Mathematical Models of Gene Expression

In this paper we analyze the equilibrium properties of a large class of stochastic processes describing the fundamental biological process within bacterial cells, {\em the production process of proteins}. Stochastic models classically used in this context to describe the time evolution of the numbers of mRNAs and proteins are presented and discussed. An extension of these models, which includes elongation phases of mRNAs and proteins, is introduced. A convergence result to equilibrium for the process associated to the number of proteins and mRNAs is proved and a representation of this equilibrium as a functional of a Poisson process in an extended state space is obtained. Explicit expressions for the first two moments of the number of mRNAs and proteins at equilibrium are derived, generalizing some classical formulas. Approximations used in the biological literature for the equilibrium distribution of the number of proteins are discussed and investigated in the light of these results. Several convergence results for the distribution of the number of proteins at equilibrium are in particular obtained under different scaling assumptions.


Introduction
The gene expression is the process by which the genetic information is synthesized into a functional product, the proteins. These macro-molecules are the crucial agents of functional properties of cells. They play an important role in most of the basic biological processes within the cell, either directly, or as a component of complex macro-molecules such as polymerases or ribosomes. The information flow from DNA genes to proteins is a fundamental process, common to all living organisms.
We analyze this fundamental process in the context of prokaryotic cells, like bacterial cells or archaeal cells. The cytoplasm of these cells is not as structured as eukaryotic cells, like mammalian cells for example, so that most of the macro-molecules of these cells can potentially collide with each other, including DNA. This key biological process can be, roughly, described as resulting of multiple encounters/collisions of several types of macromolecules of the cell: polymerases with DNA, ribosomes with mRNAs, or proteins with DNA, . . . An additional feature of this process is that it is consuming an important fraction of energy resources of the cell, to build chains of amino-acids or chains of nucleotides in particular.
The fact that the cytoplasm of a bacterial cell is a disorganized medium has important implications on the internal dynamics of these organisms. Numerous events are triggered by random events associated to thermal noise. When the external conditions are favorable, these cells can nevertheless multiply via division at a steady pace. In each cell, around 4000 different types of proteins are produced, with different concentrations: from 5 elements per cell for some rare proteins to 100,000 per cell for some ribosomal proteins. Despite of the noisy environment, the protein production process can handle such exponential growth with these constraints. The cost of the translation phase, the last step of the protein production process, is estimated to account for 50% of the energy consumption of a rapidly growing bacterial cell, see Russell and Cook [28] for example.
The key question in this context is of understanding and quantifying the role of the parameters of the cell in the production of proteins in the context of significant fluctuations. The high level of energy consumed by protein production suggests that these stochastic fluctuations are minimized or at least "controlled", to avoid a shortage of resources, so that the cell can continue to grow. From the point of view of the design, the cell can be seen as a system where the processing power is expressed in terms of numerous chemical reactions occurring in a noisy environment. An important difference with classical, non-biological, systems, like computer networks, is that there is nowhere the possibility of explicitly storing information which could be used to regulate the production process. Furthermore, strictly speaking, there are no functional units processing some kind of "information". This is a really challenging aspect of these biological systems.
1.1. The Protein Production Process. We give a quick, simplified, sketch of the steps involved in the production of proteins. See Watson et al. [35] for a much more detailed description of these complex processes.
A gene is a contiguous section of the DNA associated to a functional property of the cell. When a gene is active, a macro-molecule of the cell, an RNA polymerase, one of the macro-molecules moving in the cytoplasm, can possibly bind to the DNA at the promoter, i.e. the beginning of the section of DNA for the gene. The binding to the gene is subject to the various (random) fluctuations of the medium. The gene may also be inactive and in this case no binding is possible. The reason is that some macro-molecules, like proteins or mRNAs, of the cell can "block", via chemical bonds, the gene so that polymerases cannot access it. This is one of the regulation mechanisms of the cell. When random perturbations break these bindings, the gene becomes again active.
-Transcription. When an RNA polymerase is bound to an active gene, it starts to make a copy of this gene. The product which is a sequence of nucleotides is a messenger RNA, or mRNA. The time during which nucleotides are added sequentially is the elongation phase of the production of the mRNA. When the full sequence of nucleotides of the mRNA has been successively assembled, the mRNA is released in the cytoplasm. It has a finite life time, being degraded by other macro-molecules.
-Translation. The step is achieved through another large macro-molecule: the ribosome. A ribosome is also moving within the cytoplasm. When it encounters an mRNA, it can also bind to it via chemical reactions. In this case it builds a chain of amino-acids, a chain of amino-acids, using the mRNA as a template, producing a protein. This is the elongation phase of the production of the protein. The above description of the protein production process is clearly simplified. The processes associated to actions of polymerases and ribosomes are composed of several steps. The way ribosomes bind to mRNAs in particular, see Chapter 14 of Watson et al. [35] for example. Once a polymerase binds to the gene, the messenger RNA chain is built through a series of specific stages, in which the polymerase recruits one of the four nucleotides in accordance to the DNA template. Additionally, a dedicated proof reading mechanism takes place during this process. There is a similar description for the translation step. The binding events of polymerases to gene or of ribosomes to mRNAs are also due to a sequence of specific steps.
Because of the disorganized medium of the bacteria, the protein production process is a highly stochastic process. The randomness is partially due to the thermal excitation of the environment. It drives the diffusion of the main components, mRNAs and ribosomes within the cytoplasm and it also impacts the pairing of cellular components diffusing through the cytoplasm. It can cause the spontaneous rupture of such pairs, before either transcription or translation can start.
The problem is of understanding the mechanisms used by the cell to produce a large number of proteins with very different concentrations in a such a random, "noisy", context. The main goals of a mathematical analysis in the biological literature are generally the following.
(1) Estimate impact on variance of the number of proteins of assumptions on -Activation/Deactivation rates of the gene; -Transcription/Translation rates. Polymerases may bind more easily to some genes. A similar phenomenon holds for ribosomes and mRNAs. This is generally mathematically represented via a "rate" of binding: transcription rate for polymerases on gene and translation rate for ribosomes on mRNAs, k1 and k2 in the following; -The distributions of lifetimes of mRNAs and proteins, the death rate of mRNAs, (resp., proteins), is denoted by γ1, (resp., γ2). (2) Determine the parameters which achieve minimal variance of concentration of a type of protein with a fixed average concentration. When an analytical formula for the variance is available, one can determine, in principle, the parameters that minimize it, with the constraint of that its average is fixed. The general idea is of determining if the current parameters of the cell are adjusted, depending of the environment, to minimize this variance.
(3) Estimate biological parameters for experiments. Analytical formulas for the average and the variance of the number of proteins can also be used to estimate some biological parameters. See Taniguchi [31]. The protein production process is a fundamental biological process, a process at the basis of "life", justifying the extensive efforts to understand and to quantify, via mathematical models, the role of its different components: polymerases, ribosomes, mRNAs and proteins. It also serves as a generic example of a typical biological process: a component of type A is produced with components of type B, but via the creation of another component of type C, which may also be used in the regulation of the process.
As a general remark, the mathematician should always keep in mind that she/he is studying an incredibly complex system, the molecular biology of gene expression. It has evolved over a time span of four billion years. A mathematical model may take into account only partial aspects of the sophisticated mechanisms into play. In this presentation we focus on the basic parameters of the transcription and translation phases.

