Jérôme Martin

Pierre Robert

Fabien Vallois

Campillo

Jérôme Je

Cédric, hierry. Jacques- Henri Gabriel

Christine Nicolas

Virginie Nelly

Jim Jelena

Cyprien

Marc, Laurent Vincent H Stéphan

Mahendra J'ai

Le mécanisme de production des protéines, qui monopolise la majorité des ressources d'une bactérie, est hautement stochastique : chaque réaction biochimique qui y participe est due à des collisions aléatoires entre molécules, potentiellement présentes en petites quantités. La bonne compréhension de l'expression génétique nécessite donc de recourir à des modèles stochastiques et discrets qui sont à même de caractériser les diférentes origines de la variabilité dans la production ainsi que les dispositifs biologiques permetant éventuellement de la contrôler.

Dans ce contexte, nous avons analysé la variabilité d'une protéine produite avec un mécanisme d'autorégulation négatif : c'est-à-dire dans le cas où la protéine est un répresseur pour son propre gène. Le but est de clariier l'efet de l'autorégulation sur la variance du nombre de protéines exprimées. Pour une même production moyenne de protéine, nous avons cherché à comparer la variance à l'équilibre d'une protéine produite avec le mécanisme d'autorégulation et celle produite en e boucle ouverte u. En étudiant un modèle limite, avec une mise à l'échelle (scaling), nous avons pu faire une telle comparaison de manière analytique. Il apparaît que l'autorégulation réduit efectivement la variance, mais cela reste néanmoins limité : un résultat asymptotique montre que la variance ne pourra pas être réduite de plus de 50%. L'efet sur la variance à l'équilibre étant modéré, nous avons cherché un autre efet possible de l'autorégulation : nous avons observé que la vitesse de convergence à l'équilibre est plus rapide dans le cadre d'un modèle avec autorégulation.

Les modèles classiques de production des protéines considèrent un volume constant, sans phénomènes de division ou de réplication du gène, avec des ARN-polymérases et les ribosomes en concentrations constantes. Pourtant, la variation au cours du cycle de chacune de ces quantités a été proposée dans la litérature comme participant à la variabilité des protéines. Nous proposons une série de modèles de complexité croissante qui vise à aboutir à une représentation réaliste de l'expression génétique. Dans un modèle avec un volume suivant le cycle cellulaire, nous intégrons successivement le mécanisme de production des protéines (transcription et traduction), la répartition aléatoire des composés à la division et la réplication du gène. Le dernier modèle intègre enin l'ensemble des gènes de la cellule et considère leurs interactions dans la production des diférentes protéines à travers un partage commun des ARN-polymérases et des ribosomes, présents en quantités limitées. Pour les modèles où cela était possible, la moyenne et la variance de la concentration de chacune des protéines ont été déterminées analytiquement en ayant eu recours au formalisme des Processus Ponctuels de Poisson Marqués. Pour les cas plus complexes, nous avons estimé la variance au moyen de simulations stochastiques. Il apparaît que, dans l'ensemble des mécanismes étudiés, la source principale de la variabilité provient du mécanisme de production des protéines lui-même (bruit dit e intrinsèque u). Ensuite, parmi les autres aspects e extrinsèques u, seule la répartition aléatoire des composés semble avoir potentiellement un efet signiicatif sur la variance ; les autres ne montrent qu'un efet limité sur la concentration des protéines. Ces résultats ont été confrontés à certaines mesures expérimentales, et montrent un décalage encore inexpliqué entre la prédiction théorique et les données biologiques, ce qui appelle à de nouvelles hypothèses quant aux possibles sources de variabilité.

En conclusion, les processus étudiés ont permis une meilleure compréhension des phénomènes biologiques en explorant certaines hypothèses diicilement testables expérimentalement. Des modèles étudiés, nous avons pu dégager théoriquement certaines tendances, montrant que la modélisation stochastique est un outil important pour la bonne compréhension des mécanismes d'expression génétique. À l'issue de cete thèse, je pense que je regarderai ces trois ans de travail avec une certaine nostalgie malgré un parcours qui n'a pas toujours été aussi simple que cela (uelle thèse l'est ?). Les traditionnels remerciements sont de mise pour celles et ceux qui ont contribué directement ou indirectement à cet aboutissement.

Je tiens tout d'abord à remercier les membres de mon jury en les personnes d'Amandine Véber, Olivier

Abstract

he mechanism of protein production, to which is dedicated the majority of resources of the bacteria, is highly stochastic: every biochemical reaction that is involved in this process is due to random collisions between molecules, potentially present in low quantities. he good understanding of gene expression requires therefore to resort to stochastic models that are able to characterise the diferent origins of protein production variability as well as the biological devices that potentially control it.

In this context, we have analysed the variability of a protein produced with a negative autoregulation mechanism: i.e. in the case where the protein is a repressor of its own gene. he goal is to clarify the efect of this feedback on the variance of the number of produced proteins. With the same average protein production, we sought to compare the equilibrium variance of a protein produced with the autoregulation mechanism and the one produced in "open loop". By studying the model under a scaling regime, we have been able to perform such comparison analytically. It appears that the autoregulation indeed reduces the variance; but it is nonetheless limited: an asymptotic result shows that the variance won't be reduced by more than 50%. he efect on the variance being moderate, we have searched for another possible efect for autoregulation: it havs been observed that the convergence to equilibrium is quicker in the case of a model with autoregulation.

Classical models of protein production usually consider a constant volume, without any division or gene replication and with constant concentrations of RNA-polymerases and ribosomes. Yet, it has been suggested in the literature that the variations of these quantities during the cell cycle may participate to protein variability. We propose a series of models of increasing complexity that aims to reach a realistic representation of gene expression. In a model with a changing volume that follows the cell cycle, we integrate successively the protein production mechanism (transcription and translation), the random segregation of compounds at division, and the gene replication. he last model integrates then all the genes of the cell and takes into account their interactions in the productions of diferent proteins through a common sharing of RNA-polymerases and ribosomes, available in limited quantities. For the models for which it was possible, the mean and the variance of the concentration of each proteins have been analytically determined using the Marked Poisson Point Processes. In the more complex cases, we have estimated the variance using computational simulations. It appears that, among all the studied mechanisms, the main source of variability comes from the protein production mechanism itself (referred as "intrinsic noise"). hen, among the other "extrinsic" aspects, only the random segregation of compounds at division seems to have potentially a signiicant impact on the variance; the other aspects show only a limited efect on protein concentration. hese results have been confronted to some experimental measures, and show a still unexplained decay between the theoretical predictions and the biological data; it instigates the formulations of new hypotheses for other possible sources of variability.

To conclude, the processes studied have allowed a beter understanding of biological phenomena by exploring some hypotheses that are diicult to test experimentally. In the studied models, we have been able to indicate theoretically some trends; hence showing that the stochastic modelling is an important tool for a good understanding of gene expression mechanisms. be applied in a biological context like bacteria. he number of compounds intervening in a cellular chemical reaction can be very low (a sticking example is the DNA molecule that is potentially present in a single copy) so that the by-product of the reaction sufers from a lot of variability (see the irst chapter of [START_REF] Schrödinger | What is Life?: With Mind and Mater and Autobiographical Sketches[END_REF]). Cells display countless possible intertwined reactions, with reactants of many diferent nature, but possibly in very few amounts. hen, randomness seems at irst an obstacle to the realisation of complex biological processes: all global mechanisms of the cell are the result of potentially highly erratic reactions. Moreover the low number of entities emphasizes the discrete nature of the reactants. Understanding how this randomness is organised, or at least self-controlled, in order to be able to produce complex resilient structures, is a major issue for the global understanding of living cells.

Variability in gene expression

he question of heterogeneity in the molecular processes is especially true for the main mechanism that occurs in bacteria: gene expression. Gene expression is the process by which the genetic information is used to produce functional products: the proteins. It is the main process in the cell as it is estimated that Escherichia coli dedicates most of its energy to this usage. he production of each type of protein involves small number of entities such as DNA molecules and messenger-RNAs, and needs commonly shared macromolecules like RNApolymerases and ribosomes. his process of protein creation is therefore subject to high viability. Moreover, some proteins are known to be involved in important cellular mechanisms (like DNA-replication initiation, cell division, responds to external threat, etc.); thereby luctuations in the expression of these individual proteins can be relected in the whole cell dynamics.

Fluctuations in gene expression have indeed been highlighted since the beginning of molecular biology (for example by [START_REF] Novick | Enzyme induction as an all-or-none phenomenon[END_REF]); but it is only since the early 2000s that modern techniques of luorescent microscopy allow a new experimental highlight on this topic (pioneered by [START_REF] Elowitz | Stochastic Gene Expression in a Single Cell[END_REF], [START_REF] Swain | Intrinsic and extrinsic contributions to stochasticity in gene expression[END_REF]). hey permit quantitative measurement of the noise for particular types of proteins. Since then, variability in gene expression has become an important topic in experimental biology. hey aim to estimate the variability for diferent types of proteins and try to determine the diferent origins of this heterogeneity, the potential strategies of noise reduction for some important genes, and so on.

Yet many diferent hypotheses are not easily testable experimentally. For example, the impact of some cellular aspects on protein variability are still not well understood: some mechanisms of protein production like auto-regulation have been proposed to reduce the variability; global cellular processes like DNA-replication and division are supposed to have an important impact on protein heterogeneity; and luctuations of commons resources (like RNA-polymerases and ribosomes) in protein productions are said to be the prime source for variability of highly expressed genes. All these hypotheses are yet to be investigated.

heoretical Modelling he goal of this work is to provide new perspectives on these challenging unsolved biological questions. To do so, I have relied on theoretical models that represent the diferent steps of the production of proteins. heir analysis leads to a beter understanding of these biological mechanisms.

he aim of this modelling process is to ofer a simple but relevant representation of a particular aspect of protein production, that can be analysed using mathematical language and computational tools. he notion of "model" here completely difers from the concept of "model organism" usually used in biology: a "model organism" is an example (for instance, Escherichia coli and Bacillus subtilis are used to represent the whole realm of bacteria; Saccharomyces cerevisiae is the model of yeast and more generally to every eukaryotic cells, etc.). heoretical models, on the other hand, are rather simpliications. hey do not represent the whole reality, but only particular aspects of it.

Each of the models presented in this work is designed to address one particular biological question about protein production. hey represent only the part of the reality that is thought to be directly involved in the studied phenomenon (for example, a model adapted to study the efect of auto-regulation). herefore, the model may simplify many aspects of the real world, and even discard many others (e.g. by considering the production of only one type of protein independently from the rest of the cell for instance).

Biology

Model Simulations Mathematics

Figure 1.1: Diagram of the modelling process Figure 1.1 depicts the modelling methodology I have followed. At irst, the proposed model needs to represent what is known from biology. From the observed biological phenomenon, one has to specify the concepts, the deinitions and the questions using mathematical formalism (for instance: "What quantity to consider in order to represent 'protein noise'?"; "What do the notions of internal and external variability refer to exactly?"; etc.).

We have tried to consider at irst models that are parsimonious: we want to predict the biological feature, with a minimal amount of hypotheses. Sometimes these simple models are enough to essentially predict the experimentally observed effects. If some real aspects remain unexplained by the model, one may consider adding new features one by one to the model in order to beter relect the reality (this will be our approach in Chapter 3 and Chapter 4). hanks to this series of models of increasing complexity, one can understand the relative impact of each of these added features on the simulated phenomenon. A model with many features from the outset may well it the observations as there are more degrees of freedom, but it may also hinder the understanding of the relative impact that each of the diferent modelling hypotheses has on the phenomenon.

Two possible ways have been used to analyse models: through mathematics or through computational simulations. Both methods are complementary. Mathematical analyses are able to give analytical results (for example explicit formulas of protein variance). hese kinds of results can indicate the theoretical possibility of a phenomenon, or provide a region of parameter values that corresponds with a particular behaviour of the system. Simulations, on the other hand, can investigate more complex mechanisms, but they lack the generality of mathematical analysis: every possible situation has to be simulated with a particular set of parameters, and the cost in terms of computational time can be non-negligible.

Both the computational and mathematics analyses enrich each other, as the simulations can indicate some interesting property worth analysing mathematically and as mathematical results on simple models can provide directions to follow when simulating more complex models (as it is the case in Chapter 2 of the present work). Finally, the results obtained by these methods have to be confronted with the biological results and, in the best case, propose new orientations for experimentation. hese constant interactions between biology, the model, the mathematical and computational analyses determine the relevance of the model and enrich our understanding of the biological phenomena.

Plan of the Chapter he remaining of the chapter is an introduction to the topic of heterogeneity in gene expression. It introduces the main biological notions and modelling tools that are used in the diferent chapters of the manuscript. In Section 1.1 are presented the main notions, terms and mechanisms relative to the process of gene expression that will be used in the whole manuscript; it will be useful to the reader unfamiliar with the biological aspects. We then present experimental studies performed on the topic of gene expression and what theoretical question they raise (Section 1.2). hen, we present the classical three-stage model of protein production that is largely used in the literature and that will be our basis in the diferent models developed here (Section 1.3); the section also give a concrete example of the kind of results that can be expected from a theoretical model.

Biological Aspects of Gene Expression

We present in this section the main biological mechanisms concerning gene expression. he goal is to present to the reader unfamiliar with this topic the basic concepts and terms relative to this subject. Many aspects of protein production are not exhaustively described here. he aim is to explain the main notions that we will refer to in the manuscript. We mainly focus on bacterial mechanisms as all the models presented within the manuscript speciically take place in prokaryotic cells.

Proteins are the main functional molecules of any cell from eukaryotic to prokaryotic cells. heir function can be to transport other molecules, to catalysis reactions, to make up the structure of the cell, or to regulate other proteins. For example, in an Escherichia coli bacteria, there are about 3.6×10 6 proteins of approximately 2000 diferent types with a great variability in concentration, depending on their types: from a few dozen up to 10 5 . In total, it represents more than half of the dry mass of the bacteria (see [START_REF] Neidhardt | Chemical Composition of Escherichia coli. In Escherichia coli and Salmonella: Cellular and Molecular Biology[END_REF]). he time of the cell cycle (between the birth of the cell and its division) varies from 20 min (in the richest medium) to more than 150 min (for the poorest medium). During this time frame, the cell manages to approximately double its content; especially there is about twice more proteins in the cell just before division as there was at birth. As a consequence, it is estimated that E. coli devotes more than 67% of its energy to this usage [START_REF] Russell | Energetics of bacterial growth: balance of anabolic and catabolic reactions[END_REF], which make gene expression the main process in the bacteria.

Transcription and Translation

Particular chunks of genetic information on the DNA, the genes, can be interpreted to produce various types of proteins. he process by which the information of the gene (a sequence composed of four possible nucleobases) is transformed into proteins (a sequence of twenty possible amino acids) is called gene expression. his process is performed in two main steps: transcription and translation. Transcription he transcription is the process by which the information contained on the DNA is copied into an intermediate short sequence of nucleotides: the messenger RNA (mRNA or transcript). he catalyst of this reaction is a macromolecule called RNA-polymerase.

Initiation

At start, an RNA-polymerase, binds in a speciic position upstream of the gene called the promoter. he binding tendency of the RNA-polymerase depends on several factors: the degree of ainity with the promoter sequence, the possible presence of transcription factors (proteins that bind close to the promoter and that can repress or promote the binding of an RNA-polymerase, see below), as well as the local structure of the chromosome that might change the accessibility of the RNA-polymerase to the promoter. hen the double strand of the DNA can be opened around the RNA-polymerase in order to create the transcription bubble: this is the initiation.

Ater the initiation stage, the elongation of the mRNA begins (at this stage, the process is irreversible): the RNA-polymerase "reads" one strand of the DNA in order to create the mRNA. To each nucleobase of the DNA sequence is associated a complementary nucleobase on the mRNA (the only diference between DNA and RNA is the substitution of the nucleobase hymine in DNA, by Uracile in RNA). he RNA-polymerase creates the mRNA while advancing on the DNA nucleobase by nucleobase.

he transcription usually terminates when the RNA-polymerase elongates a speciic sequence: the local mRNA conformation (determined by its sequence) provokes the disruption of the elongation complex. Ater the termination, the newly formed mRNA is released to the medium, as well as the RNApolymerase that is anew available for transcription of another part of the DNA.

Translation he translation is the process by which the information on the mRNA is converted into a sequence of amino acids that constitutes a particular type of protein. he reaction is performed by another macro molecule, the ribosome. his process shares some similarity with the transcription process, as it involves the three main stages of initiation, elongation and termination as well.

he initiation sees the formation of the ribosome complex from diferent subunits on the mRNA. It assembles on a sequence just upstream from the beginning of the gene, called ribosomal binding site (RBS). he probability of formation of the ribosome and its ability to start elongation depends in particular on the sequence around the RBS that determines its ainity for the ribosome.

Once the initiation part is completed, the ribosome begins elongation. his process consists in associating a triplet of nucleobases (codons) to one of the twenty possible amino acids. To each codon possibly corresponds one amino acid. As for the transcription, the ribosome moves forward on the mRNA codon by codon, elongating the protein one amino acid ater another.

Speciic codons on the mRNA are responsible for the termination of translation: they are called STOP codons. Once the ribosome has reached one of them, the ribosome is disassembled into its diferent subunits and the protein is released in the medium.

It is noticeable that, contrary to eukaryotes, mRNAs are directly elongated in the cytoplasm, where ribosomes can bind on it while they are still elongated: a translation can begin on an mRNA molecule whose transcription is still ongoing.

Gene regulation

he cell has to orchestrate its protein production to be able to trigger all cellular mechanisms (like division) or to respond to environmental change. It mainly does so through transcriptional regulation: each gene sees its transcription controlled as it is prevented from or promoted to produce mRNAs during a certain period of time. Transcriptional regulation can occur in many ways, but it is usually induced by transcription factors, i.e. proteins responsible for gene elongation. In E. coli, there are up to 300 different types of diferent transcription factors [START_REF] Madan Babu | Evolution of transcription factors and the gene regulatory network in Escherichia coli[END_REF] (which represent less than 10% of the diferent types of protein produced). Transcription factors bind on designated sequences on the DNA (usually close to or overlapping with the promoter of a speciic gene). Once bound, they can promote (in this case, it is called activator) or prevent transcription (it is then called repressor) by modifying the ainity of the promoter for RNA-polymerases or by changing the local structure of the chromosome. he RNA-polymerase binding ability is afected as long as the transcription factor is present on it. A repressor can completely disable transcription as long as it is bound on the DNA; in this case, it can be in two states: it is either activated and is able to transcribe mRNAs, or inactivated and the RNA-polymerase cannot initiate transcription (see Figure 1.3).

Activated

he activity of the gene depends therefore on the nature of its transcription factors present in the cell. Transcription factors may change function depending on the environment: they may associate with other compounds (for instance nutriments or other proteins) that change their conformation and therefore afect their repressing ability, or even changing them into activators.

As transcription factors are ordinary proteins, their target promoter can control the expression of their own gene. In that case the protein can inluence its own production: this auto-regulation is called the autogenous feedback. his mechanism is the central phenomenon studied in Chapter 2. In bacteria, mRNAs have lifetimes of few minutes: [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] measured mRNA half-life of around 4 minutes.

Messenger-RNA degradation

It is much shorter than their counterparts in eukaryotic cells and shorter than the doubling time of the cell. he rate of degradation depends on the type of mRNA: their sequence and their spatial conformation can inluence their degradation speed. his rapid decay allows a quick turnover in the transcripts repartition, that is needed in the adaptation to sudden environmental changes.

During degradation, the mRNA is disassembled into individual mononucleotides which can be recycled in another translation or in the DNA replication. It is an active reaction as several types of enzymes intervene in the process: the ribonucleases (RNase) . In E. coli, this process usually requires two kinds of reactions (see Figure 1.4):

1. First, the mRNA is cleaved by a kind of ribonuclease: the endoribonuclease that intervene in the middle of the mRNA chain. he most common endoribonuclease in E. coli is RNase E which binds on regions rich in Adenine-Uracil. Once bound to mRNA, the endoribonuclease performs cleavage and the mRNA is cut into two pieces. Once it happens, the mRNA is likely to loose its translation ability. Several cleavages can occur in quick succession so that the messenger is split into multiple small mRNA fragments.

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2. Once the cleavages operated, other types of ribonuclease intervene to inish mRNA degradation: the exoribonucleases (in E. coli, the most common are PNPase, RNase II and RNase R). hey are able to atack mRNA fragments by one extremity and degrade them one nucleotide at a time.

One can refer to [START_REF] Deutscher | Degradation of RNA in bacteria: comparison of mRNA and stable RNA[END_REF] for an introduction on the subject.

Proteolysis

he protein degradation process, called proteolysis, also exists in bacteria. It has two main objectives: to degrade proteins that are misfolded, damaged or not functional; and it participates in the regulation of some functional proteins. he process shares many similarities with mRNA degradation. In particular, it relies on a type of enzymes called protease, which subdivides into two families: the endoproteases that cleave the protein from the middle of the chain, and the exopeptidases that catalyse the degradation from the extremity of the chain. One can refer to [START_REF] Miller | Escherichia coli and Salmonella: Cellular and Molecular Biology[END_REF] for a complete description.

Most of the proteins are quite stable and have a much longer lifetime than mRNAs. It oten exceeds several cell cycles [START_REF] Koch | Protein Turnover in Growing Cultures of Escherichia Coli[END_REF]. he exceptions are usually proteins that are regulated by proteolysis: for instance the protein SulA that is involved in the response to DNA damage is degraded by Lon protease in around 1 min [START_REF] Miller | Escherichia coli and Salmonella: Cellular and Molecular Biology[END_REF].

Variability in Gene Expression

Since the beginning of molecular biology, insights about the variability in gene expression have been found. For instance [START_REF] Novick | Enzyme induction as an all-or-none phenomenon[END_REF] describe the luctuation in a population of genetically identical bacteria E. coli in the expression of β-galactosidase. Experimental techniques at that time did not allow a close examination of single-cell protein production, and this lack was an obstacle to further analysis.

Eventually, this topic blossomed during the 2000s, thanks to the emergence of luorescent microscopy (whose principles were developed by chemistry Nobel prise laureates Betzig, Morner and Hell). his wide range of techniques enables the observation through microscopy of the expression of speciic genes in a given cell: the protein of interest produces luorescence that can be detected with optical microscopes (see Figure 1.5). As the luorescence in the cell depends on the given protein abundance, it is possible to estimate the number of protein with a precision of one molecule unit.

YFP Microscope image

DNA Transcription & Translation Fluorescent protein

Figure 1.5: uantitative luorescent microscopy experiment example: the sequence of a gene (called reporter gene) is introduced into the bacterial genome; this reporter gene usually codes for a luorescent protein (here the Yellow Fluorescent Protein) detectable with a luorescent microscope. Using microscopy imaging, we can deduce the quantity of proteins for each cell by the luorescence observed, and thus cell-to-cell variability in the reporter gene can be estimated. he irst two articles on this topic were [START_REF] Ozbudak | Regulation of noise in the expression of a single gene[END_REF] on Bacillus subtilis and [START_REF] Elowitz | Stochastic Gene Expression in a Single Cell[END_REF] on Escherichia coli (both are prokaryotic cells). It was quickly followed by experiments on eukaryotes such as yeast [START_REF] Raser | Control of Stochasticity in Eukaryotic Gene Expression[END_REF][START_REF] Bar-Even | Noise in protein expression scales with natural protein abundance[END_REF], Cai et al., 2006[START_REF] Newman | Single-cell proteomic analysis of S. cerevisiae reveals the architecture of biological noise[END_REF].

We present below the main experimental results, topics and questions raised in the biological literature on the subject (one can also refer to the reviews of [START_REF] Raj | Nature, Nurture, or Chance: Stochastic Gene Expression and Its Consequences[END_REF] and [START_REF] Kaern | Stochasticity in gene expression: from theories to phenotypes[END_REF] for additional information).

Importance of Noise in Bacteria

All these studies show a substantial amount of variability in gene expression: in isogenic population (where all individuals are genetically identical) and in a similar and constant environment, the production of a given type of protein shows large cell-to-cell variability. For two diferent cells in the population, a given protein may be produced in diferent concentrations; and inside the same cell, its temporal production may show large variations.

One reason for this variability is the small number of molecules that intervene directly or indirectly in the protein production. For instance, there is usually one or two copies of each gene; transcription factors can be present in small numbers is the cell (a dozen for the lac repressor [START_REF] Kalisky | Cost-beneit theory and optimal design of gene regulation functions[END_REF]), and there is usually less than one mRNA for each gene at the same time (one can refer to Table 1.A.1 at the end of this chapter). Each chemical reaction is due to random encounters between molecules in the medium through random difusion. hese small numbers of entities induce variability in the protein production.

Cell scale events are naturally source of heterogeneity. For instance, the division separates the cytoplasm and its content in two parts; every compound can be in either one of two of the daughter cells. If the molecule is present in very few copies, this can have a signiicant efect on the variance of the distribution. DNA replication is another cause of variability: as the replication fork reaches a promoter, it can unbind the transcription factors on it (thus inducing a parasite transcription for a highly repressed gene for instance); or, as the gene is replicated, its transcription rate gets doubled.

Transcriptional and Translational Bursts

When using luorescent microscopy to measure gene expression, it clearly appears in many cases that proteins are produced during short periods of times followed by long periods without any translation: proteins are produced in intermitent bursts. here are two possible explanations for such proiles: translational and transcriptional bursts. [START_REF] Ozbudak | Regulation of noise in the expression of a single gene[END_REF] conducted a series of experiments, where the expression of a reporter gene (gfp) was measured. he idea was to control the transcription rate by using an inducible promoter (so the transcription rate can be controlled with environmental conditions determined in the experiment), and to control the translation rate by changing the ribosome-RBS ainity by point mutation on the RBS. With these elegant methods, both the rates of transcription and translation varied among the experiments and the authors were able to determine the respective impact on the protein expression of each step of gene expression. he results showed that the protein relative variance strongly depends on translation eiciency: it increases linearly with the average protein abundance with stronger ribosome-RBS ainity. On the other hand, the inluence of the transcription eiciency on protein noise was much less apparent. he most probable explanation for these results is related to the translational bursts: a low number of mRNAs (possibly unique) are highly transcribed, so that the number of proteins highly depends on the small discrete number of mRNAs. Similar studies were performed in eukaryotes, showing similar results [START_REF] Blake | Noise in eukaryotic gene expression[END_REF].

But [START_REF] Golding | Real-Time Kinetics of Gene Activity in Individual Bacteria[END_REF] proposed another possible mechanism that explains the proile of protein production: the transcriptional bursts. In addition to the measure of protein production, and contrary to [START_REF] Ozbudak | Regulation of noise in the expression of a single gene[END_REF], they were able to monitor mRNA production. Using the MS2-GFP method, they managed to quantify the transcript number with a single-molecule precision: some luorescent proteins have a high tendency to bind on a speciic messenger, so this messenger can be easily monitored through luorescent microscopy.

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hey discovered that in their case, the mRNA is not synthesised uniformly in time, through uncorrelated random events, but during burst episodes, in which several mRNAs are produced in a short period of time. hese burst periods were then followed by long periods of transcriptional inactivity. he natural interpretation is to consider that these long inactivation periods are due to gene regulation. Strong repressors bind on the gene promoter for long periods of time, giving only short time windows for transcriptions. During transcription episodes, the created mRNAs are translated, thus increasing the protein abundance for a short period of time. he bursts observed in the protein proile are here explained by an underlying transcriptional burst.

It is noticeable that these two concepts are not incompatible as they can be both speciic to diferent types of genes. he translational burst can occur in a constitutive gene (without regulation) with rare mRNA transcriptions, or with a very rapid gene regulation; transcriptional burst rather occurs when the gene is inactivated for long periods of time and gets strongly transcribed when it is active. In both cases, the protein translation rate needs to be high in order to exhibit bursts. Moreover the protein production signal is not signiicantly diferent between the two kinds of burst: its abundance is still suddenly increasing. So that it is not easy to diferentiate between these two sources of noise without directly looking at mRNA production (only [START_REF] Singh | Dynamics of protein noise can distinguish between alternate sources of gene-expression variability[END_REF] proposed a protocol to distinguish these diferent sources of variability just by looking at the protein expression).

Intrinsic and Extrinsic Noise

Both transcriptional and translational noises originate from the stochastic biochemical reactions inherent to the protein production mechanism: gene activation and deactivation, mRNA production and degradation, protein elongation, etc. It is commonly referred as intrinsic noise. But many other external aspects have been proposed to add variability in protein production: division, gene-replication, resource availability, etc. All these additional sources of heterogeneity are usually denoted as the extrinsic noise. Some articles were able to propose means to quantify these two origins of variability.

One of the irst article on stochasticity in gene expression, [START_REF] Elowitz | Stochastic Gene Expression in a Single Cell[END_REF], introduced the dual reporter technique. he idea is to compare two similar genes: they are simultaneously expressed in the same cell, and they both possess an identical promoter and are hence identically regulated (see Figure 1.6). By observing correlations in the signals of the two proteins in a given cell, the authors were able to separate two possible origins of noise. he intrinsic noise is supposed to be gene-speciic, it is supposed to afect both genes independently; while the extrinsic noise, being a cell-scale luctuation, has an identical impact on both genes; as a consequence, the extrinsic noise has a correlated impact on both expressions. he authors showed with their experiments that the extrinsic contribution is predominant in gene expression and that the proportion of each noise signiicantly depends on the promoter activity. In yeast, [START_REF] Raser | Control of Stochasticity in Eukaryotic Gene Expression[END_REF], implemented the same dual-reporter technique by introducing two almost identical genes on the same locus on homologous chromosomes in the same cell. hey also obtained a high extrinsic noise resulting from a high degree of correlation between genes. [START_REF] Hilinger | Separating intrinsic from extrinsic luctuations in dynamic biological systems[END_REF] analysed the underlying theoretical idea behind the separation of extrinsic and intrinsic noise using dual reporter techniques. hey interpreted the dual-reporter decomposing method as an estimation of an environmental state decomposition: by using the notation of the article, if X represents the number of proteins of a given cell at a given time, and if Z represents the known state of the cell (its RNA-polymerase number, its volume etc.), then it is theoretically possible to decompose the variance of X into a part which is explained by Z and another part which is completely uncorrelated (this decomposition is sometimes called the law of total variance): In dual-reporter experiment, in a speciic cell, each of the two gene expressions can be interpreted as the realisation of X in a common Z environment (as they are in the same cell). In that sense, the decomposition of the signals of the two genes in the dual-reporter experiment allows to estimate the environmental state decomposition. As the authors observe, this decomposition is only possible in constant (or near constant) conditions. But in experimental work, the environment is not always constant as the promoter of the gene of interest is oten induced by external changes. hey proposed a slightly new version of the decomposition procedure that takes into account not only the current state Z, but also the history of the state in the case of non-constant conditions.

Var [X] = E [Var [X|Z]] unexplained byZ + Var [E [X|Z]]
Several years since the irst introduction of the concept by [START_REF] Elowitz | Stochastic Gene Expression in a Single Cell[END_REF], the concepts of intrinsic and extrinsic noise still remain incompletely understood at least for two reasons:

1. By considering intrinsic noise as only resulting from transcription and translation variability, the largely used dual reporter technique is still sensitive to some extrinsic contribution such as imperfect timing in replication and intracellular heterogeneity [START_REF] Kaern | Stochasticity in gene expression: from theories to phenotypes[END_REF]. As we will see in Chapter 3, this technique is for example not suited to detecting the efect of division on protein variance, as division is commonly considered as part of the extrinsic noise in this context.

2. he expression "extrinsic noise" is oten a way of denoting the unexplained part of the noise (it is exactly what is meant by environmental state decomposition in [START_REF] Hilinger | Separating intrinsic from extrinsic luctuations in dynamic biological systems[END_REF]). Many mechanisms have been proposed to explain this additional noise (partition at cell division, gene replication, luctuations in the availability of RNA-polymerases and ribosomes, uncertainty on the division etc.), but it is not easy to understand the real importance of each of these factors on protein production heterogeneity.

A complete theoretical decomposition of the diferent possible origins of noise will be the subject of Chapter 3 and Chapter 4 in the current manuscript.

Genome-wide Variability

In recent years, it has become possible to consider measuring the activity of a large number of genes, possibly of the whole genome. [START_REF] Newman | Single-cell proteomic analysis of S. cerevisiae reveals the architecture of biological noise[END_REF], by using low cytometry method monitored the expression of more than 2500 GFPtagged genes in yeasts (Saccharomyces cerevisiae) strained in a rich or a minimal media. he low cytometry technique allows, for each individual cell, to quantify the GFP-labelled protein and at the same time to have a measure of some features of the cell such as the cell size and its granularity. For each type of protein, having determined the protein abundance in each cell, the authors were able to compute the mean and the variance of the number of proteins in the population. One of the main results concerned the importance of population heterogeneity in the extrinsic noise: within a population, yeasts display a wide range of sizes and cell cycle states. his simple fact is suicient to add extra variability. Using the dual reporter technique, and considering only cells of a certain size and granularity, thus having approximately cells at the same stage in the cell cycle, they observed that the extrinsic noise was considerably reduced. Once iltered, protein variability clearly depended primarily on protein abundance. [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] performed an analogous analysis on bacteria (E. coli) by using luorescent microscopy. In each experiment, a strain of cells was considered in which a type of protein was fused with the luorescent YFP molecule. his technique allows the direct quantiication of the fused protein. hey were able to measure about 1000 diferent proteins, and for each type of protein, the measured protein abundances range from 10 -1 up to 10 4 copies. On top of that, they detect simultaneously mRNAs abundance using Fluorescence in situ hybridisation (FISH) (luorescent probes that are able to bind on a complementary speciic sequence of nucleotides). hey discovered two regimes for the protein noise: for low expressed proteins, the variability depends on protein abundance; while for highly expressed proteins, protein noise becomes independent of its abundance. hey interpreted this second regime as dominated by the extrinsic noise since the noise was not gene speciic and cannot be due to the gene expression mechanism. hey compared their results with the yeast experiment of [START_REF] Newman | Single-cell proteomic analysis of S. cerevisiae reveals the architecture of biological noise[END_REF]: they showed that a similar noise plateau due to extrinsic factors is present in both cases, but that the extrinsic noise seems larger in E. coli.

In these genome-wide studies, the "extrinsic noise" is measured at the scale of the cell. Its global impact on all the proteins seems to follow speciic trends. Nonetheless its possible origins are still unclear: diferent hypotheses have been given in these articles but without decisive arguments. But these experiments give us also measures for a large variety of proteins of the cell: proteins with diferent levels, essential and nonessential genes, etc. In particular, the simultaneous measures of mRNA and protein production in [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] could make the comparison with classical theoretical predictions possible, and thus for a majority of genes of the cell. Even more important, with these measures, it is possible to study the impact of interactions between the productions of diferent genes: we can consider the genes altogether in a single model rather than independently and check the model against these genome-wide experiments.

Detrimental and Advantageous Efects of the Noise

he consequence of the variability in gene expression can be noxious for the cell as it can corrupt the quality of protein signals. For instance, the luctuation of a transcription factor can spread over entire gene networks [START_REF] Pedraza | Noise Propagation in Gene Networks[END_REF]Oudenaarden, 2005, McAdams and[START_REF] Mcadams | Stochastic mechanisms in gene expression[END_REF]; important choice making processes are dependent on the relative concentration of particular types of proteins [START_REF] Balázsi | Cellular Decision-Making and Biological Noise: From Microbes to Mammals[END_REF][START_REF] Süel | An excitable gene regulatory circuit induces transient cellular diferentiation[END_REF]; some highly produced proteins (like the subunits of ribosomes) can have a high cost of production in terms of energy, and luctuations in their production could induce wasteful consumption. In the case of multicellular organisms, the development stages rely on precise spatial and temporal gene expression, in which case noise control is vital [START_REF] Arias | Filtering transcriptional noise during development: concepts and mechanisms[END_REF].

C 1. I 20

But in some cases, heterogeneity among cells is proposed as a possible way of adapting to changing environments [START_REF] Balaban | Bacterial persistence as a phenotypic switch[END_REF][START_REF] Acar | Stochastic switching as a survival strategy in luctuating environments[END_REF]. his strategy is called bet-hedging. hanks to heterogeneity in bacterial population phenotype, some cells may be, by chance, it to resist some external threat. Heterogeneity has been invoked to explain the selective resistance of some bacteria to antibiotics in isogenic population, or to explain the competence (the ability to take up DNA from the environment) of only a fraction of Bacillus subtilis populations growing in the same environment (see [START_REF] Raj | Nature, Nurture, or Chance: Stochastic Gene Expression and Its Consequences[END_REF] for further examples).

Strategies of Noise Reduction

As noise in bacteria seems to be oten disadvantageous, some cellular mechanisms have been proposed to be a way for the bacteria to reduce the variability of at least some speciic genes. [START_REF] Fraser | Noise Minimization in Eukaryotic Gene Expression[END_REF] showed that in yeast the expression of essential genes (i.e. which are critical for its survival) and genes coding for complex subunits have their protein production optimised in such a way that it minimises their production noise. he authors started from the idea of translational burst observed experimentally by [START_REF] Ozbudak | Regulation of noise in the expression of a single gene[END_REF] and [START_REF] Blake | Noise in eukaryotic gene expression[END_REF]: a transcription burst presupposes that a large part of the variability in protein production is due to the low number of mRNAs. Genes with lower mRNA-protein average abundance ratio should have less noisy protein production. hey ensured that the essential genes and the genes participating to the formation complexes, have a global tendency to be more transcribed than their nonessential counterparts. Of course, this noise optimisation comes with the cost of an extra mRNA production. But it stresses the idea that noise is an important aspect in the cell and that it is subject to natural selection.

Another possible way proposed to reduce the protein variability is negative feedback. As previously said, a protein can be a transcription factor of its own production. he hypothesis that this mechanism might be a way to reduce the variability of a protein has been emited by theoretical models of [START_REF] Savageau | Comparison of classical and autogenous systems of regulation in inducible operons[END_REF] or hatai and van Oudenaarden ( 2004): a protein production that luctuates above its mean is driven down as it decreases its own gene activity, and luctuations below the mean would activate the gene. his hypothesis was tested experimentally by [START_REF] Dublanche | Noise in transcription negative feedback loops: simulation and experimental analysis[END_REF] and [START_REF] Austin | Gene network shaping of inherent noise spectra[END_REF]. In these articles, several protein production circuits on plasmids (a small DNA molecule within a cell distinct from the main chromosomal DNA) have been analysed: circuits that are autoregulated, and the others that are in "open loop". hey show a decrease in the noise of the autoregulated proteins which tends to go in favour of the hypothesis. Nonetheless, it has been objected that this noise diminution mainly afects the variability induced by external changes in the number of plasmids [START_REF] Paulsson | Summing up the noise in gene networks[END_REF]. his would suggest that autoregulation has an impact only on the extrinsic noise that might come from plasmid variation, and is ineicient in reducing intrinsic noise.

he fact the autoregulation is used as a convenient way for the cell to reduce the variability of some of its proteins is still debated. Other authors like [START_REF] Camas | Autogenous and nonautogenous control of response in a genetic network[END_REF] and [START_REF] Rosenfeld | Negative Autoregulation Speeds the Response Times of Transcription Networks[END_REF] emit the hypothesis that the autogenous feedback is used mainly in genes that need to quickly change their expression in case of environmental changes. he theoretical analysis of negative feedback autoregulation is the subject of Chapter 2.

Mathematical Modelling for Protein Production

he previous experimental works have raised several questions about the diferent origins of noise in protein production, the diferent possible ways of reducing the variability on particular proteins, or how the cell globally manages to deal with luctuations to fulil its genetic program. Tackling with these questions only experimentally is diicult: experimental techniques are not suiciently advanced to allow a real time observation of every particular mechanisms in the cell, and knowing all local interactions does not directly explains how the global system behaves. he use of theoretical models has been a natural complementary means to 1.3. M M P P 21 investigate these questions. We propose in this section to describe one important classical model of protein production: the three stage model (Subsection 3.2.1). hen we will present other theoretical works of the literature oten derived from the three-stage model (Subsection 1.3.2). Finally, we present diferent limitations of these classical models to address several important biological questions (Subsection 1.3.3).

A Canonical Example: the hree-Stage Model

Let's have an insight in one canonical model of gene expression: the three-stage model, also referred as Paulsson's model as it was fully analytically described in [START_REF] Paulsson | Models of stochastic gene expression[END_REF]. It is important, as it is widely used when it comes to interpreting biological results (for instance [START_REF] Blake | Noise in eukaryotic gene expression[END_REF], [START_REF] Raser | Control of Stochasticity in Eukaryotic Gene Expression[END_REF], [START_REF] Golding | Real-Time Kinetics of Gene Activity in Individual Bacteria[END_REF], [START_REF] Bar-Even | Noise in protein expression scales with natural protein abundance[END_REF], [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF]). Moreover, it displays analytical formulas for the mean and the variance of protein expression, and thus can be used to decompose diferent causes of variability.

