution of MT length around two sister asters generate torques that rotate the centrosome pair, with a favored configuration of minimal energy that corresponds to the long axis of the cell (Fig. 3D). This model was validated in multiple studies, including by manipulating the shape of embryos, or by exploiting the natural changes in 3D blastomere cell shapes that occur in various cleaving embryos [START_REF] Bjerknes | Physical theory of the orientation of astral mitotic spindles[END_REF][START_REF] Minc | Influence of cell geometry on division-plane positioning[END_REF][START_REF] Mitchison | Growth, interaction, and positioning of microtubule asters in extremely large vertebrate embryo cells[END_REF][START_REF] Pierre | Generic Theoretical Models to Predict Division Patterns of Cleaving Embryos[END_REF][START_REF] Wuhr | A model for cleavage plane determination in early amphibian and fish embryos[END_REF]. One prime example is the cleavage pattern of fish embryos, which entails a choreography of successive cell divisions with precise 3D angular settings (Fig. 3E). The one-cell fish egg consists of a flat blastodisc located on a large yolk layer. Because the yolk blocks aster growth, the aster adopts a flat 3D shape, and as a consequence the first division axis lies parallel to the yolk interface. At telophase, asters and centrosomes are duplicated and elongate along the long sister aster shapes, perpendicular to the first division axis, giving rise to a division orthogonal to the previous plane [START_REF] Wuhr | A model for cleavage plane determination in early amphibian and fish embryos[END_REF]. This alternate orthogonal sequence of division axis and cell/aster shapes continues on until the 16-cell stage, at which point the four most-centered blastomeres, which are squeezed by their neighbors, elongate their shape perpendicular to the yolk and orient their division along this shape axis, thereby ensuring tissue layering in the 3 rd dimension [START_REF] Olivier | Cell lineage reconstruction of early zebrafish embryos using label-free nonlinear microscopy[END_REF] (Fig. 3E). Strikingly, 3D models that are solely based on shape-sensing from lengthdependent MT forces were able to account for this cleavage pattern in fish and other species [START_REF] Pierre | Generic Theoretical Models to Predict Division Patterns of Cleaving Embryos[END_REF]. Thus, the interplay between shapes and division can serve as an iterative design to produce complex patterns of oriented divisions that are instrumental to the morphogenetic development of embryos and tissues [START_REF] Pierre | Generic Theoretical Models to Predict Division Patterns of Cleaving Embryos[END_REF][START_REF] Xiong | Interplay of cell shape and division orientation promotes robust morphogenesis of developing epithelia[END_REF].

In light of these models, the factors that contribute to shape cells are thus ultimately key to understand division positioning in multicellular tissues. Cortical tension and cell-cell adhesion are prime cell-shape regulators in tissues, but tissue tension or compression forces from neighboring cells could contribute as well (Fig. 3E andF). In return, the orientation of cell division along the cell shape axis also has functional implications in tissues. It may serve to relax stresses in epithelial monolayers [START_REF] Campinho | Tension-oriented cell divisions limit anisotropic tissue tension in epithelial spreading during zebrafish epiboly[END_REF][START_REF] Wyatt | Emergence of homeostatic epithelial packing and stress dissipation through divisions oriented along the long cell axis[END_REF], or as the basis of an homeostatic mechanism regulating the topology of cell contacts [START_REF] Gibson | Control of the mitotic cleavage plane by local epithelial topology[END_REF]. One important current debate is to discern whether division orientation in tissues under tension or compression may be caused by the direct effect of external forces on the division machinery [START_REF] Fink | External forces control mitotic spindle positioning[END_REF], or is the sole result of deformed cell shapes that orient division [START_REF] Minc | Influence of cell geometry on division-plane positioning[END_REF][START_REF] Pierre | Generic Theoretical Models to Predict Division Patterns of Cleaving Embryos[END_REF].

Cell shape can also change during mitosis. Rounding is prominent in adherent vertebrate cells and in epithelia, and appears as one important mechanism temporarily erase the contribution of geometry to division axis determination [START_REF] Thery | Get round and stiff for mitosis[END_REF]). Yet, in many instances, the mitotic spindle in the rounded cell still aligns with the preceding interphase long axis. Seminal work using cells seeded on micro-patterns showed that this memory is provided by actin fibers, which read the geometry of the adhesive pattern and influence polarity cues inside the round mitotic cell to orient the spindle (Thery and Bornens, 2006;[START_REF] Thery | Experimental and theoretical study of mitotic spindle orientation[END_REF][START_REF] Thery | The extracellular matrix guides the orientation of the cell division axis[END_REF]. Studies of the protein Mud in fly epithelia (Numa in vertebrates) have recently suggested that similar shape-memory mechanisms exist in tissues [START_REF] Bosveld | Epithelial tricellular junctions act as interphase cell shape sensors to orient mitosis[END_REF]. An interesting recent study also revealed that when rounding is inhibited in the fly embryo epithelium, cell shape will largely override cortical polarity cues, yielding division orientations that are oftentimes orthogonal to the epithelial layer [START_REF] Chanet | Myosin 2-Induced Mitotic Rounding Enables Columnar Epithelial Cells to Interpret Cortical Spindle Positioning Cues[END_REF]. Thus, a tentative speculation is that any significant anisotropy in cell shape could dominate over other signals in symmetrically dividing cells.

