Stochastic models of regulation of transcription in biological cells

Abstract

In this paper we study an important global regulation mechanism of transcription of biological cells using specific macro-molecules, 6S RNAs. The functional property of 6S RNAs is of blocking the transcription of RNAs when the environment of the cell is not favorable. We investigate the efficiency of this mechanism with a scaling analysis of a stochastic model. The evolution equations of our model are driven by the law of mass action and the total number of polymerases is used as a scaling parameter. Two regimes are analyzed: exponential phase when the environment of the cell is favorable to its growth, and the stationary phase when resources are scarce. In both regimes, by defining properly occupation measures of the model, we prove an averaging principle for the associated multi-dimensional Markov process on a convenient timescale, as well as convergence results for “fast” variables of the system. An analytical expression of the asymptotic fraction of sequestrated polymerases in stationary phase is in particular obtained. The consequences of these results are discussed.

1 Introduction

The central dogma of molecular biology asserts for biological cells that genetic information flows mainly in one direction, from DNA to RNAs, and to proteins. For the two most studied bacteria Escherichia coli and Bacillus subtilis, production of proteins is a central process which can be described as a process in two main steps. In the first step, macro-molecules polymerases produce RNAs with genes of DNA. This is the transcription step. The second step, translation, is the production of proteins from mRNAs, messenger RNAs, with macro-molecules ribosomes (See Watson et al. 2007).

In bacterial cells, protein production uses essentially most of cell resources: a large number of its macro-molecules such as polymerases and ribosomes, biological bricks of proteins, i.e., amino acids, and the energy necessary to build proteins during the translation step, such as GTP.

In this paper we study an important regulation mechanism of transcription using specific RNA macro-molecules, 6S RNAs, common to a large number of bacteria. See (Wassarman 2018) for example. The functional property of this RNA is of blocking/sequestering free polymerases from producing RNAs. The general context of this regulation is related to complex mechanisms of the cell to finely tune the production of a large set of RNAs. Let us first recall the three main categories of RNAs:

• (a) RNAs used for the building of ribosomes, i.e., rRNAs, ribosomal RNAs. A ribosome is a complex assembly of around 50 proteins and, also, of several rRNAs. An rRNA is a long chain of several thousands of nucleotides, it is in particular a costly macro-molecule to produce. Reducing or speeding-up the production of ribosomes, in particular of rRNAs, has therefore a critical impact on resource management of the cell;

• (b) mRNAs, messenger RNAs, used by the translation step to produce a protein from mRNAs coding sequences;

• (c) A large set of RNAs that do not belong to the two previous categories, such as transfer RNAs, tRNAs, or Bacterial small RNAs, sRNAs, often associated to regulation mechanisms. This class includes 6S RNAs.

When the concentrations of different resources in the medium are high enough for some time, the bacterium has the ability to use them efficiently, via its complex regulatory system, to reach a stable exponential growth regime with a fixed growth rate. The growth of a bacterial population in a given medium leads therefore to an active consumption of resources necessary to produce new cells.

When resources are scarce, each bacterium of the population can adapt, to either exploit differently the available resources, or to do without some of them, as for example when some amino-acids are missing. For E. coli or B. subtilis, these bacteria use in priority resources maximizing their growth rate. In the context of this adaptation, and for reasons related to the decay of resources, each bacterial cell has to decrease its growth rate, and finally to ultimately stop its growth.

The regulatory network involved in the management of the growth rate to adapt to the environment is complex. The important point in this domain is that the bacterium has to modify the concentration of most of the agents in charge of it: number of ribosomes, concentrations of proteins in the metabolic network, transporters, ...In a first, simplified, description, the decay of a specific resource in the environment leads to a move to a state of the cell where concentrations of several components have been adapted. To study the transition between growth phases, we have chosen to focus on the action of a small RNA, 6S RNA, which plays an important, even essential, role in this domain. Note that, even if this mechanism is central, this description of the transition between growth regimes is nevertheless a simplification in our approach, since the bacterial cell has different ways to modify the steady-state level of its components.

In this article, we investigate a simplified scenario where transitions occur between two phases: an exponential growth phase and the stationary phase, where the growth rate is equal to 0. The first interest of this scenario lies in the sharp transition of the polymerase management by the cell, via the strong effect of the stringent response on the production of rRNAs: the transcription of rRNAs is completely stopped. This is where the action of 6S RNAs is crucial. See (Gottesman et al. 2006). Its second interest is experimental since its is possible in practice to create this transition by the addition of a convenient product in the medium of cell populations to induce a stringent response. Our general goal is to investigate if, with this simplified framework, the regulatory system organized around 6S RNA has the desired qualitative properties to ensure a convenient transition between these regimes. In this paper, we analyze the efficiency of the regulation by 6S RNAs with stochastic models. We investigate in particular the time evolution of the activity of polymerases in the cell under different regimes.

1.1 A simple description as a particle system

In order to explain the basic principle of the regulation mechanism investigated in this paper, we describe a simplified version in terms of a particle system. Section 1.2 describes in more depth and detail the biological context of this class of models.