Some history.
The basic principles of molecular biology, starting with Crick and Watson [36], Brenner et al. [5] and Jacob and Monod [12], have been discovered in the 1950-1960's. At that time, measuring concentrations of different types of macro-molecules within the cell was not really possible with the experimental tools available. Efficient methods to estimate these concentrations, like the fluorescence microscopy, have been available much later at the beginning of the 1990's. This is probably the main motivation of mathematical models at the end the 1970's in the pioneering works Berg [2] and Rigney [26] to study the fluctuations of the concentration of proteins within the cell. Analytical expressions for the mean and the variance of the number of proteins in a cell have been derived for some simplified models. These results were later extended in the context of Markov chain theory. They are recalled in Section 2. The formulas obtained can give some insight on the role of the parameters of the cell like the transcription rate of the gene or the average lifetime of some macro-molecules on these fluctuations. Interestingly, this is one of the few examples in applied mathematics when mathematical models have been used before measurements could be done. There is a huge literature on the stochastic analysis of this process using a large range of mathematical methods. For a detailed review of these works, see Paulsson [23]. We focus here on exact formulas that can be obtained for specific stochastic models.
State Space of the Production Process. In most mathematical analyses, the production process of proteins of a fixed gene is investigated. There is an implicit assumption in this case that the cell allocates a fixed fraction of its resources to the production of the mRNAs, proteins of this gene, independently of the production process of other proteins. With this hypotheses the production process of this type of protein can be analyzed in isolation of the other processes within the cell. This is a convenient given the complexity of the interactions. In this case, the production process is usually described as a three-dimensional Markov process (I(t), M (t), P (t)) where, for t≥0, -I(t)∈{0, 1} describes the state of the gene at time t, active or inactive; -M (t) is the number of mRNAs at this instant; -P (t) is the number of proteins. For these models, the duration of elongation of mRNAs or of proteins is not taken into account. For example, the transition M (t)→M (t)+1, the production of an mRNA, is, implicitly, associated to the binding of a polymerase to the gene. This amounts to neglect the elongation time of an mRNA. A glance at the numbers of Section A reveals that, in average, the time to build an mRNA of 3000 nucleotides is 35 seconds, which is not small compared to its lifetime, of the order of 2 minutes. Among difference types of mRNAs, the average length is 1000, see Figure 5 of Section A in the Appendix.
Another assumption, the price of the Markov property, is that the duration of the transitions have an exponential distribution. This hypothesis for the duration of time to have a binding polymerase-gene or ribosome-mRNA can be justified with the current parameters of the cell. The assumption is more questionable for lifetimes of mRNAs and, in particular, of proteins. The lifetimes of proteins are comparable to the duration of time between divisions of the cell. The interpretation of death/degradation is more difficult in such a case. See the discussion on the assumptions on the distribution of lifetimes.
The Markovian description of the protein production process gives the possibility of using classical results of Markov theory, for the convergence to equilibrium as well as for an analytical characterization of the equilibrium points.

Numbers of copies or Concentration ?
The representation of the levels of proteins and mRNAs by numbers, i.e. integers, is convenient for the mathematical analysis, in particular to investigate the stochastic fluctuations. For some aspects, it is nevertheless more natural to express the various quantities in terms of concentrations rather than numbers. For example, if R, M and P denotes respectively the chemical species ribosomes, mRNAs and proteins, the production and destruction of a protein is expressed as , using notations from chemistry, See van Kampen [34] and Chapter 6 of Murray [21] for example. The parameter kon is the rate at which a given ribosome binds to a given mRNA and ν is the exponential growth rate of the volume of the cell. The concentration of proteins decreases mostly because of volume growth. This is the dilution effect. In a Markovian context, see Section 2, the number of copies of a given type of protein is used rather than its concentration. In this case the parameter ν is interpreted as a death rate of proteins, which is less natural in some way. In this presentation, we will nevertheless use the discrete representation with numbers. It is more convenient to describe stochastic phenomena involving a finite, but not too large, number of macro-molecules, like for mRNAs. See Anderson and Kurtz [1] for an introduction on stochastic modeling of biological systems.
Organization of the Paper. Due to its historical importance, the analysis of the first moments of the equilibrium of Markovian models is presented in Section 2. Section 3 introduces a fundamental marked Poisson point process used to describe the whole protein production process. It is an extension based on the model of Fromion et al. [11]. A result of convergence to equilibrium is proved via a "coupling from the past" method. An important representation of the number of mRNAs and proteins in terms of the Poisson process is established. See Theorem 7.
In Section 4, (resp., Section 5), general formulas for the mean and variance of the number of mRNAs (resp., of proteins), at equilibrium are established, generalizing the classical formulas of Markovian models. An important part of the biological literature is devoted to the analysis of these first moments, this is the main motivation of these two sections. The results obtained are an extension, with somewhat simpler technical arguments, of results of Fromion et al. [11]. When the gene is always active, the equilibrium of the detailed description of the state of mRNAs is characterized by Proposition 12. A formula for the generating function of the equilibrium distribution of the number of proteins is established in Proposition 14. The joint distribution of the number of mRNAs and of proteins at equilibrium is also investigated in Section 5.
In Section 6 the various approximations used in the biological literature are revisited in the light of the results proved in the previous sections. They are formulated as new scaling results when one of the parameters goes to infinity: switching rate of the gene, average lifetime of proteins, translation rate, . . . generally under the constraint that the average number of proteins is fixed. Several convergence results for the equilibrium distribution of the number of proteins are obtained in this way. The methods rely on the results of Section 5, probabilistic arguments and some technical estimates. A central limit theorem, which seems to be new, is established in Proposition 18. A study on the impact of the elongation phase of proteins on the variability of the protein production process concludes this section. This topic is rarely addressed in mathematical models of the biological literature. It is shown that, under some statistical assumptions, increasing the variability of the elongation phase in a stochastic model may have the surprising effect of reducing the variance of the equilibrium distribution of the number of proteins. A consequence of this observation is that the choice of exponential distribution for the distribution of the duration of the elongation phase of proteins, as it would be natural in a Markovian context, may lead to underestimate the "real" variance.
The central result of this approach is Theorem 7, it gives an explicit representation of the number of proteins at equilibrium in terms of a functional of a marked Poisson point process. The main advantages are that (1) it is not necessary to solve invariant measure equations as in the Markovian approaches; (2) Formulas for Poisson processes, recalled in the appendix, give additional insight on the equilibrium distribution of the number of proteins; (3) General distributions, instead of exponential distributions, for the duration of elongation times and lifetimes of mRNAs and proteins can be included in the model. The material presented in this review is based on a series of studies in collaboration with Vincent Fromion (INRA) who has, convincingly, introduced the author to these highly interesting processes. A PhD thesis [17] by Emanuele Leoncini has been also part of this collaboration. The author is grateful for comments on a preliminary version of this paper by Gaëtan Vignoud, Emanuele Leoncini and Frédéric Flèche.
Mathematical Notations and Conventions. Throughout this presentation, for the sake of clarity, we will frequently do the following abuse of notations. If X is some integrable random variable on R+ (1) X(dx) will denote its distribution on R+; (2) (Xn) an i.i.d. sequence of random variables with this distribution; (3) FX a random variable whose distribution has density , with respect to Lebesgue's measure on R+. We denote by (FX,n) an i.i.d. sequence with this distribution. Note that, with Fubini's Theorem, for a≥0, where b + = max(b, 0), b∈R. Additionally, a simple calculation of the Laplace transform shows that the relation FX dist = X holds if and only if the law of X is exponential.
Throughout the paper, we will use the following notation: -For mRNAs, L1(dw) is the distribution of their lifetimes and E1(dv) the distribution of their elongation times. -For proteins, L2(dz) is the distribution of their lifetimes and E2(dy) the distribution of their elongation times. The index 1 refers to mRNAs and 2 to proteins. With our convention, items (1) and (2) above, (L1) and (E1), (resp. (L1,n) and (E1,n)), denote random variables, (resp. i.i.d. sequences of random variables) associated to the distribution L1(dw), (resp. E1(dv)). And similarly for proteins, for the random variables (L2) and (E2) and the i.i.d. sequences (L2,n) and (E2,n).