Cells lacking centrosomes such as the early blastomeres of many rodent embryos [START_REF] Courtois | The transition from meiotic to mitotic spindle assembly is gradual during early mammalian development[END_REF] can still orient their division to cell shape by mechanisms that thus far remain mostly mysterious [START_REF] Gray | First cleavage of the mouse embryo responds to change in egg shape at fertilization[END_REF]. Actin may be involved at some level, but the geometrical features that allow the actin cytoskeleton to read cell shape in this context are not known [START_REF] Chaigne | F-actin mechanics control spindle centring in the mouse zygote[END_REF]. Cortical and/or cytoplasmic protein gradients may also contribute to division positioning relative to cell shape. For instance, in adherent cells, gradients of Ran-GTP that emanate from chromosomes and diffusible signals from polo-like kinase that is located at spindle poles could serve as rulers to monitor and control the position and orientation of the spindle [START_REF] Kiyomitsu | Chromosome-and spindle-pole-derived signals generate an intrinsic code for spindle position and orientation[END_REF]. In bacterial cells, dynamic gradients of the Min proteins serve as geometry sensors to promote medial bacterial fission (see [START_REF] Kretschmer | Pattern formation on membranes and its role in bacterial cell division[END_REF] for a recent review). Bacterial systems have selforganized membrane-associated patterns that are built around a core set of three proteins, Min C, -D and -E, which exhibit pole-to-pole oscillations as a result of specific rules of their biochemical interactions and diffusion properties. Through time-averaging, this system defines a narrow mid-cell zone where the septum machinery is allowed to assemble to divide the cell. Remarkably, the Min proteins can self-assemble into dynamic patterns in vitro and probe the geometry to define a long axis even in triangular or squared shapes [START_REF] Schweizer | Geometry sensing by self-organized protein patterns[END_REF][START_REF] Wu | Symmetry and scale orient Min protein patterns in shaped bacterial sculptures[END_REF][START_REF] Zieske | Reconstitution of self-organizing protein gradients as spatial cues in cell-free systems[END_REF]. Thus a system based on reactiondiffusion can generate specific positional protein patterns with respect to the geometry of a cell and may represent an additional important feature to connect cell shape with division position [START_REF] Xiao | Mitotic Cortical Waves Predict Future Division Sites by Encoding Positional and Size Information[END_REF].

CONCLUSION AND FUTURE DIRECTIONS

The "morphobiological" mechanisms that allow cells to probe their geometry are still at their premise stage. However, by summarizing recent efforts in the literature, we identified several generic design principles, which appear to have been repeatedly used in evolution. The first one is based on MT networks, which can convert cellular geometries into defined modes of polarization or oriented division. MTs serve as prominent rulers, primarily owing to the longrange and dynamic nature of MT polymers and the forces they exert on the cortex, the cytoplasm and on nuclei and spindles. A second more local design feature is based on the differential curvature recognition that is encoded in the structural properties of molecular assemblies, such as septins, MreB filaments, and Myosin filaments. The final feature is based on reaction-diffusion and its ability to form self-assembled patterns relative to cell geometry.

An intriguing realization is that the above-mentioned systems exist in most cells, raising the question of how they may be tuned down to allow a more deterministic cue, such as for instance a chemical gradient, to override them.

Up to date, however, many of the relevant studies have only been performed at the single-cell level, oftentimes in model cell types, such as yeast, Hela cells, or invertebrate zygotes, with still only few studies having recognized the contribution of cell shape sensing in multicellular tissues. This is because tissues come with a plethora of cues that are frequently orienting cell behavior along the same direction as cell shape, making it difficult to disentangle them. In addition, cell shapes in tissues have complex 3D features, which are challenging to analyze.

The standardization of novel microscopy methodologies, such as selective plane illumination microscopy (SPIM), or two-photon microscopy, could circumvent this limitation by allowing the analysis of cell shapes in 3 dimensions deep inside of developing tissues and organs.