We consider two types of particles P and 6S. There is a fixed number of particles of type P and there are random arrivals of particles of type 6S. A particle of type P can be in three states: busy, idle, or paired with a 6S particle. Similarly, a 6S particle is either idle or paired. The possible events are:

• an idle, resp. busy, P particle becomes busy, resp. idle;

• a couple of an idle P and an idle 6S is paired;

• a pair $P{-}6S$ is broken giving two idle P and 6S;

• an idle 6S arrives/dies.

Note that only an idle 6S can die. A statistical assumption is that each couple of free P particle and free 6S particle is paired at some fixed rate and each free 6S dies at a fixed rate too. In particular if there is no idle polymerase for some time, the a free 6S will die.

We present a heuristic description of the phenomena we are interested in:

• (a) If the parameters of the P particles are such that, on average, most of particles of type P are busy. Therefore, few of them are idle, the arriving 6S particles will very likely die before they can be paired with a P particle. In this case there will be few 6S particles in the system.

• (b) Otherwise, if, on average, a significant fraction of particles of type P are idle, the arriving 6S particles will very likely pair with one of them. In particular, as long as there are many idle P particles, 6S will be quickly paired so that few of them will die. In this manner, the dynamic arrivals of 6S progressively decrease the number of idle P particles.

A pair $P{-}6S$ is seen as a sequestration of a P particle, the purpose of 6S particles is of storing “useless” P particles. The case a) corresponds to the case when most of P particles are efficient so that no regulation is required. This corresponds to the exponential growth phase of our biological process. The case b) is when there is a need of sequestration of P particles, this is the stationary growth phase of our model.

The nice feature of this mechanism is its adaptive property due to the dynamic arrivals of 6S: if they are useless, they disappear after some time. Otherwise, as it will be seen, their number builds up until some threshold of sequestration is reached.

The main goal of the paper is of understanding under which conditions on the parameters the cases a) or b) may occur. To assess the efficiency of the regulation mechanism in the case a), we study the time evolution of the number of 6S particles. In the case b), we investigate the number of sequestered P particles to determine the maximal sequestration rate of the regulation.

The model investigated in the paper is in fact a little more complicated in the sense that P particles can be “busy” in two ways: either it remains busy during a random amount of time before being idle again. The other busy state is that it joins a queue where only the particle at the head of the queue becomes idle again after a random amount of time. In our model, this is related to mRNAs and rRNAs production. The next section gives a detailed description of these aspects.

A future work in this domain should be devoted to the experimental validation of such a mathematical model. This can be done in the framework of a complete simulator of the bacterial cell such as in Fischer et al. (2021). This is an ongoing project.

1.2 Biological background

1.2.1 Transcription

In a bacterial cell, a polymerase may be associated to several specific proteins, called $\sigma$-factor to form a holoenzyme E$\sigma$. In our case we focus on the “housekeeping” $\sigma$-factor $\sigma ^{70}$. This holoenzyme binds to a large set of gene promoters to initialize the transcription. This is the initiation phase. If this step is successful, the protein $\sigma ^{70}$ is detached and the polymerase completes the elongation of the corresponding RNA, this is the phase when the nucleotids of the RNA are retrieved one by one.

This is a simplified description of course. The precise description of the mechanisms are dependent on the bacterium, it is nevertheless sufficiently accurate from our modeling perspective. Throughout the paper we do the slight abuse of using the term polymerase instead of the more biologically correct term holoenzyme. Another important aspect is that the initiation phase may fail due to random fluctuations within the cell, or to a low level of nucleotides needed for the initiation of transcription, i.e. ATP, GTP, UTP, CTP, ...When this happens the transcription is aborted. The level of GTP, for example, has an impact on the modulation of initiation of transcription with respect to the growth rate for bacterium B. subtilis, and, similarly, the level of ppGPpp for bacterium Escherichia coli. See (Geissen et al. 2010) in the case of an rRNA.

1.2.2 Regulation by small RNAs

A subset of RNAs whose sizes in nucleotides is less than 100, small RNAs or sRNAs has been shown to play an important role to regulate gene expression. The first such sRNAs were identified in the late 1960’s. See (Britten and Davidson 1969) and Zamore and Haley (2005). They were shown to turn in or turn off specific genes under convenient conditions.

Among them the sRNA 6S RNA was first discovered because of its abundance in E. coli in some circumstances. See (Hindley 1967). This has been one of the first sRNAs to be sequenced. Nevertheless, it took three decades to understand its role in the regulation of transcription.

Experimental studies have shown that 6S RNA acts in fact as a global regulator of transcription and not only for the regulation of a reduced subset of genes as most of small RNAs. A 6S RNA has a three-dimensional structure similar to a DNA promoter, so that the holoenzyme E$\sigma ^{70}$ may be bound to it and is, in some way, sequestered by it. See (Cavanagh and Wassarman 2014) and Nitzan (2014). It has been shown that during stationary phases, when the growth rate is null, the 6S RNAs accumulate to a high level, with more than 10,000 copies. During an exponential phase, when the growth is steady, the average duration time of cell division is around 40min for E. coli, there are less than 1000 copies. See (Wassarman 2018) and Steuten et al. (2014).