The Classical Markovian Three-Step Model
In this section, Markovian models of the protein production process are presented. Although these models have some limitations in terms of modeling this is an important class of models to study the stochastic fluctuations associated to gene expression. This is in fact the main stochastic model of gene expression used in the biological literature from the early works of Rigney [26], Berg [2] to more recent studies Thattai and van Oudenaarden [32], Shahrezaei and Swain [29]. See Paulsson [23] for a survey. See also Chapter 6 of Bressloff [6] and Chapter 4 of Mackey et al. [20]. These models are still popular in the biological literature.
A more general modeling of the stochasticity of gene expression is presented and discussed in Section 3. It is analyzed in the rest of the paper. In this section we will consider the time evolution of the number of mRNAs and of proteins associated to a gene G0 in a given cell (bacterium). The statistical assumptions are the following: (1) If the gene is inactive, in state 0 say, (resp., active, in state 1), it becomes active, (resp., inactive) after an exponentially distributed amount of time with parameter k+ (resp., k−).
(2) If the gene is active, the mRNAs of G0 are produced according to a Poisson process with parameter k1 and their lifetimes are exponentially distributed with parameter γ1. (3) during its lifetime an mRNA of G0 produces proteins according to a Poisson process with parameter k2 and the lifetime of a protein is exponentially distributed with parameter γ2.
For this model, the parameter k1 can be seen as the rate at which a polymerase binds on an active gene, but also the rate at which an mRNA is produced. The steps of binding and elongation are thus reduced to a single step. The same remark holds for the production of proteins. The model of Section 3 distinguishes these two steps. For t≥0, we denote I(t)∈{0, 1} the state of gene at time t and M (t)∈N, (resp., P (t)∈N), is the number of mRNAs, (resp., of proteins), at this instant. It is easily checked that the process (Z(t)) def = (I(t), M (t), P (t)) is an irreducible Markov process in the state space The non-zero elements of its Q-matrix Q=(q(z, z ), z, z ∈S0) outside the diagonal are given by, for z=(i, x, y)∈S0, where ei, i=1, 2, 3, are the unit vectors of S0. See Figure 3. As a functional operator Q, it can be defined as for some function f on S0. The starting point of analyses of the literature is always the classical system of Kolmogorov's equations associated to this Markov process for some set of test functions f , like the indicator function of x, for x∈S0. They are generally referred to as the master equation in this literature. Its fixed point is the invariant distribution of the Markov process.
Proof. The proof is skipped. This result is in fact established in the next sections in a much more general context. See the proofs of Propositions 10 and 13.
There does not seem that there exists a closed form expression for the invariant distribution (π(i, x, y)) of the equilibrium equations. See Relation (3) below for the generating function of P when the gene is always active. It turns out that, nevertheless, one can get explicit expressions for the moments of this distribution. Related formulas of the kind have been the main and essentially, the only, rigorous results of mathematical models of gene expression starting from the first studies in 1977. They are still used in the quantitative analyses of the biological literature. See the supplementary material of Taniguchi et al. [31] for example. .
The ratio of the variance and of the mean given for M and P is called the Fano factor, it can be seen as a measure of the dispersion of the probability distribution. In the biological literature it is interpreted as a measure of deviation from the Poisson distribution. Recall that the Fano factor of a Poisson distribution is 1.
Proof. The proofs in the literature are generally based on the differential equations satisfied by the generating function of the distribution of the three-dimensional process. See Leoncini [17]. A simple approach, avoiding manipulations of generating functions, is presented, it is based on some natural flow equations at equilibrium.
The equilibrium equations for the invariant distribution π of the Markov process (I(t), M (t), P (t)) can be expressed as for any function on S0 such that Q(f ) is integrable with respect to π. If, for z∈S0, by taking f the indicator function of z, this gives the usual balance equations.
We will use the class of functions f (i, x, y)=i a x b y c , with i∈{0, 1}, a, b∈{0, 1, 2} to get an appropriate system of linear equations to get these moments. Proposition 1 shows that Q(f ) is integrable for any function f of this class.
The equations give immediately the formulas for the first moments.
A second system of linear equations gives the second moments of M and P . The proposition is proved.
As it will be seen in a more general framework, when the gene is always active, i.e. k−=0, the distribution of M is Poisson. This explains in particular the identity Var(M )=E(M ) of Proposition 2 in this case. See Proposition 12 of Section 4. Note that it is not the case for P since Var(P ) by Proposition 2. This is usually interpreted in the biological literature as the fact that the distribution of P is more variable than a Poisson distribution. The above results, the first and second moments of the number of proteins at equilibrium, are essentially the main mathematical results of the current biological literature in this domain. As an example of application of the methods presented in this paper, we state several results which give some additional insight on the properties of the equilibrium distribution of the number of proteins. They are simple applications of general results which are proved in Section 3 and 6.
An explicit representation of the generating function of P is in fact available.
Proposition 3. If the gene is always active, the generating function of the equilibrium distribution of the number of proteins, for z∈(0, 1), is given by Bokes et al. [3] gives, via an analytic approach with the master equation, an alternative representation of the generating function in terms of an hypergeometric function.
The classical relation (2) for the variance shows that when the gene is always active, i.e. δ+=1, and the average lifetime 1/γ2 of proteins gets large, then The following proposition gives a more precise result, when lifetimes of proteins goes to infinity. It is a consequence of a general convergence theorem, Theorem 18 of Section 6.
Proposition 4. If the gene is always active, for the convergence in distribution, the relation holds, where N (σ) is a centered Gaussian random variable whose standard deviation is This proposition gives a Gaussian approximation result for the distribution of P as E(P )+ E(P )N (σ). The result should hold for protein types with a large number of copies in the cell.
The following proposition establishes a convergence in distribution when the death rate of mRNAs is going to infinity with the constraint that the transcription rate k2 is such that each mRNA produces a fixed average number of proteins.
Proposition 5. If the gene is always active, then, for a>0, the relation lim holds for the convergence in distribution, where N (a) is random variable whose distribution is given by, for n∈N, where Γ is the classical Gamma function.
The distribution of N (a) is the negative binomial distribution with parameters k1/γ2 and a. See Chapter 5 of Johnson et al. [13]. This distribution is frequently mentioned in the approximations of the literature, Bokes et al. [3], Friedman et al. [10] and Shahrezaei and Swain [29] for example. It is also sometimes used to fit measurements of concentration of proteins. Its advantage compared to a simple Poisson distribution may be due to the fact that it is more flexible with its two parameters.
Proof. This is a straightforward application of Proposition 20 of Section 6. The elongation time of proteins is null, i.e. E2≡0, so that the generating function of the limit N (a) is given (37) and (38) of the appendix give the representation of this distribution.