Text Box 1: Geometrical terms.

Curvature: Defines how much a surface deviates from being flat. Throughout this review, we define curvature from the perspective of the cell interior. For a spherical cell this means that its inner surface has a negative curvature. A tube-shaped protrusion of this sphere also has a negative curvature along its transverse section, but furthermore generates a region of positive curvature at its base in the direction pointing from the sphere into the protrusion. The topology at the base is called a 'saddle', where the curvature along two orthogonal axes is opposite.

Aspect ratio: Defines the proportional relation of width and height of a two-dimensional object. It can be applied to cells if depth is similar to width or width has an approximately rotational symmetry along the height axis.

Surface-to-volume ratio:

Defines the amount of surface area per unit volume of an object.

For an object with a given geometry, as it gets bigger, volume increases faster than the surface area. A spherical cell has the lowest surface-to-volume ratio, compared to e.g. flat or tube-like shapes. MT polymerizing and buckling or from pulling in the cytoplasm that is mediated by dyneincargo motion. B) Aster growth and length-dependent forces provide dynamic shape-sensing abilities to MT asters and result in aster-centering paths that depend on cell shape.

BIBLIOGRAPHY FIGURES AND FIGURE LEGENDS

Reproduced with modifications from [START_REF] Tanimoto | Shape-motion relationships of centering microtubule asters[END_REF]. C) The cellular response to the application of external forces with magnetic tweezers on centered asters suggests that asters behave as elastic springs to maintain the spindle in the cell center. Red curve: Upon force application, the distance to the center, d(t) increases and reaches a plateau value dmax , at which aster forces balance the magnetic external force. When the force is released d(t) relaxes exponentially to 0. Blue curve: By varying the externally applied forces, and measuring the resultant value of dmax , the stiffness of the centering spring can be computed. D) When an egg is shaped into a rectangular microwell, the torques and forces generated through lengthdependent MT forces will align nuclei or spindles with the long axis of a cell. E) Cleavage patterns of zebrafish embryos exemplify the iterative influence of cell shape on division orientation and vice versa. F) (Left) Cells in tissues may be exposed to external mechanical forces, such as tissue tension or compression from neighboring cells, which may influence cell shape and consequent spindle orientation with respect to external forces. (Rigth) As a consequence cells will divide with respect to those mechanical forces; which could in turn relax tissue stress, or influence the topology of cell-cell contacts.

Fig. 1 :

 1 Fig. 1: Examples of cell shape features potentially used by cells to control behavior and organization. A) Global, cell-level features, such as aspect ratio, long axis, curvature and surface-to-volume ratio, might be monitored by a cell to ensure proper polarization, signaling

Fig. 2 :

 2 Fig. 2: Mechanisms of shape sensing with relevance for cell polarity, migration and signaling. A) Fission yeast shape-sensing by MTs directly inputs on the positioning of polarity domains. B) Cell shape, MT alignment and polarization in animal cells. Left: Subapical MT alignment with cell shape in Drosophila epithelia. Right: In adherent cells, MTs organize along the long cell shape axis, defining the major polarity axis. Right panel is reproduced with modifications from (Thery et al., 2006), Copyright 2006 National Academy of Sciences. C) MTs dynamically align along the establishing long axis during symmetry breaking in cell migration. D) Bacterial shape-sensing and maintenance. E. coli cells detect shape defects through the preferential binding of MreB to positively curved membranes and

Fig. 3 :

 3 Fig. 3: Cell-shape sensing for division positioning. A) Astral Microtubule (MT) exert length-dependent forces to center nuclei and centrosomes. Length-dependency may arise from

Finally, one area that is lagging behind in this field has been the systematic identification of the gene products specifically affected by cell shape, or required for shape-sensing. Given the advent of microfabrication methods, recent studies are now beginning to describe methods to systematically screen genes required for shape sensing, for instance during spindle orientation [START_REF] Burri | TRACMIT: An effective pipeline for tracking and analyzing cells on micropatterns through mitosis[END_REF]. Another way to approach this question is by exploiting cell shape variations, which are widespread in diseases such as cancer. To this end, the recent large-scale profiling of gene expression and image analysis pipelines has begun to establish dosedependent correlations between geometry and the activation of specific pathways in some cancers [START_REF] Sailem | Identification of clinically predictive metagenes that encode components of a network coupling cell shape to transcription by image-omics[END_REF][START_REF] Sero | Cell shape and the microenvironment regulate nuclear translocation of NF-kappaB in breast epithelial and tumor cells[END_REF]. Therefore, further investigating morphobiological mechanisms may also have relevance in understanding the contribution of cell geometry to the pathophysiological behavior of cells in disease conditions.