The fluctuations of the number of copies of 6S RNAs is therefore an important indicator of the growth rate of the cell. An important question is to assess the efficiency of the regulation mechanism operated by the 6S RNAs. The impact of several parameters of the cell are investigated: The total number of polymerases, the production rate of 6S RNA and their degradation rate, initiation rates of polymerases for rRNAs and mRNAs and the sequestration rate, i.e. the binding rate of a couple 6S RNA and polymerase.

1.3 Mathematical models

Regulation of gene expression has been analyzed with mathematical models for some time now. See (Mackey et al. 2016) and also Chapter 6 of Paul (2014), and the references therein. The lac operon model is one of the most popular mathematical models in this domain, for its bistability properties in particular. See also (Dessalles et al. 2017).

Specific stochastic models of regulation by RNAs are more scarce. The regulation of mRNAs by sRNAs in a stochastic framework has been the subject of several studies recently. In (Kumar et al. 2018; Mehta et al. 2008), and Platini et al. (2011), the authors study regulation mechanisms of mRNAs by sRNAs with a two-dimensional Markov chain for the time evolution of the number of sRNAs and mRNAs. Some limiting regimes of the corresponding Fokker-Planck evolution equations are investigated and discussed. The difficulty is the quadratic dependence on the number of mRNAs and sRNAs. See also (Baker et al. 2012) and Mitarai et al. (2009). These studies can be seen as extensions of the early works on stochastic models of gene expression, see (Berg 1978; Elowitz et al. 2002) and Rigney and Schieve (1977).

1.4 The main results

In this paper, we will study the efficiency of the sequestration of polymerases by 6S RNAs. Recall that this is in fact the holoenzyme which is sequestered. We investigate the behavior of several variables associated to the regulation of the transcription phase: Number of free/sequestered polymerases and number of polymerases in the elongation phase of mRNAs and rRNAs.

1.5 Technical challenges

We assume that there are N polymerases with N large. We derive functional limiting results, with respect to this scaling parameter, of the time evolution of several stochastic processes. An important feature of our model is that the main Markov process exhibits a quadratic dependence of the state of the process, due to the use of the law of mass action for the dynamic of our model. One of the main technical difficulties is in the proof of Theorem 15 of an averaging principle. Several preliminary results have to be established as well as a convenient definition of occupation measures. This is due to the (very) fast underlying timescale, $t{\mapsto }N^2 t$, used. Formally, the diffusion component is of the order of N but should vanish for this first order result. For this reason, in a first step, the “slow” processes are included in the definition of occupation measures and not only the “fast” processes as it is done in general. In our proofs we use several coupling arguments, estimates of hitting times of rare events, stochastic calculus for stochastic differential equations driven by Poisson processes, and the framework of averaging principles.

1.6 A chemical reaction network description

For simplicity the number of polymerases is assumed to be constant. There is also a production of 6S RNAs which we will distinguish from the production of other RNAs. From the point of view of our model, polymerases can be in several states

• Free. The polymerase may bind to a gene of an mRNA, or of an rRNA, or be sequestered by a 6S RNA, $(F_N(t))$ denotes the process of the number of free polymerases.

• Transcription of an mRNA. A chain of nucleotides is produced, $(M_N(t))$ is the process associated to the number of such polymerases.

• Transcription of an rRNA. A long chain of nucleotides is produced. As it will be seen, it is described by a process with values in $(\{0,1\}{\times }{\mathbb {N}})^J$, where J is the number of types of rRNAs, it is usually small, less than ten. We denote by $({\overline{R}}_N(t))$ process of the total number of polymerases in this phase.

• Sequestered by a 6S RNA. $(S_N(t))$ is the process associated to the number of such polymerases.

Similarly a 6S RNA can be either free or paired with a polymerase, $(Z_N(t))$ denotes the process of the number of free 6S RNAs. See Sect. 2.1 for more details.

With these notations, the assumption on the constant number of polymerases gives the relation

$$\begin{aligned} F_N(t){+}M_N(t){+}S_N(t){+}{\overline{R}}_N(t){=}N,\qquad \forall t{\ge }0. \end{aligned}$$

The dynamic of this stochastic system is governed by the analogue of the law of the mass action in this context. See (Anderson and Kurtz 2015). The rate of creation of sequestered polymerases is in particular quadratic with respect to the state, it is proportional to $F_N(t)Z_N(t)$. This is one of the important features of this stochastic model.

1.7 Two Regimes

In the following, if $(X_N(t))$ is a sequence of stochastic processes, the notation

$$\begin{aligned} \lim _{N\rightarrow +\infty }(X_N(t))= \lim _{N\rightarrow +\infty }(X_N(t),t{\ge }0) \end{aligned}$$

refers to the convergence of the distribution of the process $(X_N(t), t{\ge }0)$, as in Chapter 2 of Billingsley (1999).