A General Stochastic Model
We introduce in this section the stochastic processes which will be studied for the analysis of the time evolution of the number of proteins.
The distributions of lifetimes of mRNAs and proteins are general, in particular they are not assumed to be exponentially distributed as in the Markovian model of Section 2. The general distributions of the duration of elongations of mRNAs and proteins are also incorporated in the stochastic model. Once the polymerase/ribosome is bound to the gene/mRNA, the elementary components, nucleotides for mRNAs and amino-acids for proteins, are progressively added to build these components. As it can be seen in Figure 5 in the appendix, the number of these elements can vary from ∼20 elements up to several thousands depending on the gene considered. This, of course, incurs delays which cannot be really neglected. This presentation is an extension of the framework of Fromion et al. [11]. See also Leoncini [17].
As it will be seen, a representation of the equilibrium of the protein production process in terms of marked Poisson point processes is established.
Contrary to a Markovian analysis, the resolution of the linear system of balance equations for the equilibrium, is not necessary in this case. More general assumptions on the distributions of some of the steps of the process can therefore be considered as mentioned before. It should be noted however, that even in the Markovian context of the classical three-step model, this representation simplifies much the analysis of the equilibrium. For example, asymptotic results concerning the equilibrium distribution of the number of proteins are simpler to obtain than the approach using analytic tools with hypergeometric functions. See Bokes et al. [3] and Section 6.2 for example.
3.1. Gene activation. The state of the gene for the type of protein is either active or inactive. It is activated at rate k+ and inactivated at rate k−. Time duration of these states are assumed to be exponentially distributed.
The process of activation of the gene is a Markov process (I(t), t ∈ R) with values in {0, 1} and whose Q-matrix RI is given by rI (0, 1)=k+ and rI (1, 0)=k−. Without loss of generality, it will be assumed to be stationary. For this reason (I(t)) is defined on the whole real line, in some sense, the activation/inactivation process has started at t=−∞. As it will be seen, this is a convenient formulation to describe properly the equilibrium of the protein production process.
This process can be represented as a marked point process ((ti, Xi), i∈Z) where (ti) is the increasing sequence of instants of change of state of the gene, with the convention that t0≤0<t1. The marks (Xi) in the set {0, 1} indicate the state of the gene at these instants. In particular, for i∈Z, conditionally on the event {Xi=0}, (resp., on the event {Xi=1}), the random variable ti+1−ti is exponentially distributed with rate k+, (resp., with rate k−). Because of our assumption on stationarity of the activation/deactivation process, the sequence of activation instants of the gene, (ti1 {X i =1} ), is a stationary renewal point process. See Section 11.5 of Robert [27]. We denote

Transcription and Translation: a Fundamental Poisson Process.
When a gene is active, a polymerase can be bound to it according to a Poisson process with rate k1>0. The transcription phase can start. An mRNA is then being built up by a process of aggregation of a specific sequence of nucleotides which are present in the cytoplasm. This is the elongation phase of an mRNA. Its duration has a distribution E1(dx) on R+.
Similarly a ribosome is bound to a given mRNA according to a Poisson process with parameter k2. The translation phase starts. A chain of amino-acids is created as the ribosome progresses on the mRNA. This is the elongation phase of a protein. Its duration has a distribution E2(dx) on R+. Both mRNAs and proteins have a finite lifetime. The distribution of the lifetime of an mRNA (resp., a protein) is L1(dy) (resp., L2(dy)) on R+.
Throughout the paper, it is assumed that all these distribution have a finite first moment.
The Fundamental Poisson process. These two steps are represented by a marked Poisson point process. See Section B of the Appendix.
(4) P def = (un, E1,n, L1,n, N2,n, n∈Z), on the state space Dawson [8] for example, and (1) (un) is a Poisson process on R with parameter k1, this is the sequence of possible instants when a polymerase can be bound to the gene. For n∈Z, un is indeed an instant of binding only if the gene is active at time un. The sequence (un, n∈Z) is assumed to be non-decreasing and indexed with the convention u0≤0<u1. (2) (E1,n, n∈Z) is the sequence of the duration of the elongation for these mRNAs, it is an i.i.d. sequence whose common distribution is E1. (3) (L1,n) is the i.i.d. sequence of associated lifetimes of these mRNAs, its common law is L1.
?  (4) (N2,p) is an i.i.d. sequence with the same distribution as N2, a marked Poisson point process on R×R+ with intensity k2 dx⊗E2⊗L2. For n∈Z, N2,n=(x n 2,j , E n 2,j , L n 2,j , j∈Z) is the process associated with the protein production for the mRNA with index n. (a) (x n 2,j , j∈Z) is a Poisson process on R with parameter k2, this is the sequence of possible instants when a ribosome can be bound to the mRNA with index n. Only the instants occurring during the lifetime of this mRNA matter.
is the sequence of the duration of the elongation of proteins by this mRNA, it is an i.i.d. sequence whose common distribution is E2.
sequence of associated lifetimes of associated proteins, its common law is L2. The coordinates of P(du, dv, dw, dm) are associated to the potential instants u of the binding of a polymerase on the gene, the variable v is the elongation time of the mRNA and the variable w is its lifetime. The last component m is the marked point process associated to the protein production process of this mRNA. The coordinates of m(dx, dy, dz) are, x is associated to binding instants of ribosomes on this mRNA, y is the elongation time of the corresponding protein and z its lifetime.
The intensity measure of P is where Q2, Q2(dm)=P(N2∈ dm), is the distribution on S of the random variable N2, a Poisson point process on R×R+ with intensity k2 dx⊗E2⊗L2. Equivalently, if F is a non-negative Borelian function on S, then where, with a slight abuse of notations, E1, L1 and N2 are independent random variables with respective distributions E1(dv), L1(dw) and Q2(dm).
Examples. If the gene is always active and the system starts empty at time 0, the formula gives the total number of mRNAs created up to time t. The following convention has been used. If f is non-negative Borelian function on R×R 2 + , i.e. P(du, dv, dw) is the marginal distribution of P with respect to the three first coordinates, i.e. it stands for P(du, dv, dw, Mp(R×R 2 + )). Similarly, still with a permanently active gene, is the number of mRNAs present at time t≥0.
If we include the gene dynamics into the formula, the number of messengers present at time t is If the mRNA with index n is created at time v=un+E2,n and if its lifetime is w then is the total number of proteins created by such an mRNA during its lifetime.