ACKNOWLEDGMENT

The authors acknowledge S. Dmitrieff for careful reading of the manuscript. Research in our laboratory is supported by the CNRS and grants from the "Mairie de Paris emergence" program, the FRM "amorçage" grant AJE20130426890, the ANR "CellSize" grant and the European Research Council (CoG Forcaster N° 647073).


Hirokazu Tanimoto

Nicolas Minc
email: nicolas.minc@ijm.fr

QUANTITATIVE APPROACHES FOR THE STUDY OF MICROTUBULE ASTER MOTION IN LARGE EGGS

Keywords: Microtubule aster, Dynein, Sea urchin, 3D tracking, Laser ablation 2

The proper positioning of microtubule (MT) asters underlies fundamental processes such as nuclear centration, cell polarity, division positioning and embryogenesis. In large eggs and early blastomeres, MT asters may exhibit long range motions with atypical speed and precision, to target their functional position. The biophysical mechanisms regulating such motions remain however largely unknown. The centration of sperm asters in sea urchin embryos is a stereotypical example of such aster long range motion. In this chapter, we describe methods developed in this system to (i) quantify sperm aster 3D motion with confocal microscopy and automated image analysis, and (ii) severe a portion of astral MTs with a UV laser. These methods may serve as a template to dissect the generic mechanisms of aster motion and force production in other embryos and cell types.

Introduction

The spatio-temporal control of microtubule (MT) aster position in the cell is crucial to various biological contexts and has received extensive attention in the context of cell division plane specification [START_REF] Elric | Centrosome positioning in polarized cells: common themes and variations[END_REF][START_REF] Kimura | A novel mechanism of microtubule length-dependent force to pull centrosomes toward the cell center[END_REF][START_REF] Kiyomitsu | Mechanisms of daughter cell-size control during cell division[END_REF][START_REF] Mcnally | Mechanisms of spindle positioning[END_REF][START_REF] Minc | Predicting division plane position and orientation[END_REF]. To precisely control daughter cell size and shape, cells may sense their shape to place the division plane accordingly. MT asters, which are star-shaped arrays of MTs radiating from centrosomes, may serve to convert cell shape information to division plane positioning. In small cells such as adherent animal cells, metaphase asters around the spindle attach to cell cortex and interact with molecular motors such as dynein at the cortex to mediate spindle and consequent division-plane positions [START_REF] Kiyomitsu | Cortical dynein and asymmetric membrane elongation coordinately position the spindle in anaphase[END_REF]. In contrast, in large cells such as echinoderm, amphibian, and fish eggs and early blastomeres, metaphase asters are too small to reach the cell cortex [START_REF] Mitchison | Growth, interaction, and positioning of microtubule asters in extremely large vertebrate embryo cells[END_REF]. In these cells, interphase/telophase asters fill large portion of the cytoplasm and have been proposed to mediate division plane determination through dynein pulling forces produced along MTs in bulk cytoplasm [START_REF] Minc | Influence of cell geometry on division-plane positioning[END_REF][START_REF] Tanimoto | Shape-motion relationships of centering microtubule asters[END_REF][START_REF] Wuhr | A model for cleavage plane determination in early amphibian and fish embryos[END_REF].

The functional positioning of MT asters relies on precise aster motion. While many studies have focused on the mechanisms that determine aster final position, little is known about how asters move inside cells. This question becomes particularly prominent in the context of eggs and early blastomeres given their large size and the rapidity of early embryonic cell cycles. A typical example, conserved in most species, is sperm aster centration. After fertilization sperms bring centrioles inside eggs to nucleate a sperm aster that moves to the exact center of the large egg [START_REF] Chambers | The movement of the egg nucleus in relation to the sperm aster in the echinoderm egg[END_REF][START_REF] Hamaguchi | Fertilization Process in the Heart Urchin, Clypeaster Japonicus Observed with a Differential Interference Microscope[END_REF]. These asters continuously grow in size and move at speeds of several microns per minute to reach the exact center of these large cells. The lack of information on sperm aster motion, may be in part associated to difficulties of imaging motion in 3D, in large cells, and on the limited set of quantitative methodologies to manipulate aster forces in space and time. Recently, we developed quantitative methods, based on 3D microscopy and laser ablation to address the biophysical mechanisms driving sperm aster motion in sea urchin embryos [START_REF] Tanimoto | Shape-motion relationships of centering microtubule asters[END_REF]. The first method uses confocal microscopy and MATLAB-based automated image analysis to precisely track aster motion in 3D, for normal spherical eggs and for eggs shaped in microchambers. This method serves to obtain motion characteristics such as travelling distance, speed, and velocity orientation. The second method uses a UV laser ablation setup to severe astral MTs in different positions and monitor the consequence on speeds and trajectories. Since the presented methods do not rely on any peculiar features of the organism, we emphasize they could be useful to study aster motion questions in other systems.