Our mathematical results can be described as follows. See the formal statements in Sect. 5 and 6. In Definition 1, we introduce two sets of conditions on the parameters of our model, which define the exponential regime and the stationary regime, our cases a) and b) above. We do not detail them here. Under the condition that the maximum number of polymerases simultaneously in transcription of rRNAs, resp. mRNAs, is of the order of $c_rN$, resp. $c_mN$, under some scaling conditions and appropriate initial conditions, we have:

• (1) Exponential Phase.

  For the convergence in distribution of processes

  $$\begin{aligned} \lim _{N\rightarrow +\infty } \left( \frac{{\overline{R}}_N(t)}{N},\frac{M_N(t)}{N}\right) =(c_r,1{-}c_r), \end{aligned}$$

  and, for any $t_0{>}0$, the random variable $(F_N(t_0))$ converges in distribution to a Poisson distribution and the sequence of process $(S_N(t),Z_N(t))$ is converging in distribution to a positive recurrent Markov process. See Theorem 19. In this case, the polymerases are mostly doing transcription of rRNAs or mRNAs, few of them are free or sequestered by a 6S RNA.

• (2) Stationary Phase.

  For the convergence in distribution of processes

  $$\begin{aligned} \lim _{N\rightarrow +\infty } \left( \frac{M_N(Nt)}{N}, \frac{F_N(Nt)}{N},\frac{S_N(Nt)}{N}\right) =(c_m,{\overline{f}}(t),1{-}c_m{-}{\overline{f}}(t)), \end{aligned}$$

  where $({\overline{f}}(t))$ is the solution of an ODE, such that

  $$\begin{aligned} \lim _{t\rightarrow +\infty }{\overline{f}}(t)= {\overline{f}}(\infty )>0. \end{aligned}$$

  The process $({\overline{R}}_N(t))$ is stochastically upper-bounded by a positive recurrent Markov process. See Theorem 22. In the stationary phase there are few polymerases doing transcription of rRNAs. A fraction of them remains free, asymptotically ${\overline{f}}(\infty )$, i.e. the sequestration process does not control all “useless” polymerases. This is in fact a non-trivial consequence of the dynamic creation and destruction of 6S RNAs, even if an 6S RNA paired with a polymerase cannot be degraded. The fact that sequestration phenomenon of a fraction of the N polymerases occurs on the time scale $t{\mapsto }Nt$ is intuitive given that the rate of creation of 6S RNAs is constant.

In all cases the process $(Z_N(t))$ is stochastically upper-bounded by a positive recurrent Markov processes.

1.8 Outline of the paper

Section 2 introduces in detail the complete model of transcription and also an important model, the auxiliary process. The exponential/stationary phases corresponds to super/sub critical condition for this auxiliary process. They are investigated respectively in Sect. 3 and 4. The last Sects. 5 and 6 are devoted to the exponential/stationary regimes of our model.

2 Stochastic model

The chemical species involved in the regulation process are the genes of different types of mRNAs and rRNAs and of ${6S~\text{ RNA }}$, and polymerases. The products are different types of mRNAs and of rRNAs and also 6S RNAs. We first describe our main assumptions of our stochastic model.

2.1 Modeling assumptions

• Transcription of rRNAs.

  There are J types of rRNAs and there is a promoter (binding site for polymerases) for each of them. The transcription of an rRNA of type j, $1{\le }j{\le }J$, is in two steps. The promoters of rRNAs are assumed to have a high affinity during the growth phase: If one of these promoters is empty and if there is at least one free polymerase, then the promoter is occupied right away by a polymerase. Once a polymerase is bound to the promoter of the rRNA of type $1{\le }j{\le }J$, it starts elongation at rate $\alpha _{r,j}$ if there are strictly less than $C_{r,j}^N$ polymerases in the elongation phase of this rRNA. At a given moment there cannot be more than $C_{r,j}^N$ polymerases in elongation of an rRNA of type j. For each polymerase in elongation, nucleotides are collected at rate $\beta _{r,j}$. The simplification of the model is that the polymerases in elongation are moving closely on the gene so that the duration of time to get the last nucleotide for the oldest polymerase in elongation is enough to describe the time evolution of the number of polymerases producing rRNA of type j. The polymerases associated to an rRNA of type $j{\in }\{1,\ldots ,J\}$ can then be represented as a couple $(u_j,R_j)$, where $u_j{\in }\{0,1\}$ indicates if a polymerase is on the promoter or not, and $R_j{\in }{\mathbb {N}}$ is the number of polymerases in elongation: If $R_j{\ge }1$, an rRNA of type j is therefore created at rate $\beta _{r,j}$. The assumption is reasonable in the exponential phase, since in this case the number of polymerases producing rRNA of type j is maximal, of the order of $C_{r,j}^N$. See Sect. 5. The rRNA part of the system is therefore saturated. In the stationary phase, this assumption has little impact since, as we shall see, the total number of polymerases in the elongation phase of rRNAs is small with high probability and, therefore, negligible for our scaling analysis.

• Transcription of mRNAs.