Comments on the Model
As a mathematical model, the above representation simplifies several aspects of the protein production process but the key steps of protein production are included. Furthermore, its main parameters have a clear biological interpretation.
(1) Expression Rates. The parameter k1 is the affinity of polymerases, or the transcription rate for the gene considered. The larger this rate is, the more likely polymerases will bind to this gene rather than to some other genes with a lower affinity. The same remark applies for k2, the affinity of ribosomes, the translation rate, for the corresponding mRNAs.
It should be stressed that if mRNAs and proteins are specific to a gene, polymerases and ribosomes are not. They can be used to create any type of mRNA and any type protein respectively. Polymerases and ribosomes can be seen as a kind of resources of the cell. The genes are, in some sense, competing to have access to polymerases. This approach with the parameters k1 and k2 considers that a specific gene receives a fixed portion of resources of the cell in terms of polymerases and ribosomes. In particular, it does not include the competition for access to resources between the different genes. Alternative stochastic models have to be considered to analyze these aspects.
(2) Elongation Times. The distribution of elongation E1 of an mRNA. An mRNA is a sequence of N nucleotides, then E1 can be represented as a sum of N independent random variables, each of them corresponding to the duration of time required to get each of the nucleotides within the cytoplasm. A similar situation holds for E2 for the number of amino-acids of the protein. See Figure 5 in the appendix.
(3) Lifetimes. The lifetimes of mRNAs are generally smaller than the lifetimes of proteins, mRNAs can be indeed degraded by other RNAs in the cytoplasm after a couple of minutes. This is one of the mechanisms used by the cell to regulate the protein production process. The situation is quite different for proteins, their lifetime may exceed the duration of time for cell division, of the order of 40mn for bacterium. At cell division, macro-molecules are assumed to be allocated at random into one of the two daughter cells. In this case, if one follows a specific path of cells in the associated binary tree of successive cell divisions, proteins "vanish" simply because they have been allocated in the daughter cell. This is referred to as the dilution phenomenon.
(4) Multiple Copies of the Gene. In a favorable environment, cells grow and divide very quickly. In particular a copy of the DNA is always on the way in the cell, and, therefore, another copy of the gene is present at some stage. In principle this may double the transcription rate of the gene when the corresponding part of the DNA has been duplicated. The rate is reduced at the division time. This aspect has been omitted in our model, mainly for the sake of simplicity. The main change would be that the process (I(t)) should have at least three values, depending on the number of copies of the gene which are active at a given time. See Paulsson [23] in a Markovian context. The classical Markovian model of Section 2 corresponds to the case when the elongation times of mRNAs and proteins are null and that their respective lifetimes are exponentially distributed.
How to Handle Functionals of P. We will use repeatedly several possible representations in the calculations of expected values of functionals of P. We give a quick sketch of the general approach. Let f be a non-negative Borelian function on R×R 2 + ×R×R 2 + and let, for (u, v, w, m)∈S, When the gene is always active, most of random variables of interest can be expressed under the form X= P, F def = S F (u, v, w, m)P(du, dv, dw, dm).
As it will be seen, the variable activity of the gene adds some technical complications.
(1) Calculation of the mean. Relation (35) of the appendix gives that with the same slight abuse of notations as before. For (u, v, w)∈R 3 + , since the Poisson process N2 has intensity measure k2 dx⊗E2⊗L2 on R×R 2 + , For calculation of variances, the approach is similar, via Relation (36) of the appendix and an additional trick. See the proof of Relation (13) for example.
(2) Exponential Moments of X. For more precise results on the distribution of X, one has to express E(exp(−ξX)), for some ξ≥0, the Laplace transform of X at ξ. Proposition 22 of the appendix gives the relation By using again Proposition 22 for the Poisson process N2, with the same arguments as before, we get, for (u, v, w)∈R×R 2 + , This expression for E(exp(−X)) though explicit is not simple to handle. It is nevertheless usable to get several limit results in Section 6. The trick of using independent random variables Ea, La, a∈{1, 2} or their distributions Ea(ds), La(dt) is used throughout this paper. Its main advantages are of simplifying the calculations and the expressions of the formulas obtained.
We state a simple result on some invariance properties of the Poisson process P. For m∈Mp(R×R 2 + ) and f is a Borelian function on R×R 2 + , the measurem is defined by m, f = Proof. This is a simple consequence of -the reversibility of (I(t)). It is easily checked that (I(−t)) is a Markov process with the same Q-matrix, and with same distribution at t=0 as (I(t)). See Kelly [15] for example. -If (tn) is a Poisson process on R with rate λ>0, then (−tn) is a Poisson process with the same rate. -The independence of (I(t)) and P. and therefore our proposition.
3.3. Convergence to Equilibrium. As before we denote by M (t), (resp., P (t)) the number of mRNAs, (resp., of proteins) at time t≥0. It is assumed that the initial state is as follows: there are -M (0) mRNAS with respective remaining lifetimes L 0 1,k , k=1,. . . ,M (0); -P (0) proteins with respective remaining lifetimes L 0 2,k , k=1,. . . ,P (0). The marked Poisson process associated to the creation of proteins by the kth mRNA present at time 0 is denoted by N 0 2,k .
For simplicity it is assumed that, initially, there are no mRNAs or proteins in their elongation phase. The proof of convergence in distribution does not change if the initial state includes components in their elongation phase.
The following theorem is the key result concerning the equilibrium distribution of the number of mRNAs and proteins.
(1) It shows the convergence to equilibrium without having a Markovian framework which is, usually, the classical approach to prove such a convergence. As a consequence, the distribution of elongation times, lifetimes of mRNAs and proteins can be general. where P is the marked Poisson point process on S defined by Relation (4).
The proof of the convergence relies on coupling from the past arguments. The idea consists in starting the process at time −t and to study its state at time 0, which will have the same distribution as the original process at time t. If the process has convenient properties, of monotonicity for example, it may happen that the state at time 0 of the shifted process converges almost surely as t goes to infinity. This gives then the convergence in distribution of (I(t), M (t), P (t)) when t goes to infinity. This method applies in our case. One of the earliest works using this method seems to be Loynes [19] (1962). Its use has been popularized later by Propp and Wilson [24] to study the Ising Model. See Levin at al. [18] for a survey.
Proof. We first express the variables M (t) and P (t) in terms of the point process P.
The variable M (t) is the sum of the number of initial mRNAs still alive at time t and of the number of mRNAs born before time t and still alive at time t. This gives the following formula, Similarly, P (t) is the sum of three quantities corresponding to the number of (1) the number of proteins present at time 0 and still alive at time t, For simplicity of presentation, we will prove the convergence in distribution of (P (t)). The convergence in distribution of (M (t), P (t)) is similar. Clearly, the two first terms converge almost surely to 0 when t goes to infinity. We have thus only to take care of P1(t).
The stationarity of the process P with respect to translation by −t, Proposition 6, the identity (I(s−t)) dist = (I(s)), and the fact that (I(s)) is independent of P give that P1(t) has the same distribution as P2(t) with The quantity P2(t) can be seen the number of proteins at time 0 when the process starts empty at time −t, i.e. without mRNAs or proteins. This is a non-decreasing function of t converging almost surely to P2(∞) defined by Proposition 6 gives that the quantity P2(∞) has the same distribution as it is easy to see that this last term is the second coordinate of (7). The theorem is proved.

Transcription
Relation (6) gives that the distribution of the number of mRNAs at equilibrium is given by the law of the random variable M defined by The process of activation/deactivation of the gene complicates significantly the derivation of the mathematical expressions for the mean and variance of M . Formulas are much more simple when the gene is always active. We will need some technical results on this process. We denote by F I the σ-field generated by the stochastic process (I(t)), A representation of the distribution of M in terms of a marked Poisson point process conditioned on the σ-field F I is the main tool used. For the calculation of the variance an additional work has then to be done to "remove" this conditioning. It turns out that the correlation structure of the process (I(t)) plays a role at this stage.
Define P I the marked point process by, for a non-negative Borelian function on the space S def = R×R 2 + ×Mp(R×R 2 + ), The following proposition gives the intuitive, but important, result that, conditionally on F I , the point process P I is a marked Poisson point process with intensity measure (11) ν I P (du, dv, dm) = I(u)k1 du L1(dv)Q2(dm). See Relation (5) for the unconditional case.
Proposition 9. For any non-negative Borelian function f on the state space S, the relation holds almost surely.
Proof. For the sake of rigor, despite of its intuitive content, a proof is given. The (formal) difficulty is of expressing rigorously the conditioning with respect to F I . We use the notations of Section 3.1 where the process (I(t)) is defined by the doubly infinite sequence (ti, Xi). Denote, for N ≥1, By independence of (ti, Xi) and P and by using Proposition 22, it is easily checked that, almost surely on this event, Indeed, one has to multiply both sides of this relation by some non-negative Borelian function of ((ti, Xi), −N ≤i≤N ), take the expected value of these expressions. By using the independence of ((ti, Xi), −N ≤i≤N ) and P, and the expression of the Laplace transform of a Poisson process, see Proposition 22, it is easy to check the equality of these two terms. Consequently, for N sufficiently large, almost surely, A classical result from martingale theory gives the almost sure convergence See Williams [38]. Hence the proposition holds for these class of functions f with compact support in the sense defined above. We conclude with the fact that any positive Borelian function can be expressed as a (monotone) limit of such functions.
Recall that the distribution of the elongation time of an mRNA is E1 and the duration of its life time is L1. Note that when the lifetimes L1 of an mRNA is exponentially distributed with parameters γ1, then FL 1 dist = L1, and if the elongation time of an mRNA is null, i.e. E1≡0, the above relation gives   (14) and (15) give We can rewrite the square of the integral as a double integral in the following way, If we gather these results into Relation (16), we obtain the identity which is the desired formula.
We now turn to a more detailed analysis of the invariant distribution of (M (t)) in the case when the gene is always active.