Three-dimensional tracking of microtubule asters in sea urchin embryos

This section describes a basic method to track aster centration in 3D with high precision. The method includes cell handling, confocal microscopy, PDMS chamber operation, and image analysis using a custom-made code written in MATLAB.

Detecting 3D aster position using confocal microscopy

Materials

Sea urchin (Paracentrotus lividus), Sterile water, Artificial sea water (ASW; Instant Ocean; Reef Crystals), 80 m Nitex mesh (Sefar), Hoechst 33342 (Sigma), Glass bottom dish (50 mm diameter, thickness no. 1.5; MatTek Corporation), Protamine sulfate (Sigma), PDMS chamber (prepared as described in [START_REF] Chang | Manipulating Cell Shape by Placing Cells into Micro-fabricated Chambers[END_REF]), Plasma cleaner (Harrick Plasma), Air gun, Spinning disk confocal microscopy (Nikon Ti-Eclipse combined with Yokogawa CSU-X1 spinning-head and Hamamatsu EM-CCD camera) equipped with a 40x objective (Nikon Plan Fluor, NA 1.3).

Sample preparation

a. Preparation of imaging dishes. Treat glass-bottom dishes with 1% protamine sulfate for 2 min. Rinse the dishes thoroughly with sterile water and dry them under the hood. This treatment makes the dishes positively charged, on which eggs will adhere. The coated dishes can be stored in a clean box at room temperature for at least 1 month.

b. Obtain gametes from adult sea urchins by intracoelomic injection of 0.5 M KCl. Keep sperm at 4 degree and use them within ~1 week. Rinse the eggs twice using ASW and store them at 16 degree. Eggs should be used within the day of collection. c. Remove the jelly coat of unfertilized eggs to facilitate cell adhesion by passing them three times though an 80-m Nitex mesh. Alternatively, acidic treatment may also be used (1~2 min treatment with pH 5.5 ASW at 16 degree, see [START_REF] Cheers | Rapid microinjection of fertilized eggs[END_REF]) for details).

d. Place a 6 mL solution of de-jellied eggs on a coated glass-bottom dish with adequate dilution in ASW. Add the DNA dye Hoechst 33342 at a final concentration of 1 g/mL. If the Hoechst signal intensity of the male pronucleus is too weak, one can increase Hoechst concentration up to 5 g/mL, which does not alter aster centration dynamics in our experimental condition.

Imaging

Setup

The position of MT asters is determined by tracking the center of the male pronucleus bound to centrosomes which act as microtubule organizing centers. The male pronucleus is visualized with the fluorescent DNA dye Hoechst, and its position is measured by taking confocal timelapse Z stacks (Fig. 1 A). The interval of the Z slice should be small enough to precisely determine the position of the aster in Z . However, a too small slice interval reduces the temporal resolution and can cause photo-toxicity. A Z-interval of 3 m is sufficient to catch the male pronucleus (~2 m size) in a single focal plane in one of the slice of the stack (Fig. 1 B). The number of Z slices is set so that the confocal stack covers a hemisphere of the eggs which allows to monitor the whole centration. 15 slices are sufficient to cover the hemisphere of a sea urchin egg (90 m in diameter). The source laser is a 405 nm laser. Laser intensity and exposure time are adjusted to minimize photo-toxicity. It takes ~8 sec to take one Z stack in our setup and the temporal interval, t is set to 10 sec.

Imaging in PDMS chamber

The 3D tracking framework described in the last section can also be used to monitor aster motion in eggs shaped into PDMS chambers. This subsection describes additional procedures required for imaging aster motion in this condition. Procedure a. Prepare appropriate PDMS chambers with cell-size microwells (see [START_REF] Chang | Manipulating Cell Shape by Placing Cells into Micro-fabricated Chambers[END_REF] for detailed protocols for chamber design and fabrication). f. Remove excess ASW from the slab side using kimwipe. The slab approaches to the coverslip which causes eggs to go into the microwells.

g. Place the sample under the microscope. Find a cell successfully shaped in a well and acquire time-lapse images as described in 2.1.1 (Fig. 1 G). It typically takes 3-5 min from sperm addition to the beginning of the image acquisition (steps d-g).

Quantification of aster 3D motion

Materials

Standard desktop PC with a MATLAB software (supplemented with the statistical and image analysis toolboxes; Mathworks) and dedicated scripts (available upon demand).