  It is assumed that there are $C_m^N$ different types of mRNAs and that at a given time, for any $1{\le }i{\le }C_m^N$ there is at most one polymerase in the elongation phase of an mRNA of type i. When the promoter of an mRNA of type i is free, a free polymerase may bind to this promoter at a rate $\alpha _m$. If the promoter of an mRNA of type i is occupied, an mRNA is released at rate $\beta _m$ and the corresponding polymerase leaves the promoter at that instant. The production of mRNAs have simplified in the sense that the initiation phase and the elongation phase are merged into one step. The results obtained in this paper could be obtained without too much difficulty for a model distinguishing them, but at the expense of a more complicated state variable. The main difference in our model between the rRNAs and the mRNAs is on the number of polymerases in elongation of the corresponding gene. At a given moment, under favorable growth conditions, there will be many polymerases in the elongation phase of an rRNA, due in particular to the high initiation rate of these genes. For the mRNAs, the number of polymerases in elongation phase of a given mRNA type should be small in general. Indeed, there are in each cell few copies of each messenger (from 1 to 100). Furthermore, the rate of production of each messenger is such that its small number remains on average constant during growth or stationary phases and despite the regular degradation (average of 2 min half-life in high-growth rate phase) of each of them. See Sect. 2.3. In our model we have set the maximal number of polymerases in elongation phase of a given mRNA type to 1 for simplicity, but it is not difficult to adapt our results with a maximum number D. Similarly, the initiation rates and production rate, $\alpha _m$ and $\beta _m$ are taken equal for all species of mRNA, also for the sake of simplicity. We have simplified the description of the production of mRNAs to focus mainly on the sequestration mechanisms that regulate the transcription. From our point of view, the production of mRNAs holds/stores a subset of polymerases and releases each of them after some random amount of time. It should be noted that this is in fact the usual mathematical setting to investigate the fluctuations of the production of mRNAs and proteins. See (Berg 1978) and Rigney and Schieve (1977), see also (Paulsson 2005) for a review of these models.

• Creation/Degradation of 6S RNAs.

  The creation of 6S RNAs involves, of course, polymerase. As in the case of mRNAs, it is assumed that there is at most one polymerase in elongation phase of this sRNA. A 6S RNA is created at rate $\beta _6{>}0$. A 6S RNA is free when it is not bound to a polymerase. A given free 6S RNA is degraded at rate $\delta _6{\ge }0$. Only a free 6S RNA can be degraded.

• Sequestration/de-Sequestration of Polymerases.

  A polymerase is free when it is not bound to a gene or to a 6S RNA. In our study the total number of polymerases is assumed to be constant equal to N. A free polymerase is bound to a free 6S RNA at rate $\lambda$. A complex polymerase-6S RNA breaks into a free polymerase and a free 6S RNA at rate $\eta$.

2.2 The Markov process and its Q-matrix

The vector $(\alpha _{r,j})$ introduced is the vector of initiation rates of transcription of the different types of rRNAs. The difference between a slow growth (stationary phase) and a steady growth (exponential phase) will be expressed in terms of the comparison, coordinate by coordinate, of the vectors $(\alpha _{r,j})$ and $(\beta _{r,j})$. We now give a Markovian description of our system. Convenient limiting results will be obtained for the associated Markov process in both phases.

Fig. 1

Polymerases: Transcription of mRNAs/rRNAs and Sequestration

2.3 State space

All transitions described in the last section occurs after a random amount of time with an exponential distribution. With this assumption, there is a natural Markov process to investigate the regulation of transcription. The state space is given by

$$\begin{aligned} {{\mathcal {S}}}_N{\mathop {=}\limits ^{\text { def.}}}\left\{ x{=}(f,s,z, (u_j,r_j)){\in }{\mathbb {N}}^3{\times }\prod _{j=1}^J\left( \{0,1\}{\times }\{0,\ldots ,C_{r,j}^N\}\right) :\right. \\ \left. f{+}s{+}\sum _{j=1}^J(u_j{+}r_j)\le N\text { and if } f{>}0, \text { then }u_j{=}1, \forall 1{\le }j{\le }J \right\} , \end{aligned}$$

if the state of the system is $x{=}(f,s,z, (u,r)){\in }{{\mathcal {S}}}_N$, then

• f is the number of free polymerases;

• s, the number of sequestered polymerases;

• z, the number of free 6S RNAs;

  $(u,r){=}((u_j,r_j),1{\le }j{\le }J)$,

• $u_{j}{\in }\{0,1\}$ to indicate if a polymerase is bound to the promoter of the rRNA of type j or not;

• $0{\le }r_{j}{\le }C_{r,j}^N$, number of polymerases in elongation phase of an rRNA of type j.

• In state x, the number of polymerases in elongation phase of an mRNA is given by

  $$\begin{aligned} \Psi (x) {\mathop {=}\limits ^{\text { def.}}} N{-}f{-}s{-}\sum _{j=1}^J(u_j{+}r_j). \end{aligned}$$

  (1)

Note that the condition that $f{>}0$ implies $u_j{=}1$ for all $1{\le }j{\le }J$ is a consequence of the assumption of the first paragraph of the “Transcription of rRNAs” sub-section at the beginning of this section.