Another interest is that its basic features have inspired many other modelling works (for instance [START_REF] Innocentini | Modeling stochastic gene expression under repression[END_REF], [START_REF] Shahrezaei | Analytical distributions for stochastic gene expression[END_REF], [START_REF] Fromion | Stochastic gene expression in cells: a point process approach[END_REF], [START_REF] Jansen | Stochastic equations for a self-regulating gene[END_REF], [START_REF] Fromion | A Stochastic Model of the Production of Multiple Proteins in Cells[END_REF]), including those of the next chapters of this manuscript.

Presentation of the Model

he three-stage model relies on several hypotheses that are commonly shared with other stochastic models of protein production. In particular, it is a "gene-centred model" as it aims to represent the production of a particular type of protein without considering interactions with the expressions of the other genes of the cell. herefore, there is only one type of protein in the system, produced by one type of mRNA, translated from one copy of the gene1 . Like the pioneering works of [START_REF] Berg | A model for the statistical luctuations of protein numbers in a microbial population[END_REF] and [START_REF] Rigney | Stochastic model of linear, continuous protein synthesis in bacterial populations[END_REF], and like many other stochastic models of gene expression since then, it relies on a common hypothesis: all the events (protein production, mRNA degradation, etc.) are represented as occurring at times that are exponentially distributed. he rates of these random variables may depend on the current state of the system. his naturally leads to a Markovian description: the model is "memoryless", the future of the system only depends on its current state, and not on its history.

Even if it is oten not made explicit in the literature, as the system represents inite quantities of mRNAs and proteins, one needs to consider some sort of spatial limitation. One natural way to do so is to consider that the model only represents the number of mRNAs and proteins in an arbitrary ixed volume around the considered gene. Usually, this volume may be considered as being of the order of magnitude of the cell size so that the number of compounds in the system would approximately represent the number of compounds in a cell.

he model aims at representing all the steps that intervene in gene expression (see Section 1.1). To do so it describes the evolution of three entities: I, M and P that respectively represent the state of the gene (active of inactive), the number of mRNAs and the number of proteins. he biological mechanisms of gene expression are represented: gene regulation, transcription and translation (see Figure 1.7).

Gene regulation

We consider that there are only two possible states for the gene: it can either be active (represented by I = 1), in which case the translation is possible; or inactive (represented by I = 0), in which case the translation is disabled. he repressor that inactivates the gene is independent of the system and binds on the promoter at rate λ - 1 . he repressors only leaves the promoter ater an

I = 1 M P I = 0 ∅ ∅ λ + 1 λ - 1 λ 2 µ 2 M λ 3 M µ 3 P Gene mRNAs Proteins +1 -1 +1 -1
Figure 1.7: hree-stage model presentation exponentially randomly distributed time of rate λ + 1 . Both λ + 1 and λ - 1 are ixed: it implicitly represents a case where the concentration of the repressor in the medium is constant; and that the repressorpromoter dissociation is a spontaneous event (for instance due to thermal agitation).

Transcription While the gene is active, it can be transcribed and it produces an mRNA molecule at rate λ 2 ; the number M of mRNAs is then increased by 1. Every mRNA is then degraded at rate µ 2 , so the global mRNA degradation rate is M µ 2 .

Translation Each mRNA can be translated at rate λ 3 thus creating a protein (which gives a global translation rate of λ 3 M ). A protein is considered as part of the system until its decay, which occurs with rate µ 3 . he total rate of protein decay is hence µ 3 P .

Decay Versus Degradation

he "decay" rates µ 2 for mRNAs and µ 3 for proteins are oten understood as degradation rates. As explained in Section 1.1, both the mRNAs and the proteins are broken down through active catalysed reactions. However another mechanism can be interpreted as a possible source of compound decay. During the cell cycle, the cell grows making additional space for the compounds inside the cell and as it divides, around half of the entities leaves the volume: it is dilution. As previously said, in the current case, the model takes place in a volume of about the size of a cell. As a consequence, any compound, if not degraded before, may leave the volume of interest in a time that is about the time of the cell cycle (also called doubling time). More precisely, the dilution decay has a haltime that is equal to the cell cycle. From these two perspectives, the decay rate of mRNAs and of proteins is the combined efect of degradation and of dilution. But, as generally observed, the mRNA lifetime is much smaller than the doubling time and, on the contrary, most of proteins are stable enough to subsist several cell cycles. his leads to the following distinction about the nature of rates µ 1 and µ 2 :

• he mRNA decay rate µ 1 represents a degradation rate of mRNA of the order of few minutes.

• For proteins, the decay rate µ 2 is similar to all (stable) proteins, and represents the dilution efect. It is given by the doubling time of the cell.

Analytical Expressions for the Mean and the Variance of Proteins his system is described by (I(t), M (t), P (t)) which is a Markovian process with a unique invariant distribution. he moments of the equilibrium distributions of P can be calculated recursively using equilibrium equations. In particular, we get explicit solutions for its mean and its variance:

E [P ] = δ λ 2 µ 2 λ 3 µ 3 (1.1)
Var [P ] = E [P ] ( 1 + λ 3 µ 2 + µ 3 + λ 2 λ 3 (1 -δ) (Λ + µ 2 + µ 3 ) (µ 2 + µ 3 ) (Λ + µ 2 ) (Λ + µ 3 ) ) (1.2)
with the notations δ := λ + 1 /(λ + 1 + λ - 1 ) and Λ := λ + 1 + λ - 1 . One can refer to [START_REF] Paulsson | Models of stochastic gene expression[END_REF] or [START_REF] Fromion | Stochastic gene expression in cells: a point process approach[END_REF] to know in detail how to establish these expressions.

One can at irst remark the following property: in any case, the protein variance Var [P ] is always larger than protein average E [P ], and so whatever the parameter choice. It indicates a theoretical lower bound for any protein variability: the protein signal cannot be precise beyond a certain limit. But as the three-stage model does not represent complex mechanisms like autogenous feedback, the question of this "Poisson lower bound" as being an actual biological limit is still unsolved (it will be the subject of Chapter 2).

Application to Transcriptional and Translational Bursts

We can use the formula as a way to describe the translation/transcription burst phenomena. Translational and transcriptional bursts have similar efects (burst in protein expression) but they are caused by two diferent mechanisms. he transcriptional burst efect shows bursts in the mRNA expression due to the activation of the gene during short periods of time. For the translational burst, mRNA are regularly produced during the cell cycle in very few copies and then the high translation rates provoke a sudden protein creation. he threestage model previously described gives the decomposition of these two origins of noise (see two examples in Table 1.1). We may wonder, for genes with the same average production (E [P ] is the same), for which sets of parameters transcriptional or translational bursts. For that, let's consider the relative variance deduced from Equation (1.2):

Var [P ] E [P ] = 1 + λ 3 µ 2 + µ 3 + E [P ] × µ 2 µ 3 µ 2 + µ 3 • 1 -δ δ • Λ + µ 2 + µ 3 (Λ + µ 2 ) (Λ + µ 3 )
.

(1.

3) It appears that only the third term has parameters involved in gene regulation: Λ and δ. herefore it is the only term whose contribution to the protein variance is due to the gene activation/deactivation process. For a protein with high transcriptional burst contribution, this term is predominant. In particular, low activation/deactivation phases compared to mRNA and protein decay (i.e. Λ ≪ µ 2 , µ 3 ) and short times of activation (i.e. δ ≪ 1) give a large value to this third term.

On the contrary, the second term of Equation (1.3) is independent from parameters linked to gene activity. his term represents the contribution of mRNA spontaneous luctuations to protein variability. A protein with a high translational burst contribution has this second term predominant. For instance, one can consider a protein whose gene activation/deactivation contribution to the variance is small (for instance by having δ = 1, then the gene is always activated; or Λ ≫ µ 2 , µ 3 , then the gene activation/deactivation is on a quick timescale). In that case, if the mRNA activity is high as compared to the mRNA and protein degradation rates (i.e. λ 3 ≫ µ 2 , µ 3 ), then we get translational bursts.

he three-stage model and its variants are broadly used in the literature as it represents the basic steps of protein production, and they display analytical results to express the variability of the model.

Other Models

Analytical Distributions for Stohastic Gene Expression [START_REF] Shahrezaei | Analytical distributions for stochastic gene expression[END_REF] proposed analytical solutions for the distribution of proteins taking advantage of the quick mRNA decay as compared to proteins (i.e. µ 2 ≫ µ 3 by using the notations of the previous section).

Basing their analysis on the three-stage model previously described, they considered at irst the case without gene regulation (with λ - 1 = 0). hey examined the model with the hypothesises of very short-lived mRNAs with high translational activity. In this case, the protein equilibrium distribution is shown to follow a negative binomial distribution of parameter a and b:

P [P = n] = Γ(a + n) Γ(n + 1)Γ(a) ( b 1 + b ) n ( 1 1 + b ) a
(1.4)

with P denoting the number of proteins. hey also gave biological interpretations for parameters a and b: parameter a represents the average number of mRNA created in a protein lifetime, and parameter b represents the average number of proteins create by one copy of mRNA before its degradation (using the notation of the previous section, respectively a = λ 2 /µ 3 and b = λ 3 /µ 2 ). he authors performed the same analysis on the complete three-stage model with gene activation/deactivation. Doing the same analysis, they give an analytical distribution for protein distributions in this case as well:

P [P = n] = Γ(α + n) Γ(n + 1)Γ(α) • Γ(β + n) Γ(β) • Γ(λ + 1 + λ - 1 ) Γ(λ + 1 + λ - 1 + n) × ( b 1 + b ) n ( 1 1 + b ) a × 2 F 1 ( α + n, λ + 1 + λ - 1 -β, λ + 1 + λ - 1 + n; b 1 + b )
with α and β being two parameters depending on the parameters of the system and 2 F 1 a hyper-geometric function. his more complicated distribution can display a bimodal density function in the case of very long periods of gene activation/deactivation. It also converges to the previous negative binomial distribution when λ - 1 → 0 (gene always active), or λ + 1 + λ - 1 → ∞ (activation/deactivation of the gene on a quick timescale). hese models have oten been used in the literature. In particular the negative binomial distribution Equation (1.4), oten approximated "continuous version", the gamma distribution with the same parameters a and 1.3. M M P P 25 b, has been used by [START_REF] Friedman | Linking Stochastic Dynamics to Population Distribution: An Analytical Framework of Gene Expression[END_REF], [START_REF] Cai | Stochastic protein expression in individual cells at the single molecule level[END_REF], [START_REF] Yu | Probing Gene Expression in Live Cells, One Protein Molecule at a Time[END_REF], [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] to it the experimental protein distribution. he advantage of this model is that it gives an explicit formula for the whole protein distribution and for its easily biologically interpretable parameters. Yet, this model is not as general as the three-stage model: even if they are not the majority some genes may have long mRNA lifetime, or may produce few proteins.

Changing the "Exponential Assumption"

he "exponential assumption" is an important hypothesis made in the three-stage model (and many other models): each event is supposed to happen at times that follow exponentially distributed random variables. Yet, if it is a reasonable assumption to represent random collisions between two individual molecules, it may not be the case for more complex aspects. For instance, the elongation times of RNA (or protein) chains are the result of 100-300 individual steps where nucleotides (or amino-acids) are added one by one; the exponential assumption does not seem to it in this case. herefore, some works reinterpret the three-stage model where some mechanisms were represented by more realistic distributions. [START_REF] Fromion | Stochastic gene expression in cells: a point process approach[END_REF] present a more general case for the mRNA and protein decay that does not consider the exponential assumption. Using the framework of Marked Point Poisson Possesses, they derive general formula for the three-stage model with arbitrary distributions for the decay distribution. In particular, they compare the protein variance in the case of a deterministic protein decay. hey showed that a deterministic protein decay increases protein degradation.

In Leoncini (2013, Chapter 3), this work was continued with a "four-stage model". he translation process is separated in two distinct steps: irst an initiation step, followed by an elongation step. he initiation is still represented as occurring at times that follow exponential distributions. But the elongation step is supposed to follow an Erlang distribution as it seems to be a more biologically realistic assumption: elongation results from hundreds of steps (the successive addition of amino acids). By supposing that each amino-acid requires the same amount of time to be processed, it can be represented as a inite sum of exponentially distributed times, that is to say, an Erlang distribution. hey compare the protein variance in this case and in the "classical" case where the protein elongation is represented as an exponentially distributed random variable. It appears that the Erlang distribution increases the variability, but qualitative comparisons show that this impact is small in the case of biologically relevant stable proteins.

Limitations of Classical Models

he three-stage model and its variants presented above all share a common basis: they suppose that the direct environment of the gene is not changing through time. For instance, the transcription and translation rates (respectively λ 2 and λ 3 ) are constants; as in real cells, these rates depend on the availability of RNApolymerases and ribosomes. It means that the model implicitly supposes that these entities are in constant concentration in the cell and do not luctuate through time. Similarly, the constant rate λ - 1 supposes that the concentration of repressors is not changing throughout the cell cycle. So, any luctuations in these quantities are not represented by the classical models.

More generally, diferent events in the cell cycle may have an impact on protein variability. At some point in the cell cycle, the DNA replication doubles the gene copy number, therefore instantaneously doubling the rate of transcription. At division, each protein either goes to one of the daughter cells or the other. hese two events induce additional periodic luctuations in the cell cycle that are not taken into account by classical models.

Moreover, the production of some proteins can be more complicated than what is simply described in the three-stage model: for instance, some proteins needs an extra step of maturation with the intervention of chaperones (proteins that assist in the good conformation of proteins). Also, as previously said, some proteins intervene in their own production: some are their own repressor and bind to the promoter of their gene, thus deactivating their production. In order to determine the impact that each of these mechanisms has on protein variance, we need to consider more complex models.

In this manuscript we tackle several of these limitations, trying to ofer an exhaustive description of the impact that diferent cellular mechanisms have on the protein heterogeneity.

Outline

he next three chapters present the results of my research activity during the three last years. As Chapter 2 addresses problems that are quite distinct from the remainder of the manuscript, it can be read independently from the rest. As for Chapter 3 and 4, they develop the same series of models of increasing complexity and should be read in that order.

Chapter 2: Model of Protein Production with Feedbak. In this irst chapter, we examine the variability of protein production when it is under the control of an autoregulation mechanism. he autoregulation considered here relies on a negative feedback: the considered protein is a repressor of its own gene. We propose to adapt the three-stage model to represent this mechanism and we clarify the impact of such regulation. he goal is to compare, for a same average protein production, the model with autoregulation and the classical three-stage model.

Even with a Markovian model that simply represents the feedback mechanism, there is no simple way to obtain analytical solutions for the mean and the variance of proteins at equilibrium. We therefore consider a scaling regime under which the classical three-stage model and the feedback model can be compared. In this regime, compared to the protein dynamics, the gene activation-deactivation and the mRNAs dynamics are considered to be on a quick time-scale; they both reach quickly a local equilibrium that depends on the current number of protein. We prove that the process describing the number of proteins converges then to a birth and death process where the birth rate follows a Hill repression with a hyperbolic control. In this regime, we have an explicit expression of the protein distribution. It appears in particular that the feedback indeed decreases the protein variability. But this efect is limited: an asymptotic result shows that the variance cannot be reduced of more than 50% compared to the model without feedback. With have performed simulations with parameters close to real genes, and show that in this case the variance descrease is even less import.

he limited reduction of the equilibrium variance by the autoregulation has lead us to search for other possible roles for the feedback in the cell. With additional simulations, we observe that the convergence to a new equilibrium is quicker in the case of the feedback. his feature gives a possible new role for the autoregulation: the quick adaptation of the protein production to environmental changes.

Chapter 3: Models with Cell Cycle. Usually, classical models do not explicitly represent several aspects of the cell cycle: the volume variations, the division and the gene replication. Yet these aspects have been proposed in literature to impact the protein production. In this chapter, we therefore propose a series of "gene-centred" models (that concentrates on the production of only one type of protein) that integrates successively all the aspects of the cell cycle. he goal is to obtain a realistic representation of the expression of one particular gene during the cell cycle. When it was possible, we analytically determined the mean and the variance of the protein concentration using Marked Poisson Point Process framework.

We based our analysis on a simple model where the volume changes across the cell cycle, and where only the mechanisms of protein production (transcription and translation) are represented. he variability predicted by this model is usually assimilated to the "intrinsic noise". We then add the random segregation of compounds at division to see its efect on protein variability: at division, every mRNA and every protein has an equal chance to go to either of the two daughter cells. It appears that this division sampling of compounds can add a signiicant variability to protein concentration. his efect directly depends on the relative variance (Fano factor) of the protein concentration: this efect is stronger as the relative variance is low. he dependence on the relative variance can be explained by considering a simpliied model. With parameters deduced from real experimental measures, we estimate that the random segregation of compounds can double the variability of the genes with the lowest relative variance.

Finally, we integrate the gene replication to the model: at some point in the cell cycle, the gene is replicated, hence doubling the transcription rate. We are able to give analytical expressions for the mean and the variance of protein concentration at any moment of the cell cycle; it allows to directly compare the variance with the previous model of the chapter with division. We show that gene replication has litle impact on the protein variability: an environmental state decomposition shows that the part of the variance due to gene replication represents only at most 2% of the total variability predicted by the model.

In the end, these results are compared to the real experimental measure of protein variability. It appears that the models of this chapter tend to underestimate the protein variability especially for highly expressed proteins.

Chapter 4: Multi-protein Model. In continuation of Chapter 3, we propose a model that still considers the division and the gene replication but which also integrates the sharing of common resources: the different genes are in competition for the limited quantity of RNA-polymerases and ribosomes in order to produce the mRNAs and proteins. he goal is to examine if luctuations in the availability of these macromolecules have an important impact on the protein variability, as it has been suggested in literature. As the model considers the interaction between the diferent protein productions, one needs to represent all the genes of the bacteria altogether: it is therefore a multi-protein model.

As this model is too complex to be studied analytically, we develop a procedure to estimate the parameters so that they correspond to real experimental measures. We then perform simulations in order to determine the variance of each protein and compare them with the one predicted by the models of the previous chapter. It appears that the common sharing of RNA-polymerases and ribosomes has a limited impact on the protein production: for most of proteins the variance increases of at most 10%.

In a last part, we investigate other possible sources of variability by presenting other simulations that integrate some speciic aspects: variability in the production of RNA-polymerases and ribosomes, uncertainty in the division and DNA replication decisions, etc. None of the considered aspects seems to have a signiicant impact on the protein variability.

In the last two chapters, we then have studied many of aspects that are usually suggested as possible sources of protein concentration variability. It appears that the main contribution to the protein heterogeneity is the "intrinsic noise" due to the production mechanism itself. he only important "extrinsic" contribution is due to the random sampling of mRNAs and proteins at division. All other mechanisms studied have a limited impact. New hypotheses need to be proposed in order to explain the diference of the variability predicted by the models and the one observed experimentally.

In conclusion, this work explores many hypotheses that are diicult to test experimentally. We have been able to explore unknown features of biology such the efect of the binomial division compared to the exact division. We have been able to explore important biological hypothesis such as efect of the sharing of the RNA-polymerases and ribosomes on the variance. We also give some clear theoretical limitations of some mechanisms, such the efect of the autogenous feedback on the variance. It shows that stochastic modelling is an important tool for the good understanding of gene expression mechanisms.

1.A Appendix: Useful Numbers he Table 1.A.1 regroups some biological useful numbers. hey are not meant to represent precise quantities but to give to the reader orders of magnitude for the diferent characteristics of the cell. We consider igures that corresponds to E. coli bacteria that are in slow growth (as it will be the case in Chapter 3 and Chapter 4). hen using numbers provided by the literature, we are able to get some insight in the rate of diferent events in the cell and other quantities; it is presented in Table 1 his chapter analyses, in the context of a prokaryotic cell, the stochastic variability of the number of proteins when there is a control of gene expression by an autoregulation scheme. he goal of this work is to estimate the eiciency of the regulation to limit the luctuations of the number of copies of a given protein. he autoregulation considered in this chapter relies mainly on a negative feedback: the proteins are repressors of their own gene expression. he eiciency of a production process without feedback control is compared to a production process with an autoregulation of the gene expression assuming that both of them produce the same average number of proteins. he main characteristic used for the comparison is the standard deviation of the number of proteins at equilibrium. With a Markovian representation and a simple model of repression, we prove that, under a scaling regime, the repression mechanism follows a Hill repression scheme with an hyperbolic control. An explicit asymptotic expression of the variance of the number of proteins under this regulation mechanism is obtained. Simulations are used to study other aspects of autoregulation such as the rate of convergence to equilibrium of the production process and the case where the control of the production process of proteins is achieved via the inhibition of mRNAs.

Introduction

Biological Context

he gene expression is the process by which genetic information is used to produce functional products of gene expression: proteins and non-coding RNAs. his chapter concerns itself with the production of proteins. he information low from DNA genes to proteins is a fundamental process. It is composed of three main steps: Gene Activation, transcription and translation.

1. he initiation of transcription is strongly regulated. Schematically the gene is said to be in "inactive state" if a repressor is bound on the gene's promoter preventing the RNA polymerase from binding and is in "active state" otherwise.

29

C 2. M P P F 30 2. When the gene is in active state, the RNA polymerase binds and initiates transcription that leads to the creation of a mRNA, a copy of a speciic DNA sequence.

3. he translation of the messenger into a protein is achieved by a large complex molecule: the ribosome.

A ribosome binds to an active mRNA, initiates the translation and proceeds to protein elongation. Once the elongation terminates, the protein is released in the medium and the ribosome is anew available for any another translation.

he production of proteins is the most important cellular activity, both for the functional role and the high associated cost in terms of resources. In a E. Coli bacterium for example there are about 3.6 × 10 6 proteins of approximately 2000 diferent types with a large variability in concentration, depending on their types: from a few dozen up to 10 5 . he gene expression is additionally a highly stochastic process and results from the realization of a very large number of elementary stochastic processes of diferent nature. he three main steps are the results of a large number of encounters of macromolecules following random motions, due in particular to thermal excitation, in the viscous luid of the cytoplasm. One of the key problems is to understand the basic mechanisms which allow a cell to produce a large number of proteins with very diferent concentrations and in a random context. his can be seen as a problem of minimization of the variance of the number of proteins of each type.

To study this problem, one can take a simple stochastic model, with a limited set S of parameters preferably, describing the three steps of the production of a given type of protein. Once a closed form expression of the variance of the number of proteins is obtained, it is natural to ind the parameters of the set S which minimizes the variance with the constraint that the mean number of proteins is ixed. See the survey [START_REF] Paulsson | Models of stochastic gene expression[END_REF].

A more efective way to regulate the number of proteins can be of using a direct feedback control, an autoregulation mechanism, so that the production of proteins is either sped up or slowed down depending on the current number of proteins. It should be noted that the feedback control loop can involve other intermediate proteins to achieve this goal, like the classical lac operon, but it is not considered here. See [START_REF] Yildirim | Feedback regulation in the lactose operon: a mathematical modeling study and comparison with experimental data[END_REF] for example.

he protein can regulate the gene activation simply, for example by being a repressor and tend to bind on his own gene's promoter. his is the autogenous regulation scheme. See [START_REF] Goldberger | Strategies of Genetic Regulation in Prokaryotes[END_REF] and [START_REF] Maloy | Autogenous regulation of gene expression[END_REF]. See also hatai and van [START_REF] Van Oudenaarden | Stochastic gene expression in luctuating environments[END_REF]. Other autoregulation mechanisms are possible in cells, such as an autoregulation on the mRNAs where a protein inhibits its own translation initiation by binding to the translation initiation region of its own mRNAs. It occurs for example in the production of ribosomal proteins, see [START_REF] Kaczanowska | Ribosome Biogenesis and the Translation Process in Escherichia coli[END_REF]. he idea being that a feedback mechanism may reduce signiicantly the number of large excursions from the mean. In this chapter, the mathematical analysis will mainly focus on a negative autogenous feedback, when the rate of inactivation of the gene expression grows with the number of proteins.

Literature

he classical results concerning the mathematical analysis of the variance of the number of proteins has been investigated in [START_REF] Berg | A model for the statistical luctuations of protein numbers in a microbial population[END_REF] and [START_REF] Rigney | Stochastic model of linear, continuous protein synthesis in bacterial populations[END_REF] and reviewed more recently by [START_REF] Paulsson | Models of stochastic gene expression[END_REF], see also [START_REF] Raj | Nature, Nurture, or Chance: Stochastic Gene Expression and Its Consequences[END_REF] for the biological aspects. hese references use the three stage model, the state of the system is given by three variables: the state of the promoter, the number of mRNAs and the number of proteins. Mathematically, the techniques used rely on the Fokker-Planck equations of the associated three dimensional Markov process and the observation that at equilibrium, a recurrence on the moments of the number of proteins holds. [START_REF] Fromion | Stochastic gene expression in cells: a point process approach[END_REF] investigates a more general model (elongation times are not necessarily exponentially distributed in particular) and an alternative technique to a Markovian approach is introduced.

Concerning the evaluation of autoregulation, most of mathematical models use a continuous state space, the rate of production of proteins depends linearly on the number of mRNAs and the rate of production of mRNAs is a nonlinear function k(p) exhibiting a non-linear dependence on the current number p of proteins. In [START_REF] Rosenfeld | Negative Autoregulation Speeds the Response Times of Transcription Networks[END_REF] and [START_REF] Becskei | Engineering stability in gene networks by autoregulation[END_REF], based on experiments the constant k(p) is taken a Hill repression function, i.e. k(p) = a/(b + p n ) for some constants a and b and n ≥ 1 is the Hill coeicient. See also hatai and van [START_REF] Van Oudenaarden | Stochastic gene expression in luctuating environments[END_REF]. Related models in a similar framework with further results are presented in [START_REF] Bokes | Multiscale stochastic modelling of gene expression[END_REF] and [START_REF] Yvinec | Adiabatic reduction of a model of stochastic gene expression with jump Markov process[END_REF]. For most of these models the state of the promoter, active or inactive, which is a source of variability is not taken into account, it is in some way encapsulated in the constant k(p) whose representation is rarely discussed. In [START_REF] Hornos | Self-regulating gene: An exact solution[END_REF] the state of the gene expression, on or of, is taken into account but not the number of mRNAs and therefore the luctuations generated by transcription. he parameter of activation k(p) is of course crucial in our case since autogenous regulation rely on the state of the promoter which can be inactivated by proteins. Our model includes it. See also [START_REF] Fournier | Steady-state expression of self-regulated genes[END_REF] for some simulations of these stochastic models of autoregulation as well as some experiments.

Results of the Chapter

he main goal of this chapter is to estimate the possible beneit of the autogenous regulation to control the luctuations of the number of copies of a given protein. he eiciency of a production process without feedback control is compared to a production process with an autoregulation of the gene expression, assuming that both of them produce the same average number proteins. he main characteristic used for the comparison is the standard deviation of the number of proteins at equilibrium. For this purpose, two approaches are used.

Mathematical Analysis One irst studies the distribution of the number of proteins via a stochastic model. When there is no regulation, the corresponding classical mathematical model has been investigated in detail for some time now. In particular, the standard deviation of the number of proteins at equilibrium has a closed form expression in terms of the basic parameters of the production process. See for example the survey [START_REF] Paulsson | Models of stochastic gene expression[END_REF], and also [START_REF] Fromion | Stochastic gene expression in cells: a point process approach[END_REF]. To represent the negative feedback of the autogenous regulation, a simple model is used: each protein can be bound, at some rate and for some random duration of time, on its own gene expression. In this situation the gene expression is inactive and the transcription is not possible during that time. his amounts to say that the gene expression is deactivated at a rate proportional to the number of proteins. he activation rate is constant.

As will be seen, the mathematical model of the autogenous regulation is more complicated, in particular there is no recurrence relationship between the moments of the number of proteins at equilibrium as in the classical model of protein production process. For this reason, a limiting procedure is used, it amounts to assume that the dynamics of the activation of the gene expression and of the evolution of mRNAs occur on a much faster time scale than the dynamics of the proteins. he values of the key parameters are presented in Subsection 2.5.1. he scaling parameter is the multiplicative factor describing the diference of speed of these two time scales. he main convergence result is heorem 2.2. he assumption of a fast time scale for gene expression activation and mRNAs is quite common in the literature, see [START_REF] Bokes | Multiscale stochastic modelling of gene expression[END_REF] and [START_REF] Yvinec | Adiabatic reduction of a model of stochastic gene expression with jump Markov process[END_REF]. he techniques used in these references rely on singular perturbation methods to deal with the two time scales. In our seting, a probabilistic approach is used, as will be seen, it gives precise results on the asymptotic stochastic evolution of the number of proteins.

Under this limiting regime it is shown that, asymptotically, the protein production process can be described as a birth and death process. See [START_REF] Keilson | Markov chain models-rarity and exponentiality[END_REF] for example. In state x ∈ N, the birth rate is a/(b+x) for some constants a and b. his is a contribution of the chapter that, with a simple model of the autoregulation, one can show that the repression mechanism follows indeed a Hill repression scheme with an hyperbolic control, C 2. M P P F 32

i.e. with Hill coeicient 1. he death rate is not changed by the limiting procedure, it is proportional to x. Consequently, one can get an asymptotic closed form expression of the standard deviation of the number of proteins by using the explicit representation of the equilibrium of this birth and death process. See Corollary 2.1. It is shown that, in this limiting regime, the standard deviation is reduced by 30%. he corresponding results are presented in Section 2.3 and Section 2.4 and in Section 2.A. he mathematical results are obtained via convergence theorems for sequence of Markov process, the proof of a stochastic averaging principle and a saddle point approximation result.

Simulation

We also analyze, via simulations, autogenous regulation but also other aspects related to the regulation of protein production. his is presented in Section 2.5. Simulations are used mainly because of the complexity of the mathematical models of some aspects of the autogenous regulation. By using plausible biological parameters, one gets an improvement of 15% for the standard deviation of the number of proteins can be expected. his is signiicantly less than the performances of the limiting mathematical model studied in Section 2.3. he main reason seems to be that that the scaling parameter is not, in some cases, suiciently large to have a reasonable accuracy with the limit given by the convergence result of heorem 2.2. Via simulations, one also investigates the case when the regulation is not on the gene expression but on the corresponding mRNAs: a protein can block an mRNA for some time. In this situation, it could be expected that the production process is modulated more smoothly by playing on the inactivation of a fraction of the mRNAs and not on the rough on-of control of the gene expression. It is shown that the improvement is real but not that big (less than 10%). It is nevertheless remarkable that if the average life time of mRNAs is signiicantly increased, our experiments show that the beneit of such regulation can be of the order of more than 30% on the standard deviation of the number of proteins.

Coming back to regulation on the gene expression. Our experiments show that, despite the impact of autogenous regulation on luctuations of the number of proteins can be limited, it has nevertheless a very interesting property. Starting with a number of proteins signiicantly less (or greater) than the average number of proteins at equilibrium, the autogenous regulation returns to the "correct" number of proteins much faster than the classical production process without regulation. his is a clear advantage of this mechanisms to adapt quickly when biological conditions change due to an external stress for example. See Subsection 2.5.6. his phenomenon has been observed, via experiments, in [START_REF] Rosenfeld | Negative Autoregulation Speeds the Response Times of Transcription Networks[END_REF]. See also [START_REF] Camas | Autogenous and nonautogenous control of response in a genetic network[END_REF]. Finally Subsection 2.5.5 investigates the comparison of production processes with and without a feedback on the gene expression through the estimation of their respective power spectral density.

Stohastic Models of Protein Production

We present the stochastic models used to investigate the protein production process. We will use the three stepsou model describing the activation-deactivation of the gene, the transcription phase and the translation phase. Like in most of the literature, it is assumed that the various events, like the encounter of two macromolecules, occurring within the cell have a duration with an exponential distribution. We start with the classical model used in this domain since the late 70's by [START_REF] Berg | A model for the statistical luctuations of protein numbers in a microbial population[END_REF] and [START_REF] Rigney | Stochastic model of linear, continuous protein synthesis in bacterial populations[END_REF]. See also hatai and van [START_REF] Van Oudenaarden | Stochastic gene expression in luctuating environments[END_REF] and [START_REF] Paulsson | Models of stochastic gene expression[END_REF].

he Classical Model of Protein Production

1. he inactive gene is activated at rate λ + 1 and deactivated at rate λ - 1 otherwise. 2. If the gene is active, an mRNA is produced at rate λ 2 . An mRNA is degraded at rate µ 2 .

S M P P 33

3. Given M mRNAs at some moment, a protein is produced at rate λ 3 M . Each protein is degraded at rate µ 3 .

I = 1 M P I = 0 ∅ ∅ λ + 1 λ - 1 λ 2 µ 2 M λ 3 M µ 3 P Gene mRNAs Proteins +1 -1 +1 -1
Figure = {0, 1} × N 2 , its transition rates are given by, if

(I(t), M (t), P (t)) = (i, m, p) ∈ S,      (0, m, p) → (1, m, p) at rate λ + 1 , (1, m, p) → (0, m, p) at rate λ - 1 , (i, m, p) → (i, m + 1, p) λ 2 i, (i, m, p) → (i, m -1, p) µ 2 m, (i, m, p) → (i, m, p + 1) λ 3 m, (i, m, p) → (i, m, p -1) µ 3 p. See Figure 2.1.
Lemma 2.1. he previous Markov process has a unique invariant distribution.

Proof. We can construct the coupling ( M (t), P (t)) such as M (t) ≤ M (t) and P (t) ≤ P (t) such as which corresponds to the case where the gene is always active. It is enough to prove that this process is ergodic to show the result. Using the Liapunov function f (m, p) = m + ap with a a positive number smaller than

µ 2 /λ 3 . In that case we have Qf (m, p) = λ 2 + (aλ 3 -µ 2 )m -aµ 3 p, with Q the Q-matrix of the process ( M (t), P (t)). By choosing K > max (λ 2 /(aλ 3 -µ 2 ), λ 2 /(aµ 3 ))
we have that for any (m, p) such as f (m, p) > K, it follows that Qf (m, p) < -ε with ε > 0. hen, using the Proposition 8.14 of [START_REF] Robert | Stochastic networks and queues[END_REF], it follows the result.

An explicit expression of the distribution of P at equilibrium is not known but, due to the linear transition rates, the moments of P can be calculated recursively. In the following (I, M, P ) will denote random variables whose law is invariant for (I(t), M (t), P (t)). Proposition 2.1. At equilibrium, the two irst moments of P can be expressed by

E [P ] = λ + 1 λ + 1 + λ - 1 λ 2 µ 2 λ 3 µ 3 (2.1)
Var [P ] = E [P ]
(

1 + λ 3 µ 2 + µ 3 + λ - 1 λ 2 λ 3 ( λ + 1 + λ - 1 + µ 2 + µ 3 ) (λ + 1 + λ - 1 ) (µ 2 + µ 3 ) ( λ + 1 + λ - 1 + µ 2 ) ( λ + 1 + λ - 1 + µ 3 ) )
.

(2.2)

See [START_REF] Paulsson | Models of stochastic gene expression[END_REF], [START_REF] Shahrezaei | Analytical distributions for stochastic gene expression[END_REF], [START_REF] Swain | Intrinsic and extrinsic contributions to stochasticity in gene expression[END_REF] and [START_REF] Fromion | Stochastic gene expression in cells: a point process approach[END_REF] for example. Explicit closed expressions for the moments are not that common to obtain for stochastic models of gene expression, in the continuation, we will see that it is for instance not the case for our model with autogenous regulation.

A Stohastic Model of Protein Production with Autogenous Regulation

he regulation is done via proteins which can inactivate the gene corresponding to the protein. If there are P proteins at some moment then the gene is activated at a rate proportional to P . Compared to the above model, only the irst step changes.

he inactive gene is activated at rate λ +

1 and inactivated at rate λ - 1 P otherwise. See Figure 2.2. For the sake of simplicity, we use the same notations λ + 1 and λ - 1 as for the classical model of protein production instead of λ + F,1 and λ - F,1 for example. It should be noted that in our comparisons in Section 2.5, these quantities are not necessarily the same for these two models.

he corresponding Markov process is denoted as (I F (t), M F (t), P F (t)), its transitions have the same rate as (I(t), M (t), P (t)) except for those concerning the irst coordinate.

{ (0, m, p) → (1, m, p) at rate λ + 1 , (1, m, p) → (0, m, p) at rate λ - 1 p. As before, (I F , M F , P F ) will denote random variables whose law is the invariant distribution of the Markov process (I F (t), M F (t), P F (t)). he following proposition is the analogue of Proposition 2.1 for the feedback model but with unknown quantities related to the the activity of the gene, E [I F ], and the correlation of the activity of the gene and the number of mRNAs,

E [I F M F ].
Proposition 2.2. At equilibrium, the irst moment of P F can be expressed by

E [P F ] = E [I F ] λ 2 µ 2 λ 3 µ 3 . (2.3)
Proof. We can prove that the process has a unique unique invariant distribution similarly as in Lemma 2.1. By equality of input and output for (M (t)) and (P (t)) at equilibrium, one gets the relations

λ 2 E [I F ] = µ 2 E [M F ] , λ 3 E [M F ] = µ 3 E [P F ] ,
and therefore (Equation (2.3)).

It does not seem that an expression for E [I

F ] can be obtained, the relation λ - 1 E [I F P F ] = λ + 1 (1 -E [I F ]
) of equality of lows for activation/deactivation process introduces the correlation between I F and P F . his is in fact the main obstacle to get more insight on the luctuations of the number of proteins. he next section investigates a scaling where the activation/deactivation phase is much more rapid than the production process of proteins.

I = 1 M P I = 0 ∅ ∅ λ + 1 λ - 1 P λ 2 µ 2 M λ 3 M µ 3 P Gene mRNAs Proteins +1 -1 +1 -1
Figure 2.2: hree Stage Model for Protein Production with Autogenous Regulation

A Scaling Analysis

It has been seen in the previous section that, for the feedback mechanism, an explicit representation of the variance of the number of proteins at equilibrium seems to be diicult to derive. In this section we use the fact that the time scale of the irst two steps, activation/deactivation of the gene and production of mRNAs is more rapid than the time scale of protein production. his is illustrated by the fact that the lifetime of an mRNA is of the order of 2 min. whereas the doubling time of a bacteria is around 40 min giving a lifetime of a protein of the order of one hour. See [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF], [START_REF] Li | Single molecule approaches to transcription factor kinetics in living cells[END_REF] and [START_REF] Hammar | he lac repressor displays facilitated difusion in living cells[END_REF].

As will be seen, this assumption simpliies the analysis of the feedback mechanism. We will be able to get an asymptotic explicit expression for the distribution of the number of proteins at equilibrium. A (large) scaling parameter N is used to stress the diference of time scale. When there is a feedback control, an upper index N is added to the variables so that the corresponding Markov process is denoted as

(X N F (t)) = (I N F (t), M N F (t), P N F (t))
on the state space S = {0, 1} × N 2 . he transition rates of the Markov process are given by

     (0, m, p) → (1, m, p) at rate λ + 1 N, (1, m, p) → (0, m, p) at rate λ - 1 N p, (i, m, p) → (i, m + 1, p) iλ 2 N, (i, m, p) → (i, m -1, p) µ 2 mN, (i, m, p) → (i, m, p + 1) λ 3 m, (i, m, p) → (i, m, p -1) µ 3 p.
(2.4) he initial state is constant with N given by X N F (0) = (i 0 , m 0 , p 0 ) ∈ S. he aim of this section is of proving that the non-Markovian process (P N F (t)) converges in distribution to a limiting Markov process (P F (t)). As will be seen, an averaging principle, proved in the appendix, holds: locally the "fast" process (I N F (t), M N F (t)) reaches very quickly some equilibrium depending on the current value of the "slow" variable P N F (t). It turns out that the equilibrium of this limiting process (P F (t)) can be analyzed in detail. he proof of the averaging principle relies on stochastic calculus applied to Markov processes in the same spirit as in [START_REF] Papanicolaou | Martingale approach to some limit theorems[END_REF] in a Brownian seting, see also [START_REF] Kurtz | Averaging for martingale problems and stochastic approximation[END_REF].

Notations

hroughout the rest of this chapter, we will use the following notations ρ 1 = λ + 1 /λ - 1 and, for i = 2, 3,

ρ i = λ i /µ i . C 2. M P P F 36

Scaling of the Classical Model of Protein Production

One irst states a scaling result for the classical model of protein production. he result being much simpler to prove than the corresponding result, heorem 2.2, for the feedback process, its proof is skipped. One denotes by (X N (t)) = (I N (t), M N (t), P N (t)) the corresponding Markov process, its transition rates are the same as for feedback in (Equation (2.4)) except for deactivation:

(1, m, p) → (0, m, p) at rate λ - 1 N. he following result shows that, in the limit, the evolution of the number of proteins converges to the time evolution of an M /M /∞ queue. See Chapter 6 of Robert ( 2010) for example. heorem 2.1. If X N (0) = (i 0 , m 0 , p 0 ) ∈ S, the sequence of processes (P N (t)) converges in distribution on the Skorohod space to a birth and death process (P (t)) on N whose respective birth and death rates (β x ) and (δ x ) are given by

β x = λ 3 ρ 2 ρ 1 ρ 1 + 1 and δ x = µ 3 x.
he equilibrium distribution of (P (t)) is a Poisson distribution with parameter ρ 1 ρ 2 ρ 3 /(1 + ρ 1 ).