The Equilibrium Distribution of the State of mRNAs.
It is now assumed that the process (I(t)) is constant and equal to 1. In particular δ+=1. We introduce a random measure that describes precisely the state of the mRNAs, where Ri(t) is the residual lifetime of the ith mRNA present at time t. The measure Λ1(t) is the empirical distribution associated to the residual lifetimes.
Proposition 11 (Convergence to Equilibrium of the number of mRNAs). When the gene is always active, the process (Λ1(t)) converges in distribution to a random measure (Λ * 1 ) defined by, for f a continuous function with compact support on R+, Proof. The proof is similar to the proof of Theorem 7. Theorem 3.26 of Dawson [8] is used, it is enough to show that for the convergence in distribution lim t→+∞ Λ1(t), f = Λ * 1 , f for every non-negative continuous function f with compact support on R+. By definition of the vector (Ri(t)), by invariance of the Poisson point process with respect to translation by −t, see Proposition 6. We conclude that (Λ1(t)) converges in distribution to Λ defined by by invariance with respect to the mapping (u, v, w) →(−u, v, w), again by using Proposition 6. The proposition is proved.
This proposition states that at equilibrium the number of mRNAs is Poisson with parameter k1E(L1) and the residual lifetimes of the mRNAs are independent and distributed as Fσ 1 .
Proof. We calculate the Laplace transform of Λ * 1 , by using Proposition 22 of the appendix and the fact that νP defined by Relation (5) is the intensity measure of the Poisson process P, we have By taking f ≡ − log(z) for some fixed z∈(0, 1), we obtain that the distribution of M is Poisson with parameter E(M )=k1E(L1). Additionally, since we have that Λ * 1 has the same Laplace transform, and therefore the same distribution, as the random measure where (FL 1 ,i) is an i.i.d. sequence of random variables with distribution FL 1 independent of M . The proposition is proved.

Translation
Theorem 7 shows that the distribution of the number of proteins at equilibrium has the same distribution as the random variable where P is the marked Poisson point process on S defined by Relation (4).

Mean and Variance of the Number of Proteins at Equilibrium.
We have an explicit representation of the variance of the number of proteins at equilibrium given by the following proposition. This is an extension of the results of Fromion et al. [11], see also Leoncini [17].
where Λ=k++k− and, for a=1 and 2, the random variables Ea,1 and Ea,2, are independent with respective distributions Ea, and FL a ,1 and FL a ,2 are independent random variables with density P(La≥u)/E(La) on R+.
The proof is given in Section C of Appendix. It follows the same arguments as in the proof of Proposition 10 but with a more technical framework due to the mark m, in the space Mp(R×R 2 + ), of the Poisson process P. If the average of P does not depend on the distributions of elongation times of mRNAs and of proteins, the second moment does depend on these distributions, and also on the distributions of lifetimes. Note however that if Λ is large, i.e. at least one of the states of the gene is changing rapidly, Relation (21) shows that the dependence of the distribution of the elongation time of an mRNA on the variance is small in this case.
When the elongation times are null, i.e. E1≡0 and E2≡0 and the lifetimes L1 and L2 are exponentially distributed with respective parameters γ1 and γ2, in this case, for a∈{1, 2}, the variable FL a is also exponential since FL a dist = La, a simple calculation gives the classical formula (2) of the Markovian model.
Outside the Fano parameter, the biological literature defines the noise associated to the production of proteins as the variance of P/E(P ), the quantity Var(P )/E(P ) 2 . The above formulas and Relation (12) give that it can be represented as

The Equilibrium Distribution of the Number of Proteins.
When the gene is always active, an explicit expression of the generating function of P can be obtained. As it will be seen, its form depends on the whole distribution of the lifetimes and elongation times. Nevertheless, by using appropriate scalings, it can be used to get limit results for its distribution and therefore some insight. This is the purpose of Section 6. Note that for classical Markovian models, without elongation times in particular, Relation (26) of Bokes et al. [3] gives, via an analytic approach, an explicit expression of the joint generating function of (M, P ) in terms of hypergeometric functions.
Proposition 14 (Generating Function of P ). If the gene is always active, i.e. the process (I(t)) is constant and equal to 1, and P is the random variable defined by Relation (7), the distribution of P is given by, for z∈[0, 1], where FL 2 and L2 are two independent random variables and the distribution of FL 2 has density P(L2≥u)/E(L2) on R+.
Proof. Relations (18) defining P and (22) where, as before, N2 is a Poisson process whose distribution is Q2. For (u, v, w)∈R×R 2 + , with the same arguments as in the proof of Relation (13). Hence, which gives the desired formula for the generating function.
The explicit expression of the generating function of P for Markovian models is detailed in Proposition 3 of Section 2.

The Joint Distribution of the Numbers of mRNAs and Proteins.
This topic is investigated in few references of the literature such as Bokes et al. [3] and Taniguchi et al. [31]. Its motivation lies in the fact that it could be useful if some information on the number of mRNAs could be extracted from knowledge on the number of proteins.
We begin to study the covariance function of M and P . Relations (8) and (18) give the representations where the functions G1 and G2 are defined by Relations (9) and (19). By using Proposition 9 as before on the conditional distribution of (M, P ) with respect to F I , the σ-field associated to the state of the gene. See By integrating this identity and by using the expressions of G1 and G2, we obtain that E(C I ) is equal to Using the same argument as before to express a product of integrals as a double integral, this gives the identity and which is, after the same manipulations as in the proof of Proposition 13, We have therefore proved the following proposition.
Proposition 15 (Covariance of M and P ). If M and P are the random variables defined by Relation (6) and (7), then the covariance of M and P is given by for E1,1 and E1,2, are independent random variables with distribution E1, and FL 1 ,1 and FL 1 ,2 are independent random variables with density P(L1≥u)/E(L1) on R+, FL 2 is a random variable with density P(L1≥u)/E(L1) on R+.
The lifetime L1 of the mRNAs is usually much smaller than the lifetime L2 of proteins, this implies that the same property also holds for the variables FL 1 and FL 2 . If, additionally, the average number of mRNAs E(M ) is small, the right-hand-side of Relation (24) is small, so that Cov(M, P ) should be close to 0. This property has already been noticed in some measurements, see Taniguchi et al. [31] for example. We conclude this section with an explicit representation of the generating function of the random variable (M, P ).
Proposition 16 (Joint Distribution of M and P ). When the gene is always active, for z1 and z2∈(0, 1), where FL 1 , FL 2 and E2 are independent random variables and, for a∈{1, 2}, the random variable FL a has the density P(La≥u)/E(La) on R+.
Proof. For a1, a2>0, Relation (23) gives that We have, by Relation (22) of the appendix, and, if we use Relation (22), We conclude the proof with standard calculations.