The associated Markov process is denoted by

$$\begin{aligned} (X_N(t)){\mathop {=}\limits ^{\text { def.}}} \left( F_N(t),S_N(t),Z_N(t),(U_N(t),R_N(t))\right) , \end{aligned}$$

with $(U_N(t),R_N(t)){=}((U^N_{j} (t),R^N_{j}(t)),1{\le }j{\le }J)$. The number of polymerases at time t in elongation phase of an mRNA is defined by $M_N(t){=}\Psi (X_N(t))$.

If $w{=}(w_j){\in }{\mathbb {N}}^J$, we define $\Vert w\Vert {=}w_1{+}\cdots {+}w_J$ and, for $1{\le }j{\le }J$, $e_j$ denotes the jth unit vector of ${\mathbb {N}}^J$. It is easily checked that $(X_N(t))$ is an irreducible Markov process on ${{\mathcal {S}}}_N$. Its transition rates are given by

• Transcription of rRNAs. For $1{\le }i,j{\le }J$,

  $$\begin{aligned} (f,s,z,(u,r))\longrightarrow {\left\{ \begin{array}{ll} (f{-}1,s,z,(u,r{+}e_j)) &{} \alpha _{r,j}\mathbbm {1}_{\left\{ f>0,r_j<C_{r,j}^N\right\} },\\ (0,s,z,(u{-}e_j,r{+}e_j)) &{} \alpha _{r,j}\mathbbm {1}_{\left\{ u_j=1,r_j<C_{r,j}^N,f{=}0\right\} },\\ (f{+}1,s,z,(u,r{-}e_j)) &{} \beta _{r,j}\mathbbm {1}_{\left\{ r_j>0,u_k{>}0, \forall 1{\le }k{\le }J\right\} },\\ (0,s,z,(u+e_i,r-e_j)) &{}\displaystyle \frac{\beta _{r,j}}{ J{-}\Vert u\Vert }\mathbbm {1}_{\left\{ r_j>0,u_i=0\right\} }. \end{array}\right. } \end{aligned}$$

• Transcription of mRNAs.

  $$\begin{aligned} (f,s,z,(u,r))\longrightarrow {\left\{ \begin{array}{ll} (f{-}1,s,z,(u,r)) &{} \alpha _{m}f\left( C_m^N{-}\Psi (x)\right) ,\\ (f{+}1,s,z,(u,r)) &{} \beta _{m}\Psi (x). \end{array}\right. } \end{aligned}$$

• Creation/Degradation of 6S RNAs.

  $$\begin{aligned} (f,s,z,(u,r))\longrightarrow {\left\{ \begin{array}{ll} (f,s,z{+}1,(u,r)) &{} \beta _6,\\ (f,s,z{-}1,(u,r)) &{} \delta _6z. \end{array}\right. } \end{aligned}$$

• Sequestration/de-Sequestration of Polymerases.

  $$\begin{aligned} (f,s,z,(u,r))\longrightarrow {\left\{ \begin{array}{ll} (f{-}1,s{+}1,z{-}1,(u,r)) &{} \lambda f z,\\ (f+1,s{-}1,z{+}1,(u,r)) &{} \eta s. \end{array}\right. } \end{aligned}$$

2.4 A possible extension for mRNAs

We have chosen to consider $C_m^N$ genes of mRNAs with the same parameters $\beta _m$ and $\alpha _m$ for the transcription by polymerases, for simplicity essentially. A generalization could be considered for which the $C_m^N$ types of mRNAs can be split into K sub-groups $({{\mathcal {C}}}^N_{m,k})$ of respective sizes $C^N{{\mathop {=}\limits ^{\text { def.}}}}(C^N_{m,k},1{\le }k{\le }K)$ and with the parameters $(\beta _{m,k},\alpha _{m,k},1{\le }k{\le }K)$.

It basically amounts to say that mRNAs can be partitioned according to the strengths of the affinity of their promoters and of their lengths in terms of nucleotides. See (Bremer and Dennis 2008) and Ron et al. (2010) for example. A similar assumption for translation of different types of proteins has been done in Fromion and Robert (2022). We denote by $M_{m,j}^N(t)$ the number of polymerases in the elongation phase of an mRNA whose type is in the set ${{\mathcal {C}}}_{m,j}^N$ at time t. The state process for this part of the system is

$$\begin{aligned} (M_N(t)){\mathop {=}\limits ^{\text { def.}}}(M_{m,k}^N(t)){\in }{{\mathcal {S}}}_m^N, \text { with } {{\mathcal {S}}}^N_m{\mathop {=}\limits ^{\text { def.}}}\left\{ x{\in }\prod _{k=1}^{K}\left[ 0,C_{m,k}^N\right] : |x|\le N\right\} . \end{aligned}$$

Without sequestration and transcription of rRNAs, the model is equivalent to a kind of Ehrenfest urn model, with $K+1$ urns, for $1{\le }k{\le }K$, the urn k has a maximal capacity of $C_{m,k}^N$ and the balls inside move to urn 0 at rate $\beta _{m,k}$. A ball in urn 0 go to a specific empty place of urn k at rate $\alpha _{m,k}(C_{m,k}^N{-}M_{m,k}^N(t))$

2.5 Orders of magnitude and scaling assumptions

We now discuss the orders of magnitude of the main parameters of the biological process.