Proof. he intuition of this result can be described quickly as follows. he processes (I N (t), M N (t)) live on a much faster time scale than (P N (t)) and therefore reach quickly the equilibrium. When N gets large, the process

(M N (t)) is an M /M /∞ queue with arrival rate λ 2 λ + 1 /(λ + 1 + λ - 1 )
and service rate µ 2 . See Chapter 6 of [START_REF] Robert | Stochastic networks and queues[END_REF] for example. Its equilibrium distribution is therefore Poisson with parameter ρ 2 ρ 1 /(1 + ρ 1 ). he process (P N (t)) can then be seen as an M /M /∞ queue with arrival rate λ 3 ρ 2 ρ 1 /(1 + ρ 1 ) and service rate µ 3 , i.e. a birth and death process with the transition rates of the theorem. Its equilibrium is Poisson with parameter ρ 1 ρ 2 ρ 3 /(1 + ρ 1 ).

he proof of a corresponding result in a more complicated seting, for the production process with feedback, is done below. For this reason the proof of this result is skipped.

I = 1 M P I = 0 ∅ ∅ λ + 1 N λ - 1 N λ 2 N µ 2 N M λ 3 M µ 3 P Gene mRNAs Proteins +1 -1 +1 -1 Figure 2.1: Feedback Model with Scaling Parameter N

Scaling of the Production Process with Feedbak

he following theorem is the main result of this section. As in the case of the classical model of protein production, it relies on the fact that, due to the scaling, the activation/deactivation of the gene and the production of mRNAs occurs on a fast time scale so that an averaging principle holds. See below. Some of the technical results used to establish the following theorem are presented in the Appendix.

heorem 2.2 (Hill Repression Scheme). If X N F (0) = (i 0 , m 0 , p 0 ) ∈ S, the sequence of processes (P N F (t)) converges in distribution to a birth and death process (P F (t)) on N whose respective birth and death rates (β x ) and (δ x ) are given by

β x = λ 3 ρ 2 ρ 1 ρ 1 + x and δ x = µ 3 x, with ρ 1 = λ + 1 /λ - 1 and ρ 2 = λ 2 /µ 2 . Proof. If f is a function on N with inite support then V N f (t) def. = f (P N F (t)) -f (p 0 ) - ∫ t 0 λ 3 M N F (u)∆ + (f )(P N F (u)) du - ∫ t 0 µ 3 P N F (u)∆ -(f )(P N F (u)) du,
is a local martingale. See [START_REF] Rogers | Difusions, Markov processes and martingales[END_REF] for example. he operators ∆ + and ∆ -are deined as follows, for a real-valued function f on N,

∆ + (f )(x) = f (x + 1) -f (x) and ∆ -(f )(x) = f (x -1) -f (x), x ∈ N.
With a similar method as in the proof of Assertion 1) of Lemma 2.2 in the appendix and by using the criterion of the modulus of continuity, see heorem 7.2 page 81 of [START_REF] Billingsley | Convergence of probability measures[END_REF], it is easy to show that the two processes

(∫ t 0 λ 3 M N F (u)∆ + (f )(P N F (u)) du ) and (∫ t 0 µ 3 P N F (u)∆ -(f )(P N F (u)) du ) are tight.
Because of the tightness of (P N F (t)) of Proposition 2.5 of the appendix, one can take (N k ) a subsequence such that the process

( P N k F (t), ∫ t 0 λ 3 M N k F (u)∆ + (f )(P N k F (u))du, ∫ t 0 µ 3 P N k F (u)∆ -(f )(P N k F (u))du
) converges in distribution.

Let (P F (t)) be a possible limit of (P N k F (t)), then by continuity of the mapping

(z(t)) → (∫ t 0 z(u)∆ -(f )(z(u)) du ) on D([0, T ]
) endowed with the Skorohod topology then, for the convergence in distribution. lim k→+∞

( P N k F (t), ∫ t 0 P N k F (u)∆ -(f )(P N k F (u)) du ) = ( P F (t), ∫ t 0 P F (u)∆ -(f )(P F (u)) du
) .

For t ≤ T , by using the deinition of Λ N and of E T in Subsection 2.A.2 of the Appendix, one has the relation

∫ t 0 M N k F (u)∆ + (f )(P N k F (u)) du = ∫ ET m∆ + (f )(p) [0,t] (u) Λ N k (dz),
hence, by Proposition 2.6 of Appendix, for the convergence in distribution

lim k→+∞ ∫ ET m∆ + (f )(p) [0,t] (u) Λ N k (dz) = ∫ t 0 ∑ p∈N ∆ + (f )(p) ∑ (i,m)∈{0,1}×N mℓ(i, m, p) du = ∫ t 0 ∑ p∈N ∆ + (f )(p) λ + 1 λ + 1 + λ - 1 p λ 2 µ 2 ν u (p) du C 2. M P P F 38
by Equation (2.17) of Proposition 2.6 of the Appendix. By convergence of the sequence (Λ N k ) this last expression can be expressed as

  ∫ t 0 ∑ p∈N ∆ + (f )(p) λ + 1 λ + 1 + λ - 1 p ν u (p) du   = lim k→+∞ ( ∫ t 0 ∆ + (f )(P N k F (u)) λ + 1 λ + 1 + λ - 1 P N k F (u) du ) dist. = (∫ t 0 ∆ + (f )(P F (u)) λ + 1 λ + 1 + λ - 1 P F (u) du )
for the convergence in distribution. For 0 ≤ s ≤ t, the characterisation of a Markov process as the solution of a martingale problem gives the relation

E [ f (P N F (t)) -f (P N F (s)) - ∫ t s λ 3 M N F (u)∆ + (f )(P N F (u)) du - ∫ t s µ 3 P N F (u)∆ -(f )(P N F (u)) du F s ] = 0,
from which we deduce the identity

E [ f (P F (t)) -f (P F (s)) - ∫ t s λ 3 ρ 1 ρ 2 ρ 1 + P F (u) ∆ + (f )(P F (u)) - ∫ t s µ 3 P F (u)∆ -(f )(P F (u)) du F s ] = 0.
See heorem II.2.42 of [START_REF] Jacod | Limit theorems for stochastic processes, volume 288 of Grundlehren der Mathematischen Wissenschaten[END_REF]. Consequently, a possible limit is the solution of the martingale problem associated to the birth and death process with birth rate (β x ) and death rate (δ x ) and with initial state in p 0 . One gets therefore the desired convergence in distribution of (P N F (t)). he theorem is proved.

here exist cases where the autoregulation is not achieved by the regulated protein but by a complex of this protein, e.g by a dimer (2 copies of the protein) or a tetramer (4 copies) to cite few examples. In order to handle such cases, it is necessary to add to the gene expression model, a preliminary step describing the reaction scheme of the complex formation based on the law of mass action (as it is done in [START_REF] Rosenfeld | Negative Autoregulation Speeds the Response Times of Transcription Networks[END_REF], [START_REF] Bokes | Multiscale stochastic modelling of gene expression[END_REF]). In general, the dynamics involved in the reaction scheme are (very) rapid compared to the other processes of the gene expression and leads, by a singular perturbation like argument, to represent in case of deterministic model the rate of production of mRNAs as a non-linear function of protein concentration. Furthermore, when the reaction scheme possesses suitable properties, a Hill like repression function could also be obtained. See [START_REF] Weiss | he hill equation revisited: uses and misuses[END_REF] for details. In the stochastic context, that leads to introduce a suitable scaling factor in the dynamics of the complex formation and to extend the previous derivation in the previous theorem to Hill functions, x → a/(b + x n ), with order n greater than 1.

he next section analyses, in this limiting regime, the luctuations of the number of proteins at equilibrium.

Fluctuations of the Number of Proteins

his section is devoted to the analysis of the equilibrium of the asymptotic process (P F (t)) of heorem 2.2 describing the evolution of the number of proteins with feedback. We start with a classical result for birth and death processes.

Proposition 2.3. he invariant distribution π F of the birth and death process (P F (t)) of heorem 2.2 is given by

π F (x) = 1 Z (ρ 2 ρ 3 ) x x! x-1 ∏ i=0 ρ 1 ρ 1 + i , x ∈ N, 2.4. F N P 39
where

ρ 1 = λ + 1 /λ - 1 , ρ i = λ i /µ i for i = 2
, 3 and Z is the normalization constant. he expression of π f is explicit but with a normalization constant which is not simple. he constant Z can be expressed in terms of hypergeometric functions. See [START_REF] B Abramowitz | Handbook of mathematical functions: with formulas, graphs, and mathematical tables[END_REF] for example. Even if we can get a numerical evaluation of the average and of the variance of π F , it is much more awkward to get some insight on the dependence of these quantities with respect to some of the parameters like ρ 2 or ρ 3 for example. In the following we give an asymptotic description of the ratio of the variance and the mean of the number of proteins at equilibrium when the value of the quantity ρ 1 ρ 2 ρ 3 is large. In a biological context the numerical value of this parameter is not always large but this limit results sheds some light on the qualitative behaviour of the auto-regulation mechanism. See Corollary Corollary 2.1 for example. A Laplace method is in particular used to investigate the asymptotic behaviour of the irst two moments of π F . heorem 2.1 shows that the distribution of the process (P (t)) at equilibrium is Poisson with parameter

E [P (t)] = x ρ = ρ 1 ρ 2 ρ 3 /(1 + ρ 1 )
. In particular, one has the relation

Var [P (t)] = E [P (t)].
In the rest of this section, we will be interested in the corresponding quantity for the feedback process.

For η > 0 and ρ > 0, denote by ν ρ the probability distribution on N deined by

ν ρ,η (k) = 1 Z ρ ρ k k! k ∏ i=1 1 η + i = 1 Z ρ exp ( k ∑ i=1 log ( x i(η + i) ) ) , (2.5) where Z ρ is the normalization constant. It is easily seen that π F is ν ρ,η with ρ = ρ 1 ρ 2 ρ 3 and η = ρ 1 -1.
Proposition 2.4. If, for ρ > 0 and η > -1, A ρ is a random variable with distribution ν ρ,η deined by Equation (2.5), then for the convergence in distribution

lim ρ→+∞ A ρ -a ρ √ a ρ = N ( 0, 1/ √ 2 ) ,
where

a ρ = ( √ η 2 + 4ρ -η ) /2 and N ( 0, 1/ √ 2 )
, is a centered Gaussian random variable with variance 1/2. In particular, for the convergence in distribution,

lim ρ→+∞ A ρ √ ρ = 1. Proof. If ϕ is a bounded function on R, denote ∆ ρ (ϕ) def. = 1 √ a ρ +∞ ∑ k=0 ϕ ( k -⌈a ρ ⌉ √ a ρ ) exp   k ∑ i=⌈aρ⌉ log ( ρ i(η + i) )   ,
with the following convention, to take care of the order of summation in discrete sums, if

(a n , n ∈ Z) is a sequence of real numbers, for ℓ, m ∈ Z, then ℓ ∑ i=m a i def. = - m-1 ∑ i=ℓ a i .
he deinition of ν ρ,η gives that

E [ ϕ ( A ρ -⌈a ρ ⌉ √ a ρ )] = ∆ ρ (ϕ) ∆ ρ (1) (2.6) C 2. M P P F 40
Fix ϕ some continuous function with compact support on [-K 0 , K 0 ] for some K 0 > 0. Since a ρ is the solution of the equation a ρ (η + a ρ ) = ρ, a change of variable gives the relation

∆ ρ (ϕ) = 1 √ a ρ ⌈K0 √ aρ⌉ ∑ k=-⌊K0 √ aρ⌋ ϕ ( k √ a ρ ) exp ( k ∑ i=0 log ( a ρ (η + a ρ ) (i + ⌈a ρ ⌉)(η + ⌈a ρ ⌉ + i) ) )
.

he uniform estimation k ∑ i=0 log ( a ρ (η + a ρ ) (i + ⌈a ρ ⌉)(η + ⌈a ρ ⌉ + i) ) = ∫ k 0 ( log
( a ρ (η + a ρ ) (u + ⌈a ρ ⌉)(η + ⌈a ρ ⌉ + u) ) du + O ( 1 √ a ρ ))
for all k ∈ Z with |k| ≤ K 0 √ a ρ and the fact that ϕ has a compact support give that the quantity ∆ ρ (ϕ) is equivalent to

1 √ a ρ ⌈K0 √ aρ⌉ ∑ k=-⌊K0 √ aρ⌋ ϕ ( k √ a ρ ) exp ( ∫ k 0 log ( a ρ (η + a ρ ) (u + ⌈a ρ ⌉)(η + ⌈a ρ ⌉ + u) ) du ) = 1 √ a ρ ⌈K0 √ aρ⌉ ∑ k=-⌊K0 √ aρ⌋ ϕ ( k √ a ρ ) × exp ( ∫ k/ √ aρ 0 √ a ρ log ( a ρ (η + a ρ ) (u √ a ρ + ⌈a ρ ⌉)(η + ⌈a ρ ⌉ + u √ a ρ ) ) du ) .
Again, with the uniform estimation

√ a ρ log ( a ρ (η + a ρ ) (u √ a ρ + ⌈a ρ ⌉)(η + ⌈a ρ ⌉ + u √ a ρ ) ) = -2u + O ( 1 √ a ρ ) ,
for u in some ixed inite interval, one gets that

∆ ρ (ϕ) ∼ 1 √ a ρ ⌈K0 √ aρ⌉ ∑ k=-⌊K0 √ aρ⌋ ϕ ( k √ a ρ ) exp ( -2 ∫ k/ √ aρ 0 u du ) ∼ ∫ +∞ -∞ ϕ (v) e -v 2 dv.
With similar estimations for ∆ ρ (1) (which imply in fact the tightness of the random variables (A ρ -⌊a ρ ⌋)/ √ a ρ ) and Equation (2.6), the proposition is proved.

Corollary 2.1 (Asymptotic Number of Proteins with Regulation). If P F is a random variable with distribution π F then, for the convergence in distribution

lim ρ2ρ3→+∞ E [ P F ] √ ρ 1 ρ 2 ρ 3 = 1 and lim ρ2ρ3→+∞ Var [ P F ] E [ P F ] = 1 2 . (2.7)
Furthermore, for the convergence in distribution,

lim ρ2ρ3→+∞ P F -a ρ √ a ρ = N ( 0, 1/ √ 2 ) ,
where

a ρ = ( √ (ρ 1 -1) 2 + 4ρ 1 ρ 2 ρ 3 -ρ 1 + 1 ) /2.
he equivalent of Equation (2.7) for the scaling of the classical model of protein production is

E [P ] = ρ 1 1 + ρ 1 ρ 2 ρ 3 and Var [P ] E [P ] = 1.
by heorem 2.1. it shows that a feedback mechanism reduces the variance of the number of proteins in this limiting regime by a factor 2 for the ratio of the second moment and the irst moment.

Discussion

In this section, other aspects of regulation of protein production are discussed via simulations in a plausible biological context whose parameters are going to be deined. hese simulations are performed using the [START_REF] Gillespie | Exact stochastic simulation of coupled chemical reactions[END_REF] algorithm. Simulation follows the models in Subsection 2.2.2 and simulates the variables I F , M F , and P F , not their scaling limits.

Numerical Values of Biological Parameters

For the model with feedback, there are six parameters to determine. By using the literature one can estimate the common orders of magnitude of these parameters in a biological context. We therefore propose a set of parameters corresponding to an "ordinary" gene.

1. Gene regulation. he parameter λ - 1 gives the rate at which a given protein reaches its own promoter. It has been shown that this motion combines a three-dimensional difusion in the cytoplasm and onedimensional sliding along the DNA, see [START_REF] Halford | An end to 40 years of mistakes in DNA-protein association kinetics?[END_REF].

Experiments on the lac repressor, using live-cell single-molecule imaging techniques, show that this time is of the order of 5 min, see [START_REF] Li | Single molecule approaches to transcription factor kinetics in living cells[END_REF] and [START_REF] Hammar | he lac repressor displays facilitated difusion in living cells[END_REF]. For this reason we will take

λ - 1 = 3.3 × 10 -3 s -1 .
he parameter λ + 1 can be quite variable, depending on the ainity of the protein to the DNA sequence, we set λ + 1 = 1 s -1 .

2. mRNAs. he lifetime of an mRNA is µ -1 2 ≃ 4 min, see [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF]. When the gene expression is always active (corresponding to the case where our variable I remains equals to 1), there is an average of 2 messengers, that is to say λ

-1 2 = µ -1 2 /m = 120 s which gives λ 2 = 8.3 × 10 -3 s -1 .
3. Proteins. A doubling time for the cell of t 1/2 ≃ 40 min gives a protein decay of around one hour. For this reason one takes µ 3 = log 2/t 1/2 = 2.8×10 -4 s -1 for the rate of protein decay. It is assumed that a give type of protein that is produced in p = 300 copies when the gene expression is always active. From one messenger, a protein should be produced in a duration of time of the order of

λ -1 3 = m × µ -1 3 /p which gives λ 3 = 4 × 10 -2 s -1 .
hese parameters may correspond to an "ordinary bacterial" gene: in a E. Coli genome of 4300 genes, there are around 3.6 × 10 6 proteins and 1.4 × 10 3 mRNAs per gene, see Table 1 of Chapter 3 of [START_REF] Neidhardt | Chemical Composition of Escherichia coli. In Escherichia coli and Salmonella: Cellular and Molecular Biology[END_REF], the number of messengers and proteins is of the order of magnitude of our numerical estimation of the parameters.

Impact of Autogenous Regulation on Gene Expression

We have compared two mechanisms: the classical model without regulation and the autogenous regulation process. he mean number of proteins is the same as well as the mean number of mRNAs produced

E [M ] = E [M F ].
Parameters λ + 1 and λ - 1 are adapted in the classical model to fulil these conditions. he other parameters are as deined in the previous section.

he comparison is shown in Figure 2.1. he mean number of proteins is 178, as can be seen that the curve for the autogenous regulation is slightly more concentrated around the mean but not that much. he values of the corresponding standard deviations are not really diferent √ Var [P ] = 42.2 and √ Var [P F ] = 35.8. he impact of the autogenous regulation on the variability of the number of proteins is non-trivial but not really spectacular for the set of parameters associated to a "typical" gene. his is signiicantly less than the performances of the limiting mathematical model studied in Section 2.3. he main reason seems to be that the scaling parameter is not, in some cases, suiciently large to have a reasonable accuracy with the limit given by the convergence result of heorem 2.2.

he Limiting Scaling Regime as a Lower Bound

Roughly speaking, heorem 2.1 and Corollary 2.1 give that for N and ρ 2 ρ 3 large, then the ratio Var

[ P N F ] /E [ P N F ]
converges to 1/2. In Figure 2.2, one considered a simulation with ixed product ρ 2 ρ 3 = 71.43 with N varying. he interesting feature is that the ratio is decreasing with N , this suggests that the variance of the limit of the scaling procedure should provide a lower bound for the variance of the real model. We have not been able to show rigorously this phenomenon. For N = 250, the value of the ratio Var

[ P N F ] /E [ P N F ] = .
7964 which is quite far from its limiting value 1/2 given by Corollary 2.1. his can be explained by the fact that the quantities N and ρ 2 ρ 3 are not very large.

Regulation of the Production Process on mRNAs

he regulation on the gene has the efect of an ON/OFF mechanism. When the gene is active, it is producing mRNAs at full speed and no mRNA is produced when it is inactive. his suggests that the production of proteins follows roughly the same patern: steady production rate at some instants and litle is produced otherwise. his scheme can consequently increase the variability of the production process of proteins. A possible idea to reduce the variance due to the activation/inactivation of the gene is to transfer the activation/inactivation process at the level the mRNAs. his possibility is investigated in this section. Each mRNA can be inactivated by a protein at rate λ - 2 , in this state it cannot produce proteins. An inactivated mRNAs becomes active at rate λ + 2 . In this way the production process can, hopefully, be modulated more smoothly by playing on the inactivation of a fraction of the mRNAs. In this way at time t, if the number of active [resp. inactive] mRNAs is M (t) [resp. M * (t)], the process (M (t), M * (t), P (t)) is Markov with transition rates, for

(m, m * , p) ∈ N 3 ,      (m, m * , p) → (m + 1, m * , p) at rate λ 2 , (m, m * , p) → (m -1, m * + 1, p) at rate λ - 2 mp, (m, m * , p) → (m + 1, m * -1, p) at rate λ +
2 m * , the other transitions are as before, active of inactive mRNAs die at rate µ 2 and proteins are produced at rate λ 3 m and die at rate µ 3 .

To compare the two regulation processes, either on the gene or on mRNAs, simulations have been done with the following constraints: the average number of proteins is ixed around 1400. 1 To have a fair comparison, we add the constraint that the number of mRNAs produced should be the same in all simulations. he numerical values have been estimated by using similar methods as in Section 2.3 but for this seting. Experiment (3) considers the case of an average lifetime of an mRNA of 40 min, if this is far from a "normal" biological seting, as it will be seen, this scenario has the advantage of stressing the importance of this parameter in this coniguration.

Numerical Values of Parameters

Regulation on the gene.

1 For the model with regulation on the gene, we determined the parameters by using Equation (2.3) and by ixing E [P ] = 1400. We also make the approximation that

E [I] ≃ λ + 1 /(λ + 1 + λ - 1 E [P ]
). he resulting simulations show a relatively precise (the mean around 1403). A similar strategy to determine the parameters of the model with regulation on mRNAs. C 2. M P P F 44

λ + 1 λ - 1 λ 2 µ 2 λ 3 µ 3 0.21" 5' 12" 4' 25" 1h.
Regulation on mRNAs (I). For this experiment, the expected lifetime of an mRNA is twice the corresponding value of case (1).

λ 2 λ + 2 λ - 2 µ 2 λ 3 µ 3 23" 2" 45' 8' 25" 1h.
Regulation on mRNAs (II). For this second experiment on the regulation of mRNAs, the expected lifetime of an mRNA is 10 times than in case (1).

λ 2 λ + 2 λ - 2 µ 2 λ 3 µ 3 23.8" 2"
45' 40' 25" 1h.

Results of the Experiments

Impact of Feedbak on Frequency

In this section, we study the nature of the luctuations of the number of proteins at equilibrium from the point of view of signal processing or automatic control. he aim of a feedback is oten of changing the nature of the signal, atenuating disturbances by reducing, for instance, high frequencies. In these cases, spectral analysis gives a characterisation of the nature of changes.

By analogy, we consider our model as a system that has to achieve a command (the production of a given mean number of proteins) and where the resulting signal P (t) or P F (t) is altered by some noise. In this framework, one can study if the efect of the feedback has an impact on the signal, by rejection of some frequency ranges. To do so, consider the signals (P (t)) and (P F (t)) of two simulations with or without autogenous regulation. he analysis of these signals is done by estimating the power spectral density, that describes the spectral characteristics of stochastic process. We estimate the power spectral density for each signal, using classical estimator of smoothed periodogram. See [START_REF] George | Processing of Random Signals[END_REF] and Chapter 10 of [START_REF] Miller | Probability and Random Processes: With Applications to Signal Processing and Communications[END_REF] for example.

he result is shown in Figure 2.4. Both spectra seem to represent a low-pass ilter with a cut of frequency in the order of magnitude of the dilution factor µ 3 = 2.8 × 10 -4 s -1 . he two power spectral densities do not seem to exhibit signiicant diferences. he feedback has therefore no noticeable efect in terms of reduction of frequency disturbances.

10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8

Versatility of the Protein Production Process

his section is devoted to the impact of autogenous regulation on another aspect of protein production. Up to now, we have considered the production process of proteins at equilibrium, by assuming that the production rate of a given protein has to be ixed. It may happen nevertheless that, due to an external stress, such as antibiotics, DNA damage by UV, see [START_REF] Camas | Autogenous and nonautogenous control of response in a genetic network[END_REF], or nutriment absorption, see [START_REF] Schleif | Regulation of the L-arabinose operon of Escherichia coli[END_REF], the cell has to change rapidly its production rate to quickly produce a large amount of proteins for example. he ainity of the transcription factor for the promoter of the gene can be adapted for that purpose. Conversely, when the external stress disappears, the production of the protein has to be quickly reduced to minimize the consumption of resources.

We consider the situation when the two production processes, with and without autogenous regulation, give the same average output of proteins at equilibrium. Two cases are investigated: when the initial number of proteins is below the value equilibrium, see Figure 2.5a, or above this value, see Figure 2.5b. As it can be seen, the autogenous production process converges more rapidly to equilibrium in both cases. Our simulations show that when the initial value is 290, the autogenous production process is 40% faster than the process without feedback to reach the level 1300 (the equilibrium is at 1400 in this case). A similar result holds in the other case.

hese interesting properties are related to the modulation of the gene activity. In the experiment of Figure 2.5a, for the autogenous process the rate of activity of the gene is of the order of 50% at the beginning and it is only of the order of 0.1 later at equilibrium. Without regulation this rate is constant throughout the simulation. his explains the "fast start" of the autogenous process. An analogous explanation holds for the experiment of Figure 2.5a, in the autogenous process. he gene is rapidly switched of due to the large number of proteins, thereby decreasing rapidly the number of proteins. his is consistent with experiments described in [START_REF] Camas | Autogenous and nonautogenous control of response in a genetic network[END_REF] and especially [START_REF] Rosenfeld | Negative Autoregulation Speeds the Response Times of Transcription Networks[END_REF] where the improvement has been estimated at 80% in some cases.

2.A Appendix: Convergence Results

We irst introduce some notations that will be used throughout this section.

2.A.1 Evolution Equations

We will use the Skorohod's topology for convergence in distribution in the space D([0, T ], R + ) of càdlàg processes. See Chapter 3 of [START_REF] Billingsley | Convergence of probability measures[END_REF] for example. To simplify the presentation, all our processes will be deined on the same probability space in the following way.

Let N + i , N - i , i = 1, 2, 3 be independent Poisson processes on R 2 + with rate 1 deined on a probability space (Ω, F, P).

If A ∈ B(R 2 + ) is a Borelian subset of R 2 + and (i, c) ∈ {1, 2, 3} × {+, -}, N c i (A)
denotes the number of points of the process N c i in the subset A. For t ≥ 0, one denotes by F t the σ-ield generated by the random variables

N c i (B × [0, t]) for B ∈ B(R + ) and (i, c) ∈ {1, 2, 3} × {+, -}.
It is easily seen that the process (X N F (t)) has the same distribution as the solution of the following stochastic diferential equations (SDE)

dI N F (t) = {I N F (t-)=0} N + 1 ([0, λ + 1 N ] × [dt]) (2.8) -{I N F (t-)=1} N - 1 ([0, λ - 1 N P N F (t-)] × [dt]) dM N F (t) = {I N F (t-)=1} N + 2 ([0, λ 2 N ] × [dt]) -N - 2 ([0, µ 2 N M N F (t-)] × [dt]) (2.9) dP N F (t) = N + 3 ([0, λ 3 M N F (t-)] × [dt]) -N - 3 ([0, µ 3 P N F (t-)] × [dt]) (2.10)
with the same initial condition. For any N ≥ 1, (X N F (t)) is a F t -Markov process adapted to the iltration

F t deined as σ ( X N F (0); N c i (A × [0, s[), s ≤ t, A ∈ B(R 2 + ), (i, c) ∈ {1, 2, 3} × {+, -}
) . hese SDE can be rewriten as, for some function f with inite support on S,

f (X N F (t)) = f (X N F (0)) + ∫ t 0 λ + 1 N (1 -I N F (u))∆ 1 (f )(X N F (u)) du + ∫ t 0 λ - 1 N P N F (u)I N F (u)∆ 1 (f )(X N F (u)) du + ∫ t 0 λ 2 N I N F (u)∆ + 2 (f )(X N F (u)) du + ∫ t 0 µ 2 N M N F (u)∆ - 2 (f )(X N F (u)) du + ∫ t 0 λ 3 M N F (u)∆ + 3 (f )(X N F (u)) du + ∫ t 0 µ 3 P N F (u)∆ - 3 (f )(X N F (u)) du + W N f (t) (2.11)
where, for x = (i, m, p) ∈ S, the operators ∆

+/- • are deined by      ∆ 1 (f )(x) = f (1 -i, m, p) -f (x) ∆ + 2 (f )(x) = f (i, m + 1, p) -f (x), ∆ - 2 (f )(x) = f (i, m -1, p) -f (x) ∆ + 3 (f )(x) = f (i, m, p + 1) -f (x), ∆ - 3 (f )(x) = f (i, m, p -1) -f (x), C 2. M P P F 48 and (W N f (t)
) is a local martingale whose previsible increasing process is given by

⟨ W N f ⟩ (t) = ∫ t 0 [ λ + 1 N (1 -I N F (u)) + λ - 1 N P N F (u)I N F (u) ] [ ∆ 1 (f )(X N F (u)) ] 2 du + ∫ t 0 λ 2 N I N F (u) [ ∆ + 2 (f )(X N F (u)) ] 2 du + ∫ t 0 µ 2 N M N F (u) [ ∆ - 2 (f )(X N F (u)) ] 2 du + ∫ t 0 λ 3 M N F (u) [ ∆ + 3 (f )(X N F (u)) ] 2 du + ∫ t 0 µ 3 P N F (u) [ ∆ - 3 (f )(X N F (u)) ] 2 du.
(2.12)

See [START_REF] Rogers | Difusions, Markov processes and martingales[END_REF] for example.

Deinition 2.1. Let (M N (t), P N (t)) be the Markov process with transition rates given by

{ (m, p) → (m + 1, p) at rate λ 2 N, (m, p) → (m -1, p) " µ 2 mN, (m, p) → (m, p + 1) " λ 3 m, (m, p) → (m, p -1) " µ 3 p (2.13)
and initial state (M N (0), P N (0)) = (m 0 , p 0 ).

he process (M N (t), P N (t)) is simply the analogue of our process (M N F (t), P N F (t)) when the gene is always active.

Lemma 2.2.

1. For the convergence in distribution for the uniform norm on compact sets

lim N →+∞ (∫ t 0 M N (u) du ) = (ρ 2 t).
2. For T > 0,

sup N ≥1 E [ sup 0≤t≤T P N (t) ] < +∞.
Proof. From Equation (2.13), it is easily seen that the process (M N (t)) can be expressed (L 1 (N t)) where

(L 1 (t)
) is an M /M /∞ queue with arrival rate λ 2 and service rate µ 2 with L 1 (0) = m 0 . See Chapter 6 of Robert (2010) for example. Elementary stochastic calculus gives, for t > 0,

L 1 (N t) = m 0 + λ 2 N t -µ 2 ∫ N t 0 L 1 (u) du + M N 1 (t), (2.14)
where (M N 1 (t)) is a local martingale whose previsible increasing process is given by

⟨ M N 1 ⟩ (t) = λ 2 N t + µ 2 ∫ N t 0 L 1 (u) du.
It is possible to show that (M N 1 (t)) is a martingale: we have that for every t > 0

M N 1 (t) < m 0 + N 2 [0, N t[+λ 2 N t + ∞ ∑ i=0 ∫ N t 0 i≤L1(u) N i µ2 (ds) + µ 2 ∫ N t 0 L 1 (u) du, 2.A. A : C R 49 with ( N i µ2 ) independent point Poisson processes of rate µ 2 . It follows that E [ sup s≤t M N 1 (t) ] ≤ m 0 + 2λ 2 N t + 2λµN 2 t 2 ,
so with heorem A.7 of [START_REF] Robert | Stochastic networks and queues[END_REF], it comes that (M N 1 (t)) is a martingale. By applying Doobs' inequality, it shows that the process (M N 1 (t)/N ) vanishes for the convergence in distribution as N gets large. For ε > 0 and x ∈ N, if [START_REF] Robert | Stochastic networks and queues[END_REF] shows the convergence in distribution

T x = inf{t ≥ 0 : L 1 (u) ≥ x}, Proposition 6.10 of
lim x→+∞ ρ x 2 (x -1)! T x = E 0
where E 0 is an exponential random variable with parameter µ 2 exp(-ρ 2 ). his shows in particular the process (L 1 (N t)/N ) converges in distribution to 0 for the uniform convergence on compact intervals since

P     ∑ 0≤t≤T L 1 (N t) N ≥ ε     ≤ P [( T ⌊εN ⌋ ≤ N T )] .
From Equation (2.14), one gets

∫ t 0 M N (u) du = 1 N ∫ N t 0 L 1 (u) du = ρ 2 t + 1 µ 2 ( m 0 N - L 1 (N t) N + M N 1 (t) N
) and therefore assertion 1) of the lemma.

For the last assertion, the method is similar: one irst write the evolution equation

P N (t) = p 0 + λ 3 ∫ t 0 M N (u) du -µ 3 ∫ t 0 P N (u) du + M N 2 (t),
where (M N 2 (t)) is a local martingale whose previsible increasing process is given by

⟨ M N 2 ⟩ (t) = λ 3 ∫ t 0 M N (u) du + µ 3 ∫ t 0 P N (u) du.
As for (M N 1 (t)), it is possible to show that the local martingale (M N 2 (t)) is indeed a martingale by showing that for every t, E

[

sup s≤t M N 2 (t) ] is inite. Deine P N * (t) = sup{P N (u) : 0 ≤ u ≤ t}, then for 0 ≤ t ≤ T E [ P N * (t) ] ≤ p 0 + λ 3 E [ ∫ T 0 M N (u) du ] + E [ sup 0≤u≤t |M N 2 (u)| ] + µ 3 ∫ t 0 E [ P N * (u) ] du. (2.15) Doob's Inequality gives, for t ≤ T , E [ sup 0≤u≤t |M N 2 (u)| ] ≤ 2λ 3 ∫ T 0 E [ M N (u) ] du + 2µ 3 ∫ t 0 E [ P N * (u) ] du, C 2. M P P F 50
and from the ergodic theorem for (L

1 (t)) (recall that M N (t) = L 1 (N t)) one gets lim N →+∞ E [ ∫ T 0 M N (u) du ] = ρ 3 T.
One concludes by using Gronwall's Lemma.

Proposition 2.5. he sequence (P N F (t)) is tight for the convergence in distribution of càdlàg processes. Proof. Aldous' criterion for tightness is used. See heorem 4.5 page 320 of [START_REF] Jacod | Limit theorems for stochastic processes, volume 288 of Grundlehren der Mathematischen Wissenschaten[END_REF] for example. For T > 0, one denotes by T T the set of stopping times associated to the iltration (F t ) which are bounded by T . For η > 0, let τ 1 , τ 2 ∈ T T be such that τ 1 ≤ τ 2 ≤ τ 1 + η. he respective probabilities that, on the time interval [τ 1 , τ 2 ], no protein is made or that no protein is degraded are respectively given by

E [ exp ( -λ 3 ∫ τ2 τ1 M N F (u) du )]
and E

[ exp

( -µ 3 ∫ τ2 τ1 P N F (u) du )]
By using the strong Markov property, one gets the relation

P [( |P N F (τ 1 ) -P N F (τ 2 )| ≥ 1 )] ≤ 1-E [( exp
( -λ 3 ∫ τ2 τ1 M N F (u) du ))] +1-E [( exp
( -µ 3 ∫ τ2 τ1 P N F (u) du ))] .
With a simple coupling using the same Poisson processes N +/-2/3 of Equation (2.9) and Equation (2.10) gives a process as in Deinition 2.1 on the same probability space such that the relations M N F (t) ≤ M N (t) and

P N F (t) ≤ P N (t)
hold almost surely for all t ≥ 0. From the last relation, one gets the inequality

P [( |P N F (τ 1 ) -P N F (τ 2 )| ≥ 1 )] ≤ 1 -E [ exp ( -λ 3 ∫ τ1+η τ1 M N (u) du )] + 1 -E [ exp ( -µ 3 η sup 0≤t≤T P N (t) )] ≤ 1 -E [ exp ( -λ 3 sup 0≤t≤T ∫ t+η t M N (u) du )] + 1 -E [ exp ( -µ 3 η sup 0≤t≤T P N (t) )]
.

Lemma 2.2 gives the relation lim

N →+∞ sup τ1∈TT E [ exp ( -λ 3 ∫ τ1+η τ1 M N (u) du )] = e -λ3ρ2η
and, for ε > 0, the existence of K > 0 such that sup

N ≥1 P [( sup 0≤t≤T P N (t) ≥ K )] ≤ ε. Consequently lim η→0 lim N →+∞ sup τ1,τ2∈TT τ1≤≤τ2≤τ1+η P [( |P N F (τ 1 ) -P N F (τ 2 )| ≥ 1 )] = 0,
hence, by Aldous' criterion, the tightness of the sequence (P N F (t)) is established. he proposition is proved.

2.A. A

: C R 51

2.A.2 Convergence of Occupation Measures

For N ≥ 1 and T > 0, one deines the random measure

Λ N on E T def. = {0, 1} × N 2 × [0, T ] as follows, for a non-negative Borelian function G on E T , Λ N (G) = ∫ T 0 G(X N F (u), u) du. If A is a Borelian subset of E T , Λ N (A) denotes Λ N ( A ).
Proposition 2.6. he sequence Λ N of random measures is tight and any of its limiting points Λ can be writen as

Λ(F ) = ∑ (i,m,p)∈S ∫ T 0 G(i, m, p, u)π p (i, m)ν u (p) du.
where, for any u ≤ T , ν u is a positive measure on N such that, almost surely,

∫ t 0 ν u (N) du = t, ∀t ≤ T,
and, for p ∈ N, π p is the invariant distribution of the Markov process on {0, 1} × N whose transition rates are given by, for

(i, m) ∈ {0, 1} × N,      (i, m) → (1 -i, m) at rate λ + 1 i + λ - 1 p(1 -i), (i, m) → (i, m + 1) λ 2 i, (i, m) → (i, m -1) µ 2 m.
(2.16)

Additionally, one has

∑ (i,m)∈{0,1}×N mπ p (i, m) = λ + 1 λ + 1 + λ - 1 p λ 2 µ 2 .
(2.17)

Proof. For K > 0, if K K is the compact subset {0, 1} × [0, K] 2 × [0, T ] of E T , then E [ Λ N (E T \ K K ) ] ≤ ∫ T 0 P [( M N F (u) ≥ K )] du + T P [( sup 0≤u≤T P N F (u) ≤ K )]
.

By using the same coupling as in the proof of Proposition 2.5, one gets that

E [ Λ N (E T \ K K ) ] ≤ ∫ T 0 P [( M N (u) ≥ K )] du + T P [( sup 0≤u≤T P N (u) ≤ K )]
.

By Lemma 2.2, for ε > 0, there exists some K such that sup

N ≥1 E [ Λ N (E T \ K K ) ] ≤ ε.
Consequently, the sequence (Λ N ) of random Radon measures on E T is tight. See Dawson (1993, Lemma 3.28, page 44) for example.

C 2. M P P F 52

Proof. Let Λ be a limiting point of some subsequence (Λ N k (•)). By using Radon-Nikodym's heorem, see Chapter 8 of [START_REF] Rudin | Real and complex analysis[END_REF] for example, it is not diicult to see that there exists some non-negative random variables (ℓ u (x)(ω), (ω, x, u) ∈ Ω × S × [0, T ]) such that (ω, x, u) → ℓ u (x)(ω) is measurable and Λ can be expressed as

Λ(G) = ∑ x∈S ∫ T 0 G(x, u)ℓ u (x) du.
From the domination relation of Lemma 2.2, one gets that, almost surely, there is no loss of mass, i.e.

∫ t 0 ℓ u (S) du = t, ∀t ≤ T, (2.18)
holds almost surely. Now take a function f with bounded support on S and let's use Equation (2.12). As previously, we can apply the Doob's Inequality to this process (⟨W N f ⟩(t ∧ T )) and show that the process (⟨W N f ⟩(t)) satisies the relation lim

N →+∞ 1 N 2 E [⟨ W N f ⟩ (T ) ] = 0.
It implies that the martingale (W N f (t)/N ) converges in distribution to 0 for the uniform norm on [0, T ]. By dividing Equation (2.11) by N , one gets that, for the convergence in distribution, the relation lim

N →+∞ (∫ t 0 λ + 1 ( 1 -I N F (u) ) ∆ 1 (f )(X N F (u)) du + ∫ t 0 λ 1 P N F (u)I N F (u)∆ 1 (f )(X N F (u)) du + ∫ t 0 λ 2 I N F (u)∆ + 2 (f )(X N F (u)) du + ∫ t 0 µ 2 M N F (u)∆ - 2 (f )(X N F (u)) du ) = 0.
holds almost surely for all 0 ≤ t ≤ T and for all indicator functions of elements S. Now, for p ∈ N and g a function with inite support on {0, 1} × N, deine f (i, m, p) = g(i, m), the above relation gives

∑ x=(i,m,p)∈S ℓ u (i, m, p)λ + 1 (1 -i)∆ 1 (g)(i, m) + ℓ u (i, m, p)λ - 1 ip∆ 1 (g)(i, m) + ℓ u (i, m, p)λ 2 i∆ + 2 (g)(i, m) + ℓ u (i, m, p)µ 2 m∆ - 2 (g)(i, m) = 0 (2.19
) holds almost surely for all u ∈ A ⊂ [0, T ] and [0, T ] -A is negligible for Lebesgue measure. Equation (2.19) shows that for u ∈ A, the vector (ℓ u (i, m, p)) is proportional to the invariant distribution π p of the Markov process on {0, 1} × N whose transition rates are given by Equation (2.16).