Image analysis framework

The 3D position of the male pronucleus is determined from a fluorescence time-lapse Z stack of images using home-built MATLAB scripts. Aster Z position is determined by detecting the plane in which the male pronucleus is in focus (Fig. 1 A andB). Aster XY position is then determined by detecting the 2D center of male pronucleus within the focal plane. The method enables a precise positional detection: the precision of the position in XY is estimated to be below 1 m (using immobile sperm on the dish), and that in Z to be ~ 3 m (set by the Z slice interval) (Fig. 1 C).

The aster 3D position X ⃗ ⃗ (𝑡) is used to extract aster motion characteristics (Fig. 1 D-F).

Aster travelling distance 𝐿(𝑡) is defined as the distance between aster initial and current positions 𝐿(𝑡) = |X ⃗ ⃗ (𝑡) -X ⃗ ⃗ (0)|. Aster instantaneous velocity is defined as V ⃗ ⃗ (𝑡) = {X ⃗ ⃗ (𝑡 + 𝜏) -X ⃗ ⃗ (𝑡)} 𝜏 ⁄ , where =30 sec, and |𝑉| is the aster speed. Aster 3D orientation is defined using spherical coordinate;

V ⃗ ⃗ = ( 𝑉 𝑥 𝑉 𝑦 𝑉 𝑧 ) = ( |𝑉| sin 𝜃 cos 𝜑 |𝑉| sin 𝜃 sin 𝜑 |𝑉| cos 𝜃 ),
where  and  are polar and azimuthal angles, respectively.

Since aster motion involves fluctuations around a persistent directed motion, it may be non-trivial to define a travelling distance for asters undergoing a non-straight motion. In the case of sperm asters' motion in shape-manipulated cells, aster rotations are very sharp so the migrating path consists of multiple straight lines and small part connecting them (Fig. 1 e. The scripts automatically find the Z plane in which the male pronucleus is in focus at successive time points, and detect the XY position in the plane.

f. The script allows to check the tracking results in an interactive way. The script generates images of detected outline of male pronucleus superimposed on the real image of the male pronucleus at the focal plane. The tracking results are checked manually based on these images. If an error is found at a given time point, the aster position at this point may be updated by changing the initial position estimation and/or the threshold value for male pronucleus shape detection.

Output

The scripts generate an aster 3D trajectory (Fig. 1 C) and computes aster motion characteristics such as travelling distance (Fig. 1 D) and amplitude and orientation of the instantaneous velocity (Fig. 1 E andF). These outputs are saved both as plot figures and as MatLab files which can be exported to excel. These quantifications can be used to characterize complicated aster motion such as rotation(s) in shape-manipulated cells (Fig. 1 G-K).

Laser ablation of astral microtubules and in-situ immunostaining

This section describes: (i) a method to photo-ablate MTs on different parts of migrating sperm asters, and (ii) a method to perform rapid in-situ immunostaining of ablated asters.

Materials

In addition to the materials required for tracking: the confocal microscopy is equipped with a high-power pulsed 355-nm UV laser system (iPulse, Roper Technologies) operated by iLas2 software (Roper Technologies). Chemicals for immunostaining, as listed in [START_REF] Minc | Influence of cell geometry on division-plane positioning[END_REF].

Laser ablation of astral microtubules

Procedure a. Prepare a sample and place it on the microscope as described in 2.1.1.

b. Fertilize eggs under the microscope by adding activated sperm.

c. Select an egg in which the sperm enters close to the equatorial plane so that the aster moves mostly in plane.

d. Start a time-lapse acquisition to monitor the aster motion as described in 2.1.2.

Since the aster moves almost in-plane, a Z stack is not always necessary for the measurement. We often reduce the number of Z slice to 5 and Z to 1~2 m.

e. Perform laser ablation when the aster has moved ~ 25 m away from the cortex.

The size of the target aster is a critical parameter to obtain a significant and reproducible effect of ablation on aster motion. The ablation position should be far enough from both the aster center and cell cortex. If the ablation position is too close to the male pronucleus, the laser irradiation bleaches the Hoechst signal and prevents subsequent tracking of the male pronucleus.

In addition, ablating too close to the aster center oftentimes stops aster motion for unknown reason(s). If the ablation position is too close to the cell cortex, the irradiation affects cell integrity and oftentimes leads to cell bursting.

We perform laser ablation when the aster has moved ~ 25 m from the cortex, when the aster size is around 50 m [START_REF] Tanimoto | Shape-motion relationships of centering microtubule asters[END_REF]. The irradiated line is set to be ~30 m long.