• The scaling variable used in our analysis is N, the total number of polymerases in the cell. It is assumed that this number is constant during the growth phase investigated, this quantity is quite large, between 2000 and 10000 for E. coli, depending of the environment. See (Bakshi et al. 2012).

• The number J of different types of rRNA is small, of the order of 10. See (Bremer and Dennis 2008).

• We shall assume that the maximal number of polymerases in transcription of an rRNA of type j, $C_{r,j}^N$, is of the order of N, the total number of polymerases. Indeed, in a steady growth phase a given rRNA gene can accommodate a significant number of polymerases. Recall that the length in nucleotides of an rRNA is large, of the order of 5000.

• Similarly, the total number of different types of mRNAs is also of the order of N, several thousands, of the order of 3500 for E. coli.

See also (Ron et al. 2010; Karp et al. 2019) and Neidhardt and Umbarger (1996) for the estimation of the numerical values of these quantities in various contexts.

Due to the modeling assumptions of Sect. 2.1, we assume that the relations

$$\begin{aligned} \lim _{N\rightarrow +\infty } \frac{C_{r,j}^N}{N}=c_{r,j}>0,\ 1{\le }j{\le }J, \text { and } \lim _{N\rightarrow +\infty } \frac{C_m^N}{N}=c_m, \end{aligned}$$

(2)

hold and that, in order to cope with the production of rRNAs during a steady growth phase, the total number of polymerases N is larger than the total maximal number of polymerases in elongation phase of rRNAs, i.e. that $C_{r,1}^N{+}\cdots {+}C_{r,J}^N$ and, also that there are not too many polymerases for the transcription, i.e.

$$\begin{aligned} \max \left( C_m^N,\sum _{j=1}^JC_{r,j}^N\right)< N < C_m^N{+}\sum _{j=1}^JC_{r,j}^N. \end{aligned}$$

In view of (2), these assumptions are expressed by the following conditions on the scaled parameters $(c_{r,j})$ and $c_m$,

$$\begin{aligned} \max \left( c_m, \sum _{j=1}^J c_{r,j}\right)< 1 < \sum _{j=1}^J c_{r,j}{+}c_m. \end{aligned}$$

(3)

We can now introduce the two regimes of interest in our paper.

Definition 1

• (a) The Exponential Phase is defined by the relation

  $$\begin{aligned} \min _{1{\le }j{\le }J}\frac{\alpha _{r,j}}{\beta _{r,j}}>1. \end{aligned}$$

  (4)

  The initiation rate $\alpha _{r,j}$ of type j rRNAs is greater than its production rate.

• (b) The stationary phase is defined by the relation

  $$\begin{aligned} \max _{1{\le }j{\le }J}\frac{\alpha _{r,j}}{\beta _{r,j}}<1. \end{aligned}$$

  (5)

  The initiation rate $\alpha _{r,j}$ of type j rRNAs is less than its production rate.

2.6 An auxiliary model

In Sects. 3 and 4, we study a process which can be interpreted as a model similar to $(X_N(t))$ but with only transcription of mRNAs and sequestration by 6S RNAs but without rRNAs. The introduction of the auxiliary process is essentially motivated for the mathematical analysis. It is a technical tool used for the proofs of convergence for the biological system. The intuitive reasons to study this case are two-fold:

• (a) Exponential Phase. If Condition (4) holds, as we shall see, “most” of the $J{+}C_{r,1}^N{+}C_{r,2}^N{+}{\cdots }{+}C_{r,J}^N$ available places for transcriptions of rRNAs will be occupied by polymerases. Provided that this situation holds on a sufficiently large time scale, under Condition (3), there are $A_N$ available polymerases for sequestration and transcription of mRNAs, with

  $$\begin{aligned} A_N{\mathop {=}\limits ^{\text { def.}}}N{-}J{-}\sum _{j=1}^JC_{r,j}^N\sim \gamma N{<} C_m^N. \end{aligned}$$

  (6)

  The system works as if there were $A_N$ polymerases available for the transcription of mRNAs. With Condition (3), we have $A_N{<}C_m^N$.

• (b) Stationary Phase. When Condition (5) holds, then, roughly speaking, the total number of polymerases in the elongation phase of rRNAs is O(1), so that this part of the system is in some way negligible. In this case the total number of polymerases available for transcription of mRNAs is essentially N and thus greater than $C_m^N$ under Condition (3).

The mathematical analysis of exponential phase, resp. stationary phase, is done in Sect. 5, resp. Section 6.