One gets therefore the existence of a constant ν u (p) such that ℓ u (i, m, p) = ν u (p)π p (i, m) for all (i, m, p) ∈ S. Equation (2.18) gives the relation

∫ t 0 ν u (N) du = t, ∀t ≤ T.
Hence one has ν u (N) = 1 almost surely for all u ∈ A 1 ⊂ [0, T ] and [0, T ] -A 1 is negligible for Lebesgue measure.

Straightforward calculations as in the proof of Proposition 2.2 complete the proof of the proposition to give Equation (2.17 In particular, the article Taniguchi et al. ( 2010) presents a comprehensive study of messengers and proteins production in E. coli. It describes the behaviour of a large number of proteins, not only in terms of average expression but also in terms of variability: in populations of cells, the means and the variances of many types of mRNAs and proteins are measured. In total, about 1000 gene are considered. hese data can be confronted with stochastic models of production of proteins which exist since the 1970s: [START_REF] Berg | A model for the statistical luctuations of protein numbers in a microbial population[END_REF], [START_REF] Rigney | Stochastic model of linear, continuous protein synthesis in bacterial populations[END_REF] (see a review of [START_REF] Paulsson | Models of stochastic gene expression[END_REF]). he usual model presented in [START_REF] Paulsson | Models of stochastic gene expression[END_REF] is a three-stage model where gene regulation, transcription and translation are represented. All events occur at exponentially distributed times: the activation and deactivation of the gene, mRNA transcription and degradation, protein production and degradation. he rates of these events depend on the current state of the model.

Yet, these classical models are not considering many aspects that may yet have an impact on the protein variability. For instance, they do not integrate events of the cell cycle such as gene replication and division. Moreover each of them is based on the production of one particular type of protein: only one gene is considered and it produces only one type of protein. No interaction between the diferent protein production processes are considered (like the common sharing of common resources like RNA-polymerases and ribosomes to produce mRNA and proteins). he aim of the two following chapters is to determine the impact on the protein variability of these diferent aspects that are not considered by the classical models.

In this Chapter, we begin by considering successively three diferent origins of protein variability: the noise that directly originated from the transcription and translation mechanisms, then the efect of division and inally the impact that has gene replication. he next chapter will provide a new step toward the global understanding of the whole protein production as we will be interested in interactions between production of diferent classes of proteins.

Plan of the Chapter Section 3.1 will present in detail the experimental study of [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] on which will be based all the comparisons with our models. he techniques, the results and the interpretations of the article will be displayed. In Section 3.2, we will discuss the pertinence to use classical models to reproduce 53 C 3. M C C 54 the experimental measures. Because they do not represent explicitly the growth, we show that they are unit for quantitative comparisons with [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] measures of protein variability. To address this problem, one will need new kinds of models that take into account this aspect. he Section 3.3 will present our irst two models that integrate volume growth. he irst model aims to represent aspects only due to the production of proteins, so that the noise predicted is only due the gene expression process. he second model includes the random segregation of mRNAs and proteins at division. We will show that even if both these aspects are important sources of variability, they are not suicient to reproduce the results observed in [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF]. In Section 3.4 we continue our study by providing a model that also introduces gene replication at some point in the cell cycle. In this section, we propose the main theoretical results as will be able to give explicit solutions for the mean and the variance of every proteins. We will show that the noise induced by the replication of the gene is negligible compared the two previous sources of variability.

Taniguhi et al. Measures

In the article [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF], an overall experimental study of mRNA and protein expression of 1018 genes was performed using single-molecule luorescence microscopy in bacteria (Eschierichia coli). A series of experiments was conducted; and each of them considers a strain of cells where a particular gene was fused with the sequence coding for the luorescent YFP molecule. It gives a population of cells, for which the luorescence abundance denotes the speciic protein quantity. For each strain, the emited luorescence was hence measured in each cells. he obtained global luorescence was normalised by the luorescence emited and by one single protein and divided by the cell size.

In each experiment, what result is the concentration of the considered protein in each cells. So its distribution among the population could then be deduced. In particular, for each type of proteins, the value of the mean µ p and the standard-deviation σ p of the concentration of proteins was deduced from these distributions (see Figure 3.1a). For each type of protein, the obtained concentration of proteins range from 10 -1 and 10 4 copies per om 3 (1 om 3 is in the order of magnitude of the volume of a cell).

On top of that, the article shows measures in the mRNA abundance performed by two techniques. First 137 mRNA types (the most expressed mRNAs) were detected, using Fluorescence in situ hybridisation (FISH) technique: in that case, mRNA expression is measured at the same time as the corresponding proteins and provides the mean µ m and the standard-deviation σ m of the concentration of each mRNA type. he other method is the mRNA-sequencing technique that allows the measure of 841 average mRNA concentration (but this method does not determine the cell to cell variability in the population). his analysis was completed with the measure of average mRNA lifetime τ m . he analysis of the article considers the coeicient of variation (sometimes called "noise") of each mRNA and protein concentration. he coeicient of variation (CV) is used in biology literature as a way to normalise the variance: it is deined as the the variance divided by the mean squared. For instance, Figure 3.1b and Figure 3.1c depict, for every gene, the CV of mRNA concentration (deined as σ 2 m /µ 2 m ) and protein concentration (deined σ 2 p /µ 2 p ) as a function of the average of mRNA and protein concentration. Among other things, these graphs allow, for every gene, to compare the distribution of mRNA and protein with a Poisson distribution: for a Poisson distribution, for an average expression µ, the noise would be 1/µ; the noise would scale inversely with the mean.

For the mRNAs, the noise in Figure 3.1b appears indeed to scale inversely with mRNA mean concentration. But it is higher than expected for Poisson distributions: for a mean µ m the noise here appears to be around 1.7/µ m instead of 1/µ m . he Figure 3.1c also shows the noise of proteins as a function of the average concentration of proteins. It clearly appears that there are two regimes for the protein CV, regimes that the article [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] denotes as "intrinsic noise" and "extrinsic noise" regimes. For low expressed m /µ 2 m ) as a function of the average mRNA expression µ m for every gene. he CV is inversely proportional to mRNA mean concentration, but it is higher than expected for Poisson distributions (red dashed lines). (Measures have been made using FISH technique on 137 mRNA types.) (c): Protein production CV function of the average protein copy number for every gene. For low expressed proteins (mean protein number < 10), the CV is inversely proportional to the average protein production, this part is considered lowered by an "intrinsic noise limit" (red dashed line). For genes with higher protein expression, the CV becomes in independent of the protein expression level, protein expression is here denoted as dominated by the "extrinsic noise" (blue dashed line). proteins (mean protein concentration < 10) the CV roughly scales inversely with the average concentration, the protein variability is dominated by the "intrinsic noise". For genes with higher protein production (mean protein concentration > 10), the CV becomes independent of the average protein production level, the plateau is around 10 -1 ; this regime of gene expression is denoted as dominated by the "extrinsic noise". hese terms of "intrinsic and extrinsic noise" were irstly introduced by [START_REF] Elowitz | Stochastic Gene Expression in a Single Cell[END_REF] and [START_REF] Swain | Intrinsic and extrinsic contributions to stochasticity in gene expression[END_REF] (see Section 1.1) to diferentiate the noise coming from the protein production mechanism itself through translation and transcription (intrinsic noise) and the impact of global luctuations of the cells on the whole gene expression eiciency (extrinsic noise). he authors of [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] use these terms because in the "intrinsic noise" regime, the luctuations seem gene-speciic, as the CV depends on the average protein production; as a consequence, the noise in this area seems to only depend on variables intrinsically speciic to the considered gene, and not to any external other factors. In the second "extrinsic noise" regime on the contrary, luctuations in the protein concentration are gene-independent and are therefore supposed to have an origin not directly linked to the protein production mechanism itself. hey presumed that this external heterogeneity is the result of low luctuations of "global" cellular components such as such as metabolites, ribosomes, and RNA-polymerases. he stochastic behaviour of these compounds is said to have a similar global impact on all protein productions; in particular, it is said to dominate the noise for highly expressed proteins.

v 1 p 1 v 2 p 2 v 3 p 3 v 4 p 4 v 5 p 5 Empirical mean µ p = 1 N N ∑ i=1 p i v i Empirical variance σ 2 p = 1 N N ∑ i=0 ( p i v i ) 2 -µ 2 p (a)
Several arguments in the article are brought to justify the extrinsic nature of the noise observed in the lower plateau of protein noise of Figure 3.1c. In particular they show large heterogeneity in the protein production between cells of the same population, while dynamic luctuations inside a cell are very low (the timescale is in the order or several cell cycles). Nevertheless, it is not fully dismissed that the origin of this noise can lie in the mechanism of protein production itself or some global cell events like division and gene replication. 1 In this Chapter we want to estimate the relative contributions of the mechanism of protein production, the division and the gene replication to the protein variance, and check that they cannot take into account the observed two regimes of noise observed in [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF]. To do so we irst show in the next section that the classical models are not it to be quantitatively compared with the measures; we will show the need for new kinds of models that take into account the cell cycle.

Inadequacies of Classical Models

M P ∅ ∅ λ 1 σ 1 M λ 2 M σ 2 P mRNAs Proteins +1 -1 +1 -1
Figure 3.1: Classical model: Gene constitutive model 1 Comparisons with standard models are indeed made in [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] , but to explain the poor correlation between mRNA and their protein number inside a cell (section 17 of S.I.), not to explain the lower plateau of the noise for highly produced proteins of igure Figure 3.1c.

I q

C M 57

In this section we propose to investigate the interpretation of experiment measures by classical models, by taking the example of gene constitutive model. Ater presenting the gene constitutive model (the other models of the chapter will be based on it), we show why it is not adequate to represent real experiments as they lack the notion of cell growth and division.

Constitutive Gene Model

Let's consider one of the simplest models that describes the production of one type of protein: the gene constitutive model, also referred as the two-stage model. It is a particular case of the three-stage model described in Subsection 1.3.1 (without the gene regulation part), but as it serves as a base for all models of this chapter, it is useful to recall its main mechanisms.

We consider the productions of each protein as being independent from each other: for instance, the production of the protein Adk has no inluence on the production of YjiE. It is a "gene-centred" model. Every event (transcription, degradation, etc.) is supposed to happen at times that follow exponential distribution. Moreover it represents the number of mRNAs and proteins in an arbitrary ixed volume V around the considered gene. In literature, the value of V is oten not explicitly given. But a reasonable value for it would be 1 om 3 in order to directly interpret the quantity of compounds described by the model as the actual concentration expressed in copies per om 3 (we will discuss the consequences of this choice in Subsection 3.2.2).

he gene constitutive model considers that each gene is continually available for transcription. Hence, for a given gene, two types of entities are considered (see Figure 3.1):

mRNAs mRNAs are transcribed at rate λ 1 ; the number M of mRNAs is then increased by 1. Each mRNA exists for a certain time determined by the rate σ 1 until it disappears; as there is M mRNAs, the total rate of mRNA disappearance is σ 1 M . When a disappearance occurs, M is decreased by 1.

Proteins Each mRNA can be translated into a protein at rate λ 2 ; since the number of mRNA is given by M , the total rate of protein production is λ 2 M . When a translation occurs, the number of proteins P is increased by 1. Analogously to messengers, each protein exists during a certain time until its decay which occurs at an exponential time with rate σ 2 ; the total rate of protein decay is hence σ 2 P .

he "decay" rates σ 1 for mRNAs and σ 2 for proteins are not representing the same efect:

• he mRNA decay is mainly due to the rapid degradation through an active catalysed reaction with enzymes. hrough this mechanism, the cell ensures the quality control of the molecules which can be denatured through time. For the mRNAs, this degradation is of the order of few minutes; it is much quicker than the cell cycle (the median lifetime of mRNAs is about 5 min in [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF], while the cell cycle is around 150 min). herefore, in that case the mRNA decay rate σ 1 represents a degradation rate; it is speciic to the type mRNA since each of them has diferent ainity with the degradation enzymes.

• For the proteins, the active degradation also exists (in this case, it is called proteolysis), but it usually occurs only in very long period of times, much higher than the duration of the cell cycle (see [START_REF] Koch | Protein Turnover in Growing Cultures of Escherichia Coli[END_REF]). As the model takes place in a ixed volume V , the protein will certainly leave it before being degraded and is considered to never return inside: it is the decay by dilution. In the case of proteins, the decay rate σ 2 hence represents the dilution efect; this rate is similar for all proteins and represents the time for a compound to leave the considered volume.

Diferent properties of this model are known (it is a particular case of the three-stage model presented in Subsection 1.3.1). In particular, the mRNA copy number at equilibrium is known to follow a Poisson distribution and explicit expressions for the mean and the variance of the number of proteins at equilibrium are C 3. M C C 58 known and can possibly be used to it the diferent measurements of [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] experiments: it will make predictions for the variances of mRNAs and proteins that can be compared to experimental measures. But as we show in the next subsection, this quantitative comparison may not be relevant as this model (and by extension all classical models) does not take into account the real volume of the cell.

Impact of the Considered Volume in Classical Models

As previously said, in classical models, a ixed volume V is considered surrounding the gene of interest. In this volume, only one copy of the gene is considered, and when one compound (either a protein or a mRNA) leaves the volume, it is assumed that it never returns inside (see Figure 3.2).

Figure 3.2: Volume in classical models are based on dilution. he volume V is ixed, once a compound (mRNA or proteins in green) leaves the volume, it does not return.

At a time t, from the number M t of mRNAs and P t of proteins inside the volume V , it is possible to consider their concentration as their number per unit of volume V : at time t, the concentration of mRNAs and proteins are respectively

M t V and P t V .
With this deinition of the concentration, the mean and the variance of the concentration of proteins is interpreted as:

E [P t /V ] = E [P t ] /V, Var [P t /V ] = Var [P t ] /V 2 ,
and similarly for the mean and the variance of mRNA concentration. he choice for V of 1 om 3 permits directly to interpret the mean E [P t ] and the variance Var [P t ] of the number of proteins directly as the mean E [P t /V ] and the variance Var [P t /V ] of their concentration per om 3 . Nonetheless, one can wonder if this particular choice of V has an impact on the obtained values of the mean and the variance of the concentration. he following example shows that it is indeed the case.

Example 3.1. Let's consider two volumes of size V . In each of these volumes two independent but with identical dynamics processes occur: (P 1,t ) and (P 2,t ) would be the processes that represent the number of proteins in respectively the irst and in the second volume. Concentration being an extensive quantity, one expects that the behaviour of the concentration in one volume would be similar as in both volumes taken altogether. But the mean and the variance of the concentration in the large volume are:

E [(P 1,t + P 2,t )/(2V )] = E [P 1,t /V ] , Var [(P 1,t + P 2,t )/(2V )] = Var [P 1,t /V ] /2.
he mean is indeed identical if the volume 2V is considered instead of V , but not the variance.

he previous example shows that, with a classical models, the concentration is not an extensive quantity in terms of variance, and the distribution of the concentration of each compounds depends on the considered volume V . his has important consequences and it raises diiculties when it comes to interpreting experimental results with those predicted with classical models.

he usual comparison between a classical model and the measures are done as following: in order to deduce the parameters of the model, the equilibrium mean E [P /V ] of the model is interpreted with the 3.3. M

C

C 59 empirical mean µ p of the measures; once all the parameters known, one compares the obtained variance

Var [P /V ] with the empirical variance σ 2 p of the measures. Figure 3.1a explains schematically how empirical mean and variance are computed: in each cell of the population, concentrations were computed using the speciic volume of the cell and not an abstract volume V . In the model, on the contrary, one can ix E [P /V ] and still have diferent Var [P /V ] depending on the volume V chosen. To sum up, the predicted variance Var [P /V ] is volume V dependent, therefore it cannot be directly be interpreted as the empirical variance σ 2 p . Comparisons between theoretical variances obtained with classical models and the empirical variances measured in real experiments are hence problematic. hey do not represent the same thing: Var [P /V ] denotes the variance of proteins inside an abstract volume V , as the empirical variance is computed using real volume of cells. In order to represent what are exactly the empirical mean and the empirical variance described in Figure 3.1a, one needs to propose a model with a quantity that depicts the actual volume of the cell, and this volume changes across time as the cell grows and divides.

Model with Cell Cycle

he model of this section is close to the gene constitutive classical model, but we introduce the notion of volume of the cell that changes across the cell cycle. his aspect will enable quantitative comparisons with experimental dataset of [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF]. his model only considers aspects due to the protein production; as a consequence, the variability predicted here arises from protein synthesis mechanism itself and not from external factors. First, in Subsection 3.3.1, we present this new feature, as it will be common in all the remaining models of this chapter and also the model of Chapter 4. In Subsection 3.3.2 is the presented the model in detail; the Subsection 3.3.3 is dedicated to its theoretical analysis in order to it parameters to the measures. We will analyse the simulations of the model and show that the protein coeicient of variation globally follows what the authors [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] denotes as the "intrinsic noise limit". In Subsection 3.3.6 we will interest in the efect of the random segregation of each cell compounds at division and show that it has a signiicant impact on some protein variability.

Division

New Feature: A Time Dependent Volume

he baseline of all the models of this chapter and the next is the following two key features: the cell growth and the division. hey are the minimum notions to introduce in order to somewhat take into account the cell cycle. So we describe in this subsection aspects that will remain true for all the models to come.

At any time t, the volume V (t) considered is the entire volume of the cell which is increasing as the cell grows (see Figure 3.1). From this volume, concentrations can be considered: if M t and P t respectively denote the number of mRNAs and proteins at time t, it comes the concentrations M t V (t) and P t V (t) .

At periodic times, divisions occur; it corresponds to the step between t 2 and t 3 in Figure 3.1. Two daughter cells are created and the model only focuses on one of them; the volume considered then is the one of this C 3. M C C 60 newborn cell. During the event of division, each mRNA and protein either goes in the next considered cell or not. In a irst step, we consider that this segregation is exact, that is to say that exactly one half of mRNAs and proteins goes to the considered cell (in Subsection 3.3.6 we will consider the case where this segregation is random: each mRNA and protein has an equal chance to be in the considered daughter cell or not).

From this perspective, the notion of dilution of compounds introduced for classical models (the phenomenon by which a compound can spontaneously leave the volume of the model) is no longer used. It is replaced by the molecule segregation at division where the number of compounds is halved. he spontaneous "decay" of compounds will only be due to their hydrolysis, the active catalysed degradation of mRNAs or proteins. mRNAs In classical model Subsection 3.2.1, in a ixed volume of 1 om 3 , mRNAs are spontaneously created at a constant rate. In this model, we keep this concept of spontaneous creation per volume unit as the rate of mRNA creation is λ 1 V (t), with V (t) the volume of the cell at time t; hence per volume unit, the rate of production will remain constant. When an mRNA creation occurs, the number of mRNAs M is increased by one. As for the gene-constitutive model, each mRNA degrades at an exponential time given by the rate σ 1 .

Presentation of the Model

Proteins As in the model with constitutive gene, each mRNA can be translated into a protein at rate λ 2 ; the number of proteins P is then increased by 1. But here there is no disappearance rate: since the proteolysis occurs in a timescale much longer than the cell cycle, its decay is dominated by protein segregation that occurs at division.

Division Periodically, every time τ D , a division occurs. At this instant, the number of mRNAs and the number of proteins undergo an exact division to keep only half of the molecules that are in the considered daughter cell.

On top of that, one also considers the volume growth of the cell independently. In real life experiments, bacteria volume grows exponentially (see [START_REF] Wang | Robust growth of Escherichia coli[END_REF]) and approximately doubles its volume at the time of division τ D . As a consequence, the model considers that, if s is the time spent since the last division, then the volume grows as V (s) = V 0 2 s/τD with V 0 being the typical size of a cell at birth. Remark 3.1. his model only considers aspects that are speciic to protein production, so that randomness induced by the model is only due to the transcription and translation mechanisms:

• mRNA rate of production is proportional to the volume so that there is no efect of gene replication. As a result, the average concentrations of mRNAs and proteins remain constant across the whole cell cycle (it will be proven in the next subsection).

• exact segregation at division minimises the efect of division: as the volume is strictly halved as well as the number of mRNAs and proteins, their concentrations remain unchanged during the division process.

he goal is to have a basic model with the notion of the cell cycle that only consider sources of noise that are speciic to the mechanism of protein production: this model is a way to estimate the "intrinsic noise" of the gene expression variability. It is our starting point for the analysis of the protein variability as, later in the chapter, external features will be added one by one to the model. Now, our aim is to analyse the variability predict by this model. To do so, in the next subsection, we conduct at irst a theoretical analysis of the average mRNA and protein production by the model. his analysis will permit to it parameters to the experimental measures of [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF].

Dynamic of the Number of mRNAs and Proteins

he content of this section is technical: the goal is to justify the Proposition 3.3 of the next subsection. he reader more interested in biological aspects can skip this and go directly to Subsection 3.3.5, while admiting its irst proposition.

Messenger-RNA Dynamic

For any time t ∈ R + , M t denotes the number of mRNAs at this instant. Let's depict the distribution of M t for any t ∈ R + . We suppose that the initial time t = 0 is a time of division; in this case, at each time i • τ D with i ∈ N are moments of division. For any i ∈ N , M iτD denotes the number of mRNAs at the beginning of i-th cell cycle and M iτ D -the number of mRNAs in the (i -1)-th cell cycle just before division. We suppose that a lot of cell divisions already occurred even before time t = 0, and hence the considered cell cycle occurs when the embedded Markov chain (M iτ D ) i has already reached its equilibrium: it means that the distribution of M iτD is the same as the distribution of M (i+1)τ D . If the equilibrium is already reached at time 0, it implies that the distribution of any M iτD+s for any i ∈ N and s ∈ [0, τ D [ is equal to the distribution of M s . As a consequence, we can only consider the irst cell cycle s ∈ [0, τ D [ to fully characterise the behaviour of M s at any time s ∈ R + .

he dynamic of M s for s ∈ [0, τ D [ resembles the classical constitutive gene model (Subsection 3.2.1); but in the classical model, equilibrium properties were used to describe its behaviour. Here we need to describe the evolution of (M s ) between times 0 and τ D (during this period of time, the number of mRNA approximately doubles).

As previously said, in our case, the number of messengers M s ater a time s does not reach its equilibrium; we therefore need, not to describe the equilibrium, but the dynamics of M s .

To do so, for any time s of the cell cycle, let's group mRNAs two categories:

• First group: mRNAs which were present at the birth of the cell. Each mRNA i of the irst group is characterised by E i σ1 , its lifetime given by an exponential random variable of rate σ 1 . he i-th mRNA still exists at time s if and only if E i σ1 > s. As a consequence, the number of mRNAs of this group still exists at time s is given by

M0 ∑ i=1 {E i σ 1 >s} .
(3.1)

• Second group: mRNAs which have been created since the birth of the cell. he description of the number of mRNA of this group is more complicated. It is necessary to resort to the framework of Marked Poisson Point Processes (MPPP). A MPPP is a two-dimensional process. It is based on a Poisson process where each of its random point is "marked" with another random variable; each point of a MPPP is a couple (x, y) where x is part of a Poisson point process and y is the mark distributed according to a certain distribution. One can refer to the irst Chapter of [START_REF] Robert | Stochastic networks and queues[END_REF] or [START_REF] Kingman | Poisson Processes. Number 3 in Oxford Studies in Probability[END_REF] for the main results concerning MPPP.

We use this tool to characterise the number of mRNAs of the second group. In our case, the irst variable

x represents the time at which the mRNA is created and the second variable y represents the mRNA lifetime. Let's deine N an MPPP of intensity

ν(dx, dy) = λ 1 V (x)dx ⊗ σ 1 e -σ1y dy.
It is noticeable that the underlying Poisson Process of this MPPP is not homogeneous. If the i-th mRNA of this group is born at time x i and its lifetime is y i , then it exists at time s if and only if (x i , y i ) ∈ ∆ s with

∆ s = { (x, y) ∈ R 2 + , 0 < x < s, y > s -x } .
One can refer to Figure 3.3. herefore the number of mRNA of this group still present at time s is given by

N (∆ s ) = ∫ ∫ R 2 + (x,
(x i , y i ) is in the set with ∆ s = { (x, y) ∈ R 2 + , 0 < x < s, y > s -x } .
By summing the number of mRNAs for each group (Equation (3.1) and Equation (3.2)), it comes the total number of mRNAs present at time s ∈ [0, τ D [:

M s = M0 ∑ i=1 {E i σ 1 >s} + N (∆ s ) . ( 3
E [M s ] = V (s) λ 1 τ D σ 1 τ D + log 2 .
Proof. By taking the mean of Equation (3.3), it follows for any time s of the cell cycle:

E [M s ] = E [ M0 ∑ i=1 {E i σ 1 >s} ] + E [N (∆ s )] .
Since all ( E i σ1 ) i are i.i.d. and independent of M 0 , the irst term is given by

E [ M0 ∑ i=1 {E i σ 1 >s} ] = E [M 0 ] e -sσ1 .
For the second term, one has to remark that as N is a MPPP, N (∆ τ D -) is a Poisson random variable (Proposition 1.13.a of [START_REF] Robert | Stochastic networks and queues[END_REF]). he parameter of this Poisson random variable is given by

ν (∆ s ) = ∫ ∫ ∆s ν (dx, dy) . C 3. M C C 64 Since, in the deinition of ∆ s ν (∆ s ) = ∫ s 0 ∫ ∞ s-x λ 1 V (x)σ 1 e -σ1y dy dx = V 0 λ 1 σ 1 ∫ s 0 2 x/τD ∫ ∞ s-x
e -σ1y dy dx

= V 0 λ 1 e -σ1s ∫ s 0 exp ( x ( log 2 τ D + σ 1 )) dx = V 0 λ 1 σ 1 log 2 + σ 1 τ D ( 2 s/τD -e -σ1s
) .

As a consequence, it comes that for any time s in the cell cycle:

E [M s ] = E [M 0 ] e -sσ1 + V 0 λ 1 τ D τ D σ 1 + log 2 • ( 2 s/τ D -e -sσ1
) .

We still have to specify the mean number of mRNAs at birth E [M 0 ]. At the end of the cell cycle, for s = τ D -, the average number of mRNAs is given by

E [M τD-] = E [M 0 ] e -τ D σ1 + V 0 λ 1 τ D τ D σ 1 + log 2 • ( 2 -e -τ D σ1 ) ,
and since at equilibrium,

E [M τD ] = E [M 0 ] = E [M τD-] 2 . Hence E [M 0 ] ( 2 -e -τ D σ1 ) = V 0 λ 1 τ D τ D σ 1 + log 2 • ( 2 -e -τ D σ1
) , which gives the result.

Protein Dynamic

he mean number of mRNAs is now determined for any moment of the cell cycle. Each of the mRNAs potentially produces proteins at rate λ 2 . As for the mRNAs, we describe the number of proteins at time s by grouping them into two categories:

• he P 0 proteins that were present at birth and which remain in the bacteria during all the cell cycle (as said in Subsection 3.3.2 the proteolysis is not considered in this model).

• he proteins that have been created during the current cell cycle. he rate of production is depending on the current number of mRNAs. We consider N i λ2 (for i ∈ N and i ≥ 1) independent Poisson Point Processes of intensity λ 2 . If the i-th mRNA exists at time s (that is to say if i ≤ M s ), then the number of proteins produced by this mRNA between s and s + ds is N i λ2 (ds). To sum up, the number of proteins at a time s of the cell cycle is:

P s = P 0 + ∞ ∑ i=1 ∫ s 0 i≤Mu N i λ2 (du) .
(3.4) he irst term is the number of proteins at birth, and the second take into account all the proteins created between times 0 and s. Based on that, one can determine the mean number of proteins at any time s of the cell cycle:

3.3. M C C 65
Proposition 3.2. At equilibrium, the mean number of proteins at any time s ∈ [0, τ D [ of the cell cycle is

E [P s ] = V (s) • λ 2 τ D log 2 • λ 1 τ D σ 1 τ D + log 2 .
Proof. Taking the average of Equation (3.4) gives

E [P s ] = E [P 0 ] + ∞ ∑ i=1 E [∫ s 0 i≤Mu N i λ2 (du) ] = E [P 0 ] + ∞ ∑ i=1 E [∫ s 0 i≤Mu λ 2 du ] = E [P 0 ] + λ 2 ∫ s 0 E [ ∞ ∑ i=1 i≤Mu ] du = E [P 0 ] + λ 2 ∫ s 0 E [M u ] du.
As we know the mean number of mRNAs E [M u ] at any time u of the cell cycle with Proposition 3.1:

E [P s ] = E [P 0 ] + λ 2 τ D log 2 • λ 1 τ D σ 1 τ D + log 2 • (V (s) -V 0 ) .
Since the system is at equilibrium, we have for time τ D -, E [P τD-] = 2E [P 0 ]; so

E [P 0 ] = λ 2 τ D log 2 • λ 1 τ D σ 1 τ D + log 2 • (V (τ D -) -V 0 ) = λ 2 τ D log 2 • λ 1 τ D σ 1 τ D + log 2 • V 0 .
Consequently, for any time s of the cell cycle,

E [P s ] = λ 2 τ D log 2 • λ 1 τ D σ 1 τ D + log 2 • V 0 ( 1 + 2 s/τD -1 ) ;
hence the result.

Expressions for the variance of the number of mRNAs and proteins are not easily obtained for this model, we will hence determine them with simulations. (In Section 3.4, will be presented a model, that beter represent the real dynamic of the cell, and from which we have then analytical expressions of the mean and the variance of mRNAs and proteins).

Parameter Computations

In this subsection, we determine the parameters λ 1 , σ 1 and λ 2 so that the average production of mRNAs and proteins correspond to the measure of [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF]. To do so, we will use the previous analytical results on the mean number of mRNAs and proteins.

Mean and Variance of Concentration over the Cell Cycle

his model represents mRNAs and proteins in cells with a volume that changes across time: we have described mRNA and protein number at any time of the cell cycle. But experimental measures are not done at a particular time in the cell cycle. So we need to characterise the mean and the variance of compound concentration, not at a given time s in the cell cycle, but over the cell cycle. At any moment of the cell cycle, the concentration of any compound (either mRNAs or proteins) is deined as its current number divided by the current volume : for instance, in the case of mRNAs, the concentration at time s ∈ [0, τ D [ is M s /V (s) (with V (s) = V 0 2 s/τ D ). One can consider the global average concentration over the cell cycle E [M /V ], as simply the mean E [M s /V (s)] averaged over the cell cycle for s from 0 to τ D . E [M /V ] is then deined as:

E [M /V ] := 1 τ D ∫ τD 0 E [ M s V (
[ (M s /V (s)) 2 ] - E [M /V ] 2
. hen let's denote the global variance Var [M /V ] as the average over the cell cycle of this deviation:

Var [M /V ] := 1 τ D ∫ τD 0 [ E [ ( M s V (s) ) 2 ] -E [M /V ] 2 ]
ds.

(3.6)

Equivalently for the proteins, let's deine:

E [P /V ] := 1 τ D ∫ τD 0 E [ P s V (s) ] ds, (3.7
)

Var [P /V ] := 1 τ D ∫ τ D 0 [ E [ ( P s V (s) ) 2 ] -E [P /V ] 2 ]
ds.

(3.8) Remark 3.2. Let's consider a population of independent cells, where each cells have speciic mRNA and protein concentrations (as it is the case in real experiments). If the ages of cells of the population are uniformly distributed in the interval [0, τ D [, the mean and the variance of mRNAs and proteins concentration would be equivalent to Equations (3.5), (3.6), (3.7) and (3.8). In reality, the experimental population considered are in exponential growth, which means that the population age distribution is not uniformly distributed, but we will see in the next Section (subsubsection 3.4.5.4), that this has litle impact on the population distribution.

In the case of this model, we have determined in the previous subsection the average mRNA and protein number for any time s of the cell cycle in Proposition 3.1 and Proposition 3.2, in particular, the concentrations at any time s remain constant and are given by

E [ M s V (s) ] = λ 1 τ D σ 1 τ D + log 2 and E [ P s V (s) ] = λ 2 τ D log 2 • λ 1 τ D σ 1 τ D + log 2 .
With the deinition of E [M /V ] and E [P /V ] (Equation (3.9) and Equation (3.10)), it follows the next proposition.

Proposition 3.3. he average mRNA and protein concentrations over the cell cycle are:

E [M /V ] = λ 1 τ D σ 1 τ D + log 2 (3.9)
and

E [P /V ] = λ 2 τ D log 2 • λ 1 τ D σ 1 τ D + log 2 (3.10) 3.3. M C C 67

Parameters Deduced from Experimental Dataset

For each gene, we want to identify the parameters λ 1 , σ 1 and λ 2 . We also need to determine the "global" quantities τ D and V 0 . To do so, we use measurements of [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF]. First, let's ix the parameters common to all genes: the division time τ D is said to be 150 min in the article and the volume at birth V 0 is taken equal to 1.3 om 3 . 2 hen we have to determine for each gene the three gene-speciic parameters λ 1 , σ 1 and λ 2 . We consider the genes of the article for which was measured the empirical mean of messengers µ m and proteins µ p concentrations, as well as the mRNA half-life time τ m .

First we determine the rate of mRNA degradation for each gene with the measured mRNA half-life time τ m : a half-life τ m indicates that a mRNA has a probability 1/2 to disappear within a duration τ m , hence e -σ1τm = 1/2. From that, we can compute the rate σ 2 (speciic for each type of mRNA):

σ 1 = log 2/τ m .
hen we can identify the averages of mRNA and protein concentrations of the model (respectively E [M /V ] and E [P /V ]) with the empirical averages µ m of mRNA concentration and µ p of protein concentration of the article. As a consequence, with Equation (3.9) and Equation (3.10), the parameters λ 1 and λ 2 are: A summary of the diferent parameters can be seen in Table 3.1. Having determined all the parameters allows to perform simulations of the model using stochastic algorithm in order to assess the variability of every protein and compare them with those experimentally obtained in [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF]. When performing simulations, one needs to take care of the non-homogeneity of the Poisson processes describing mRNA creation times: the rate protein production λ 1 V (s) is not a homogeneous rate as it changes with time. hat does not allow a direct application of [START_REF] Gillespie | Exact stochastic simulation of coupled chemical reactions[END_REF] algorithm and a more complex algorithm need to be used. One can refer to appendix Section 3.A for more information.

λ 1 = µ m • σ 1 τ D + log 2 τ D , λ 2 = µ p • log 2 τ D • σ 1 τ D + log 2 λ 1 τ D . Param Median Mean Maximum Minimum λ -

Results of the Model with Cell Cycle

For each gene, we have performed a simulation using the parameters previously described. In each case, we regularly recorded the protein concentration at diferent moments of the cell cycle for thousands of generations. From these results, the behaviour of each protein concentration distribution during the whole cell cycle can be deduced.

he results of the protein variance are shown in Figure 3.4, where the variability of the protein concentration in the simulations of the model and of the experiments are shown. In the irst igure is shown the proiles 2 he value of V 0 is not explicitly given in [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF]. But it can be estimated via the area range of cells and the typical width given in the supplementary material of [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF]

. C 3. M C C 68 -1 0 1 2 3 Prot. Adk -1 0 1 2 3
Normalised protein conc.

Prot. FabH 0 of three diferent representative proteins (Adk, FabH and YjiE) across the cell cycle; these three proteins are respectively highly, moderately and lowly expressed. As predicted, the mean protein concentration remains constant during the cell cycle: there is no average periodic luctuation due to cell cycle in this model. he igure also shows that the relative standard deviation decreases as the average production increases. As previously said, the protein variance of this model is only due to the protein production mechanism. herefore the results presented in Figure 3.4 conirm that the variability due to protein production itself (the intrinsic noise) cannot explain all the protein variability experimentally observed. In the next subsection we add the irst contribution to the "extrinsic" noise: the efect of random distribution of compounds (either mRNA or proteins) in daughter cells during division.

¿ D =4 ¿ D =2 3¿ D =4 ¿ D -1 0 1 2 3 Prot. YjiE

It is conirmed by

M C

C 69

Model with Cell Cycle and Binomial Division

In this subsection, we propose our irst model extension. In the model presented in Figure 3.2b, the division performed is considered as exact: the numbers of mRNAs and proteins are halved. In reality, this division is not exact as the position of any compound in the dividing cell is random. By supposing that the size of the two daughter cells are equivalent, every compound has an equal chance to be in the next considered bacteria or not. Given the number of mRNAs M τD-and proteins P τD-just before the division, the total number of mRNAs and proteins ater division is no longer deterministic, it is the result of a random sampling. We called this sample the binomial sampling: for instance, knowing M τD-, the number of mRNAs M τD ater division follows a binomial distribution B (n, p) with parameters n = M τD-(the number to distribute) and p = 1/2 (equal chance to be in the considered cell). Every other aspects remain identical to the previous model (see Figure 3.5a).

Even if the sampling at division is now random, on average we still have

E [M τD ] = 1 2 E [M τD-] and E [P τD ] = 1 2 E [P τD-]
and it does not change the results for the mean of mRNA and protein production: the proofs of Proposition 3.1 and Proposition 3.2 remain correct even with the binomial sampling. In particular, we still have

E [M /V ] = λ 1 τ D σ 1 τ D + log 2 and E [P /V ] = λ 2 τ D log 2 • λ 1 τ D σ 1 τ D + log 2 ,
which allows to determine parameters λ 1 and λ 2 based on [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] dataset.

Comparative simulations of the two models have been performed: with exact division and with binomial division. On Figure 3.5b is shown the protein concentration CV in the model with exact division divided by the variance in the CV with binomial sample. his ratio allows us to know the proportion of noise that is due to the binomial sampling. As this ratio is below 1 for every gene, it shows that, as expected, the binomial sampling indeed adds variability. For most of proteins around 10% of the noise is due to binomial sampling. his proportion can increase up to 50%. It corresponds to proteins that have the lowest Fano factor (deined as Var [P /V ] / E [P /V ] ) such as OmpC. It is noticeable that, as low expressed genes tend to have a low Fano factor (not shown), these genes tend to be more sensitive to the binomial efect of division.

One can explain this clear dependence on the relative variance with a toy model (in dashed cyan line in the igure) which is explained with further details in Section 3.C. In Figure 3.5c is shown the proile of the protein OmpC, with a high contribution of the binomial sampling to the variance: a higher variability at the beginning of the cell cycle that tends to diminish during the cell cycle due to dilution.

Even if the additional noise can be important for some genes, the variability imposed by protein production still prevails. In particular, as it has an impact primarily on less expressed proteins, this efect is not able to explain the "extrinsic noise" lower plateau observed for highly expressed proteins in [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF]. he protein CV as a function of the average protein concentration shows behaviour not that diferent as in Figure 3.4b

In the next section we propose a more complete model where the mRNAs are no longer created spontaneously in every unit of volume, but their creations depend on the number of gene copies in the cell.

Model with Cell Cycle and Gene Replication

his section presents the main model of the chapter. It takes into account all the basic features that can be expected for the production of a type of protein inside a cell cycle: the transcription, the translation, the gene replication and the division. Unlike the previous models, it also represents the gene as an entity that is replicated at some point in the cell cycle, hence doubling the transcription rate at some point in the cell cycle. he goal of the section will be to quantify its contribution to the protein noise. To do so, and contrary to the previous models, we will be able to give analytical expressions for the mean and the variance of the mRNAs and proteins; so that we will not need simulations to estimate the variability each gene expression. In Subsection 3.4.1 we will present the model and its mechanisms in detail. he two subsections that follow contain the main theoretical results of this section: the explicit distribution for the number of mRNA is given by heorem 3.3 in Subsection 3.4.2 and the mean and the variance of the number of proteins is given by heorem 3.4 in Subsection 3.4.3. hese two analytical results will be helpful in the last part Subsection 3.4.4 to determine the parameters, and in Subsection 3.4.5 to predict the protein variance of the parameters. In this last subsection, we will show that the gene replication has a low impact on the protein variability.