The irradiated position is set to be 10 ~15 m away from the aster center (Fig. 2 A). This setup enables us to perform laser ablation either at the front or on the side of MT asters. However, this setup does not work for rear MT ablation as the laser line tends to cross the cell cortex, and yields to cell bursting typically 1 min after ablation.

The laser power is set to be 50%. Each irradiated line consists of three 350-nm beam lines spaced by an interval of 1 m (by setting the "line thickness" parameter in the iLas2 software to be 2 in the setting for the 40x objective). Each line irradiation takes 5-7 ms and is iterated 400 times. To ablate the MT aster in 3D, the ablation is performed at three different heights: at the aster center and 5 μm above and below. The total time needed to complete the ablation is ~ 10 sec, thus if the experiments do not need fine temporal resolution, ablations of several eggs may even be performed in parallel using a motorized XY stage. This setup is sufficient to significantly reduce MT length and density at the ablated region (Fig. 2 B).

f. Continue time-lapse acquisition (Fig. 2 C). Analyze aster motion as described in 2.2 to quantify the effects of ablation assay.

Laser ablation may sometimes causes small egg movement. In these cases, we determine the cell movement from the auto-fluorescence of the cytoplasm or DIC and subtract it from the aster motion to get the aster motion in the cell referential frame.

Output

The laser ablation assay of astral MTs serves to dissect the direction of MT forces that drives aster motion. In the case of sperm asters of sea urchin eggs, side ablation shifts the aster centration path away from the ablation side (Fig. 2 D and E), and front ablation leads to transient reverse of aster velocity orientation [START_REF] Tanimoto | Shape-motion relationships of centering microtubule asters[END_REF]; both suggesting that aster motion is driven by pulling forces mediated by MTs. f. Perform fixation immediately after laser ablation, by rapidly introducing the fixative from the hole, following a fixation procedure described in [START_REF] Minc | Influence of cell geometry on division-plane positioning[END_REF]. With this setup one can fix samples within less than 10 sec after ablation.

In-situ immunostaining of ablated asters

g. Perform successive steps of immunostaining by exchanging solutions under the microscope, to ensure that the eggs do not move nor deform during the process.

Output

Since the re-growth of ablated astral MTs may be very quick [START_REF] Hyman | Centrosome movement in the early divisions of Caenorhabditis elegans: a cortical site determining centrosome position[END_REF], visualizing the effects of laser ablation requires rapid in-situ immunostaining. The presented protocol enables rapid in-situ immunostaining of ablated MT asters that is crucial to assess the effects of laser ablation on aster size and density (Fig. 2 B).

Conclusions

The methods described in this chapter serve to study aster motion in sea urchin embryos and possibly for other species for which characterization of aster motion remains to be analyzed and understood [START_REF] Stewart-Savage | The temporal and spatial relationships between cortical contraction, sperm trail formation, and pronuclear migration in fertilizedXenopus eggs[END_REF]. The dynamics of aster motion reflects the underlying mechanisms of aster force production (A. [START_REF] Kimura | Computer simulations and image processing reveal lengthdependent pulling force as the primary mechanism for C. elegans male pronuclear migration[END_REF][START_REF] Kimura | Local cortical pulling-force repression switches centrosomal centration and posterior displacement in C. elegans[END_REF]. However, our studies suggest that MT asters in different model organisms could exhibit qualitatively different motions even if they may rely on common biological principles [START_REF] Tanimoto | Shape-motion relationships of centering microtubule asters[END_REF]. These observations should stimulate the developments of unified biophysical theories, which will likely require further quantitative experiments of aster motion in various cell types.

Laser ablation is a powerful tool to dissect the direction of MT forces, and may serve to obtain more quantitative information on the functional form of aster force dependency on aster size and shape; which are key for constructing quantitative mathematical models.

Acknowledgments:

We thank Akatsuki Kimura, François Rousset, and members of the Minc Laboratory for technical help and discussion. This work is supported by the CNRS and grants from the "Mairie de Paris emergence" program, the FRM "amorçage" grant AJE20130426890 and the ERC (CoG Forcaster N° 647073). The aster migrates from left to right, and black and red colors mark the aster center before and after ablation, respectively. The side ablation leads to an aster moving in a direction away from the MT ablation site, indicating the aster may be driven by outward forces. Bars: 50 m (B-D)

and 10 m (E).