Convergence Results
The purpose of this section is of revisiting, via rigorous convergence theorems, several classical research topics of the literature on the stochasticity of the gene expression in a general framework concerning the distributions of lifetimes and elongation times. Most of studies in this domain take place in the Markovian setting of Section 2 without the elongation of mRNAs and proteins. Kolmogorov's equations associated to the Markov process, the "master equation" as it is usually presented, are the starting point of these studies.
As the equilibrium equations cannot be solved explicitly, several scenarios are investigated, via scalings, to get insight on the equilibrium distribution of the number of proteins: Fast Switching rates between active and inactive states, mRNAs with short life times, proteins with long lifetimes, gene with large expression rates, . . . See Friedman et al. [10], Raj et al. [25], Shahrezaei and Swain [29], Swain et al. [30] and Thattai and van Oudenaarden [32]. See Bokes et al. [3] for a quite extensive analysis of the Markovian model. We will look at three different scaling situations in the light of the results derived in the previous sections.
The distribution of lifetimes of proteins is an exponential distribution with parameter γ2, and its mean is converging to infinity. A central limit theorem is proved for the equilibrium of the number P of proteins. It is shown that, in distribution, P ∼E(P )+ E(P )N , where N is a centered Gaussian random variable.
(2) Short-lived mRNAs and Long-lived Proteins. The distribution of lifetimes of mRNAs and proteins are exponential with respective parameters γ1 and γ2, the mean lifetime 1/γ1 of mRNAs is converging to 0 and the death rate γ2 of proteins is fixed so that k1k2/(γ1γ2), the mean number of proteins at equilibrium, is fixed.
(3) Short-Lived mRNAs and High Expression Rate of Proteins. The distribution of lifetimes of mRNAs and proteins are exponential with respective parameters γ1 and γ2. The mean lifetime of mRNAs is converging to 0 and the translation rate k2 of proteins is fixed so that the quantity k2/γ1, the average number of proteins produced by an mRNA, is fixed.
In most cases, the gene will be assumed to be always active. The reason is that, if some results could be obtained on the distribution of P , we have not been able to get usable expressions to get some insight in the scaling regimes analyzed here. The next result shows that, in practice, a model with a permanently active gene is reasonable, provided some parameters are adjusted. This is a folk result of the biological literature: if the state of the gene switches more and more rapidly, then, in the limit, the model is equivalent to a model with a permanently active gene but with a reduced transcription rate.
6.1. Fast Switching Rates of Gene between Active and Inactive States. Proposition 11 shows that when the gene is always active then the equilibrium distribution of the number of mRNAs is Poisson. This is minimal from the point of view of the Fano factor in this class of models. If the gene switches quickly between the two states active/inactive, one can expect an averaging effect. The following proposition establishes this intuitive result.
Proposition 17. If PN is the random variable defined by Relation (7) with the activation/deactivation rates are respectively given by k+N and k−N for some scaling parameter N ≥1, then, for the convergence in distribution, where the distribution of P is the equilibrium distribution of the number of proteins for a model of gene expression for which the gene is always active and with the same parameters except for k1 which is replaced by k1k+/(k++k−).
Proof. The stationary activation/deactivation process (IN (t)) can clearly be expressed as (I(N t)) where (I(t)) is a stationary process with switching rates k+ and k− Let f be an integrable function on R+ and we now prove the convergence in distribution Note that the convergence is certainly true for the first moment of XN , since E(I(t))=δ+ for all t≥0.
By writing the square of XN as a double integral and with Proposition 2, we obtain the relation Therefore, in particular (XN ) converges in distribution to the limit of the sequence (E(XN )).
Denote by F I N the filtration associated to the process (I(N t)). Relations (18) and (19) for PN , and (22) of the appendix for the Laplace transform of Poisson point processes give where N2 is a Poisson process whose distribution is Q2 defined by Relation (5).
By using the convergence result, we obtain that which is the Laplace transform of the variable P associated to a Poisson process with the same characteristics as P except that the transcription rate is δ+k1 and with the situation that the gene is always active.

Asymptotic Behavior of the Equilibrium Distribution.
In this section, the invariant distribution of the number of proteins when the gene is always active is analyzed under some scaling conditions. Several of them rely on the fact that the average lifetime of a protein is much larger than the lifetime of an mRNA, which is of the order of 2mn for an mRNA, where, for a protein, it is at least 30mn. See the numbers of Section A of the appendix. This approach is used in the literature to get a further insight on the distribution of P , i.e. with more information than the first two moments that have an explicit expression. It is usually done via approximations on the equation satisfied by the generating function of P . See Bokes et al. [3] or Shahrezaei and Swain [29] where this is done via an expansion of an hypergeometric function. The representation of Proposition 14 of the generating function of P will allow to get convergence results for the distribution of P in a quite general case and without too much technicality.

A Limiting Gaussian Distribution for the Number of Proteins
The scaling considered here assumes that the average lifetime of proteins goes to infinity and the other parameters are fixed. In particular these proteins should be numerous within the cell. This setting is well suited for protein types having a large number of copies, of the order at least of 10,000 for example. The following result shows that a central limit theorem holds in this context.

Theorem 18 (Central Limit Theorem). Under the conditions
(1) the gene is always active; (2) the distribution L2 of lifetimes of proteins is exponential with parameter γ2; (3) the parameters k1, k2 are fixed as well as the distribution L1 of the lifetimes of mRNAs and the respective distributions E1 and E2 of the elongation times of mRNAs and proteins. The distribution L1 of lifetimes of mRNAs has a finite third moment, E(L 3 1 )<+∞, if P is the random variable defined by Relation (7), then for the convergence in distribution, and, for a>0, N (a) is a centered Gaussian random variable with standard deviation a.
Despite this framework is quite natural, curiously it does not seem to have been investigated in the biological literature, even for the classical Markovian three-step model. It gives in fact a Gaussian approximation for the number of proteins at equilibrium when the average lifetime 1/γ2 of proteins is large. It should be noted that, in this scaling regime, the second order does depend on the distribution of the elongation times of proteins.
Proof. By using Proposition 14, the relation FL 2 dist = L2 due to the exponential assumption for L2, and the calculation of the proof of Relation (20) of Proposition 13, we get that, for ξ>0, with, for (u, w)∈R 2 + , The elementary inequality gives the relation After integration with respect to the measure k1 duL1(dw) on R 2 + and by using the relation we get the inequality We now study the behavior of each of these four terms when γ2 goes to 0. For the first term A1, this is is straightforward since The second term A2(γ2) is handled in the same way as in the derivation of Relation (21) of Proposition 13. First note that Lebesgue's Theorem gives therefore the relation We now analyze the upper bound of Relation (28) expressed with the terms B1(γ2) and B2(γ2). The last one is easy to handle it converges to 0 as γ2 goes to 0. For the term B1(γ2), Jensen's Inequality and Fubini's Theorem give With simple calculations, we get that, for w>0, the quantity is the sum of two terms, C1 and C2, with The elementary inequality (26) gives the relations C1 ≤ γ 2 2 w 3 3 and |C2| ≤ 4γ 2 2 w 3 .
By using the fact that L1 has a finite third moment, we deduce the relation Relations (28), (29) and (30) give therefore the convergence The theorem is proved.

Short-Lived mRNAs and Long-Lived Proteins
For the scaling considered in this section the death rate γ2 of proteins goes to 0 and with the constraint that the death rate γ1 of mRNAs is such that the average number of proteins is kept fixed and equal to a>0, In particular the average lifetime 1/γ1 of an mRNA goes to 0.

Proposition 19. Under the conditions
(1) the gene is always active; (2) the distribution of lifetimes of mRNAs and proteins are exponential with respective parameters γ1 and γ2; (3) the transcription and translation parameters k1 and k2>0 are fixed as well as the distributions of the elongation times, if P is the random variable defined by Relation (7), then for a>0, the convergence in distribution, lim where Pois(a) is the Poisson distribution with parameter a.
Note that Proposition 2 gives readily that From the point of view of the Fano factor, the fluctuations of the number of proteins are minimal. A (rough) picture of that is that short lived mRNAs and almost permanent proteins minimize fluctuations of gene expression. The proposition is more precise since it states that the number of proteins at equilibrium is in fact asymptotically Poisson.
Proof. In view of Proposition 14 and, since L2 is exponentially distributed, L2 dist = FL 2 , all we have to do is that the quantity converges to (1−z)a, as γ1 goes to infinity, with the constraint γ1γ2=k1k2/a.

With the elementary inequality (26), we get
We now take care of these two quantities. By invariance by translation, we have .
The Cauchy-Shwartz Inequality gives for the second term hence it converges to 0 as γ1 goes to infinity with γ1γ2 being constant. The proposition is proved by gathering these results.
The above result can be explained quite simply as follows. We assume that the elongation times are null for simplicity. With the "coupling from the past" approach of Section 3.3, if (un) denotes the non-decreasing sequence of instants of binding instants before time 0, since the distribution of the sequence of lifetimes (L1,n) of mRNAs is converging to 0, with high probability there should at most one protein created by the mRNA with index n, so that 1 {N2([un+E1,n,un+E1,n+L1,n]) =0,un+E 1,n +L 2,n >0} , and a simple calculation gives as γ2 goes to 0 with γ1γ2 constant. The distribution of the variable R is asymptotically Poisson with parameter k2k1/(γ2γ1). It is not difficult to see that, with convenient estimates, a rigorous proof could be obtained by using this observation.