Presentation of the Model

In the models of the previous section, every gene sees its mRNAs spontaneously created in every volume unit in the cell. It would represent a case where, the gene quantity increases continuously with the volume thus keeping the gene concentration constant. In reality the gene quantity is a discrete number that doubles with DNA-replication. In this section, we propose an extension of the model: cell growth and binomial sampling are still considered but we add the notion of gene replication during the cell cycle.

As for the previous model, it still focuses on one particular gene, the cell volume V (s) is still increasing exponentially during the cell cycle until time τ D ; at division, all compounds undergo a binomial sampling before beginning the new cycle. he diference here is in the rate of mRNA production: it is no longer proportional to the volume, but it remains constant until it doubles at the deterministic time of gene replication τ R (with 0 < τ R < τ D ) and remains anew constant until the time of division. 3he model represents four aspects of that intervene in protein production (see Figure 3.1b):

mRNAs Messenger-RNAs are transcribed at constant rate λ 1 before the replication and at constant rate 2λ 1 ater gene replication. When transcription happens, the number M of mRNAs then increased by 1. As in the previous model, each mRNA has a lifetime of rate σ 1 (so the global mRNA degradation rate is

σ 1 M ).
Proteins Each protein has the same dynamic as presented in the model with cell division gene (Subsection 3.3.2).

Gene replication At deterministic times τ R ater each division (with τ R < τ D ), it occurs the gene replication. he gene responsible for the mRNA transcription is replicated, hence doubling the mRNA transcription rate until next division.

Division Divisions still occur periodically at deterministic times τ D . he efect is a binomial sampling that only keeps molecules that are in the considered daughter cell. Moreover, as there is only one copy of the gene in the newborn cell, the mRNA transcription rate is anew set to λ 1 until the next gene replication. For one particular type of protein, the number of mRNAs and proteins are respectively M and P ; events occur at stochastic times that depend on parameters λ 1 , σ 1 , λ 2 and on the current state of the system (see main text for more details).

OriC YjiF

Adk

On top of that, the growth of the cell volume is still considered as deterministic: at any time s of the cell cycle the volume of the cell is

V (s) = V 0 2 s/τD
with V 0 being the typical size of a cell at birth. In this section we will be able to have explicit expressions to describe the mean and the variance for both mRNAs and proteins. It is helpful as it allows to determine the parameters that corresponds to the experimental genes and also that the variability of each protein can be computed directly, without resorting to simulations. he next two subsections gather technical proofs that are needed in order to have analytical results for the protein production. he reader more interested in biological interpretations of the model can directly go to Subsection 3.4.5 and admit the analytical expressions of heorem 3.3 (depicts the mRNA distribution) and heorem 3.4 (depicts the protein mean and variance).

Dynamics of mRNA number

he aim of this section is to prove heorem 3.3 which states that at any time of the cell, the mRNA number follows a Poisson distribution. To do so, we irst give a description of the number mRNAs at any time in the cell cycle using Marked Poisson Point Process. Using this description, we will be able to show Proposition 3.4, that the distribution of M 0 at the beginning of the cell cycle is a Poisson distribution. his proposition will allow to inally prove the main theorem of the subsection. Let's consider that time s = 0 is the beginning of a new cell cycle and that the system is already at equilibrium in the same sense as the previous models (see Subsection 3.3.3). We consider that M 0 , the number of mRNAs at birth is known. As in Subsection 3.3.3, we assort mRNAs in independent groups; here let's consider three categories:

• mRNAs which were present at the birth of the cell. Each of them is characterised by its lifetime given by an exponential time of rate σ 1 . he i-th mRNA is still present at time s if and only if

E i σ1 > s, with ( E i σ1
) being i.i.d. exponential random variables of parameter σ 1 .

• mRNAs created since the birth of the cell by the irst copy of the gene. he i-th mRNA of this group is characterised by the time of creation t i given by a Poisson Process of rate λ 1 and its lifetime δ i given by a exponential time of rate σ 1 .

• mRNAs created since the gene replication by the second copy of the gene. As in the previous group, the i-th mRNA is characterised by the time of creation t i given by a Poisson Process of rate λ 1 and its lifetime δ i given by an exponential time of rate σ 1 . But here, the Poisson Process of rate λ 1 begins at time τ R , the time of replication of the gene.

As in Subsection 3.3.3, one can represent the number of mRNAs of the second and the third group as two independent MPPPs N and N ′ . he irst variable x of each of these MPPPs will represent the time. he intensity of each of the MPPP is the same:

ν(dx, dy) = λ 1 dx ⊗ σ 1 e -σ1y dy.

he only diference between N and N ′ is the fact that they begin at time 0 for N and at time τ R for N ′ (see Figure 3.2). As a consequence, if we consider an mRNA of either group, the conditions of its existence at time s ∈ [0, τ D [ are respectively:

• if it is in the second group:

(t i , δ i ) ∈ ∆ s with ∆ s = {(x, y), 0 < x < s, y > s -x} ,
• if it is in the third group:

(t i , δ i ) ∈ ∆ ′ s with ∆ ′ s = {(x, y), τ R < x < s, y > s -x} .
Hence, we can describe the number of mRNAs at any time s ∈ [0, τ D [ as follows:

M s = M0 ∑ i=1 {E i σ 1 >s} + N (∆ s ) + s≥τR N ′ λ1 (∆ ′ s ) . (3.11)
Each term corresponds to each group of mRNAs previously described. At irst we want to characterise the distribution of M 0 , the number of mRNAs at the birth of the cell. To do so, we use the equilibrium hypothesis that implies that M 0

D = M τD : C 3. M C C 74 0 ¿ D Time 0 mRNA lifetime Birth Replication Division s ¢ s t i t i + ± i ± i (a) Representation of N 0 ¿ D Time 0 mRNA lifetime Birth Replication Division s ¢ 0 s t i t i + ± i ± i (b) Representation of N ′ Figure 3
∆ s = {(x, y), 0 < x < s, y > s -x}); the process N ′ , in (b)
, begins at replication (in particular an mRNA is still present at time s, if and only if the point

(t i , δ i ) is in the set ∆ ′ s = {(x, y), τ R < x < s, y > s -x}).
Proposition 3.4. At equilibrium, the number of mRNAs at birth M 0 follows a Poisson distribution of parameter:

x 0 = λ 1 σ 1 [ 1 - e -(τD-τR)σ1 2 -e -τDσ1
] .

Proof. When s = τ D -, there is by Equation (3.11):

M τD-= M0 ∑ i=1 {E i σ 1 >τD-} + N λ1 (∆ τD-) + N ′ λ1 ( ∆ ′ τD-
) .

he irst term corresponds to initial messengers that were not degraded ater the time τ D . Let's suppose that M 0 is distributed according to a Poisson distribution of parameter x 0 . hen the random variable

M0 ∑ i=1 {E i σ 1 >τD-} also follows a Poisson distribution of parameter x 0 e -τDσ1 .
he second term corresponds to mRNAs that were created by the irst copy of the gene and which are still present at division. Since N is a MPPP, N (∆ τD-) is a Poisson random variable (Proposition 1.13 of Robert ( 2010)) with parameter ) is a Poisson random variable of parameter

ν (∆ τD-) = ∫ τD 0 ∫ ∞ τ D -x λ 1 σ 1 e -σ1y dy dx = λ 1 σ 1 ( 1 -e -τDσ1
ν ( ∆ ′ τD- ) = ∫ τD τ R ∫ ∞ τ D -x λ 1 σ 1 e -σ1y dy dx = λ 1 σ 1 ( 1 -e -(τ D -τ R )σ1
) .

As M τD-is the sum of three independent Poisson random variables, it comes that

M τD-∼ P ( x 0 e -σ1τ D + λ 1 σ 1 ( 1 -e -τ D σ1 ) + λ 1 σ 1 ( 1 -e -(τ D -τ R )σ1
) )

∼ P

( x 0 e -σ1τD + λ 1 σ 1 ( 2 -e -τDσ1 -e -(τD-τR)σ1
) ) .

Between τ D -and τ D , with the binomial sampling, each mRNA has an equal chance to stay or to disappears, therefore

M τD = Mτ D - ∑ 1=0 B 1/2,i with ( B 1/2,i ) i.i.d
M τD ∼ P ([ x 0 e -σ1τ D + λ 1 σ 1 ( 2 -e -τ D σ1 -e -(τ D -τ R )σ1
) ] /2

) .

As the system is at equilibrium, it comes that M 0 D = M τD , therefore

x 0 = ( x 0 e -σ1τD + λ 1 σ 1 ( 2 -e -τDσ1 -e -(τD-τR)σ1
) ) /2

which gives:

x 0 = λ 1 σ 1 [ 1 - e -(τD-τR)σ1 2 -e -τDσ1
] .

Since the equilibrium distribution is unique, the number of mRNAs at birth follows a Poisson distribution of parameter x 0 at equilibrium. We have determined the equilibrium distribution of the embedded Markov Chain (M iτD ) i∈N . Now, let's look at the distribution of mRNA number at any time s of the cell cycle: heorem 3.3. At equilibrium, at a time s in the cell cycle, the mRNA number M s follows a Poisson distribution of parameter

x s = λ 1 σ 1 [ 1 - e -(s+τD-τR)σ1 2 -e -τDσ1 + s≥τR ( 1 -e -(s-τR)σ1
) ] .

In particular, the mean and the variance of mRNA number are known at any time s of the cell cycle:

E [M s ] = λ 1 σ 1 [ 1 - e -(s+τD-τR)σ1 2 -e -τDσ1 + s≥τR ( 1 -e -(s-τR)σ1
) ] ,

Var [M s ] = λ 1 σ 1 [ 1 - e -(s+τD-τR)σ1 2 -e -τDσ1 + s≥τR ( 1 -e -(s-τR)σ1 ) ] . C 3. M C C 76
Proof. At a moment s of the cell cycle, let's consider the moment-generating function of M s with ξ < 0:

E [exp (ξM s )] = E [ exp ( ξ ( M0 ∑ i=1 {E i σ 1 >s} + N (∆ s ) + s≥τR N ′ λ1 (∆ ′ s ) ))]
.

Since M 0 , E i σ1 , N λ1 and N ′ λ1 are all independent, it follows that

E [exp (ξM s )] = E [ exp ( M0 ∑ i=0 ξ {E i σ 1 >s} )] • E [exp (ξN (∆ s ))] • E [exp (ξ s≥τ R N ′ (∆ ′ s ))] .
For the irst factor, since all the random variables {E i σ 1 >s} are i.i.d. Bernoulli variables of parameter e -sσ1 and independent of M 0 , it comes then

E [ exp ( M0 ∑ i=0 ξ {E i σ 1 >s} )] = E [ E [ exp ( ξ {E 1 σ 1 >s} ) |M 0 ] M0 ] = E [ exp ( 1 + e -sσ1 (e ξ -1) ) M0 ] .
With Proposition 3.4, M 0 is known to be a Poisson random variable of parameter x 0 , hence, with the probabilitygenerating function of a Poisson random variable, it holds:

E [ exp ( M0 ∑ i=0 ξ {E i σ 1 >s} )] = E [ exp ( x 0 e -sσ1 ( e ξ -1 ))]
For the second factor, one can recall that N (∆ s ) is a Poisson random variable. As in Proposition 3.4, its parameter can be calculated:

ν (∆ s ) = ∫ s 0 ∫ ∞ τD-x λ 1 σ 1 e -σ1y dy dx = λ 1 σ 1 ( 1 -e -sσ1
) .

Identically for the third factor, N ′ (∆ ′ s ) is a Poisson random variable of parameter:

ν (∆ ′ s ) = ∫ s τR ∫ ∞ τD-x λ 1 σ 1 e -σ1y dy dx = λ 1 σ 1 ( 1 -e -(s-τ R )σ1
) .

As a consequence, the moment-generating function of M s is

E [exp (ξM s )] = E [( x 0 e -sσ1 + λ 1 σ 1 ( 1 -e -sσ1 ) + s>τR λ 1 σ 1 ( 1 -e -(s-τR)σ1 ) ) ( e ξ -1 ) ] = E [( x 0 e -sσ1 + λ 1 σ 1 ( 1 -e -sσ1 + s>τR ( 1 -e -(s-τ R )σ1 )) ) ( e ξ -1 ) ]
which is the moment-generating function of a Poisson random variable of parameter

x 0 e -sσ1 + λ 1 σ 1 ( 1 -e -sσ1 + s>τ R ( 1 -e -(s-τR)σ1
)) .

In this subsection, we have managed to obtain explicit expressions for the mean and the variance of mRNA number at any time s of the cell cycle at equilibrium (and in particular, its mean and its variance are known). In the next subsection, we are interested in obtaining the same kind of results for proteins.

M C

C G R 77

Dynamics of protein number

As for the previous analysis of the mRNA number, we search an expression for the protein production through the cell cycle. his case is more complicated than the mRNA case: the protein distribution is not as simple as a Poisson distribution, and we will only calculate analytical expressions only for the irst two moments of P s . heorem 3.4 is the main theoretical result of this section: for any time s of the cell cycle, it gives explicit expressions for the mean E [P s ] and the variance Var [P s ] of the protein number. his result is important as in the next section, it will be used to calculate directly the mean E [P /V ] and variance Var [P /V ] of the protein concentration averaged across the cell cycle without using simulations: only with the parameters of the model (λ 1 , σ 1 , λ 2 , τ R and τ D ), we will be able to know the behaviour of the protein concentration in terms of variance.

In order to prove the heorem 3.4, we will characterise E [P s ] and Var [P s ] in the two following cases:

1. In a irst step, we consider the case before replication (s < τ R ). We begin by considering that the state of the cell at birth (M 0 , P 0 ) is known and we calculate the irst two moments of P s for any time s < τ R (Corollary 3.1). hen, we integrate over all the possible initial states (M 0 , P 0 ) to determine expressions for E [P s ] and Var [P s ] for any time s < τ R (Proposition 3.6). hese expressions are dependant of the irst moments of (M 0 , P 0 ):

they depend on E [M 0 ], E [P 0 ], Var [M 0 ], Var [P 0 ] and Cov [M 0 , P 0 ].
2. In a second step, we consider the case ater replication (s ≥ τ R ). Similarly the irst case, we will consider that the state of the cell at replication (M τR , P τR ) is known and we calculate the irst two moments of P s for any time τ R ≤ s < τ D (Proposition 3.8). Ater integration, expressions for E [P s ] and Var [P s ] for any time s ater replication are determined, these expressions depend on

E [M τR ], E [P τR ], Var [M τR ],
Var [P τR ] and Cov [M τR , P τR ] (Proposition 3.8). In the end, in heorem 3.4, are presented the mean and variance of protein number at any time s of the cell cycle, only depending on the irst moments of (M 0 , P 0 ) and (M τR , P τR ). Additional results are presented in the appendix Section 3.B, which determine explicitly the irst moments of (M 0 , P 0 ) and (M τR , P τR ) so that the mean and variance of protein number will be fully characterised.

Description of the Protein Number Process

Before beginning, let's at irst have a description for the number of proteins P s for any time s. We will use this description in the upcoming proofs. Similarly to mRNA case (Equation (3.11)) , we group them into two categories:

• he proteins that were there at birth and which remain in the cell during all the cell cycle (as said in Subsection 3.4.1 the proteolysis is not considered in this model).

• he proteins that were created during the cell cycle. he rate of production depends on the current number of mRNAs. For that we consider

( N i λ2
) i∈N , a sequence of i.i.d. Poisson Point Processes of intensity λ 2 ; if the i-th mRNA exists at time s (that is to say if i ≤ M s ), then the number of proteins produced by this mRNA between s and s+ds is N i λ2 (ds). Hence, the total number of proteins produced between s and s + ds is then ∑ ∞ i=1 i≤Mu N i λ2 (ds). To sum up, the number of proteins at a time s of the cell cycle is:

P s = P 0 + ∞ ∑ i=1 ∫ s 0 i≤Mu N i λ2 (du) .
(3.12) he irst term is the number of proteins at birth, and the second takes into account all proteins created between times 0 and s. Let's beginning with the case before replication, where s < τ R . he random variables M 0 and P 0 are supposed to be known; so that we use the notation

E M0,P0 [ ] := E [ | (M 0 , P 0 )].
At irst, we want to characterise the irst two moments of P s for any time s of the cell cycle conditionally to (M 0 , P 0 ). To do so, as for the mRNAs, we determine at irst the moment-generating function of P s .

Proposition 3.5. For any s ∈ [0, τ R [, by supposing the birth state (M 0 , P 0 ) known, it comes that the moment generating function of P s is:

E M0,P0 [exp (ξP s )] = exp (ξP 0 ) • h 1,s ( λ 2 ( e ξ -1
))

for any ξ < 0 and such as h 1,s is the moment generating function of ∫ s 0 M u du. he expression of h 1,s is:

h 1,s (ξ) := exp [ M 0 log [ σ 1 -ξe -(σ1-ξ)s σ 1 -ξ ] + λ 1 ξ σ 1 -ξ ( s - 1 -e -(σ1-ξ)s σ 1 -ξ )] .
Proof. With Equation (3.12), it is is easy to show that

E M0,P0 [exp (ξP s )] = exp (ξP 0 ) • E M0,P0 [ ∞ ∏ i=1 E M0,P0
[ exp

( ξ ∫ s 0 i≤Mu N i λ2 (du) ) | (M u ) u≤s ] ] .
We then consider the Laplace functional for the Poisson process N i λ2 :

E M0,P0 [ exp ( ξ ∫ s 0 i≤Mu N i λ2 (du) ) | (M u ) u≤s ] = exp [ λ 2 ∫ ∞ 0 (exp (ξ i≤Mu u≤s ) -1) du ] = exp [ λ 2 ( e ξ -1 ) ∫ s 0 i≤Mu du ] .
By making the product for i from 1 to ininity, it comes that:

∞ ∏ i=1 E M0,P0
[ exp

( ξ ∫ s 0 i≤Mu N i λ2 (du) ) | (M u ) u≤s ] = exp [ λ 2 ( e ξ -1 ) ∫ s 0 M u du ] .
As a consequence, it indeed follows that

E M0,P0 [exp (ξP s )] = exp (ξP 0 ) • h 1,s ( λ 2 ( e ξ -1 )) .
What remains is to show the expression of h 1,s (ξ). he expression of M s in Equation (3.11) integrated between time 0 and time s < τ R gives

∫ s 0 M s du = ∫ s 0 M0 ∑ i=1 {E i σ 1 >u} du + ∫ s 0 N (∆ u ) du.
As h 1,s is the moment-generating function of ∫ s 0 M s du, it follows that for any ξ < 0:

h 1,s (ξ) = E M0,P0 [ exp ( ξ ∫ s 0 M0 ∑ i=1 {E i σ 1 >u} du )] E [ exp ( ξ ∫ s 0 N (∆ u ) du )] = E [ exp ( ξE 1 σ1 ∧ s )] M0 E [ exp ( ξ ∫ s 0 N (∆ u ) du )] . 3.4. M C C G R 79
We recall that N a MPPP of intensity ν = λ 1 dx⊗σ 1 e -σ1y dy and that ∆ s is deined as ∆ s = {(x, y), 0 < x < s, y > s -x}.

Let's begin with the irst term of h 1,s (ξ):

E [ exp ( ξE 1 σ1 ∧ s )] M0 = (∫ s 0 e ξu σ 1 e -σ1u du + ∫ ∞ s e ξs σ 1 e -σ1u du ) M0 = ( σ 1 -ξe -(σ1-ξ)s σ 1 -ξ )M 0 .
Let's continue with the second term of h 1,s (ξ). he integration of N (∆ u ) on [0, s[ gives:

∫ s 0 N (∆ u ) du = ∫ s 0 ∫ ∫ R 2 x<u • y<u-x N (dx, dy) du = ∫ ∫ R 2 ∫ s 0 x<u<x+y du N (dx, dy) = ∫ ∫ R 2 ((x + y) ∧ s -x ∧ s) N (dx, dy) .
We then consider the Laplace functional for this MPPP:

E [exp (N (g))] = exp [ λ 1 ∫ ∞ 0 ∫ ∞ 0 ( e g(x,y) -1 ) dx σ 1 (dy) ]
with σ 1 (dy) the density distribution of an exponential random variable of parameter σ 1 . In our case g(x, y) :=

ξ ((x + y) ∧ s -x ∧ s): E [ exp ( ξ ∫ s 0 N (∆ u ) du )] = exp [ λ 1 ∫ ∞ 0 ∫ ∞ 0 (exp [ξ ((x + y) ∧ s -x ∧ s)] -1) dx σ 1 (dy) ] = exp [ λ 1 (∫ s 0 ∫ ∞ 0 (exp [ξ (y ∧ (s -x))]) σ 1 (dy) dx -s )] = exp [ λ 1 (∫ s 0 ∫ s-x 0 exp [ξy] σ 1 (dy) dx + ∫ s 0 ∫ ∞ s-x exp [ξ (s -x)] σ 1 (dy) dx -s )] = exp [ λ 1 ξ σ 1 -ξ ( s - 1 -e -(σ1-ξ)s σ 1 -ξ )] .
As a consequence the moment-generating function of ∫ s 0 M s du is given by:

h 1,s (ξ) = exp [ M 0 log [ σ 1 -ξe -(σ1-ξ)s σ 1 -ξ ] + λ 1 ξ σ 1 -ξ ( s - 1 -e -(σ1-ξ)s σ 1 -ξ )]
As the moment generating function of P s has been characterised, it is possible to deduce, by derivation, the irst two moments of P s knowing (M 0 , P 0 ) for any time s before the gene replication. C

M C C 80

Corollary 3.1. At equilibrium, for a time s ∈ [0, τ R [, knowing the state of the cell at birth (M 0 , P 0 ), the irst two moments of P s are:

E M0,P0 [P s ] = P 0 + λ 2 ( λ 1 σ 1 s + ( M 0 - λ 1 σ 1 ) 1 -e -σ1s
σ 1

) ,

E M0,P0 [ P 2 s ] = (E M0,P0 [P s ]) 2 + M 0 λ 2 σ 1 ( 1 -e -σ1s + λ 2 σ 1 [ 1 -e -σ1s ( e -σ1s + 2sσ 1 )] ) + λ 1 λ 2 σ 2 1 [ sσ 1 -1 + e -σ1s + 2 λ 2 σ 1 ( σ 1 s ( 1 + e -σ1s ) -2 ( 1 -e -σ1s
))

]

Proof. From Proposition 3.5, it follows that the irst two moments of P s can be obtained by derivation of the moment-generating function:

E M0,P0 [P s ] = lim ξ→0 d dξ [ exp (ξP 0 ) h 1,s ( λ 2 ( e ξ -1 ))] = lim ξ→0 [ exp (ξP 0 ) • ( P 0 h 1,s ( λ 2 ( e ξ -1 )) + λ 2 e ξ h ′ 1,s ( λ 2 ( e ξ -1 )
))]

= P 0 + λ 2 h ′ 1,s (0)
and

E M0,P0 [ P 2 s ] = lim ξ→0 d 2 dξ 2 [ exp (ξP 0 ) h 1,s ( λ 2 ( e ξ -1 ))] = lim ξ→0 [ exp (ξP 0 ) × ( P 2 0 h 1,s ( λ 2 ( e ξ -1 )) + 2P 0 λ 2 e ξ h ′ 1,s ( λ 2 ( e ξ -1 )) +λ 2 e ξ h ′ 1,s ( λ 2 ( e ξ -1 )) + ( λ 2 e ξ ) 2 h ′ 1,s ( λ 2 ( e ξ -1 )
) )]

= P 2 0 + (2P 0 + 1) λ 2 h ′ 1,s (0) + (λ 2 ) 2 h ′′ 1,s (0) = (E M0,P0 [P s ]) 2 + λ 2 h ′ 1,s (0) + (λ 2 ) 2 ( h ′′ 1,s (0) -h ′ 1,s (0) 2 )
Consider the expression of h 1,s of Proposition 3.5, and let's search its derivatives:

h 1,s (ξ) = exp ( M 0 log [ σ 1 -ξe -(σ1-ξ)s σ 1 -ξ ] + λ 1 ξ σ 1 -ξ ( s - 1 -e -(σ1-ξ)s σ 1 -ξ )) , h ′ 1,s (ξ) = h 1,s (ξ) { M 0 [ 1 (σ 1 -ξ) - (1 + ξs) e -(σ1-ξ)s ( σ 1 -ξe -(σ1-ξ)s ) ] +λ 1 ( σ 1 (σ 1 -ξ) 2 s - 1 -e -(σ1-ξ)s (σ 1 -ξ) 2 + ξ • se -(σ1-ξ)s (σ 1 -ξ) 2 -2ξ • 1 -e -(σ1-ξ)s (σ 1 -ξ) 3 )} , h ′′ 1,s (ξ) = h 1,s (ξ) { h ′ 1,s (ξ) 2 + M 0 [ 1 (σ 1 -ξ) - (1 + ξs) e -(σ1-ξ)s ( σ 1 -ξe -(σ1-ξ)s ) ] +λ 1 ( σ 1 (σ 1 -ξ) 2 s - 1 -e -(σ1-ξ)s (σ 1 -ξ) 2 + ξ • se -(σ1-ξ)s (σ 1 -ξ) 2 -2ξ • 1 -e -(σ1-ξ)s (σ 1 -ξ) 3 )} . 3.4. M C C G R 81
By taking the limit for ξ → 0, it follows: ] . In the next proposition, we integrate these expressions over all birth states (M 0 , P 0 ) to ind formulas for E [P s ] and Var [P s ] for any time s < τ R before replication. hese expression depends on joint moments of M 0 and P 0 .

h ′ 1,s (0) = M 0 1 -e -σ1s σ 1 + λ 1 σ 1 ( s - 1 -e -σ1s σ 1 ) , h ′′ 1,s (0) = h ′ 1,s (0) 2 + M 0 σ 2 1 [ 1 -e -σ1s ( e -σ1s + 2sσ 1 )] + 2 λ 1 σ 3 1 ( σ 1 s ( 1 + e -σ1s ) -2 ( 1 -e -σ1s
Proposition 3.6. Let's consider the functions:

f 1 (s) := ( λ 1 σ 1 s + ( x 0 - λ 1 σ 1 ) 1 -e -σ1s σ 1 ) , g 1 (s) := ( λ 2 1 -e -σ1s σ 1 ) 2 x 0 +x 0 λ 2 σ 1 ( 1 -e -σ1s + λ 2 σ 1 [ 1 -e -σ1s ( e -σ1s + 2sσ 1 )] ) + λ 1 λ 2 σ 2 1 [ sσ 1 -1 + e -σ1s + 2 λ 2 σ 1 ( σ 1 s ( 1 + e -σ1s ) -2 ( 1 -e -σ1s
))

] with x 0 deined in Proposition 3.4. At any time s ∈ [0, τ R [ before replication, depending on joint moments of P 0 and M 0 , the mean and the variance of P s are given by:

E [P s ] = E [P 0 ] + λ 2 f 1 (s), Var [P s ] = Var [P 0 ] + 2λ 2 1 -e -σ1s σ 1 Cov [P 0 , M 0 ] + g 1 (s).
Proof. By considering the mean of the random variable E [P s |(M 0 , P 0 )] in Corollary 3.1, it directly comes the result for E [P s ]. For the variance, let's consider the expression of

E [ P 2 s |(M 0 , P 0 ) ] E [ P 2 s ] = E [ (E M0,P0 [P s ]) 2 ] + E [M 0 ] λ 2 σ 1 ( 1 -e -σ1s + λ 2 σ 1 [ 1 -e -σ1s ( e -σ1s + 2sσ 1 )] ) + λ 1 λ 2 σ 2 1 [ sσ 1 -1 + e -σ1s + 2 λ 2 σ 1 ( σ 1 s ( 1 + e -σ1s ) -2 ( 1 -e -σ1s
))

]

Var [P s ] = E [ E M0,P0 [P s ] 2 ] -E [P s ] 2 + E [M 0 ] λ 2 σ 1 ( 1 -e -σ1s + λ 2 σ 1 [ 1 -e -σ1s ( e -σ1s + 2sσ 1 )] ) + λ 1 λ 2 σ 2 1 [ sσ 1 -1 + e -σ1s + 2 λ 2 σ 1 ( σ 1 s ( 1 + e -σ1s ) -2 ( 1 -e -σ1s )) ] . C 3. M C C 82 Now let's consider the expression of E [ E M0,P0 [P s ] 2 ] -E [P s ] 2 : E [ E M0,P0 [P s ] 2 ] -E [P s ] 2 = E [ P 2 0 ] + E [ ( λ 2 ( λ 1 σ 1 s + ( M 0 - λ 1 σ 1 ) 1 -e -σ1s σ 1 )) 2 ] +2E [ P 0 × λ 2 ( λ 1 σ 1 s + ( M 0 - λ 1 σ 1 ) 1 -e -σ1s σ 1 )] -E [P 0 ] 2 + E [ λ 2 ( λ 1 σ 1 s + ( M 0 - λ 1 σ 1 ) 1 -e -σ1s σ 1 )] 2 -2E [P 0 ] E [ λ 2 ( λ 1 σ 1 s + ( M 0 - λ 1 σ 1 ) 1 -e -σ1s σ 1 )] = Var [P 0 ] + Var [ λ 2 ( λ 1 σ 1 s + ( M 0 - λ 1 σ 1 ) 1 -e -σ1s σ 1 )] +2Cov [ P 0 , λ 2 ( λ 1 σ 1 s + ( M 0 - λ 1 σ 1 ) 1 -e -σ1s σ 1 )] .
Finally, one has just to remark that due to Proposition 3.4

E [M 0 ] = Var [M 0 ] = x 0 .

Protein Number Ater Replication

Let's continue to the case ater replication, for a time s such as τ R ≤ s < τ D . We adopt a similar approach as for the previous case: let's consider that the state just ater replication (M τR , P τR ) is known, and we want to determine the irst two moments of P s for any time s ater the replication.

Proposition 3.7. At equilibrium, for a time s ∈ [τ R , τ D [, knowing the state of the cell at replication (M τR , P τR ), the irst two moments of P s are:

E Mτ R ,Pτ R [P s ] = P τR + λ 2 ( 2 λ 1 σ 1 (s -τ R ) + ( M τR -2 λ 1 σ 1 ) 1 -e -σ1(s-τR) σ 1
) ,

E Mτ R ,Pτ R [ P 2 s ] = ( E Mτ R ,Pτ R [P s ] ) 2 +M τR λ 2 σ 1 ( 1 -e -σ1(s-τ R ) + λ 2 σ 1 [ 1 -e -σ1(s-τ R ) ( e -σ1(s-τ R ) + 2 (s -τ R ) σ 1 )] ) +2 λ 1 λ 2 σ 2 1 [ (s -τ R ) σ 1 -1 + e -σ1(s-τR) +2 λ 2 σ 1 ( σ 1 (s -τ R ) • ( 1 + e -σ1(s-τR) ) - 2
( 1e -σ1(s-τR)

)) ] .

Proof. Ater the replication, the rate of mRNA production is doubled, but otherwise, the dynamic is identical as it was before the replication. One can hence easily adapt the proofs of Proposition 3.5 and Corollary 3.1, by replacing the initial state by the state at replication (M τR , P τR ), by considering that the mRNA production rate is 2λ 1 , and that the time spent since the initial state is sτ R .

We can then integrate the previous expressions on all possible states at replication (M τ R , P τ R ). It follows that:

3.4. M C C G R 83
Proposition 3.8. Let's consider the functions:

f 2 (s) := ( 2 λ 1 σ 1 (s -τ R ) + ( x τR -2 λ 1 σ 1 ) 1 -e -σ1(s-τR) σ 1 ) , g 2 (s) := ( λ 2 1 -e -σ1(s-τR) σ 1 )2 x τR +x τR λ 2 σ 1 ( 1 -e -σ1(s-τR) + λ 2 σ 1 [ 1 -e -σ1(s-τR) ( e -σ1(s-τR) + 2 (s -τ R ) σ 1 )] ) +2 λ 1 λ 2 σ 2 1 [ (s -τ R ) σ 1 -1 + e -σ1(s-τR) +2 λ 2 σ 1 ( σ 1 (s -τ R ) ( 1 + e -σ1(s-τ R ) ) -2 ( 1 -e -σ1(s-τ R )
)) ] .

with x τR as deined in heorem 3.3. At any time s ∈ [τ R , τ D [ ater replication, depending on joint moments of P τR and M τR , the mean and the variance of P s are given by:

E [P s ] = E [P τ R ] + λ 2 f 2 (s), Var [P s ] = Var [P τR ] + 2λ 2 1 -e -σ1(s-τR) σ 1 Cov [P τR , M τR ] + g 2 (s).
Proof. It is similar to the proof of Proposition 3.6.

Protein Number in the Whole Cell Cycle

Now we can gather the two previous cases, we are able to propose an expression for E [P s ] and Var [P s ] for any time s of the cell cycle.

heorem 3.4. At any time s of the cell cycle, the mean and the variance of the protein number P s are

E [P s ] = E [P 0 ] + λ 2 (f 1 (τ R ∧ s) + s≥τR f 2 (s)) Var [P s ] = Var [P 0 ] + 2λ 2 1 -e -σ1s∧τR σ 1 Cov [P 0 , M 0 ] + g 1 (s ∧ τ R ) + s≥τR ( 2λ 2 1 -e -σ1(s-τR) σ 1 Cov [P τR , M τR ] + g 2 (s)
)

with f 1 and g 1 deined in Proposition 3.6 and f 2 and g 2 deined in Proposition 3.8.

Proof. For s < τ R , the expressions correspond to those of Proposition 3.6. For τ R ≤ s < τ D , one can remark that a direct consequence of this proposition gives expressions for = P τR , one can consider the limit for s → τ R in the case of s ∈ [0, τ R [ in Proposition 3.6:

E [P τR ] and
E [P τR ] = E [P 0 ] + λ 2 f 1 (τ R ) , (3.13
)

Var [P τR ] = Var [P 0 ] + 2λ 2 1 -e -σ1τR σ 1 Cov [P 0 , M 0 ] + g 1 (τ R ) . (3.14)
Consequently, it is possible to write expressions of E [P s ] and Var [P s ] as depending only on E [P 0 ], Explicit expressions for these quantities are still unknown. But as we are at equilibrium, it comes that (P 0 , M 0 ) D = (P τD , M τD ) which allows to ind expressions for all these quantities that explicitly only depend on the model parameters: λ 1 , σ 1 , λ 2 , τ R and τ D . hese expressions are given in the appendix Section 3.B: the expressions of E [P 0 ], Var [P 0 ] are given in Proposition 3.9, as the expressions of Cov [P 0 , M 0 ] and Cov [P τR , M τR ] are determined in Proposition 3.10.

Var [P 0 ], Cov [P 0 , M 0 ] and Cov [P τR , M τR ].

Parameter Computation

In this subsection, we explain how to ix the parameters λ 1 , σ 1 , λ 2 and τ R to make them correspond to the exponential measures. As in Subsection 3.3.5, we characterise the mean and the variance of either mRNA or protein concentration, not at a given time s, but over the whole cell cycle. To do so, we deine the global average and global variance of mRNAs as

E [M /V ] := 1 τ D ∫ τD 0 E [ M s V (s) ] ds, (3.15
)

Var [M /V ] := 1 τ D ∫ τD 0 [ E [ ( M s V (s) ) 2 ] -E [M /V ] 2 ] ds; (3.16)
and consider analogue deinitions for the proteins:

E [P /V ] and Var [P /V ].
As in the previous models, we set the doubling time τ D to 150 min and the volume at birth V 0 = 1.3 om 3 . For each gene, we have to determine four diferent parameters λ 1 , σ 1 , λ 2 and τ R . We have considered the genes of [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] for which the empirical mean of messengers µ m and proteins µ p concentrations, as well as the mRNA half-life time τ m have been measured. We still deduce the mRNA degradation rate σ 1 with the mRNA half-life time τ m (such that σ 1,i = log 2/τ M,i ). To determine the time τ R of replication of each gene, we have looked at the gene position. We irst estimate the time at which the DNA begins its replication in the cell cycle; as the speed of replication is relatively constant, we determine the time τ R only with the distance from the gene to the origin of DNA replication (for more details, see the appendix Section 4.A).

We still have to determine the transcription rate λ 1 and the translation rate λ 2 . One can interpret the empirical average mRNA and protein concentration of the experiment (respectively µ m and µ p ) as the global average of mRNA and protein concentrations of the model (respectively E [M /V ] and E [P /V ]). he global averages are known through the integration, over the cell cycle, of the mean formulas of heorem 3.3 and heorem 3.4:

E [M /V ] = λ 1 σ 1 1 τ D ∫ τD 0 1 V 0 2 s/τD ( 1 - e -(s+τ D -τ R )σ1 2 -e -τ D σ1 + s≥τR ( 1 -e -(s-τR)σ1
) )

ds,

E [P /V ] = λ 2 1 τ D ∫ τD 0 1 V 0 2 s/τD (f 1 (τ R ) + f 2 (τ D ) + f 1 (τ R ∧ s) + s≥τR f 2 (s)) ds. 3.4. M C C G R 85
As a consequence, parameters λ 1 and λ 2 can be expressed as follows

λ 1 = σ 1 τ D µ m (∫ τD 0 1 V 0 2 s/τD ( 1 - e -(s+τD-τR)σ1 2 -e -τDσ1 + s≥τR ( 1 -e -(s-τR)σ1
) ) ds

)-1 , λ 2 = τ D µ p (∫ τD 0 1 V 0 2 s/τD (f 1 (τ R ) + f 2 (τ D ) + f 1 (τ R ∧ s) + s≥τR f 2 (s)) ds ) -1 .
For each gene, all the parameters can be hence determined.

Biological Interpretation of the Results

In this subsection we determine the variability added by the gene replication and we compare the results with the variability measured in [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF]. Unlike the previous model of Section 3.3, we can directly compute the protein variance thanks to the explicit expressions of heorem 3.3 and heorem 3.4. he proile during the cell cycle and the global noise of each protein concentration are then analytically computed.

Environmental State Decomposition in Proile of Protein Concentration

he analytical expressions of the mean and the variance allow to study the evolution of every protein production during the whole cell cycle. In Figure 3.3a, we take the example of the protein Adk: we show its average concentration (thick line) and its standard deviation (blue area) during the cell cycle. It appears that the mean concentration at a given time s of the cell cycle E [P s /V (s)] is not constant during the cell cycle, as it was the case for the irst two models. he curve of E [P s /V (s)] luctuates around 2% of the global average protein production E [P /V ]. Experimental measures of average protein expression during the cell cycle show similar results: the article [START_REF] Walker | Generation and iltering of gene expression noise by the bacterial cell cycle[END_REF] for instance measures the expression of genes at diferent positions on the chromosome and shows a similar proile during the cell cycle and depicts a luctuation also around 2% of the global average (see igure 1.d and igure S6.b of the article). his proile shows that there is, in this model, two origins for the global variability Var [P /V ]: one which is induced by the production mechanism itself, and the other that is due to cell cycle efect. he irst one is represented in the igure by the standard deviation at any time of the cell cycle √ Var [P s /V (s)], the other is represented by the distance of E [P s /V (s)] at any time s of the cell cycle around the global mean E [P /V ]. We can have a quantitative description of these two contributions to the variability through the notion of environmental state decomposition that is used in the literature (see [START_REF] Hilinger | Separating intrinsic from extrinsic luctuations in dynamic biological systems[END_REF]). For any type of protein, one can decompose the global variance Var [P /V ] (as it is deined in Equation (3.16)) as:

Var [P /V ] = Var 1 [P /V ] + Var 2 [P /V ]
with Var 1 [P /V ] and Var 2 [P /V ] representing the quantities

Var 1 [P /V ] := 1 τ D ∫ τD 0 Var [ P s V (s)
] ds, (3.17)

Var 2 [P /V ] := 1 τ D ∫ τD 0 ( E [ P s V (s) ]) 2 ds -E [P /V ] 2 . (3.18)
he two terms represent the two natures of the global variance:

Var 1 [P /V ] represents the deviation of P s /V (s) around its "local mean" E [P s /V (s)] for any time s of the cell cycle. his variability is the direct result of the stochastic events that occurs in the protein production mechanism: stochastic changes in the mRNA number as well as events of translation.

C

M C C 86

Var 2 [P /V ] represents the impact of the cell cycle on the global variability. It takes into account the distance between E [P s /V (s)] and the global average production E [P /V ]. his distance is deterministic and is due to periodic external events of the cell cycle (in our case, the gene replication) that change "local mean" E [P s /V (s)]. his term is hence interpreted as the contribution of the external inluence of the cell cycle to the global variability.

Remark 3.5. Such decomposition was not considered for the previous models. he reason is that if such decomposition is used back then, the term Var 2 [P /V ] would be null (because in these models, the means E [P /V (s)] remain constant across the cell cycle). We will discuss again the use of the environmental state decomposition in these case in subsubsection 3.4.5.3.