  Procedure a. Place the sample on the microscope stage. b. Dilute dry sperm 1000-fold in ASW by putting 4 L sperm into 4 mL ASW. Activate them by vigorous aeration. Add the sperm solution dropwise to the dish under the microscope. c. Detect the Z position of a sperm entering an egg. Choose the initial Z position of the timelapse movie so that the Z stack volume includes both initial male pronucleus position and the egg center. d. Start time-lapse acquisition. The entire aster centration takes ~10-15 min in normal conditions (Fig. 1 C).

  b. Clean PDMS chambers and glass cover slips. The cover slips should be larger than the PDMS slabs. Treat them with Acetone, Isopropanol and then rinse with sterile water. Dry them thoroughly using an air gun and keep them on kimwipes. c. Activate PDMS chambers, with micro-wells facing up with a plasma cleaner. Check if they become hydrophilic by putting a drop of ASW. If not, dry them and repeat plasma activation possibly by increasing treatment time. Keep activated chambers in ASW until use. The activation lasts ~ 4 hours. d. Go to the microscope. Fertilize eggs in 15 mL falcon tubes. Wait ~ 1 min until the eggs accumulate at the bottom of the falcon tube by sedimentation. Place 40 L of a dense egg solution on the clean coverslip. e. Remove ASW from the activated PDMS slab with a kimwipe. Slowly apply the PDMS slab on the top of the egg solution with microwells facing down. Do not press the slab.

  G). The travelling distance of these asters can be defined as a summation of the passed paths and the distance from the last rotating position. 10 Procedure a. Open MATLAB and run the script. b. Choose the Z slice in which the male pronucleus is in focus at the first time point. c. The script opens the image of the first time point. Select manually a rough position of the male pronucleus in the image. d. The script updates the pronucleus position in the first time point by detecting the male pronucleus shape based on thresholding.

  Procedure a. Prepare PDMS slabs with a wide and straight perfusion channel and two inlet holes. The dimension of the channel is 8 mm*25 mm and 70~75 m in height. The height is ~10% smaller than the egg diameter to slightly flatten the eggs. The slight flattening stabilizes cell position and forces sperms to enter around the cell equatorial plane. Pierce holes at the two extremes of the channel to allow liquid perfusion. b. Choose coverslips larger than the slab dimension. Clean the PDMS slabs as well as coverslips using acetone and isopropanol. Rinse them with water and thoroughly dry with an air gun. c. Place a 25 L solution of unfertilized egg onto the PDMS channel facing up. Put the coverslip onto the PDMS slab, and push onto the periphery to seal the channel. d. Fertilize the eggs in the chamber by introducing activated sperm from the channel inlet hole. e. Perform laser ablation as described in 3.1.1.

  Figure 1: Three-dimensional tracking of centering MT asters. (A) Sperm pronuclei at the aster center are imaged at each time point by acquiring confocal zslices. The slice interval (Z) is set to be 3 m and 15 slices are acquired so that the Z stack covers the hemisphere of a sea urchin embryo whose radius is 45 m. Each Z stack takes ~8 sec for the acquisition and the time interval (t) is set to be 10 sec. (B) Confocal images of the sperm pronucleus visualized with a DNA dye Hoechst. Images in which the sperm pronucleus is in focus, and 3 μm above/below are shown. The Z position of the center of the sperm aster is determined as the plane with maximum Hoechst signal. The XY position is determined as the center of mass of the 2D shape of the nucleus extracted by thresholding (red line). (C-F) Quantification of aster 3D centration. (C) 3D trajectory of a centering MT aster in a normal spherical cell. X=Y=Z=0 is the center of the egg (the same for (G)). The color indicates time. The trajectory consists from initial concaved penetration and long straight centration. The result shows that the precision of the tracking method is sufficient to confirm that the aster centration path is straight not only in XY but also in XZ and YZ. (D-F) Aster travelling distance (D), speed (E), and velocity orientation (F) are plotted as a function of time. See main text for definition.(G-K) Non-straight aster centration in an egg shaped into a microfabricated chamber. The aster rotates during centration (indicated by a black arrow). Note that the tracking method identifies aster rotation in 3D, which is characterized by changes in both polar and azimuthal angles (K).

Figure 2 :

 2 Figure 2: Laser ablation assay for MT asters. (A) Schematic of the laser ablation assay. The setup for ablation at the aster side is shown. The red broken line indicates where the laser is applied (same for B-D) Intact and severed MTs are shown by green and gray colors, respectively. The ablation is performed in three Z heights: the aster focal plane and 5 m above/below. (B) Tubulin immunostaining of an ablated aster. An aster was fixed in situ immediately after side laser ablation. Both density and length of MTs at the ablation side are significantly reduced. (C-E) Aster side ablation. (C) Time-lapse confocal images of the male pronucleus during aster side laser ablation. Time 0 corresponds to the moment when laser ablation is done. Arrowhead: male pronucleus. (D) Time projection of fluorescent images shown in (C). (E) The position of the ablated aster is shown every 10 sec.