Short Lived mRNAs and High Expression Rate of Proteins
In this section the death rate γ1 of proteins goes to infinity but it is assumed in addition that the quantity k2/γ1, the average number of proteins translated by an mRNA, is constant. In particular k2, the expression rate of proteins, is large and the average number of proteins is fixed.
Proposition 20. Under the conditions (1) the gene is always active; (2) the distribution of lifetimes of mRNAs and proteins are exponential with respective parameters γ1 and γ2; (3) the parameters k1 and γ2>0 are fixed as well as the distributions of the elongation times, if P is the random variable defined by Relation (7) , then for a>0, the relation lim holds for the convergence in distribution, where N (a) is a discrete random variable whose generating function is given by, for z∈(0, 1), The distribution of the random variable N (a) belong to the class of generalized Neyman Type A distributions. See Section 9 of Chapter 9 of Johnson et al. [13] for example. In the Markovian case of Section 2, this is a negative binomial distribution. See Proposition 5.
Proof. For z∈(0, 1), the quantity − log(E(z P ))/k1 is, since FL 2 dist = L2, With similar arguments as in the proof of Proposition 19 and Lebesgue's Theorem, we obtain the relation lim The proposition is proved.
6.3. The Impact of Elongation on Variability. In this section we discuss the impact of the elongation of proteins on the variability of the protein production process. This aspect has been rarely considered in the mathematical models of the literature. Note that the first moments of M and P do not depend on the distributions of the elongation times. The distribution of the random variable M and the second moment of P do not depend on the distribution of the elongation phase of mRNAs when the gene is always active.
To simplify the presentation, we assume in this section that the lifetimes of mRNAs and proteins are exponentially distributed with respective parameters γ1 and γ2 and that the gene is always active.
If the protein is composed of N amino-acids, it is natural to assume that the average of the elongation time is proportional to N , i.e. given by N/α for some α>0. An index N is added to the variables E2 and P defined by Relation (7). Relation (21) of Proposition 13 gives that the variance of the number of proteins at equilibrium is given by, (32) Var by the equality in distribution FL a dist = La for a∈{1, 2} due to the exponential distribution assumption, with S N 2 =E N 2,1 −E N 2,2 and Y2=L2,1−L2,2. Two assumptions on the distribution of the elongation time of a protein are now considered.
(1) Markov model. The variable E N 2 is an exponential random variable given by where Vα is an exponential random variable with parameter α. The variable defined by Relation (7) is denoted as P N M . A similar assumption can be done for the elongation of mRNAs, i.e. that its distribution is also exponential. If M 0 (t), (resp., P 0 (t)), denotes the number of mRNAs, (resp., proteins), being built at time t≥0, then the process (I(t), M 0 (t), M (t), P 0 (t), P (t)) has clearly the Markov property. This is a natural Markovian extension of the classical Markovian model including elongation. Vα,i where (Vα,i) is an i.i.d. sequence of exponential random variables with parameter α. This is also a natural assumption since it considers that it takes a random amount of time with an exponential distribution to find each amino-acid of the protein within the cytoplasm. The variable defined by Relation (7) is denoted as P N A .
A non-intuitive phenomenon. A simple calculation gives Var(E N 2,M ) = N 2 α 2 and Var(E N 2,A ) = N α 2 As it can be seen the variability of the elongation time is larger for the Markov model. Intuitively, this could suggest that the same property holds for the number of proteins, that the variance of P N M is larger than the variance of P N A . But a glance at Relation (32) suggests in fact the contrary. Indeed, the variable S N 2 +Y2 will be more variable for the Markov model and, therefore, due to this relation, the variance should be smaller. The next proposition gives a more formal formulation of this observation. See also Leoncini [17] for related numerical results of this kind.
Furthermore, if γ1<γ2, there exists a constant C>0 such that This proposition shows that, if the second moment of P depend on the distribution of the elongation times, its dependence is somewhat limited for both models, Markov and additive. It converges to some constant as the average (and the variance) of the elongation time goes to infinity. Remember that the average number of proteins does not depend at all on the distribution of the elongation time.
The last inequality of the proposition shows that the variance of the additive has a much slower convergence rate to p compared with the Markov model. The condition γ1<γ2 is just a technical condition which does not seem to be significant.

Proof. With the notations of Relation (32), we have the elementary relations
Since γ1<γ2, one has E e γ 1 |Y 2 | <+∞, hence one has only to take care of the limiting behavior of E exp −γ1|S N 2 | as N goes to infinity. For the Markov model, a simple calculation gives that |S N 2,M | is exponentially distributed with parameter α/N , .
For the additive model, S N 2,A is a sum of N i.i.d. centered random variables where (V 1 α,i ) and (V 2 α,i ) are i.i.d. sequences of exponential random variables with parameter α. Let Berry-Esseen's theorem, See Feller [9] for example, gives that, for all N ∈N, where N (1) is a centered Gaussian random variable with standard deviation 1. From Fubini's Theorem, we have and, with the Relation (33), this gives Let aN =γ1σ √ N , then With Lebesgue's Theorem, we obtain we conclude the proof by using Inequality (34) and Equation (32).

Appendix A. Some Numbers for Escherichia coli Bacterial Cells
We give some numbers and some statistics for a single cell, to give an idea of the orders of magnitude of the different elements of this production process. It is also used to justify some of the assumptions in the mathematical models introduced. The sources of these numbers can be found at the address www.bioNumbers.org. The references Bremer and Dennis [4] and Chen et al. [7] have also been used.  (2) Polymerases and Ribosomes. A ribosome is a complex macro-molecule composed of ribosomal proteins, of the order of 50 units, and ribosomal RNAs corresponding to a total of ∼3000 nucleotides. Polymerases and ribosomes can be seen, for the production process, as resources of the cell. They are used by all types of genes of the cell, in some way, genes (resp., the mRNAs) of all types are competing to have access to polymerases (resp., ribosomes) to produce mRNAs (resp., proteins).
Note that the range of variations of these numbers depends on the growth rate of the cells, i.e. of the environment.
-Number of mRNAs in a cell: ∼7800.
-Diffusion coefficient of a protein: ∼10 −2 sec to cross cell.

Appendix B. Poisson Point Processes
The main results concerning Poisson processes used in this chapter are briefly recalled, in the following H is a separable locally compact metric space. See Kingman [16], Neveu [22], or Robert [27]. Definition 1. If λ>0, µ is a probability distribution on H, a marked Poisson process on R×H with intensity λ dx⊗µ is a sequence N λ =((tn, Xn), n∈Z) of elements of R×H where -(tn, n∈Z) is a (classical) Poisson process on R with rate λ, with the convention that tn≤tn+1 for any n∈Z and t0≤0<t1. -(Xn, n∈Z) is an i.i.d. sequence of random variables with values in H and whose distribution is X(dx).
The sequence N λ can also be seen as a marked point process on R×H, if F is a nonnegative Borelian function on R×H, then N λ , F = R×H F (u, x)N λ (du, dx) = n∈Z F (tn, Xn).
The following important proposition characterizes marked Poisson point processes. Appendix C. Proof of Proposition 13 As in section 4, F I denotes the σ-field generated by the stochastic process (I(t)), by using again Proposition 9 , we have We now take care successively of the two terms of the right hand side of the last equality. First we study E G2 (u, E1, L1, N2) 2 . For u, v, w∈R+, the distribution of the random variable G2 (u, v, w, N2) =