In the case of Figure 3.3a, the term Var 1 [P /V ] of the environmental state decomposition is higher than

Var 2 [P /V ], meaning that most of the noise is explained by the protein production itself. It is the case for all genes of the set, the ratio

Var 2 [P /V ]
Var [P /V ] is very small (99% of the cell have such ratio below 2%). It conirms the previous results: the gene cycle has almost no efect on the variability, its contribution to the global variability is negligible compared to the efect of the stochastic nature of the protein synthesis mechanism and the binomial sampling at division.

Proteins with Higher Cell Cycle Efect

he previous result shows that for the genes considered, there is no signiicant contribution of the cell cycle to the protein variability with the parameters obtained through [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF]. Yet, some proteins have been proposed to have cycle-dependent concentrations and could trigger periodic events such as DNA replication initiation, or division. We investigate what range of parameters of our model would give such proteins. We show below that with our model of gene replication and division, such protein can hardly be obtained with realistic biological parameters.

In order to have such cycle-dependent proteins, one needs at least to have a reliable periodic signal: the concentration should follow a predictable path across the cell cycle, with minimal luctuations around this path. In our case, it means that protein concentration across cell cycle P s /V (s) should be close to its mean protein production E [P s /V (s)]. To have so, the term Var 1 [P /V ] of the environmental state decomposition should as low as possible. As we have analytical solutions for protein concentration mean and variance, we can investigate which range of parameters indeed decrease Var 1 [P /V ].

Based on the protein Adk, while keeping the global average concentration E [P /V ] constant, we have analysed the following efects on the ratio Var 2 [P /V ] / Var [P /V ] (see Figure 3.3c):

• Gene position: we have changed the time of gene replication, by changing the gene position from close to the origin of replication up to the termination. We have adapted the gene activity λ 1 in order to keep the same average protein production E [P /V ]. Changes on this parameter make no changes in the ratio

Var 1 [P /V ] / Var [P /V ]: the variability is still largely due to the protein production mechanism and not the cell cycle.

• mRNA number: we have increased the mRNA number by increasing the gene activity λ 1 , while decreasing the mRNA activity λ 2 in order to keep the average E [P /V ] constant. It appears that a high mRNA number indeed decreases the ratio: 50 times more mRNAs can gain give a a proile where 20% of the variance is due to the term Var 2 [P /V ]. In this version, there is a higher number of mRNAs (approximately ten times more) that last less time. he efect is a higher term Var 2 [P i /V ] in the environmental state decomposition (main igure), but the it is not enough to clearly separate between the distributions at birth (at time s = 0) and at the replication of the gene (at time s = τ R ) (inset). (c): Show the ratio

3.4. M C C G R 87 0 ¿ D =4 ¿ D =2 3¿ D =4 ¿ D Time in cell cycle
Var 2 [P /V ] / Var [P /V ] while varying successively the gene position, the mRNA number and the mRNA lifetime while keeping

E [P /V ] constant. C 3. M C C 88
• mRNAs lifetime: we have increased the mRNA degradation rate σ 1 while increasing mRNA activity λ 2 in order to keep the average E [P /V ] constant. he ratio Var 2 [P /V ] / Var [P /V ] change of few percents but the efect has much less impact on the outcome compared to the mRNA number.

It appears that only a higher mRNA number, and to a lesser extent, a lower mRNA lifetime can increase the ratio Var 2 [P /V ] / Var [P /V ]. he protein production of such protein is shown in Figure 3.3b: this protein is based on Adk but with approximately ten times more mRNAs that last ten times shorter (we also diminished the mRNA activity rate λ 2 in order to keep the same average protein production E [P /V ]). It represents around one transcription every 4 seconds (which is among the speediest transcription rates). Even if the proile is more gathered around the mean concentration E [P s /V (s)] curve, it is still not providing a reliable enough signal of protein concentration. Indeed, as we consider the protein concentration at time 0 and at time τ R (times where the distribution are the most distant from each other), the two distributions are still greatly overlapping (see inset of Figure 3.3b).

his part shows that, with biologically relevant parameters, it is not diicult to have a cycle-dependent protein with reliable enough signal to be able to trigger periodic events. As our model only represents gene replication and division, it is possible that other mechanisms, such as complex formation, feedback or proteolysis might give a more precise signal. In all cases, these observations support the previous results: gene replication seems to play a limited role in the protein variability.

Environmental State Decomposition and Intrinsic/Extrinsic Decomposition

he introduction of the environmental state decomposition in this section, largely used in literature, brings us to the following comment as it oten used as a way to distinguish the extrinsic to the intrinsic noise. We show here that this decomposition does not separate exactly what is usually considered as extrinsic noise from the intrinsic noise.

he environmental state decomposition is used in literature [START_REF] Swain | Intrinsic and extrinsic contributions to stochasticity in gene expression[END_REF], Elowitz et al., 2002[START_REF] Hilinger | Separating intrinsic from extrinsic luctuations in dynamic biological systems[END_REF] as a way to decompose the two natures of the protein variability: the intrinsic noise due to the stochastic nature of birth and death of mRNAs and proteins and the extrinsic due to randomising external efect from the biological environment. In our case, the environmental state decomposition states that the average global variance of a protein Var [P /V ] is the sum of the two terms Var 1 [P /V ] and Var 2 [P /V ] respectively deined by Equation (3.17) and Equation (3.18). With the usual interpretation of literature, Var 1 [P /V ] would be interpreted as the "intrinsic variance" and Var 2 [P /V ] as the "extrinsic variance".

But it is noticeable that the second term only captures a part of what is generally accepted as the extrinsic noise. he binomial sampling (studied in Subsection 3.3.6) for instance, is not directly due to the protein production mechanism and can hence naturally be considered as having an external efect on the protein noise. And yet, the additional variance of this mechanism is not added in the second term Var 2 [P /V ]: indeed the binomial division has no efect on E [P s /V (s)], it only afects the variance Var [P s /V (s)]. As a consequence, by deinition of Var 1 [P /V ] and Var 2 [P /V ], the binomial division only increases the irst term in the environmental state decomposition.

In the model of the present section, the gene replication is only external efect to be separate in the environmental state decomposition as it is the only mechanism that makes the mean E [P s /V (s)] change across the cell cycle.

Efect of the Population Distribution

As noticed, in heorem 3.2, the deinitions of E [P /V ] and Var [P /V ] implicitly represent the mean and the variance of protein concentration in a population of cells whose ages are uniformly distributed. In real

3.4. M C C G R 89
experimental populations of cells, like in [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF], the number of cells in the population is exponentially growing: every dividing cell gives birth to two daughter cells. he distribution of ages is therefore not uniform.

To take into account this efect, one needs to correct the deinitions of E [P /V ] and Var [P /V ] by weighting them according to a typical age distribution of exponentially growing populations. Let's consider ν the distribution in age of such population (i.e., the probability that the age of the cell is between s and s + ds is given by ν(ds)), then we can propose new deinitions for the global mean and variance:

E [P /V ] := ∫ τD 0 E [ P s V (s)
] ν(ds),

Var [P /V ] := ∫ τD 0 [ E [ ( P s V (s) ) 2 ] -E [P /V ] 2 ] ν(ds).
he distributions of age and length in exponentially growing populations have been studied in the literature (see [START_REF] Collins | Rate of Growth of Bacillus cereus Between Divisions[END_REF], [START_REF] Sharpe | Bacillus subtilis cell cycle as studied by luorescence microscopy: constancy of cell length at initiation of DNA replication and evidence for active nucleoid partitioning[END_REF], [START_REF] Robert | Division in Escherichia coli is triggered by a size-sensing rather than a timing mechanism[END_REF] for instance), and we have deduced the typical age distribution ν from it.

By remaking the analyses using these deinitions, we compute, for every gene, the variance in the case an uniform population divided by the one obtained in an exponentially growing population. his ratio is centred around 1 with a standard deviation of 8 • 10 -3 . he distribution considered has therefore not signiicant impact on the variance of the model. his is due to the fact that the mean concentration of every gene remains approximately constant during the cell cycle, there is therefore no diference of protein concentration dosage at the beginning or at the end of the cell cycle.

As this efect seems negligible, for further models, we will still consider the mean and the variance as uniformly averaged over the cell cycle when we will have to estimate the parameters.

Comparison with experimental results

he proile during the cell cycle and the global noise of each protein concentration are analytically computed and can be compared with the previous models and [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] measures. As previously said, the gene replication has litle contribution to the protein noise; therefore it is not surprising that litle changes appears in Figure 3.4a compared with the previous models. Figure 3.4a shows the proile of the three representative genes : yjiE, adk, fabH (the non-normalised proile of adk was shown in Figure 3.3a). hese three genes are respectively close, distant and opposite to the origin of replication (besides having very diferent average productions). he mean concentration seems still having less relative variability for the highest produced protein. Even if we have shown that the mean concentration of each protein (thick lines) changes across the cell cycle, it is only in barely perceptible proportions. Figure 3.4b shows, for each protein, the global coeicient of variation of protein concentration (deined as

Var [P /V ] / E [P /V ] 2
) as a function of the average concentration. he global tendency is still approximately inversely scaling the average protein concentration, and there is still no lower bound limit as it is the case in [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] experiment. It is conirmed with the inset of Figure 3.4b where the ratio between the variances for the protein concentration of this model is compared with the model with binomial sampling of Figure 3.5a. his ratio is always around 1, indicating that the replication does not contribute signiicantly to the global variance.

3. M C C 90 -1 0 1 2 3 Prot. Adk -1 0 1 2 3
Normalised protein conc.

Prot. FabH 0 he proile shows no signiicant diferences with the previous models (see Figure 3.4a for the more details on Adk). Figure 3.4b: coeicient of variation (CV) of the protein concentration as a function of the average protein concentration. here are, once again, no signiicant diferences as before (see Figure 3.4b). In particular, there is no lower "extrinsic" noise plateau for highly expressed proteins, as it is the case for Taniguchi et al. ( 2010) (yellow area). Inset: ratio between the CV of protein concentration of the model with binomial sampling (Subsection 3.3.6) divided by the one of this section. Gene replication adds no particular noise compared with the model with only binomial sampling.

¿ D =4 ¿ D =2 3¿ D =4 ¿ D -1 0 1 2 3 Prot. YjiE

Conclusions on Models with Cell Cycle

In this chapter, we have produced a series of models that have taken into account the cell cycle, with a growing volume and division. Unlike classical models, this feature permits a quantitative comparison between theoretical models and experiments in terms of variability. It clearly appears in this chapter analysis that the main contributor to the noise of protein expression so far is the protein production mechanism itself, what literature refers as the "intrinsic noise". he only signiicant external efect for the noise of some protein expressions the binomial segregation at division. he efect of gene replication was studied in the last section. he analytical expressions for the mean and the variance for this last model allow direct comparisons with experimental measures without simulation. It clearly appears that the variability added by the gene replication represents only a very litle proportion of the global variability. Proteins with a signiicant proportion of the variability due to gene replication are diicult to obtain in a biological context. We have also determined in the environmental state decomposition the binomial division efect on the variance is not separate from the term usually atributed to intrinsic variability. Some aspects of the protein production process are not considered neglected in these models. In particular, like classical models they "gene-centred": the production of a type of protein has no inluence on the others. In reality, it is not the case. To produce mRNAs, genes sequester an RNA-polymerases during the elongation; and

3.A. A : G

A N -H P 91 so does the mRNAs with ribosomes to produce proteins. Both RNA-polymerases and ribosomes are common resources shared among all genes; and luctuations of these quantities may have repercussions on protein variability. In the next chapter will propose a model production of all types of proteins that takes into account this aspect.

3.A Appendix: Gillespie Algorithms for Non-Homogenous Process [START_REF] Gillespie | Exact stochastic simulation of coupled chemical reactions[END_REF] describes an algorithm to simulate stochastic trajectories such as the quantities of diferent chemical species interacting together. he main idea is to consider the state of a system (for instance the number of each chemical compounds) and to compute the irst reaction to occur, as well as the time when it happens. Once both pieces of information computed, one change the current state of the system accordingly with the reaction, and update the time.

One important hypothesis is that all reactions occur at exponential times (even if the rates of these exponential times may depend on the current state of the system). In this work, we have encountered a case where it was not the case. In the model with cell cycle of Subsection 3.3.3, at any time s, the state is described by (M s , P s ) (respectively, the number of mRNAs and proteins), and the rate of mRNA production is Λ(s) = λ 1 V (s) with λ 1 a parameter and V (s) the non-constant volume of the cell. he parameterΛ(s) does not depend on the state (M s , P s ) but is time dependent through V (s); for this reason, it is not an exponential time.

In this case, the time laps T until the next mRNA production is characterised by

∀x > 0, P [T > x] = exp ( - ∫ x 0 Λ(s) ds )
which is not a exponential distribution as Λ is non-constant. To compute T , let's consider that Λ(s) is strictly positive for any s ∈ R + , as a consequence F (x) := ∫ x 0 Λ(s) ds is strictly increasing. Let's sample the exponential random variable E of parameter 1. We have hence ∀y > 0 P [E > y] = exp (-y) .

If we consider the case of y = F (x), since F is strictly increasing, it comes

P [E > y] = exp (-F (x))
and

P [E > y] = P [ F -1 (E) > x ] .
As a consequence the random variable F -1 (E) has the same distribution as T .

Based on that we can propose a new version of the algorithm of Gillespie that can take into account nonexponential times such as T : Algorithm 1. he equivalent of Gillespie algorithm that considers non-homogenous events is 1. Initialisation: Initialise time of molecules in the system and the time.

2. Next exponential event: determine the next event that occurs at exponential time as in Gillespie algorithm.

3. Next non-homogeneous event: determine the next event that occurs at non-homogenous rates with the method previously described. In this appendix, we propose to give explicit expression for these quantities if we are at equilibirum. Between times τ D -and τ D , the proteins undergo a binomial segregation, and since the system is at equilibrium, the distribution of the number of proteins ater division P τD is the same as the distribution of proteins at birth P 0 . As a consequence:

Pτ D - ∑ i=1 B i,1/2 D = P 0 with ( B i,1/2
) being independent Bernoulli random variables of parameter 1/2 and being all independent of P τD-. It comes the following Lemma: Lemma 3.1. he mean and the variance of P 0 depend on the mean and the variance of P τD-such as:

E [P τD-] = 2E [P 0 ] Var [P τD-] = 4Var [P 0 ] -2E [P 0 ] .
Proof. With the moment-generating function of P 0 , it comes that

E [exp [ξP 0 ]] = E   Pτ D - ∏ 1=1 E [ exp [ B i,1/2 ]]   = E [ ( 1 + e ξ 2 ) Pτ D - ] = E [ exp [ log ( 1 + e ξ 2 ) P τ D - ]]
As a consequence, by calling η(ξ) := E [exp [ξP τD-]] the moment generating function of P τD-, it follows:

d dξ E [exp [ξP 0 ]] = e ξ 1 + e ξ • η ′ ( log ( 1 + e ξ 2
))

d 2 dξ 2 E [exp [ξP 0 ]] = e ξ (1 + e ξ ) 2 • η ′ ( log ( 1 + e ξ 2
))

+ ( e ξ 1 + e ξ ) 2 • η ′′ ( log ( 1 + e ξ 2
)) .

As ξ goes to 0, one inds:

E [P 0 ] = E [P τD-] 2
E [ P 2 0 ] = 1 4 • E [P τD-] + 1 4 • E [ P 2 τD- ] .
Hence comes the result.

Using this Lemma allows to determine the mean and the variance of P 0 :

Proposition 3.9. At equilibrium, the mean and the variance of the protein number at birth are:

E [P 0 ] = λ 2 (f 1 (τ R ) + f 2 (τ D )) , Var [P 0 ] = 1 3 { 2E [P 0 ] + 2 λ 2 σ 1 [ ( 1 -e -σ1τR ) Cov [P 0 , M 0 ] + ( 1 -e -σ1(τD-τR) ) Cov [P τR , M τR ] ] +g 1 (τ R ) + g 2 (τ D ) } 3.B. A : M , V C (M 0 , P 0 ) (M τ R , P τ R ) 93
with f 1 , f 2 , g 1 and g 2 deined in Proposition 3.6 and Proposition 3.8.

Proof. By considering the expressions of Proposition 3.8 for s = τ D -, it comes:

E [P τD-] = E [P τR ] + λ 2 f 2 (τ D ) , Var [P τD-] = Var [P τR ] + 2λ 2 1 -e -σ1(τD-τR) σ 1 Cov [P τR , M τR ] + g 2 (τ D ) .
By Equation (3.13) and Equation (3.14), we have that

E [P τD-] = E [P 0 ] + λ 2 (f 1 (τ R ) + f 2 (τ D ))
,

Var [P τD-] = Var [P 0 ] + 2 λ 2 σ 1 [ ( 1 -e -σ1τR ) Cov [P 0 , M 0 ] + ( 1 -e -σ1(τD-τR) ) Cov [P τR , M τR ] ] +g 1 (τ R ) + g 2 (τ D ) .
Lemma 3.1 describes the efect of the binomial sampling between τ D -and τ D on the mean and the variance of P . Since, we are at equilibrium of the cell cycles, it comes that

E [P 0 ] = λ 2 (f 1 (τ R ) + f 2 (τ D )) 3Var [P 0 ] = 2E [P 0 ] + 2 λ 2 σ 1 [ ( 1 -e -σ1τ R ) Cov [P 0 , M 0 ] + ( 1 -e -σ1(τ D -τ R ) ) Cov [P τR , M τR ] ] +g 1 (τ R ) + g 2 (τ D ) .
he only two remaining quantities to determine are Cov [P 0 , M 0 ] and

Cov [P τ R , M τ R ] .
Proposition 3.10. Let's deine the functions

k 1 (s) := λ 1 λ 2 σ 2 1 E [M 0 ] e -sσ1 ( sσ 1 - ( 1 -e -σ1s )) + λ 1 σ 1 E [P 0 ] ( 1 -e -sσ1 ) + λ 1 λ 2 σ 2 1 E [M 0 ] ( 1 -e -sσ1 ) 2 + λ 2 σ 1 e -sσ1 (( E [ M 2 0 ] -E [M 0 ] ) ( 1 -e -σ1s ) + σ 1 sE [M 0 ] ) + λ 1 λ 2 σ 2 1 [ λ 1 σ 1 ( 1 -e -sσ1 ) ( sσ 1 - ( 1 -e -σ1s )) + ( 1 -e -σ1s (sσ 1 + 1) ) ] . C 3. M C C 94 and k 2 (s) := 2λ 1 λ 2 σ 2 1 E [M τR ] e -(s-τ R )σ1 ( (s -τ R ) σ 1 - ( 1 -e -σ1(s-τ R )
))

+ 2λ 1 σ 1 E [P τR ]
( 1e -(s-τR)σ1

)

+ 2λ 1 λ 2 σ 2 1 E [M τR ] ( 1 -e -(s-τR)σ1 ) 2 + λ 2 σ 1 e -(s-τR)σ1 ( ( E [ M 2 τR ] -E [M τR ] ) ( 1 -e -σ1(s-τR) ) + σ 1 (s -τ R ) E [M τR ] ) + 2λ 1 λ 2 σ 2 1 [ 2λ 1 σ 1 ( 1 -e -(s-τR)σ1 ) ( (s -τ R ) σ 1 - ( 1 -e -σ1(s-τR)
))

+ ( 1 -e -σ1(s-τR) ((s -τ R ) σ 1 + 1) )] .
hen comes the covariances:

Cov [M 0 , P 0 ] = 1 (4 -e -τDσ1 ) { k 1 (τ R ) e -(τD-τR)σ1 + k 2 (τ D ) } -E [M 0 ] E [P 0 ]
and

Cov [M τR , P τR ] = (Cov [M 0 , P 0 ] + E [M 0 ] E [P 0 ]) e -τRσ1 + k 1 (τ R ) -E [M τR ] E [P τR ] .
Proof. Let's irst determine E [P s M s ] for any time 0 < s < τ R before replication. For this proof, we consider another description of the protein production that the one proposed in Equation (3.12). We consider three groups of proteins :

• Proteins present at birth.

• Proteins created during the cell cycle by mRNAs present at birth. Each of the M 0 mRNAs present at birth is able to produce proteins at rate λ 2 until its degradation that occurs in an exponential time of parameter σ 1 .

• Proteins created during the cell cycle by mRNAs also created during the cell cycle. Each of the N ([0, s[×R + ) mRNAs created since birth is able to create proteins at rate λ 2 during its existence that lasts an exponential time of parameter σ 1 .

he protein number hence decomposed can be writen as

P s = P 0 + M0 ∑ i=0 N 0,i λ2 ( [0, θ 0,i ∧ s[ ) + N ([0,s[×R+) ∑ i=1 N 1,i λ2 ( [0, θ 1,i ∧ ( s -t 1,i ) [ ) .
with for any i ∈ N and l ∈ {0, 1}, t l,i being the mRNA birth time (they are uniformly distributed in [0, τ R [), θ l,i being the lifetime of mRNAs (θ l,i ∼ E (σ 1 )), and N l,i λ2 denote Point Poisson Process of parameter λ 2 . Furthermore, we recall the process M s as described in Equation (3.11) for s < τ R :

M s = M0 ∑ i=1 {E i σ 1 >s} + N (∆ s ) . 3.B. A : M , V C (M 0 , P 0 ) (M τ R , P τ R ) 95
When making the mean of the product M s P s , there is three crossed terms that are the product of independent variables:

A = M0 ∑ i=1 {θ 0,i >s} × N ([0,s[×R+) ∑ i=1 N 1,i λ2 ( [0, θ 1,i ∧ ( s -t 1,i ) [ ) , B = N (∆ s ) × P 0 , C = N (∆ s ) × M0 ∑ i=0 N 0,i λ2 ( [0, θ 0,i ∧ s[ ) ;
and three crossed terms that are the product of non-independent random variables:

D = M0 ∑ i=1 {θ 0,i >s} × P 0 , E = M0 ∑ i=1 {θ 0,i >s} × M0 ∑ i=0 N 0,i λ2 ( [0, θ 0,i ∧ s[ ) , F = N (∆ s ) × N ([0,s[×R+) ∑ i=1 N 1,i λ2 ( [0, θ 1,i ∧ ( s -t 1,i ) [ ) .
he crossed terms with independent variable are calculated with the mean of each term which does not present any diiculties. It gives

E [A] = λ 2 λ 1 σ 1 E [M 0 ] e -sσ1 ( s - 1 -e -σ1s σ 1
) ,

E [B] = λ 1 σ 1 E [P 0 ] ( 1 -e -sσ1
) ,

E [C] = λ 1 λ 2 σ 2 1 E [M 0 ] ( 1 -e -sσ1 ) 2 .
Let's now consider the three remaining terms. he term D gives

E [D] = E [ M0 ∑ i=1 E [ {θ 0,i >s} | (M 0, P 0 ) ] × P 0 ] = E [M 0 P 0 ] e -sσ1 . C 3. M C C 96
he term E gives:

E [E] = E   M0 ∑ i=1 M0 ∑ j̸ =i {θ 0,i >s} N 0,j λ2 ( [0, θ 0,j ∧ s[ )   +E [ M0 ∑ i=1 {θ 0,i >s} N 0,i λ2 ( [0, θ 0,i ∧ s[ ) ] = E   M0 ∑ i̸ =j E [ {θ 0,i >s} |M 0 ] E [ N 0,j λ2 ( [0, θ 0,j ∧ s[ ) |M 0 ]   +E [ M0 ∑ i=1 {θ 0,i >s} E [ N 0,i λ2 ( [0, θ 0,i ∧ s[ ) |M 0 , θ 0,i ] ] = E   M0 ∑ i̸ =j e -sσ1 λ 2 σ 1 ( 1 -e -σ1s )   + E [ M0 ∑ i=1 {θ 0,i >s} E [ N 0,i λ2 ([0, s[) |M 0 , θ 0,i ] ] = λ 2 σ 1 ( E [ M 2 0 ] -E [M 0 ] ) e -sσ1 ( 1 -e -σ1s ) + λ 2 sE [M 0 ] e -sσ1 = λ 2 σ 1 e -sσ1 [( E [ M 2 0 ] -E [M 0 ] ) ( 1 -e -σ1s ) + σ 1 sE [M 0 ] ] .
he case of the last term F is more complicated; we separate the sum on N ([0, s[×R + ) into two sums: one on N (∆ s ) and the other on N ( ∆ s ) with ∆ s := {(x, y), 0 < x < s, y < s -x} (in order to have ∆ s ∪ ∆ s = [0, s[×R + and N (∆ s ) independent of N ( ∆ s )). Hence it follows that:

E [F ] = E   N (∆ s ) × N (∆s) ∑ i=1 N 1,i λ2 ( [0, θ 1,i ∧ ( s -t 1,i ) [ )   +E [N (∆ s )] × E   N ( ∆s) ∑ i=1 N 1,i λ2 ( [0, θ 1,i ∧ ( s -t 1,i ) [ )   = E   N (∆ s ) × N (∆s) ∑ i=1 E [ N 1,i λ2 ( [0, θ 1,i ∧ ( s -t 1,i ) [ ) |θ 1,i , t 1,i , N (∆ s ) ]   +λ 2 λ 1 σ 1 ( 1 -e -sσ1 ) × E   N ( ∆s) ∑ i=1 θ 1,i ∧ ( s -t 1,i )   3.B. A : M , V C (M 0 , P 0 ) (M τ R , P τ R ) 97 E [F ] = λ 2 E   N (∆ s ) × N (∆s) ∑ i=1 E [ θ 1,i ∧ ( s -t 1,i ) | ( θ 1,i , t 1,i ) ∈ ∆ s ]   +λ 2 E [N (∆ s )] × ∫ ∫ ∆s θ ∧ (s -t) λ 1 dt ⊗ σ 1 e -σ1θ dθ = λ 2 E [ N (∆ s ) 2 ] 1 ν (∆ s ) E [ θ 1,i ∧ ( s -t 1,i ) | ( θ 1,i , t 1,i ) ∈ ∆ s ] +λ 2 E [N (∆ s )] × ∫ ∫ ∆s θ ∧ (s -t) λ 1 dt ⊗ σ 1 e -σ1θ dθ = λ 2 E [ N (∆ s ) 2 ] 1 ν (∆ s ) ∫ ∫ ∆s θ ∧ (s -t) λ 1 dt ⊗ σ 1 e -σ1θ dθ +λ 2 E [N (∆ s )] × ∫ ∫ ∆s θ ∧ (s -t) λ 1 dt ⊗ σ 1 e -σ1θ dθ.
We know that N (∆ s ) is a Poisson random variable of parameter ν

(∆ s ) = λ 1 /σ 1 × (1 -e -sσ1 ), hence E [N (∆ s )] = ν (∆ s ) and E [ N (∆ s ) 2 ] = ν (∆ s ) (ν (∆ s ) + 1
). As a consequence:

E [F ] = λ ( 1 + λ 1 σ 1 ( 1 -e -sσ1 ) ) ∫ ∫ ∆s θ ∧ (s -t) λ 1 dt ⊗ σ 1 e -σ1θ dθ +λ 2 λ 1 σ 1 ( 1 -e -sσ1 ) × ∫ ∫ ∆s θ ∧ (s -t) λ 1 dt ⊗ σ 1 e -σ1θ dθ. = λ λ 1 σ 1 ( 1 -e -sσ1 ) ∫ ∫ [0,s[×R+ θ ∧ (s -t) λ 1 dt ⊗ σ 1 e -σ1θ dθ +λ 2 ∫ ∫ ∆s θ ∧ (s -t) λ 1 dt ⊗ σ 1 e -σ1θ dθ = λ λ 1 σ 1 ( 1 -e -sσ1 ) λ 1 σ 1 ( s - 1 -e -σ1s σ 1 ) + λ 2 ∫ s 0 ∫ ∞ s-t (s -t) σ 1 e -σ1θ dθ λ 1 dt = λ ( λ 1 σ 1 ) 2 ( 1 -e -sσ1 ) ( s - 1 -e -σ1s σ 1 ) + λ 2 ∫ s 0 t ∫ ∞ t σ 1 e -σ1θ dθ λ 1 dt = λ ( λ 1 σ 1 ) 2 ( 1 -e -sσ1 ) ( s - 1 -e -σ1s σ 1 ) + λ 2 λ 1 ∫ s 0 te -σ1t dt = λ ( λ 1 σ 1 ) 2 ( 1 -e -sσ1 ) ( s - 1 -e -σ1s σ 1 ) + λ 2 λ 1 σ 1 ( -e -σ1s s + 1 -e -σ1s σ 1 ) = λ λ 1 σ 2 1 [ λ 1 σ 1 ( 1 -e -sσ1 ) ( sσ 1 - ( 1 -e -σ1s )) + ( 1 -e -σ1s (sσ 1 + 1) ) ]
so, it comes that for any s before replication that:

E [M s P s ] = k 1 (s) + E [M 0 P 0 ] e -sσ1 . (3.19)
Similarly for τ R ≤ s < τ D ater replication, one can show that:

E [M s P s ] = k 2 (s) + E [M τR P τR ] e -(s-τR)σ1 . (3.20) C 3. M C C 98
With these two relations, one can determine the expression of E [M τD-P τD-]:

E [M τD-P τD-] = k 2 (τ D ) + E [M τR P τR ] e -(τ D -τ R )σ1
= E [M 0 P 0 ] e -τDσ1 + k 1 0 (τ R ) e -(τD-τR)σ1 + k 2 τR (τ D ) . Since M τD-and P τD-undergo a binomial sampling between τ D -and τ D , and since at equilibrium (M 0 , P 0 ) D = (M τD , P τD ), by considering (B k,i ) k∈{1,2},i∈N i.i.d. Bernoulli random variables of parameter 1/2, it comes

E [M 0 P 0 ] = E   Mτ D - ∑ i=0 B 1,i Pτ D - ∑ i=0 B 2,i   = E   Mτ D - ∑ i=0 Pτ D - ∑ i=0 E [B 1,i B 2,i | (M τD-, P τD-)]   = 1 4 E [M τD-P τD-] .
As by deinition of the covariance

Cov [P 0 , M 0 ] := E [M 0 P 0 ] -E [M 0 ] E [P 0 ] , it

3.C Appendix: Simple Model to Predict the Efect of Binomial Sampling

In Figure 3.5b, we have considered the ratio of protein noise: the noise in the model where the division is exact, divided by the noise with the case where the division is binomial. We have igured in cyan dash line the prediction of a simpliied model of such a ratio. In this appendix, we described the model used.

Let's consider a quantity P that goes through a division. he division can be performed by two means, either through exact division, or through binomial sampling (see Subsection 3.3.6). he result of these divisions will respectively denoted by P e and P b . During division, the volume is divided by two, changing from 2V 0 to V 0 . In order to be ploted in Figure 3.5b, we need to consider the coeicient of variation of protein concentration ater division

η := Var [P e /V 0 ] /E [P e /V 0 ] 2 Var [P b /V 0 ] /E [P b /V 0 ] 2 as a function of x := Var [P /(2V 0 )] E [P /(2V 0 )] .
Proposition 3.11. he coeicient of variation ratio η as a function of x is given by

η = 2V 0 x 2V 0 x + 1 .
Proof. We have the quantity before division P . Let's observe the efect of exact division on its concentration. Since by deinition, we have that P e = P /2, it comes that

E [P e /V 0 ] = E [P /(2V 0 )] ,
Var [P e /V 0 ] = Var [P /(2V 0 )] .

3.C. A

: S M P E B S 99

For the efect of binomial division, one can refer to Lemma 3.1, that describes the efect of the binomial division on the mean and the variance of any number. With it, it comes that

E [P b ] = E [P ] 2 , Var [P b ] = Var [P ] + 2E [P b ] 4 .
We then divide by the volume in order to observe the concentrations:

E [P b /V 0 ] = E [P ] 2V 0 = E [P / (2V 0 )]
and

Var [P b /V 0 ] = Var [P b ] V 2 0 = Var [P ] + 2E [P b ] 4V 2 0 = Var [P / (2V 0 )] + E [P /(2V 0 )] 2V 0 .
As a consequence, it comes that

η = Var [P e /V 0 ] Var [P b /V 0 ] = Var [P /(2V 0 )] Var [P / (2V 0 )] + E [P /(2V 0 )] /(2V 0 ) = 2V 0 x 2V 0 x + 1 M - M
he previous chapter has considered models with cell cycle, that allow the comparison of the predicted variance with real experimental measures of [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF]. But it seems that the noise obtained in these models do not reproduce the protein variability, especially for highly expressed genes. It has been proposed in [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF], and even earlier in the literature [START_REF] Elowitz | Stochastic Gene Expression in a Single Cell[END_REF], Swain et al., 2002), that luctuations of commonly shared resources in the protein production, such as RNA-polymerases and ribosomes (macromolecules required respectively for every transcription and translation) can add signiicant variability in gene expression. We describe in this chapter a model that extends the previous models with the introduction of this key feature: the limited amount of RNA-polymerases and ribosomes for the production of every protein. As the models of the previous chapter were "gene-centred", each class of proteins was considered independently from each other; the common sharing or RNA-polymerases and ribosomes advocates here for the consideration of a multi-protein model where all the genes are considered altogether. Models that consider multi-protein production are rare in literature. Two examples are [START_REF] Mather | Translational Cross Talk in Gene Networks[END_REF] that describes a production of two types of proteins, and [START_REF] Fromion | A Stochastic Model of the Production of Multiple Proteins in Cells[END_REF] that includes the translation of a large number of classes of proteins, both consider with a limited number of ribosomes available for translation. Both articles carried out mathematical analysis, but they both focus on the translation part, considering the number of mRNAs as constant, or (in the case of [START_REF] Mather | Translational Cross Talk in Gene Networks[END_REF]) as independently luctuating. In reality, mRNAs production is neither constant, neither independent, as it depends on RNA-polymerase dynamic. Moreover, the models of the articles implicitly take place in a ixed volume; there is no notion growth of the cell volume, nor replication and division, which is also the case for classical models. As we have said in Subsection 3.2.2, it is diicult to quantitatively compare their results with experimental measurements in this case.

We hence propose a model that is in the direct continuation of the previous models: with cell cycle, binomial sampling and gene replication; but we also introduce the notion of limited amount of RNA-polymerases and ribosomes. We will investigate the potential impact they can have on gene expression variability. At irst, in Section 4.1, the model is described in detail. As it appears that its complete mathematical analysis is complex, we will propose to examine a simpliied description in Section 4.2 that helps to ix the parameters on experimental measures. In Section 4.3, we will examine the impact on the protein variance of free ribosomes and free RNA-polymerases. Globally, it will appear litle additional noise in protein production compared M M 103

• the gene copy number of this gene

• the free RNA-polymerase concentration

• a parameter that takes into account the speciicity of the promoter such as the promoter ainity with RNA-polymerase, the chromosome architecture or the propensity of initiation

Once the binding-initiation event previously described is inished, the mRNA production can begin. In our model, the process of elongation and the termination are considered together as a single event. During this time, the RNA-polymerase is considered as sequestered on the DNA. In real cells, the speed of elongation is relatively constant in stable environmental conditions; as a consequence, in our model, we consider that it depends only to the gene length. In the model, there is no notion of operon and each transcript contains only one gene, we choose to consider the gene length as representative of the length of the transcript. herefore the rate of the event is considered in the model as only depending on the length of the gene. Ribosome mechanic has a lot in common with RNA-polymerase in the model. hey are also grouped into two categories: the non-allocated ribosomes (or free ribosomes) that evolve in the medium and the allocated ribosomes (or sequestered ribosomes) which are bound on mRNA involved in a translation process. he translation is considered as occurring in two steps. At irst, the ribosome binds on the RBS and initiate the elongation; the rate at which this event occurs depends on the free ribosome concentration, on the mRNA copy number and on some mRNA speciic parameter that takes into account aspects like RBS-ribosome ainity for instance. Secondly, once the elongation initiated, the process of elongation and termination is considered as only depending on the length of the gene.

In the models of the previous chapter, the volume was deined as an external and deterministic object: the idea was to study the behaviour of one gene immersed into a "background environment" where the cell grow and divide. One key assumption in this case is that the gene of interest has no inluence on the overall bacteria dynamic. In the current model, it is not any more the case: all the genes are considered simultaneously and their production represents the production of all proteins. It is not possible to consider that the production of a single type of protein has no efect on the global performance of the cell in this model. he volume growth depends now on the global production of proteins, and not as an independent and deterministic feature.

To represent the volume growth we rely on the "density constraint": it appears that the cell tends to maintain constant its density of cell components in order to have an eicient intracellular difusion [START_REF] Marr | Growth rate of Escherichia coli[END_REF]. So the total mass of compounds in the cell (proteins, metabolites, DNA, etc.) can be considered as proportional to the cell volume. Most of the dry mass of the cell is due the total amount of proteins [START_REF] Neidhardt | Chemical Composition of Escherichia coli. In Escherichia coli and Salmonella: Cellular and Molecular Biology[END_REF], we therefore consider that the volume is proportional to the total mass of proteins (each protein contributes in proportion to its own mass to the total protein mass). 1

Model Presentation

Let's present more exhaustively the mechanisms and the notations of the model. he model has some global variables such as the volume, the number of free RNA-polymerases and ribosomes; and some variables that are gene speciic, like the number of mRNAs, of proteins, etc. One can refer to Figure 4.1 for an overview of the model.

Units of Production

In this model, we consider K types of proteins; each protein is produced in a single production unit, with a particular type of mRNAs and a speciic gene associated with. In each unit of production 1 Ribosome components also represent an important part of the cell dry mass [START_REF] Neidhardt | Chemical Composition of Escherichia coli. In Escherichia coli and Salmonella: Cellular and Molecular Biology[END_REF]. But as it will be seen in the following section, the concentration of ribosomes is considered as constant in the model; therefore, the total mass of ribosomes and proteins taken altogether is still proportional to the volume. It is still consistent with the "density constraint" hypothesis.

F Y F R G i E Y,i M i E R,i P i ∅ Divisions at 2V 0 λ 1,i G i F Y /V µ 1,i E Y,i +1 -1 σ 1,i M i λ 2,i M i F R /V µ 2,i E R,i +1 -1 B(M, 1/2) B(P, 1/2) (b)
Production of production unit of the i-th protein with the common pools of free RNA-polymerases and ribosomes. i ∈ {1, ..., K} we denote by G i (s), M i (s) and P i (s) respectively the number of gene copies, of messengers and of proteins at time s.

Volume Increase As previously said, the volume V (s) is no longer deterministic at it was the case in the models of the previous chapter and it is considered as proportional to the current total mass of proteins in the cell. We denote by β P represents ratio mass-volume and by w i the mass of a type i protein. In that case, we have by deinition

V (s) = K ∑ i=1 w i P i (s)/β P . (4.1)
hat means that each protein of type i created increases the total volume of the cell with respect to the factor

w i /β P .
Global Variables he total number of RNA-polymerases and ribosomes (whether allocated or not) are respectively denoted by N Y (s) and N R (s). In a irst step, we consider that the both these quantities are in constant concentration, that is to say

N Y (s) = ⌊β Y V (s)⌋ and N R (s) = ⌊β R V (s)⌋ ,
with β Y and β R constant parameters and where ⌊ ⌋ is the notation for the loor function. It means that as the cell grows, new RNA-polymerases and ribosomes are added to the system in the corresponding proportion. In Subsection 4.4.2 we will consider the case more complex where both RNA-polymerases and ribosomes are directly produced through a gene expression process. At any time s, we denote by F Y (s) the random variable that represents the number of free polymerases, that is to say RNA-polymerases that are not speciically sequestered on a messenger. In the same way, let's denote by F R (s) the random variable that represents the number of free ribosomes at time s.

Reaction Rates Let's then deine all the reactions that are speciic to each gene. As previously said, the rate at which an RNA-polymerase binds on a speciic promoter and initiate elongation depends on the concentration of free RNA-polymerases F Y (s)/V (s) and the copy number of the gene G i (s). he rate is therefore λ 1,i G i (s)F Y (s)/V (s) where λ 1,i accounts for the speciicity of the promoter (its ainity for the RNApolymerase etc.). As elongation begins, the RNA-polymerase is considered as sequestered (decreasing the number of free polymerases F Y by one unit) on the DNA until the termination. he total number of RNApolymerases currently elongating a messenger of type i is denoted by the random variable E Y,i (s). As a consequence, the total amount of RNA-polymerases N Y is given by

N Y = F Y + K ∑ i=1 E Y,i .
he elongation time is given by an exponential random variable of rate µ 1,i . Once the elongation terminates, the RNA-polymerase is released in the cytoplasm (increasing the number of free RNA-polymerase F Y by one unit). A messenger is considered created as soon as its elongation begins: the reason for it is that in bacteria (unlike eukaryotes), since transcription and translation happen in the same medium, a translation can begin on an mRNA on which the transcription is not inished. Each messenger of type i has a lifetime given by an exponential random variable of rate σ 1,i .

Similarly to transcription, the rate at which a ribosome encounters an mRNA of type i and initiate translation depends on the number of mRNAs M i (s) and on the ribosome concentration F R (s)/V (s) . he rate for translation initiation is therefore λ 2,i M i (s)F R (s)/V (s) where λ 2,i will account for mRNA speciic aspects (RBS ainity for ribosome, etc.). he total number of ribosomes sequestered on messengers of type i is E Y,i (s) and each elongation time follows an exponential distribution of rate µ 2,i . Here we consider that the protein is created ater termination (since the protein is usually fully functional once its translation is completed); the number of proteins P i (s) is then increased by one unit. As in the models of the previous chapter, we do not consider protein proteolysis since it usually occurs at much longer timescale than cell cycle.

Remark 4.1. It can be remarked that in the models of the previous chapter, the mRNA initiation rate per gene was considered as constant. It implicitly meant that we have considered that the concentration of free RNA-polymerases remains constant across the cell cycle. Similarly with the translation part, free ribosomes were also considered in constant concentrations. Now, the consequence common pool of non-allocated RNA-polymerases and ribosomes make these rates explicitly depends on these varying concentrations.

DNA Replication and Division of the Cell At a time s, each gene i ∈ {1, ..., K} is characterised by the gene copy number G i (s). As in the gene-centred model of Section 3.4, there is only one replication per gene in the cell cycle: as a consequence, the for each i ∈ {1, ..., K}, G i (s) is constantly equal to 1 until the gene is replicated; from this instant until the division, it is set to 2. here is two modelling choice for when the DNA replication is initiated: it can occur at a ixed time ater the last division or it can or when the cell reaches a certain volume V I . he irst simulations are made by considering the volume-dependent initiation event, but as we will see in Subsection 4.4.5, simulations with the two possibilities show no noticeable diference. he volume V I is ixed to 1.8 om 3 (see [START_REF] Wallden | Fluctuations in growth rates determine the generation time and size distributions of E. coli cells[END_REF] and the appendix Section 4.A about this choice). We consider the speed of DNA replication as constant; as a consequence, once known the replication time τ I , the delay until the replication of i-th gene is ixed, and is given by the gene position.

For the division, we considered at irst that, like in the previous models, the division occurs when the reaches exactly the volume 2V 0 (with V 0 = 1.3 om 3 as it was the case for the previous models). We will consider in Subsection 4.4.4 the case where the division is not as precise. As in the previous models, the efect of septation is a binomial sampling of messengers and proteins: each of them as an equal chance to be in the next cell or not. he volume of the new cell is proportional to the total mass of the remaining proteins. Moreover, at division, all gene copy number are anew set to one; ribosomes and RNA-polymerases will anew set accordingly to the new volume. he model of this section is more complex than its counterparts of the previous chapter. It is due in part to the feedback loop that proteins have on their own production: the more proteins, the more the volume increases, thereby increasing the total amount of ribosomes and hence the translation rates. his complexiies the complete analytical description of mRNA and protein mean productions, which has a detrimental impact on the search of parameters based on [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] experiments. he next section proposes a model that mimics the average behaviour of our stochastic model: the goal is to be able to it parameters to real measures and use them for stochastic simulations.

Simple Deterministic Model for Protein Production

Expressions for the mean of mRNA and protein concentrations were used in the models of the previous chapter as a way to ix the model parameters based on experimental measures. But as the multi-protein model of this chapter is diicult to describe analytically, the exact mean of protein and mRNA concentrations are unknown. As a consequence, it is not possible to directly adjust the parameters to make the protein productions of the model correspond to those of [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF].

D M P P 107

To address this problem, we propose a representation to relect the average behaviour of the stochastic model; for that, we use the classical framework of systems of Ordinary Diferential Equations (ODEs)2 . his framework is usually used in literature to describes the average behaviour associated to gene expression (see [START_REF] Borkowski | Translation elicits a growth rate-dependent, genome-wide, diferential protein production in Bacillus subtilis[END_REF], [START_REF] Goelzer | Cell design in bacteria as a convex optimization problem[END_REF] for instance): the evolution of mRNA and protein concentrations are described through the chemical kinetics representation. he goal is to have a relatively correct representation of the average production, so that the parameters of the stochastic model can be correctly ited.

We will present this simpliied model in Subsection 4.2.1 and describe its dynamics in Subsection 4.2.2. We will use these results to deduce a set of parameters that corresponds to [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] experiment as it will be explained in Subsection 4.2.3. Finally, in Subsection 4.2.4, we will validate the global correspondence between the average behaviour of the stochastic model and model presented in this section.

Presentation of the Deterministic Production Model

he model chosen to relect the average behaviour of the stochastic model previously described is a system of ordinary diferential equations (ODEs) that describes the kinetics of each compound concentration of the system.

We still consider K genes, each of them associate with a particular type of mRNA and protein. For a gene of type i, the concentrations of gene copies is given by g i (s); mRNAs and protein concentrations are denoted by m i (s) and p i (s). Similarly, f Y (s) and f R (s) respectively represent the concentrations of free RNApolymerases and free ribosomes; while e Y,i (s) and e R,i (s) denote the concentrations of RNA-polymerases and ribosomes currently sequestered in the i-th protein production unit. It is important tot note that, contrary to the stochastic model of the previous section, all these quantities correspond to concentration and not numbers of entities (their stochastic counterparts would be the concentrations G i (s)/V (s), M i (s)/V (s), P i (s)/V (s), etc.).

he reactions between diferent compounds are given by the law of mass action, that is to say that the rate of chemical reaction is proportional to the reactant abundance and their activities. We are interested for instance in the evolution of m i that denotes the concentration of mRNAs of type i . he creation of a type i mRNA is the result of a reaction between a free RNA-polymerase (whose concentration is f Y (s)) and the gene i (whose concentration is g i (s)); λ 1,i is interpreted as the ainity constant of the reaction. he type i mRNA degradation is the result of a reaction that occurs at rate σ 1,i .

As in the usual description of the cell (see [START_REF] Goelzer | Cell design in bacteria as a convex optimization problem[END_REF] for instance), one also must consider the dilution: without any molecule creation, the concentration of the compound still decreases as the cell grows due to dilution. If we consider that the cell is growing exponentially, doubling of volume in a time τ D , then the rate of dilution is log 2/τ D . he exponential growth corresponds to the volume dynamics of real bacteria [START_REF] Wang | Robust growth of Escherichia coli[END_REF], and we will see in Subsection 4.2.4 that it is a good approximation of the growth of cells in stochastic simulations.

All these aspects considered altogether, the kinetics of the concentration of mRNAs of type i is given by the ODE:

dm i ds (s) = λ 1,i g i (s) • f Y (s) -σ 1,i m i (s) - log 2 τ D • m i (s). (4.2)
he irst term represents the mRNA creation; the second, the mRNA degradation; and the last, the dilution. Similarly, for the other reactions, it comes for i ∈ {1, ..., K}, at any time s:

C 4. M - M 108
dp i ds (s) = µ 2,i e R,i (s) - log 2 τ D • p i (s), (4.3) de Y,i ds (s) = λ 1,i g i (s) • f Y (s) -µ 1,i e Y,i (s) - log 2 τ D • e Y,i (s), (4.4) de R,i ds (s) = λ 2,i m i (s) • f R (s) -µ 2,i e R,i (s) - log 2 τ D • e R,i (s). (4.5)
Similarly to the stochastic model of the previous section, we consider that the RNA-polymerases (whether allocated or not) are still considered in constant concentration β Y . It means that

β Y = f Y (s) + K ∑ i=1 e Y,i (s), (4.6)
since ∑ i e Y,i and f Y represent the concentrations of respectively the allocated and non-allocated RNApolymerases. It is similar to the ribosomes as we have:

β R = f R (s) + K ∑ i=1 e R,i (s).
(4.7)

he classical strategy in literature to study such system (an analogue model is done in Borkowski et al. ( 2016)) is to consider the system in steady state growth: the gene concentration g i is considered as constantly equal to its average value during the cell cycle, and then one can calculate the concentrations of m i , p i , e Y,i and e R,i at steady-state by writing the equation Equations (4.2) to (4.7) with the derivative term as null. We have tried such methods to determine the concentrations. But, even if the protein concentrations then predict are around the stochastic simulation; they are not precise enough: there is a clear shit between the stochastic protein concentration and the one predicted by this method. In fact, it would be a good approximation if the gene concentration is constant during the cell cycle, as it was the case in the irst model of the last chapter (Section 3.3).

So we have decided to describe more precisely the cell cycle with a non-constant gene concentration. We place ourselves in one cell cycle: at a time s such as 0 ≤ s < τ D . We also consider known all times τ R,i of each gene replication; it means in particular that for every time s, the i-th gene copy number is known:

g i (s) = ( 1 + s≥τR,i
) /(V 0 2 s/τD ) (the factor V 0 2 s/τD represents the volume). By analogy with the equilibrium condition presented in Subsection 3.3.2, we expect that a large number of cell cycles have already occurred, so that the concentration of any entities is the same at the beginning and at the end of the cell cycle. It means that, for each unit of production, the concentrations m i , p i , e Y,i and e R,i are such as ∀i ∈ {1, ..., K}

{ p i (0) = p i (τ D ), m i (0) = m i (τ D ), e Y,i (0) = e Y,i (τ D ), e R,i (0) = e R,i (τ D ). (4.8)
With these considerations, we have a system of ODEs that aims to emulate the average behaviour of stochastic model of Section 4.1 during the cell cycle. In the next section, under some simpliications, we propose to gives expressions for m i (s), p i (s), e Y,i (s), e R,i (s), f Y (s) and f R (s) as a function of all the parameters (λ 1,i , σ 1,i , etc.) and g i (s).

Dynamics of the Average Production Model

In order to estimate the parameters, one needs to have expressions for m i , e Y,i , p i , e R,i , f R and f Y of the previous ODEs for any time s of the cell cycle. But the interdependence between e Y,i and f Y on one hand and e R,i and f R on the hand raise diiculties when integrating these equations. Explicit solution for the dynamics m i , e Y,i , p i , e R,i , f R and f Y are therefore not easy to obtain directly. In order to have expressions for these quantities anyway, we choose to make some biologically reasonable simpliications that permit to give explicit expressions for f Y and f R . In the next subsections, the stochastic simulations will show a good correspondence between their average concentration of free RNA-polymerase and ribosomes and the ones predicted here; it will therefore justify a posteriori the simpliications that we make in this section.

Let's consider at irst the RNA-polymerases. We denote by ⟨µ 1 ⟩ := ∑ i µ 1,i /K the mean elongation rates of transcription and the function h such as

h(s) := K ∑ i=1 e Y,i (s) ⟨µ 1 ⟩ µ 1,i .
he dynamic of h is given by summing the equations Equation (4.4) for i from 1 to K, and by using Equation (4.6):

d ds h(s) = f Y (s) • ⟨µ 1 ⟩ ( 1 + K ∑ i=1 λ 1,i µ 1,i g i (s) ) -β Y ⟨µ 1 ⟩ - log 2 τ D • h(s). (4.9)
he h is simply a weighted sum of the allocated RNA-polymerases e Y,i . We decided to consider that such weighting has litle inluence, and that h does not greatly difer from the uniform sum ∑ i e Y,i , that is to say:

h(s) = K ∑ i=1 e Y,i (s) ⟨µ 1 ⟩ µ 1,i ≃ K ∑ i=1 e Y,i (s) = β Y -f Y (s).
It would be in particular true if all elongation rates µ 1,i are identical for every genes (i.e. if µ 1,i ≡ ⟨µ 1 ⟩ for all i). With this simpliication, from Equation (4.9), on obtain a diferential equation on f Y :

d ds f Y (s) = ⟨µ 1 ⟩ β Y ( log 2 ⟨µ 1 ⟩ τ D + 1 ) -⟨µ 1 ⟩ ( 1 + log 2 ⟨µ 1 ⟩ τ D + K ∑ i=1 λ 1,i µ 1,i g i (s) ) f Y (s). (4.10)
One can remark that the concentrations of free RNA-polymerases is on a quick timescale. Indeed, Table 1.A.2 in Chapter 1 gives the total numbers of transcriptions and translations in the whole cell: there are a dozen of transcriptions, and hundreds of translations per second. As a consequence, one can expect that that f Y quickly reach their equilibrium compared to the cell cycle. his consideration will be justify a posteriori with the correspondence with the stochastic simulations.

With these considerations, we consider that the derivative term of Equation (4.10) is null and it comes:

f Y (s) = β Y 1 + log 2 ⟨µ 1 ⟩ τ D K ∑ i=1 λ 1,i µ 1,i g i (s) + 1 + log 2 ⟨µ 1 ⟩ τ D . C 4. M - M 110
Moreover, as the elongation rates µ 1,i will be determined in the next section, it will appears that log 2/(⟨µ 1 ⟩× τ D ) ∼ 10 -3 ≪ 1. herefore, it is possible to neglected the contribution of this term. Taking this aspect into consideration, and with the same logic for the free ribosomes, it follows:

f Y (s) = β Y 1 1 + K ∑ i=1 λ 1,i µ 1,i g i (s) and f R (s) = β R 1 1 + K ∑ i=1 λ 2,i µ 2,i m i (s)
.

With the global quantities f Y and f R known, we are able to give expression for gene-speciic variables. For each i ∈ {1, ..., K}, the number of mRNAs of type i, one can integrate Equation (4.2) and ind that:

dm i ds (s) = λ 1,i g i (s) • f Y (s) -σ 1,i m i (s) - log 2 τ D • m i (s).
With the boundary conditions (Equation (4.8)), it is easy to deduce that:

m i (s) = λ 1,i e -σ1,is 2 s/τD [ ∫ s 0 2 u/τD e σ1,iu g i (u)f Y (u) du + ∫ τD 0 2 u/τD e σ1,iu g i (u)f Y (u) du 2e σ1,iτD -1 ] . (4.11)
Since the quantities g i , f Y are known, we have an explicit solution for m i .

Similarly for e Y,i (s) and e R,i (s), it comes

e Y,i (s) = λ 1,i e -µ1,is 2 s/τ D [ ∫ s 0 2 u/τD e µ1,iu g i (u)f Y (u) du + ∫ τD 0 2 u/τD e µ1,iu g i (u)f Y (u) du 2e µ1,iτD -1 ] , e R,i (s) = λ 2,i e -µ2,is 2 s/τD [ ∫ s 0 2 u/τD e µ2,iu m i (u)f R (u) du + ∫ τD 0 2 u/τD e µ2,iu m i (u)f R ((u) du 2e µ2,iτD -1 ] .
Let's now consider the type i protein concentration. By integrating the equation Equation (4.2), and by considering the Equation (4.8), it comes:

p i (s) = µ 2,i 2 s/τD ∫ τD 0 (1 + u<s ) 2 u/τD e R,i (u) du.
(4.12)

As in the models of the previous chapter, we are interested in the average concentrations over the cell cycle. Since, in the system of ODEs, we deine the average concentrations over the cell cycle of free RNApolymerases and ribosomes respectively as

f Y = 1 τ D ∫ τD 0 β Y 1 K ∑ i=1 λ 1,i µ 1,i g i (s) + 1 ds and f R = 1 τ D ∫ τD 0 β R 1 K ∑ i=1 λ 2,i µ 2,i m i (s)+1
ds.

(4.13)

We deined similarly the concentrations m i and p i averaged over the cell cycle. By integrating Equation (4.12) and Equation (4.12), it follows:

m i = λ 1,i σ 1,i τ D + log 2 ∫ τD 0 g i (u)f Y (u) du and p i = λ 2,i µ 2,i τ D log 2 (µ 2,i τ D + log 2) ∫ τD 0 m i (u)f R (u) du. (4.14)
Now we have expressions of the average concentrations of m i , p i , f R and f Y for any time s in the cell cycle that will be used in the next subsection to determine the parameters.

Parameters Estimation

Parameters that will be used in the stochastic simulations are determined in this section. In total, we have to determine all reaction rates for every protein type (λ 1,i , µ 1,i , σ 1,i , λ 2,i and µ 2,i for i ∈ {1, ..., K}) as well as concentration parameters of RNA-polymerases, and ribosomes (respectively β Y and β R ), the proportion between the volume and the proteic mass β P , the mass of each proteins w i and the copy number g i of every gene.

To do so, the idea is to use the measures of [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF]: the average concentration of mRNAs and proteins for each gene, as well as the mRNA haltime. As explained in the previous chapter (Section 3.1), 1081 genes were considered in the experiment, among which 841 have their mRNA production measured. he genome of E. coli is about approximately 4000 expressed genes so the measures permit to represent only a part of the total protein production mechanism. In a irst step, we only take into account the 841 genes with protein and mRNA production measured and consider that it would represent the whole genome; in Subsection 4.4.1 we will study the case of a simulation with a completed set of genes. [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] gives no measures about the quantities of non-allocated RNA-polymerases or ribosomes. So, to be able to completely determine a set of parameters, we ix the average concentration of free RNA-polymerases and ribosomes. It means we can have multiple sets of parameters depending on this choice. During the simulations, we will examine several simulations with diferent values for average free RNA-polymerase and ribosome concentrations to see their impact on the dynamic of the model Section 4.3

As in the models of the previous chapter, the rate σ 1,i of mRNA degradation of type i is still deduced from its half-life measured in [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] τ m,i through the expression σ 1,i = log 2/τ m,i (for more information, refer to subsubsection 3.3.4.2); the doubling time τ D = 150 min is given by the article. he gene copy number g i (s) at time s is deduced from the position of the gene position of the i-th gene (see Section 4.A for more details). he rates µ 1,i , µ 2,i of mRNAs and protein elongation rates can be deduced from the gene length of the i-th gene. In the description of model, we have considered that the length of the mRNA is characterised by its length; so a rate the parameter µ 1,i is given by the mRNA elongation speed (39 Nucl/s in [START_REF] Bremer | Modulation of Chemical Composition and Other Parameters of the Cell at Diferent Exponential Growth Rates[END_REF] for slow growing cells) divided by the length of the i-th gene. Similarly, µ 1,i is given by the protein elongation speed (12 aa/s in [START_REF] Bremer | Modulation of Chemical Composition and Other Parameters of the Cell at Diferent Exponential Growth Rates[END_REF] for slow growing cells) divided by the number of amino-acid coded by the i-th gene divided . he mass of each protein w i is also deduced from the length of the gene as it determines the number of amino-acids of the protein.

What remains to determine are the concentration parameters of RNA-polymerases, and ribosomes (β Y and β R ), the proportion between the volume and the mass of proteins β P , as well as the activities of the gene and the mRNA (respectively λ 1,i and λ 2,i ) in each unit of production i ∈ {1, ..., K}. To do so, we interpret the mRNA and protein concentration of each type measured in [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] as the average concentration of each mRNA and proteins over the cell cycle of this model (respectively m i and p i ). Moreover, as previously said, the average concentrations of free RNA-polymerases f Y and free ribosomes f R are ixed.

We want now to compute β Y , β R , λ 1,i and λ 2,i based on known values for f Y , f R , m i and p i . Let's irst determine the parameter β P . In the description of the stochastic model, Equation (4.1) states that at every moment, the volume is considered to be proportional to the total mass of proteins. Interpreting p i as the average concentration of the protein of type i leads by integration of Equation (4.1) to

β P = K ∑ i=1 w i p i .
Let's continue with the parameters relevant to the transcription: λ

           β Y = f Y   1 τ D ∫ τD 0 ( K ∑ i=1 λ 1,i µ 1,i g i (s) + 1 ) -1 ds   -1 λ 1,i = m i • (σ 1,i τ D + log 2) • (∫ τD 0 g i (u)f Y (u) du ) -1 ∀i ∈ {1, ...,
[β R , λ 2,1 , ..., λ 2,K ], it is solution of the system            β R = f R ×   1 τ D ∫ τD 0 ( K ∑ i=1 λ 2,i µ 2,i m i (s) + 1 ) -1 ds   -1 λ 2,i = p i × ( µ 2,i τ D log 2 (µ 2,i τ D + log 2) ∫ τD 0 m i (u)f R (u) du ) -1 ∀i ∈ {1, ..., K}.
(4.16)

By ixing the average amount of free RNA-polymerases and ribosomes, it is possible, through this procedure to setle sets of parameters that are ited to the experimental measures.

Validation of the Average Production Model

he description of the average production through the system of ODE make the computation of parameters of the stochastic model possible. Yet the good correspondence between the model of average production of this section and the average behaviour of the stochastic model of Section 4.1 has to be supported. For instance, one has to check that the stochastic simulations are producing the good quantities of mRNAs and proteins (which is the main purpose of this section).

Here, we present the results of a particular simulation with parameters determined using the previous protocol of parameters. he quantitative description of the set of parameters is given in Table 4.1. Its average behaviour will be compared with expressions of the expression derived from the system of ODEs. he simulation presented here takes only the 841 genes with protein and mRNA production described in [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF], and we have ixed the number of free RNA-polymerases and ribosomes in order to compute the parameters; but the results of this subsection remain true with the other sets of parameters later presented.

he system of ODEs supposes that the volume growth is exponential with rate log 2/τ D . In Figure 4.1a, the volume of the cell indeed seems to grow exponentially in the simulation; the growth rate is centred around which corresponds to the expected a doubling time of τ D .

For each type of gene, Figure 4.1b shows the ratio between the protein production observed in the simulation divided by the protein production expected. It appears that the correspondence is correct, especially for the highly expressed proteins. It is less precise for the protein less expressed (which may be due in part to the longer time needed for their empirical mean to converge). Globally, the correspondence between the productions seems good enough.

he stochastic simulation displays relative quick timescale for the evolution of free RNA-polymerases (of the order of the second) and even quicker for the free ribosomes (insets of Figure 4 Computed from the stochastic simulations, the main Figure 4.1c and Figure 4.1d present the mean number of free RNA-polymerases and ribosomes as a function of the cell volume. he mean of each free entity is not constant during the cell cycle. he dashed lines represent the expected value of free entities given by the model of average production of this section (Equation (4.13)). It is indeed a good approximation for the behaviour of free RNA-polymerases and ribosomes. All these results support the idea that the expressions deactivated from the system of ODEs are a good way to describe the average behaviour of the stochastic model. In the next sections, interest in the results of the simulation in terms of distribution.

Impact of Free RNA-polymerases and Ribosomes

As said in Subsection 4.2.3, the parameter computation supposes the ixation of the average concentration of free RNA-polymerases and ribosomes. In this section we propose several simulations where the average concentration of these free entities are changed and see the inluence it has on their distribution and on the protein variability.

Few Free Ribosomes and Many Free RNA-polymerases

We begin with a simulation with a low concentration of non-allocated ribosomes as it seems a reasonable biological assumption. Indeed since ribosomes are composed of multiple subunits which comes with high costs for the cell, they are present in limited amount. Consequently, they are subject to a large competition between transcripts (see [START_REF] Warner | Economics of Ribosome Biosynthesis[END_REF] in the case of the yeast); therefore it is reasonable to take a low concentration of free ribosomes. At the same time, the parameters are setled in such a way that most of the RNA-polymerases are non-allocated: it appears in real cells at every instant, most of the RNA-polymerases are not speciically bound on the DNA [START_REF] Klumpp | Growth-rate-dependent partitioning of RNA polymerases in bacteria[END_REF]. In that, this simulation aims to represent a case tat is close to what happens in real cells.

Free RNA-polymerase and Ribosome Distributions

In these simulation, we look at the distributions of free RNA-polymerases and ribosomes. In Figure 4.1a and Figure 4.1b we show these distributions at three diferent phase in the cell cycle: we have selected cells of a given volume (either 1.40 om 3 ,1.95 om 3 or 2.50 om 3 , which correspond to the beginning, middle and end

P I P V 121 F Y D Y E Y,i E Y,1 E Y,K … … λ + F Y G/(K • V ) λ -• D Y λ 1,i G i F Y /V µ 1,i E Y,i
Non-speciically bound Free Speciically bound We have created an alternative version of the stochastic model where is introduced a third possible class for RNA-polymerases: RNA-polymerases can also bound non-speciically on the DNA. he binding rate is modelled as follows: at any time s, a free RNA-polymerase bind on the DNA at a rate that depends on the number of free RNA-polymerases F Y (s) and on the DNA concentration G(s)/(K • V (s)); the global rate is hence λ + F Y (s)G(s)/(K • V (s)) where λ + is a parameter that represents the natural ainity of RNApolymerases for the DNA. Once an RNA-polymerase is bound, it is released in a time represented by an exponential random variable of rate λ -(see (Figure 4.3)).

We performed a simulation where the parameters λ + and λ -are chosen such that around 90% of the RNA-polymerases are sequestered on the DNA at any time (as it was the case in [START_REF] Klumpp | Growth-rate-dependent partitioning of RNA polymerases in bacteria[END_REF]). he protein noise does not seem to be impacted in this case either.

Uncertainty in the Replication Initiation and Division

In the stochastic model as it was initially described, replication initiation and division occur when the cell reaches the respective volumes of V I and 2V 0 . In reality, these cell decisions are not exact. We propose here a modiication of the stochastic model that takes into account this aspect.

he replication and division decision is still a topic of research (for example, see [START_REF] Tyson | Sloppy size control of the cell division cycle[END_REF][START_REF] Wang | Robust growth of Escherichia coli[END_REF][START_REF] Soifer | Single-cell analysis of growth in budding yeast and bacteria reveals a common size regulation strategy[END_REF][START_REF] Osella | Concerted control of Escherichia coli cell division[END_REF]). For the division, one hypothesis (referred as "sizer model") is that the division decision depends on the current size of the cell (the size can refer to the mass or the volume, but as explained in Subsection 4.1.1, the density constrain [START_REF] Marr | Growth rate of Escherichia coli[END_REF] ensures the close proportionality between these two quantities). With this hypothesis, at every instant, the instantaneous probability to divide depends only on the current cell size. It appears that, at least in a irst approximation, the cell size distributions observed experimentally can be explained by this "size model" [START_REF] Robert | Division in Escherichia coli is triggered by a size-sensing rather than a timing mechanism[END_REF][START_REF] Osella | Concerted control of Escherichia coli cell division[END_REF]. It is therefore this framework that we have considered to represent the cell division decision.

At every moment s of the simulation, with a cell of volume V (s), we introduce an instantaneous division rate b D (V (s)) with b D is a positive function (it means that the probability to divide between times s and s+ds is given by b D (V (s)) ds ). he division decision is hence only volume dependent. he function b D is chosen so that the division occurs around the volume 2V 0 with V 0 = 1.3 om 3 and division precision can be ixed (for more information about the function b D , see Section 4.B).

Similarly for the replication initiation decision, the stochastic model initially described consider a ixed volume V I at which the DNA-replication is initiated. We introduce variability in this cell decision, in the same way as we do for the division: at every moment s, we consider a replication initiation rate b I (V (s)) such as the function b I is chosen in order to have a replication initiation that occurs around volume V I . We perform several simulation where we consider diferent function b D and b I in order to have diferent precision in the division and replication initiation decisions. All these aspects did not seem to have a determinant inluence on the protein variability.

Deterministic Time for Replication

When the stochastic model has been initially presented (in Subsection 4.1.2), we have proposed two ways to model the time of DNA replication initiation τ I . It can either occurs at a deterministic time ater the last division, or it can happen when the cell reaches the speciic volume V I . We have checked that this modelling choice has no signiicant inluence on the global dynamic of the system, in particular in the protein noise.

Conclusions on the Diferent Sources of Variability

In the two last chapters, we have investigated a series of models of increasing complexity that incorporate diferent cellular mechanisms that interfere with gene expression. he goal has been to propose a large description of many diferent possible contributions to the protein variability. Below, we sum up all the results we have obtained in these last two chapters.

• Transcription and translation (Section 3.3): to begin with, we have proposed a gene-centred model where only the mechanisms of transcription and translation are considered. he protein variability predicted is only due internal to gene expression mechanism itself, it is usually referred as "intrinsic noise". It has given us the basic model on which diferent external aspects have been added. Globally, the noise predicted by this model globally it the irst "intrinsic regime" predicted by [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] (characterised by a coeicient of variation, deined as the variance divided by the mean squared, inversely proportional to the mean). But the noise of highly expressed proteins seems still underestimated for this simple model.

• Division (Subsection 3.3.6): then the efect of division has been introduced. he binomial sampling of each protein appears to have potentially substantial additional variability for some proteins. Proteins with a low Fano factor (deined as the variance divided by the mean) of protein concentration have a signiicant increase in their variability: for the set of proteins studied, this efect can double the noise coming from the transcription and translation processes.

• Gene replication (Section 3.4): the third model has considered the replication of each gene at a certain point in the cell cycle. he consequence is, for each type of protein, to have the mean protein concentration that changes across the cell cycle. But the additional variability due to this efect is very small in regards to the heterogeneity induce by the protein production mechanism and the division.

• Fluctuations of ribosomes and RNA-polymerases (Subsection 4.3.1): in this chapter, we have observed the inluence of the global sharing of limited amount of RNA-polymerases and ribosomes in the production of proteins. An additional variability is observed but seems limited: the protein concentration increases its variance of at most 10% compared to the gene-centred model with gene replication. But globally, the gene-centred model approximates correctly the behaviour of the protein production. It is analogue to a mean-ield property, such as the one that was demonstrated in the simpler model of [START_REF] Fromion | A Stochastic Model of the Production of Multiple Proteins in Cells[END_REF].

• Other sources (Section 4.4): other possible sources of potential variability such as RNA-polymerases and ribosomes produced through gene expression, the non-speciic binding and random division and D S V 123 replication decision have been studied. It appears that none of these efects inluence signiicantly the protein production variability.

hrough this work, we have examined the most usual external sources of variability proposed in the literature. Of all the possible origin of extrinsic heterogeneity tested, the binomial sampling seems the prevalent one. As for the free ribosomes and RNA-polymerase luctuation, yet oten proposed as being the principal source of external noise [START_REF] Kaern | Stochasticity in gene expression: from theories to phenotypes[END_REF], Swain et al., 2002[START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF], their impact seems quite limited. he results of this chapter seems indeed to show that the only noticeable additional variability is due to the low concentration of free ribosomes and high concentration of RNA-polymerases, and that, in any cases, their contributions is inferior to the protein mechanism itself or on the binomial division.

-1 0 1 2 3

Prot. Adk At the end, it is not surprising that the protein proile and the global protein coeicient of variation for the multi-protein model (Figure 4.1) has litle diference with their counterparts of the previous models. In particular, the additional variability induced by ribosome luctuations cannot explain the second regime observed in [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] experiment: the protein coeicient of variation (the variance divided by the mean squared) still globally inversely scales the average production and there is no lower bound limit.

To understand this decay, we can propose two possible explanations: either an experimental bias when measuring the data, or another process not consider in our model. he exhaustive measures of [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] have not been fully replicated and covers a large range of luorescence intensities. As this efect C 4. M -M 124 mainly afects proteins with the highest luorescence, it is possible that some saturation induces a bias in the estimation of variance of highly produced proteins. Another possibility is to suggest that some important cellular aspect, not modelled here, can have an important impact on protein variance. Even if we have represented the mechanisms that are usually referred in literature as potential important sources of noise, the model proposed here is far from taking into account all the aspects of gene expression. Many other possible sources of heterogeneity can be proposed. For instance, changes in the availability of amino-acids in the medium can induce luctuations in the translation speed. Also, in the model of this chapter, we have considered the binding and initiation as a single event for both the transcription and translation. A more precise representation would be to describe them as two diferent processes [START_REF] Siwiak | Transimulation -Protein Biosynthesis Web Service[END_REF] gives for instance a median transcriptional initiation time of 15 s which is of the same order of magnitude as the elongation time). Another aspect not modelled here is the gene regulation procedure: the activation and deactivation of the gene can induce an additional variability in the protein production; it is even more true since the transcription factor is itself a protein and, as a consequence, is itself subject to variability. he assumption that every event occurs at exponential times can also be discuss: for instance, the elongation time would be beter represented as having an Erlang distribution (see Chapter 2 of Leoncini ( 2013)).

4.A Appendix: Gene Replication Times

In simulations, the time at which a gene is replicated is estimated as follows: we irst determine the time of DNA replication initiation (the time τ I in the cell cycle); the, as we consider that the DNA-polymerase replicates DNA at constant speed, we can deduce the time of replication of each gene only by knowing its position.

he article [START_REF] Wallden | Fluctuations in growth rates determine the generation time and size distributions of E. coli cells[END_REF] investigate the replication initiation; it appears that the initiation occurs at a relative ixed volume per replication origin, and thus independently from the time since the previous division. his relative volume seems, furthermore, constant for diferent conditions. For slow growing bacteria (with only one DNA replication per cell-cycle), such as those in [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF], the volume at which DNA replication initiation occurs is V I = 1.8 om 3 . he time of replication initiation τ I can hence either be considered as a deterministic or stochastic quantity:

• if the volume growth is considered as deterministic and exponential, such as V (s) = V 0 2 s/τD , then the time of initiation can be considered as being equal to

τ I = τ D log 2 log V I V 0 .
• we can also consider that the replication initiation is only volume dependent. One can trigger the initiation when the volume of the cell is around V I (as it is suggested in [START_REF] Wallden | Fluctuations in growth rates determine the generation time and size distributions of E. coli cells[END_REF] and which corresponds to the classical model suggested by [START_REF] Donachie | Relationship between Cell Size and Time of Initiation of DNA Replication[END_REF]). In that case, the moment of initiation τ I is stochastic: the probability to divide between s and s + ds is given by b I (V (s)) ds (with V (s) the volume at time s of the cell cycle). he function b I is chosen such as the average volume of initiation is indeed V I . his mechanism is comparable to the one use to determine the moment of division, in particular, b I is similar to b D (see Section 4.B).

In the previous chapter, in the model Section 3.4, as the volume growth is considered as deterministic, we use the irst method. In the current chapter, both of these possibilities have been tested without any signiicant diferences (see Section 4.4).

4.B. A : S D 125

Once known the initiation of DNA replication τ I , the remaining delay to gene replication of the gene i is considered as deterministic as we consider the speed of DNA replication as relatively constant. We consider that the whole chromosome is replicated in around 40 min [START_REF] Grant | DnaA and the timing of chromosome replication in Es-cherichia coli as a function of growth rate[END_REF]. As a consequence, the distance of the gene from the origin of replication is suicient to determine the time it takes for the DNA-polymerase to replicate it. he position of each gene was determined with Ecogene database [START_REF] Zhou | EcoGene 3.0[END_REF].

4.B Appendix: Stohastic Division

he previous simulations considered that division as occurring exactly when the cell reaches the volume 2V 0 . In real cells, the division is not exact. he cell division decision has been studied in the literature [START_REF] Tyson | Sloppy size control of the cell division cycle[END_REF][START_REF] Sharpe | Bacillus subtilis cell cycle as studied by luorescence microscopy: constancy of cell length at initiation of DNA replication and evidence for active nucleoid partitioning[END_REF][START_REF] Wang | Robust growth of Escherichia coli[END_REF][START_REF] Osella | Concerted control of Escherichia coli cell division[END_REF], Robert et al., 2014[START_REF] Soifer | Single-cell analysis of growth in budding yeast and bacteria reveals a common size regulation strategy[END_REF].

In this literature, it clearly appears the need for a division decision that takes into account, at least partially, the cell volume in order to maintain robust size distributions. If the division mechanism only relies on time in the cell cycle, small variations in the growth produce generations of cells with unstable size distribution.

We propose in the model of the chapter a division that only depends on the volume: at a time s, if the cell is of volume V (s) the rate of division is given by b D (V (s)). We want to choose b D such as the distribution of cell size at division would be a log-normal distribution centred around 2V 0 . he parameters of this lognormal distribution are referred as µ and σ. he parameter σ that indicates the spreading of the log-normal distribution is considered as setled (we have studied the impact on protein variability of this parameter by making several simulation with diferent value for it, see Section 4.4). As a consequence, the parameter µ can be determined in order to have a distribution centred around 2V 0 .

Let's now propose a function of division rate b D in order to obtain such log-normal distribution for the cell division that gives such log-normal distribution for the volumes at division. To do so, let's place in a particular cell and we consider that it grows exponentially at rate α with v(0) its volume at birth: in that case we would have v(s) = v(0)e αs (in the case of this chapter, we take α := log 2/τ D ). With the (Equation (4.17)), it follows that

d D (v) = b D (v) αv .
As the function d D is known to be a log-normal distribution of parameters µ and σ, we now have an expression for b D .

4.C Appendix: Simple Models for Transcription and Translation

he article [START_REF] Fromion | A Stochastic Model of the Production of Multiple Proteins in Cells[END_REF] proposes a multi-protein model for translation, with a shared limiting number of ribosomes, where each type of mRNA is supposed to be in constant quantities and where the maximum number of ribosomes on one single mRNA is limited. he system also evolves in a ixed volume as it is the case for classical models.

In order to have a prediction for the number of respectively free RNA-polymerases and free ribosomes; we consider two analogue models that are slightly simpliied versions of the model of [START_REF] Fromion | A Stochastic Model of the Production of Multiple Proteins in Cells[END_REF]. he two analogue models respectively represent the transcription and translation part; they are completely independent.

he goal is, for each of the model, to provide to reproduce the equivalent of the irst results of [START_REF] Fromion | A Stochastic Model of the Production of Multiple Proteins in Cells[END_REF] and we will show that the expected distribution of free RNA-polymerases (or free ribosomes) is binomial in these simpliied cases.

Model for Transcription

As explained in Subsection 4.3.1, one can interpret the model of [START_REF] Fromion | A Stochastic Model of the Production of Multiple Proteins in Cells[END_REF] as taking place in a ixed volume V (it would correspond to a small portion of the cell cycle in the stochastic model of this chapter, a portion where the volume of the cell does not change much). We also consider that the gene copy of each unit of production remains constant; as a consequence, the gene copy number of the i-th gene G i is constant and known. As in the stochastic model of the chapter, and contrary to the model of [START_REF] Fromion | A Stochastic Model of the Production of Multiple Proteins in Cells[END_REF], we consider that there is no limiting number of elongating RNA-polymerases on one gene.

In a pool of K genes, let's denote by N Y the constant total number of polymerases. We consider the random variables E Y,i for i ∈ {1, ..., K} be the number of RNA-polymerases atached to the i-th gene. As a consequence, the random variable

F Y := N Y - K ∑ i=1 E Y,i (4.18)
is the number of free RNA-polymerases in the system. he process X(t) = (E Y,i (t), i ∈ {1, ...K}) takes place in the state place S the subset of N K such as

S := { x ∈ N K , K ∑ i=1 x i ≤ N Y } .
It means that there is at most N Y RNA-polymerases that can be atached to genes at the same time. We can describe the Markov process transition by the following Q-matrix: by seting the vector e i = (δ i ′ =i ) i ′ ∈{1,...,K}

4.C. A

: S M T T 127

(δ is used here as the Kronecker delta), for any x, y ∈ S,      q(x, x + e i ) = λ 1,i G i λ 1,i f (x)/V for any i ∈ {1, ...K}, q(x, xe i ) = µ 1,i x i for any i ∈ {1, ...K}, if x i > 0, q(x, y) = 0 if ∥x -y∥ > 1.

where

f Y (x) := N Y - K ∑ i=1
x i the number of free RNA-polymerases. Equation (4.18) leads in particular to f (xe i ) = f (x) + 1 for all i.

We search here an invariant reversible probability measure π for the Markov process, that is to say a measure that veriies for any i ∈ {1, . . . , K} :

π(x)µ 1,i x i = π(x -e i ) • λ 1,i G i (f Y (x) + 1) /V. (4.19)
Proposition 4.2. he invariant measure π of the number of RNA-polymerases in each gene has the following form

π(x) = 1 Z • 1 f Y (x)! K ∏ i=1 (G i λ 1,i /(V µ 1,i )) xi x i !
for any x ∈ S and with Z > 0 the normalisation constant.

Proof. Let's consider such a distribution π and verify that it veriies Equation (4.19). For a gene i ∈ {1, ..., K}, we take x ∈ S such that xe i ∈ S .

π(x)µ 1,i x i = 1 Z • 1 f Y (x)! K ∏ i ′ =1 (G i ′ λ 1,i ′ /(V µ 1,i ′ )) x i ′ x i ′ ! µ 1,i x i = 1 Z • 1 f Y (x)! K ∏ i ′ =1 (G i ′ λ 1,i ′ /(V µ 1,i ′ )) x i ′ x i ′ ! • x i G i λ 1,i /(V µ 1,i ) • G i λ 1,i V = 1 Z • 1 (f Y (x) + 1)! K ∏ i ′ ̸ =i ( (G i ′ λ 1,i ′ /(V µ 1,i ′ )) x i ′ x i ′ ! ) • (G i λ 1,i /(V µ 1,i )) (xi-1) (x i -1)! • G i λ 1,i V (f Y (x) + 1) = π(x -e i ) • G i λ 1,i (f Y (x) + 1)/V.
So the measure π indeed veriies Equation (4.19).

We can now derive from the previous proposition the equilibrium distribution of F Y , the number of free RNA-polymerases of the process. with Λ deined such as

Λ Y := K ∑ i=1 G i λ 1,i /(V µ 1,i ).
It means that F Y follows a binomial distribution B (ϕ, N ) for which ϕ = (1 + Λ Y ) -1 and N = N Y .

Remark 4.2. For a given volume V , the average number of free polymerases is

E [F Y ] = N Y 1 + Λ Y = N Y 1 + K ∑ i=1 G i λ 1,i /(V µ 1,i )
.

It is identical to the number of free RNA-polymerases obtained in a cell of volume V in the deterministic model (see Equation (4.13)).

Proof of Proposition 4.3. Now we search the distribution of the random variable F Y , the number of free RNApolymerases. Indeed, with Equation (4.19), it follows that for a given n ∈ {1, ..., N Y }:

P [F Y = n] = ∑ x∈S π(x) ∑ i xi=NY -n = ∑ x∈S 1 Z • 1 n! K ∏ i=1 (G i λ 1,i /(V µ 1,i )) xi x i ! ∑ i xi=NY -n = 1 Z • 1 n! ∑ x∈S K ∏ i=1 (G i λ 1,i /(V µ 1,i )) xi x i ! ∑ i xi=NY -n = 1 Z FY • 1 n! P [ K ∑ i=1 Gi ∑ k=1 C i,k = N Y -n ]
with ∀i ∈ {1, . . . , K} and ∀k ∈ {1, . . . , G i }, C i,k ∼ P(λ 1,i /(V µ 1,i )).

Since the random variables C p,k are following a Poisson distribution, so does their sum with the parameter Λ := ∑ K i=1 G i λ 1,i /(V µ 1,i ). We can, moreover, ind an expression for Z FY :

1 = NY ∑ n=0 P [F Y = n] = 1 Z FY • NY ∑ n=0 1 n! P [C 1,1 = N Y -n] = 1 Z FY • N Y ∑ n=0 1 n! e -Λ Λ NY -n (N Y -n)! = 1 Z FY • 1 N Y ! e -Λ • NY ∑ n=0 ( N Y n ) Λ NY -n so Z FY = e -Λ NY ! • (1 + Λ) NY .

  Aspects of Gene Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Variability in Gene Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Mathematical Modelling for Protein Production . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 A Canonical Example: the hree-Stage Model . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Other Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Limitations of Classical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendices 1.A Appendix: Useful Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  Figure 1.2: Schematic steps of transcription and translation

  Figure 1.3: Gene activation and deactivation through a repressor.

  Figure 1.4: mRNA degradation process in E. coli.

  Figure 1.6: Dual reporter technique principle. (a): on the same chromosome, there are two genes with the same promoter, each coding for a luorescent protein emiting a diferent wave length. (b): uncorrelated production give cells that express more of one of the luorescent molecules. Above: a hypothetical situation showing fully correlated protein production; below: uncorrelated protein production (inspired by Elowitz et al. (2002)).

Figure 2

 2 Figure 2.1: Simulations: Protein distribution with and without autogenous regulation with a ixed mean number of proteins of 178.

]

  Figure 2.2: Simulations: Evolution of the ratio Var [ P N F ] /E [ P N F ] as a function of N .

  Figure 2.3: Simulations: Probability Distribution of the Number of Proteins with Regulation on Gene or on mRNAs, µ -12 is the average lifetime of an mRNA. he average number of proteins is 1400.

  Figure 2.4: Power spectral density estimation of signals with and without regulation

  Initial Point at 1400, equilibrium at 290.

Figure

  Figure 2.5: Simulations: Evolution of the Mean Number of Proteins

  of the 2000s, luorescent microscopy experiments permit to quantitatively measure cell by cell gene expression (see for instance[START_REF] Elowitz | Stochastic Gene Expression in a Single Cell[END_REF],[START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF][START_REF] Valgepea | Escherichia coli achieves faster growth by increasing catalytic and translation rates of proteins[END_REF]).

  Protein production coeicient of variation.

Figure 3 . 1 :

 31 Figure 3.1: Results on mRNA and protein production in the article Taniguchi et al. (2010). (a): In the experimental dataset, a large population of cells are considered; each cell i of the population has a speciic number of proteins p i and a speciic volume v i ; these values are used to compute empirical mean µ m and variance σ 2 m . (b): mRNA production coeicient of variation (σ 2m /µ 2 m ) as a function of the average mRNA expression µ m for every gene. he CV is inversely proportional to mRNA mean concentration, but it is higher than expected for Poisson distributions (red dashed lines). (Measures have been made using FISH technique on 137 mRNA types.) (c): Protein production CV function of the average protein copy number for every gene. For low expressed proteins (mean protein number < 10), the CV is inversely proportional to the average protein production, this part is considered lowered by an "intrinsic noise limit" (red dashed line). For genes with higher protein expression, the CV becomes in independent of the protein expression level, protein expression is here denoted as dominated by the "extrinsic noise" (blue dashed line).

Figure 3 . 1 :

 31 Figure 3.1: Volume in models with cell division. he volume V increase as the cell grows. At division, the model follows only one cell, segregation occurs on the compounds (mRNA or proteins in green) to know if there are in the considered daughter cell.

  of one type of protein.

Figure 3 . 2 :

 32 Figure 3.2: Model with cell cycle. (a) he model of this section considers genes independently from each other; the three mechanism of transcription, translation and division are represented. (b) For one particular type of protein, the number of mRNAs and proteins are respectively M and P (see main text for more details).

  .3) his description of the dynamic of M s , together with the equilibrium hypothesis which implies that M 0 D = M τ D , allows to prove the next proposition. Proposition 3.1. At equilibrium, the mean number of mRNAs at time s ∈ [0, τ D [ of the cell cycle is

  then deine the global variance of mRNA concentration Var [M /V ]. On can look at how much the concentration M s /V (s) deviates from the global average E [M /V ] at any time s of the cell cycle: E

  Figure 3.4: Result of simulations of the model with cell cycle, compared with Taniguchi et al. (2010) experiments. (a): Normalised protein concentration proile over the cell cycle for three representative proteins. he thick line represents the normalised mean concentration over the cell cycle E [P s /V (s)], and the coloured areas represents the standard deviation. As predicted, the mean protein concentration remains constant over the cell cycle. Furthermore, we observe that the relative variance is higher for the less expressed proteins such as YjiE. (b): coeicient of variation (CV) of the protein concentration (deined as Var [P /V ] / E [P /V ] 2 ) as a function of the average protein concentration. It appears that the CV predicted by the simulation scales approximately inversely with the average protein production. hough, unlike Taniguchi et al. (2010) experiments, indicated by the yellow area (corresponding to the point cloud of Figure 3.1c), there is no lower plateau for highly expressed proteins.

  Figure 3.4b. he igure shows the protein CV (deined as Var [P /V ] / E [P /V ]2) against the average protein concentration E [P /V ]. It appears that the noise approximately scales inversely the average protein concentration like in the irst "intrinsic noise" regime of[START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF]. But unlike in[START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] experiment, there is no lower plateau for highly expressed proteins.

  Protein concentration proile for one gene in the two models.

Figure 3 . 5 :

 35 Figure 3.5: Efect of the binomial sample on the protein concentration noise. (a): he model difers from the model with cell cycle (Figure 3.2b) only through division: at division the mRNAs and proteins undergo a binomial sampling. (b): Fractions of the coeicient of variation (CV) of protein concentration in simulations with exact division over simulations with binomial division. he efect of the binomial division can represent up to 50% of the total CV (proteins with low Fano factor). A simpliied model of the ratio of noise (cyan dashed line) reproduces this behaviour (see Section 3.C). (c): proile example of the gene OmpC, the central line represents the mean protein concentration over the cell cycle, and the coloured areas represent the standarddeviation in both models.

  Figure 3.1: Model with cell cycle and gene replication. (a) Four biological mechanisms are represented: DNAreplication, transcription, translation and cell division. (b) For one particular type of protein, the number of mRNAs and proteins are respectively M and P ; events occur at stochastic times that depend on parameters

  E [P s ] and Var [P s ] of the theorem still depend on E [P 0 ], Var [P 0 ], Cov [P 0 , M 0 ] and Cov [P τR , M τR ].

  Figure 3.3: Protein proile. (a): Protein proile concentration of Adk. he mean concentration E [P s /V (s)]is not constant across cell cycle and luctuates across the global average E [P /V ] (in red). he large standard deviation of P s /V (s) (coloured area) indicates a large term Var 1 [P i /V ] in the environmental state decomposition. (b): proile of a modiied version of AdK. In this version, there is a higher number of mRNAs (approximately ten times more) that last less time. he efect is a higher term Var 2 [P i /V ] in the environmental state decomposition (main igure), but the it is not enough to clearly separate between the distributions at birth (at time s = 0) and at the replication of the gene (at time s = τ R ) (inset). (c): Show the ratio

C

  Figure 3.4: Result of simulations of the model with cell replication, compared with Taniguchi et al. (2010) experimentation. Figure3.4a: normalised protein concentration proile over the cell cycle for three representative proteins. he thick line represents the mean concentration over the cell cycle, and the coloured area represents the standard deviation. he proile shows no signiicant diferences with the previous models (see Figure3.4a for the more details on Adk). Figure3.4b: coeicient of variation (CV) of the protein concentration as a function of the average protein concentration. here are, once again, no signiicant diferences as before (see Figure3.4b). In particular, there is no lower "extrinsic" noise plateau for highly expressed proteins, as it is the case forTaniguchi et al. (2010) (yellow area). Inset: ratio between the CV of protein concentration of the model with binomial sampling (Subsection 3.3.6) divided by the one of this section. Gene replication adds no particular noise compared with the model with only binomial sampling.

  4. Update: choose between events of Step 2 or Step 3 that happen irst. Update the time and the molecule count accordingly. 5. Iterate: Consider again the Step 2 unless it has reached the end of the simulation. Appendix: Means, Variances and Covariances of (M 0 , P 0 ) and (M τ R , P τ R ) he Subsection 3.4.3 was considering protein number the model with cell cycle and gene replication. he subsection ends up to the heorem 3.4 that gives expressions for the mean and the variance of P s for any time s of the cell cycle depending on E [P 0 ], Var [P 0 ], Cov [P 0 , M 0 ] and Cov [P τR , M τR ].

  comes the result for Cov [P 0 , M 0 ]. For Cov [P τR , M τR ], one can simply use the expression for Equation (3.19) with s = τ R since the quantity E [M 0 P 0 ] is now known.

  Biological mechanism for three typical genes.

Figure 4 . 1 :

 41 Figure 4.1: Multi-protein model. (a) he model of this chapter considers interdependent genes through the common sharing of RNA-polymerases and ribosomes. (b)For i-th type of protein, the number of genes, mRNAs and proteins are respectively G i , M i and P i ; elongation event depends on free RNA-polymerase concentration F Y /V and free ribosome concentrations F R /V (see main text for more details).

  Ratio of average protein concencentration(b) Protein production in simulations and in experiments.

  Figure 4.1: Average correspondence of the stochastic model to the system of ODEs. (a): a simulation sample that shows that cells grow exponentially from around V 0 up to around 2V 0 (inset); the growth rate distribution is centred around the expected growth rate log 2/τ D (main igure). (b): Ratio between the average protein concentration in simulation and in experiments. (c) and (d):the mean of respectively of free RNA-polymerases and ribosomes at each moment of the cell cycle in the simulations (solid lines) and is predicted by the system of ODEs (dashed lines). Insets: example of dynamics of respectively free RNA-polymerases and ribosomes for one simulation.

C

Figure 4

 4 Figure 4.3: RNA-polymeses dynamics in the case of non-speciic binding. An RNA-polymerases can be either among the speciically bound on the DNA (in E Y,i for i ∈ {1, ..., K}) or free (among F Y ) or non speciically bound on the DNA (among D Y ).

Figure 4 . 1 :

 41 Figure 4.1: Results of simulations of the multi-protein model, compared to Taniguchi et al. (2010) experimentation (dataset of (Subsection 4.3.1), but similar for all model variations). (a) Normalised protein concentration proile over the cell cycle for three representative proteins. (b) Coeicient of variation (deined as the variance divided by the mean squared) of the protein concentration as a function of the average protein concentration. here is no particular diference with the model with cell cycle and gene replication (see Figure 3.4). It does not replicate Taniguchi et al. (2010) experiments, indicated by the yellow area (corresponding to the point cloud of Figure 3.1c), especially for highly expressed proteins.

  Let's denote by T D and V D respectively the time and the volume at division. he goal is to determine the distribution of V D depending on the division function b D . Proposition 4.1. For an exponentially growing cell, the distribution of V D , the volume at division, has its probability density function d D that depends on the division function b D such as d D (v) = b D (v) αv at any volume v. Proof. A rate of division b D (v(s)) is inhomogeneous, the distribution of the time of division T D is therefore given by ∀t > 0 P [T D > t] V D = v(0) exp (αT D ). As d D denotes the probability density function of V D , it comes that for any v > v(0) d D (v) = d (P [V D < v]) dv =

  Proposition 4.3. he number of free polymerases F Y followsP [F Y = n] = ( N Y n ) Λ N Y -n Y (1 + Λ Y ) NY ,

  Stochastic Models of Protein Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 he Classical Model of Protein Production . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 A Stochastic Model of Protein Production with Autogenous Regulation . . . . . . . . . 2.3 A Scaling Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Scaling of the Classical Model of Protein Production . . . . . . . . . . . . . . . . . . . 2.3.2 Scaling of the Production Process with Feedback . . . . . . . . . . . . . . . . . . . . . 2.4 Fluctuations of the Number of Proteins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Numerical Values of Biological Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Impact of Autogenous Regulation on Gene Expression . . . . . . . . . . . . . . . . . . 2.5.3 he Limiting Scaling Regime as a Lower Bound . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Regulation of the Production Process on mRNAs . . . . . . . . . . . . . . . . . . . . .

	C	1. I	10

2 Model of Protein Production with Feedbak 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Biological Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Results of the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 2.5.5 Impact of Feedback on Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  .A.2.

	C	2
		Name Number of coding gene Total number of proteins Total number of RNA-polymerases Total number of ribosomes mRNA elongation speed Total number of mRNAs M P	Symbol K P M N Y N R c Y F	Value 4000 2.6 × 10 6 39 Nucl/s 6.8 × 10 3 1.5 × 10 3 Bremer and Dennis (1996), Table 3, Source Blatner et al. (1997) for a time of division of 100 min (the closest to the doubling time of Taniguchi et al. (2010)) 1.4 × 10 3 Neidhardt and Umbarger (1996), Table1 P
		Protein elongation speed	c R	12 aa/s
		mRNA average lifetime Doubling time	τ m τ D	4 min 150 min	Taniguchi et al. (2010)
		Table 1.A.1: Useful numbers in E. coli.
		Name		Expression	Value
		Transcriptions per second		M /τ m	6 s -1
		Translations per second Average mRNA number per genes	P /τ D M /K	3 × 10 2 s -1 0.32
		Average proteins number per genes mRNA number produced in one cell cycle M • τ D /τ m P /K	6.0 × 10 2 5.2 × 10 4
		Table 1.A.2: Rate of events and other quantities deduced from Table 1.A.1.

Table 2

 2

	Regulation on	Gene	mRNAs/8 min mRNAs/40 min
	Mean number of mRNAs	10.33	19.74	99.04
	Mean number of Active mRNAs	10.33	9.77	9.81
	Mean number of Proteins	1403.63	1400.29	1403.36
	Standard Deviation of number of Proteins	92.66	84.22	59.04

Table

Table 2

.1 shows that the mean number of mRNAs produced per unit of time is essentially the same in all experiments as well as the mean number of active mRNAs. It should be noted the impact of regulation on mRNAs for the standard deviation of the number of proteins when the mean life time is 8 min is not really signiicant (10% gain) than the regulation on the gene. When the mean lifetime is 40 min the improvement, 36%, of the standard deviation becomes signiicant, showing that in this case the production process is "smoothed" by this mechanism. he three distributions of the number of proteins of these experiments are presented in Figure

2

.3. .1: Comparison of Regulation Processes on Gene or on mRNAs with Diferent Lifetimes

  y)∈∆s N (dx, dy) .

	3.3. M	C	C			63
				Birth	¢ s	Division
			mRNA lifetime	± i
				0 0	Time t i	s	t i + ± i	¿ D
	Figure 3.3: Illustration of the Marked Point Poisson Process describing the dynamic of mRNAs: each mRNA is characterised by the point (x
						(3.2)

i , y i ) (with (x i , y i ) following the MPPP N , whose distribution is of intensity ν). he random variable x i represents the time at mRNA creation and y i its lifetime, hence this mRNA exists from volume x i up to volume x i + y i ; that is to say that mRNA is still present at time s, if and only if the point

  .2: he illustration of the Marked Point Poisson Processes that describe the dynamic of mRNAs: each mRNA is characterised by the point (t i , δ i ), with (t i , δ i ) following MPPP N or N ′ . Both MPPPs are of intensity ν. he random variable t i represents its birth time and δ i its lifetime, hence this mRNA exists from time t i up to time t i + δ i . he only diference for the two processes is the starting time: the process N in (a) begins at birth (in particular, an mRNA is still present at time s, if and only if the point (t i , δ i ) is in the set

  ) . corresponds to mRNAs that were created by the second copy of the gene (replicated at time τ R ) and which are still present at division. As before, N ′ ( ∆ ′

	3.4. M	C	C	G	R	75
	he third term τD-

  Var [P τR ] only depending on E [P 0 ] and Var [P 0 ]. Indeed, since almost surely it comes that

	P τR-	a.s.

Table 3 .

 3 1: uantitative summary of the parameters in min (*: this value of the gene yjiY is biologically unrealistic ; maybe due to an error on the measure of its mRNA in[START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF]

  1,i and β Y . With Equation (4.13) and , if we consider the vector [β Y , λ 1,1 , ..., λ 1,K ], it can be considered as a solution of the system

	C	4. M	-	M	112
	Equation (4.14)

  K}.Since f Y , m i and g i (s) have already been setled, we can use a ixed point optimisation procedure to determine β Y and all λ 1,i . hen, as these parameters are determined, we now have an explicit expression for f Y (s) for any time s of the cell cycle.Let's inish with parameters relevant to translation, that is to say λ 2,i and β R . Here again, we use a ixed point optimisation procedure to deliver the result. With Equation (4.13) and the expression of p i in Equation (4.14), if we consider now the vector

	(4.15)

  .1c and Figure 4.1d). Average protein conc. [copies/µm 3 ] in sim.

	D		M	P	P							113
	Distribution		Time [min] 0 200 400 600 800 1.2 1.6 2.0 2.4 Volume [µm 3 2.8 ]
	0.96 • log(2) τD	log(2) τD	1.04 • log(2) τD	1.08 • log(2) τD	10 -2	10 -1	10 0	10 1	10 2	10 3	10 4
		Growth rate
	(a) Growth rate distribution and volume

  Table4.1: uantitative summary of the parameters in min. (*: show litle changes with other choice of f Y and f R ; **: this value of the gene yjiY is biologically unrealistic, maybe due to an error on the measure of one type of mRNA in[START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF]; removing this aberrant value does not change the simulations).

	4. M	-	M				114
		uantity		Param.	Median	Mean	Maximum	Minimum
	Waiting time for a transcription per gene*	(λ 1,i f Y ) -1	65.1	3.90 • 10 2	3.22 • 10 4	0.69
	Waiting time for a translation per mRNA*	(λ 2,i f R ) -1	0.57	9.75	1.75 • 10 3	8.79 • 10 -3 **
		mRNA lifetime	σ -1 1,i	5.15	6.63	52.1	0.91
		mRNA elongation	µ -1 1,i	0.41	0.49	1.97	7.05 • 10 -2
		Protein elongation	µ -1 2,i	0.44	0.53	2.14	7.64 • 10 -2

[START_REF] Paulsson | Models of stochastic gene expression[END_REF] considers a case with possibly multiple copies of the same gene, but for the sake of simplicity, we consider the case of one gene copy.

Only one replication during the cell cycle is considering here, as it is the case for the slowly growing bacteria of[START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF]. But the work of this section can be generalised for more than one replication during the cell cycle.

he model presented here would rather be a luid limit model. Since the stochastic model is non-linear it is theoretically not corresponding to the average production of the stochastic model. Nonetheless, we will see in Subsection 4.2.4 that it is still a good prediction of the behaviour of the average protein production.

Remerciements

with the models of the previous section and that this small contribution will appear mainly due to the low number of free ribosomes and high number of RNA-polymerases. In Section 4.4 we will present the results of simulations with diferent diferent modelling choice; we will show that they have globally no apparent impact on the protein heterogeneity.

Description of the Main Model

he aim of this model is to integrate the production of all proteins of the cell and their interactions. It is therefore a model at the scale of the whole bacteria: all genes, mRNAs and proteins are considered, as well as all ribosomes and RNA-polymerases. We are interested in the intertwined efects between local units of production of a particular type of proteins, and the global behaviour of common quantities such as the number of free RNA-polymerases and ribosomes. In this section, we present the model: irstly in Subsection 4.1.1 are presented the main biological aspects included in the model, then in Subsection 4.1.2 are described in detail all the mechanisms and notation used in the model of the chapter.

Main Features of the Main Model

he introduction of RNA-polymerases and ribosomes has several consequences on the model: several features have to be added or changed in order to have a consistent representation of the cell. We present below these diferent aspects. First of all, RNA-polymerases and ribosomes are explicitly present in the model. As these macromolecules are shared among all types of proteins, we cannot consider each gene as independent from each other as it was the case in the previous chapter. One therefore has to take into account all the diferent types of genes, mRNAs and proteins of the cell altogether; and the production of each type of proteins depends on the availability of RNA-polymerases and ribosomes. In the model, we will suppose that there is no notion of operons (an operon is a single mRNA strand on which several genes are coded): each gene is therefore considered as having its own speciic promoter.

In the model, RNA-polymerases can be allocated to a gene or not: if it is an allocated (or sequestered), then it is speciically bound on a gene in a transcription process; if it is non-allocated (or free), then it is either moving freely in the cytoplasm or is sliding on the DNA non-speciically (the sliding on the DNA has been proposed as taking part in the kinetics of promoter binding [START_REF] Kabata | Visualization of single molecules of RNA polymerase sliding along DNA[END_REF]). In the irst part of the chapter, we will gather all these non-allocated RNA-polymerases into one single group of free RNA-polymerases (in Subsection 4.4.3 we will study the case where there are two separate cases for the cytoplasmic and the nonspeciically bound polymerases).

As explained in Section 1.1, in order to produce an mRNA, the RNA-polymerase has to bind on the gene promoter, initiate the transcription; then elongation occurs in which the mRNA chain is polymerised; inally, the termination releases both the RNA-polymerase and the mRNA in the medium. In the model of this chapter, we separate this process in two parts: the binding and initiation on one side and the elongation and termination on the other side.

As the binding and the initiation are gathered in one single event in the model, one has to represent its rate of occurrence. he probability to bind on a speciic promoter depends on the concentration of free RNApolymerases: the same number of free RNA-polymerases has a lower tendency to bind on a promoter as the volume of the cell is higher. It also depends on promoter speciic aspects such as its sequence ainity for RNA-polymerases or the chromosome architecture. Also, as the DNA is replicated, the gene has twice more promoters, hence increasing the occurrence this encounter event. As a consequence, the binding and initiation of an RNA-polymerase on a speciic gene is gathered in the model as a single event whose rate is considered as depending on three quantities: In order to interpret the observed distributions of free RNA-polymerases and ribosomes at a certain extend, we can propose a simpliied model of RNA-polymerase and ribosome allocation (it is greatly inspired by the model described in [START_REF] Fromion | A Stochastic Model of the Production of Multiple Proteins in Cells[END_REF]). It is a simpliication of the stochastic model of the chapter, mainly in that translation and the translation are considered separately, and that there is no notion of cell growth. he idea would be to approach the "local" equilibrium of RNA-polymerases and ribosomes before any signiicant change in the volume.

his simpliied description predicts that for a given volume V , the distribution of free RNA-polymerases and ribosomes would be both a binomial distribution (see their parameters in Section 4.C). hese predicted binomial distributions are ploted in Figure 4.1 in thick lines. In the RNA-polymerase case, the binomial distribution globally it the histograms. he ribosome distribution is denatured: the parameters of the binomial distribution (N, ϕ) are such that ϕ ≪ N . It is due to the low concentration of free ribosomes chosen for the parameters computation. But even this denatured case shows a good correspondence between the binomial distribution and the simulation histograms.

Noise of Proteins

By performing the simulations, the global noise of each protein concentration is measured. In order to estimate the variance added by the interactions between the diferent protein production units, we compare this noise with the one obtained in the gene-centred model of Section 3.4 (with cell-cycle, binomial division and gene replication). his subsection compares these two models.

Figure 4.2a shows, for each gene, the variance of protein concentration in the gene-centred model, divided by the one in the multi-protein model. It appears that 90% of the genes have a variance ratio above 0.9 (the mean of the ratio is 0.96 in the set of genes). It means that the interactions between protein productions only represents at most 10% of variability.

his good concordance between the two models in terms of protein average expression (Figure 4.1a) and the protein variance (Figure 4.2b) is conirmed by the general aspect of the protein proiles. Taking the example of protein FabH, Figure 4.2b shows a comparison of its proiles between both models: the igure shows, for each volume of the cell cycle, the mean and from either side the standard deviation of protein concentration. he evolution of mean protein production (the thick lines in the igures) difers: the efect of gene replication In blue is the proile obtained in the multi-protein model through simulation; in cyan the proile obtained in the model of section aSection 3.4. Even if the standard deviation seems similar, the proile seems less sensible to the cell-cycle than in the case of the previous model. his result seems to support the idea that globally, the gene-centred model is a correct irst approximation of the dynamic of protein concentration during the cell cycle. his reassembles to a mean-ield property where the interdependent productions of protein can be approximated by independent processes [START_REF] Fromion | A Stochastic Model of the Production of Multiple Proteins in Cells[END_REF] proved such result in the case of their own model).

his simulation with low abundance of free ribosomes and a large concentration of free RNA-polymerase seems, as previously explained, relatively biologically pertinent. In what follows, all the simulations take the set of parameters of this subsection and change one particular simulation aspect for each of them.

Inluence of Free RNA-polymerase Concentration

In this subsection, we interested in the efect of the abundance of free RNA-polymerases on the protein variability. We have produced a series of parameters where the average concentration of free RNA-polymerases was ixed successively to 1, 10, 100 and 1000 copies/om 3 . In each case, we have deduced a set of parameters, where the ainity constants λ 1,i , λ 2,i are still calculated in such a way that average mRNA and protein concentrations still correspond to the experimental measures.

In Figure 4.3 are shown the results for a very low free RNA-polymerase concentration. As the number of free RNA-polymerases is low, its distribution reassembles the distribution of ribosomes (see Figure 4.3b) and is still well predicted by the simpliied model (presented in Section 4.C). he variance of protein seems to decrease as the free RNA-polymerase concentration is lower. he gap between between the multi-protein model and the gene-centred model is reduced: now, on average the variance ratio is 0.98 (90% of the genes have a variance ratio above 0.92).

One the contrary, a very high number of free RNA-polymerases show similar results as in Subsection 4.3.1.

Inluence of Free Ribosome Concentration

Analyse similar to the previous section has been performed for the case of free ribosomes: we computed a set of parameters based on average concentrations of non-allocated ribosomes of 1, 10, 100 and 1000 copies/om 3 . It can be irst remarked that for very high free concentrations, the binomial it of the simpliied model (described in Section 4.C) is not relevant to describe the free ribosome distribution.

In this case again, changes to the average concentration of free ribosomes show a litle but noticeable diference. As the average concentration of free ribosomes increases, the variance of each protein decreases. As shown in Figure 4.4a, for a concentration of 1000 copies/om 3 , the variance of the multi-protein represent on average 0.98 of the one predicted by the gene-centred model (90% of the genes have a variance ratio above 0.93).

Fluctuations in the number of free ribosomes seem the be the main source of the additional variability observed in the multi-protein model; and this efect seems less important as the number of free ribosomes is high. But in real bacteria, the number of free ribosomes usually seems quite low due to the high cost of ribosome production; then, the a low number of free ribosomes (like in the simulation of Subsection 4. seems more plausible than this simulation.

To conirm the speciic inluence of luctuations of ribosomes on the protein variability, we have performed a simulation with a modiied version of the model. he multi-protein model has been changed in such a way that the concentration of non-allocated ribosomes is ixed as constant during the whole simulation (meanwhile the free RNA-polymerases are still luctuating). Results about protein variability are similar of what is shown in Figure 4.4a: the variance of each protein concentration is equivalent to what was described by the genecentred model. he conclusion of this section is that the interaction between the diferent productions of proteins add litle additional noise to the model: in the best case, the gene have an increase of 10% of variability compared to the case where all the production are considered independently. his additional variability seems to be less important as the concentration of free RNA-polymerases is low and the concentration of free ribosomes is high.

Other Possible Inluence on Protein Variability

In this section, based on the set of parameters of the simulation Subsection 4.3.1 (with few free ribosomes and more RNA-polymerases), we make variations on some modelling choices for some cellular mechanisms: a large set of genes, RNA-polymerases and ribosomes as a result of gene expression, the introduction of RNApolymerase non-speciic binding on the DNA, considering uncertainty in the division and DNA replication 4.4. O P I P V 119 processes, etc. We will show that the protein variability is quite robust to any of these changes. We will show that most of these changes do not seem to bring a signiicant addition variability source for the protein noise: as for the results presented in Subsection 4.3.1, the protein variance is still increased by at most 10% compared to the gene-centred model.

Additional Genes

he genome of E. coli has approximately 4000 expressed genes. But the measures of [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] take into account only a part of it. Only 1018 protein types were considered in the article, and among them, only 841 types have the mRNA production estimated. In order to beter represent the complete genome of the bacteria, we have created a set of parameters with an extended pool of additional randomly created genes so that the total number of genes would be 4000. For each new gene, we have sampled an average protein and mRNA concentration, an mRNA lifetime and a gene position. By studying the data of [START_REF] Taniguchi | uantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF], we have investigated all possible statistical correlations between these quantities; it appears that only the mRNA and protein concentration are positively correlated. We therefore have sampled the mRNAs lifetime and the gene position and length independently from the two other quantities.

As in the dataset, the genes appears evenly distributed on the chromosome; we have sampled the gene position uniformly. he empirical mRNA lifetime distribution ited a log-normal distribution; we have chosen the mRNA lifetime accordingly.

For the mRNA and the protein, we have taken into account their correlation. he dataset was binned according to the protein production (the different colours in Figure 4.1). At irst, the protein production is sampled according to its empirical distribution. Depending on which bin the obtain protein production falls in, the corresponding mRNA production is sampled according to the mRNA empirical distribution in the bin. By these procedures, the created genome seems representative to the original dataset (see (Figure 4.1)).

Simulations with the completed genome show no signiicant diference in terms of protein variability. In particular, the variance ratio between protein concentration of the gene-centred model and the multi-protein model is not diferent as in Subsection 4.3.1.

Production of RNA-polymerase and Ribosomes

In the stochastic model of the chapter, all ribosomes and all RNA-polymerases are supposed to have constant concentrations (respectively β R and β Y ). In reality, both RNA-polymerases and ribosomes are composed of diferent subunits, each subunit is either a protein or, in the case of ribosomes, a functional RNA. he variability of the production of these subunits can have an overall impact on the global production.

We have performed a preliminary simulation that takes into account this aspect: the goal is not to have a precise description of mechanisms of RNA-polymerase and ribosome production, but rather to have an insight in the magnitude of additional variability it can induce. In this version of the model, the expression of one gene represents RNA-polymerase production and the expression of another gene represents the ribosome production. It refers to a case where the RNA-polymerases and ribosomes would be composed of only one proteic subunit.

We therefore created two genes, whose protein production was ixed to correspond to the wanted concentration of RNA-polymerases and ribosomes. he mRNA production and lifetime, the gene position and length have been chosen by the same procedure as described in to the previous subsection. his simulation brings an additional variability in the growth rate: the cell growth is more luctuant. hese luctuations are directly correlated with the number of ribosomes in the cell (Figure 4.2a). But surprisingly, these additional variability has no signiicant impact in the protein variability. he Figure 4.2b shows the distribution of the protein FabH for cells of diferent volumes. his case does not difer from the case where the total amount of RNA-polymerases and ribosomes were in constant concentration.

We can propose a possible interpretation of these results. he luctuations in the total number of ribosomes seems inluence primarily the speed of growth (as shown in Figure 4.2a): when the ribosomes are produced, it accelerates the global production of every types of proteins thus increasing the volume. As a consequence, both the production of each type of protein and the volume are co-regulated. Fluctuations in the total number of ribosomes afect the volume growth and the production of the i-th protein in the same way such as in a cell of a given volume, the i-th protein distribution is relatively unchanged.

Non-speciically Bound Polymerases

In the stochastic model of this chapter, as described in Subsection 4.1.2, RNA-polymerases are either on the DNA involved in a transcription process, or is among the F Y free RNA-polymerases that freely evolve in the cytoplasm. But it has been shown that a lot of RNA-polymerases can bind non-speciically on the DNA, without initiating transcriptions. For instance, [START_REF] Klumpp | Growth-rate-dependent partitioning of RNA polymerases in bacteria[END_REF] estimated that around 90% of the RNA-polymerases are non-speciically bound to the DNA.

: S M T T 129

Model for Translation he model for translation considered here is completely analogue to the transcription case. We still consider that the volume V is ixed and that for each gene, the number M i of mRNA of type i is known and constant (because of these, the process describe here is independent from transcription). As in the stochastic model of the chapter, and contrary to the model of [START_REF] Fromion | A Stochastic Model of the Production of Multiple Proteins in Cells[END_REF], we consider that there is no limiting number of elongating ribosomes on one mRNA.

Similarly to the transcription, we can deine N R (the total number of ribosomes), E R,i (the number of ribosomes elongating an mRNA of type i) and F R (the number of free ribosomes) such as

he rate at which a ribosome is sequestered on a type i mRNA is therefore M i λ 2,i /V , and the rate at which an elongation terminates on a type i mRNA is µ 2,i E R,i .

As this model is analogue to the transcription case, we can also prove that Proposition 4.4. he number of free ribosomes F R follows

It means that F R follows a binomial distribution B (ϕ, N ) for which ϕ = (1 + Λ R ) -1 and N = N R .

Titre : Modèles stochastiques pour la production des protéines : l'impact de l'autorégulation, du cycle cellulaire et des interactions entre les productions de protéines sur l'expression génétique Keywords : Expression génétique, modèle stochastique, production des protéines, autorégulation Résumé : Le mécanisme de production des protéines, qui monopolise la majorité des ressources d'une bactérie, est hautement stochastique : chaque réaction biochimique qui y participe est due à des collisions aléatoires entre molécules, potentiellement présentes en petites quantités. La bonne compréhension de l'expression génétique nécessite donc de recourir à des modèles stochastiques qui sont à même de caractériser les diférentes origines de la variabilité dans la production ainsi que les dispositifs biologiques permettant éventuellement de la contrôler.

Dans ce contexte, nous avons analysé la variabilité d'une protéine produite avec un mécanisme d'autorégulation négatif : c'est-à-dire dans le cas où la protéine est un répresseur pour son propre gène. Le but est de clariier l'efet de l'autorégulation sur la variance du nombre de protéines exprimées. Pour une même production moyenne de protéine, nous avons cherché à comparer la variance à l'équilibre d'une protéine produite avec le mécanisme d'autorégulation et celle produite en « boucle ouverte ». En étudiant un modèle limite, avec une mise à l'échelle (scaling), nous avons pu faire une telle comparaison de manière analytique. Il apparaît que l'autorégulation réduit efectivement la variance, mais cela reste néanmoins limité : un résultat asymptotique montre que la variance ne pourra pas être réduite de plus de 50%. L'efet sur la variance à l'équilibre étant modéré, nous avons cherché un autre efet possible de l'autorégulation : nous avons observé que la vitesse de convergence à l'équilibre est plus rapide dans le cadre d'un modèle avec autorégulation.

Les modèles classiques de production des protéines considèrent un volume constant, sans phénomènes de division ou de réplication du gène, avec des ARN-polymérases et les ribosomes en concentrations constantes. Pourtant, les variation au cours du cycle de chacune de ces quantités a été proposée dans la littérature comme participant à la variabilité des protéines. Nous proposons une série de modèles de complexité croissante qui vise à aboutir à une représentation réaliste de l'expression génétique. Dans un modèle avec un volume suivant le cycle cellulaire, nous intégrons successivement le mécanisme de production des protéines (transcription et traduction), la répartition aléatoire des composés à la division et la réplication du gène. Le dernier modèle intègre enin l'ensemble des gènes de la cellule et considère leurs interactions dans la production des diférentes protéines à travers un partage commun des ARN-polymérases et des ribosomes, présents en quantités limitées. Pour les modèles où cela était possible, la moyenne et la variance de la concentration de chacune des protéines ont été déterminées analytiquement en ayant eu recours au formalisme des Processus Ponctuels de Poisson Marqués. Pour les cas plus complexes, nous avons estimé la variance au moyen de simulations stochastiques. Il apparaît que, dans l'ensemble des mécanismes étudiés, la source principale de la variabilité provient du mécanisme de production des protéines lui-même (bruit dit « intrinsèque »). Ensuite, parmi les autres aspects « extrinsèques », seule la répartition aléatoire des composés semble avoir potentiellement un efet signiicatif sur la variance ; les autres ne montrent qu'un efet limité sur la concentration des protéines. Ces résultats ont été confrontés à certaines mesures expérimentales, et montrent un décalage encore inexpliqué entre la prédiction théorique et les données biologiques, ce qui appelle à de nouvelles hypothèses quant aux possibles sources de variabilité.

En conclusion, les processus étudiés ont permis une meilleure compréhension des phénomènes biologiques en explorant certaines hypothèses diicilement testables expérimentalement. Des modèles étudiés, nous avons pu dégager théoriquement certaines tendances, montrant que la modélisation stochastique est un outil important pour la bonne compréhension des mécanismes d'expression génétique.

Université Paris-Saclay

Espace Technologique / Immeuble Discovery Route de l'Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France Title: Stochastic models for protein production: the impact of autoregulation, cell cycle and protein production interactions on gene expression Keywords: Gene expression, stochastic model, protein production, autoregulation Abstract: The mechanism of protein production, to which is dedicated the majority of resources of the bacteria, is highly stochastic: every biochemical reaction that is involved in this process is due to random collisions between molecules, potentially present in low quantities. The good understanding of gene expression requires therefore to resort to stochastic models that are able to characterise the diferent origins of protein production variability as well as the biological devices that potentially control it. In this context, we have analysed the variability of a protein produced with a negative autoregulation mechanism: i.e. in the case where the protein is a repressor of its own gene. The goal is to clarify the efect of this feedback on the variance of the number of produced proteins. With the same average protein production, we sought to compare the equilibrium variance of a protein produced with the autoregulation mechanism and the one produced in "open loop". By studying the model under a scaling regime, we have been able to perform such comparison analytically. It appears that the autoregulation indeed reduces the variance; but it is nonetheless limited: an asymptotic result shows that the variance won't be reduced by more than 50%. The efect on the variance being moderate, we have searched for another possible efect for autoregulation: it havs been observed that the convergence to equilibrium is quicker in the case of a model with autoregulation. Classical models of protein production usually consider a constant volume, without any division or gene replication and with constant concentrations of RNA-polymerases and ribosomes. Yet, it has been suggested in the literature that the variations of these quantities during the cell cycle may participate to protein variability. We propose a series of models of increasing complexity that aims to reach a realistic representation of gene expression. In a model with a changing volume that follows the cell cycle, we integrate successively the protein production mechanism (transcription and translation), the random segregation of compounds at division, and the gene replication. The last model integrates then all the genes of the cell and takes into account their interactions in the productions of diferent proteins through a common sharing of RNA-polymerases and ribosomes, available in limited quantities. For the models for which it was possible, the mean and the variance of the concentration of each proteins have been analytically determined using the Marked Poisson Point Processes. In the more complex cases, we have estimated the variance using computational simulations. It appears that, among all the studied mechanisms, the main source of variability comes from the protein production mechanism itself (referred as "intrinsic noise"). Then, among the other "extrinsic" aspects, only the random segregation of compounds at division seems to have potentially a signiicant impact on the variance; the other aspects show only a limited efect on protein concentration. These results have been confronted to some experimental measures, and show a still unexplained decay between the theoretical predictions and the biological data; it instigates the formulations of new hypotheses for other possible sources of variability. To conclude, the processes studied have allowed a better understanding of biological phenomena by exploring some hypotheses that are diicult to test experimentally. In the studied models, we have been able to indicate theoretically some trends; hence showing that the stochastic modelling is an important tool for a good understanding of gene expression mechanisms.

Université Paris-Saclay

Espace Technologique / Immeuble Discovery Route de l'Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France