We denote $(X^0_N(t)){=}(F^0_N(t),S^0_N(t),Z^0_N(t))$ the system defined in Sect. 2.2 but without the part of the model for rRNAs. For $(f,s,z){\in }$, the transition rates are:

$$\begin{aligned} (f,s,z)&\longrightarrow {\left\{ \begin{array}{ll} (f{-}1,s,z) &{} \alpha _{m}\left( C_m^N{-}(N{-}f{-}s)\right) f,\quad \\ (f{+}1,s,z) &{} \beta _{m}(N{-}f{-}s),\\ (f,s,z{+}1) &{} \beta _6,\\ \end{array}\right. } \nonumber \\ (f,s,z)&\longrightarrow {\left\{ \begin{array}{ll} (f,s,z{-}1) &{} \delta _6z,\\ (f{-}1,s{+}1,z{-}1) &{} \lambda f z,\\ (f+1,s{-}1,z{+}1) &{} \eta s. \end{array}\right. } \end{aligned}$$

(7)

Using the classical formulation in terms of a martingale problem, see Theorem (20.6) in Section IV of Rogers and Williams (1994) for example, the Markov process $(X^0_N(t))$ whose Q-matrix is given by Relation (7), as $(F^0_N(t),S^0_N(t),Z^0_N(t))$, the solution of the SDEs,

$$\begin{aligned} \mathop {}\mathopen {}\textrm{d}F^0_N(t)&={{\mathcal {P}}}_1\left( \left( 0,\eta S^0_N(t{-}\right) , \mathop {}\mathopen {}\textrm{d}t\right) {-}{{\mathcal {P}}}_3\left( \left( 0,\lambda F^0_N(t{-})Z^0_N(t{-})\right) ,\mathop {}\mathopen {}\textrm{d}t\right) \end{aligned}$$

(8)

$$\begin{aligned}&\quad {+}{{\mathcal {P}}}_2\left( \left( 0,\beta _m\left( N{-}F^0_N(t{-}){-}S^0_N(t{-})\right) ,\mathop {}\mathopen {}\textrm{d}t\right) \right) \nonumber \\&\quad {-}{{\mathcal {P}}}_4\left( \left( 0,\alpha _mF^0_N(t{-})\left( C_m^N{-}\left( N{-}F^0_N(t{-}){-}S^0_N(t{-})\right) \right) \right) ,\mathop {}\mathopen {}\textrm{d}t\right) \nonumber \\ \mathop {}\mathopen {}\textrm{d}S^0_N(t)&={-}{{\mathcal {P}}}_1\left( \left( 0,\eta S^0_N(t{-}\right) , \mathop {}\mathopen {}\textrm{d}t\right) {+}{{\mathcal {P}}}_3\left( \left( 0,\lambda F^0_N(t{-})Z^0_N(t{-})\right) ,\mathop {}\mathopen {}\textrm{d}t\right) \end{aligned}$$

(9)

$$\begin{aligned} \mathop {}\mathopen {}\textrm{d}Z^0_N(t)&={{\mathcal {P}}}_5\left( \left( 0,\beta _6\right) ,\mathop {}\mathopen {}\textrm{d}t\right) {-}{{\mathcal {P}}}_6\left( \left( 0,\delta _6Z^0_N(t-)\right) ,\mathop {}\mathopen {}\textrm{d}t\right) \nonumber \\&\quad {+}{{\mathcal {P}}}_1\left( \left( 0,\eta S^0_N(t{-}\right) , \mathop {}\mathopen {}\textrm{d}t\right) {-}{{\mathcal {P}}}_3\left( \left( 0,\lambda F^0_N(t{-}) Z^0_N(t{-})\right) ,\mathop {}\mathopen {}\textrm{d}t\right) , \end{aligned}$$

(10)

with the convenient initial conditions, where ${{\mathcal {P}}}_i$, $i{\in }\{1,2,3,4\}$ are independent Poisson point processes on ${\mathbb {R}}_+^2$ with intensity $\mathop {}\mathopen {}\textrm{d}s{\otimes }\mathop {}\mathopen {}\textrm{d}t$, where $\mathop {}\mathopen {}\textrm{d}s$ and $\mathop {}\mathopen {}\textrm{d}t$ refer to Lebesgue’s measure.

An integral representation of these SDEs is given in Section A.1 together with the expression of the previsible increasing processes of the associated martingales.

We will study two regimes of this stochastic model:

• Sub-critical case, when $c_m{>}1$, i.e. $N{<}C_m^N$ for N sufficiently large.

• Super-critical case, when $c_m{<}1$.

As it will be seen these two regimes are respectively associated to the exponential and stationary phases.

2.7 Notations

We define a filtration common to all our processes, as follows, for $t{\ge }0$

$$\begin{aligned} {{\mathcal {F}}}_t=\sigma \left( {{\mathcal {P}}}_i(A{\times }[0,s]): A{\in }{{\mathcal {B}}}({\mathbb {R}}_+), i{\in }\{1,\cdots ,6\}, s\le t\right) . \end{aligned}$$

(11)

From now on, all notions of stopping time, adapted process, martingale, refer to this (completed) filtration. A càdlàg process is an adapted process such that with probability one, it is right-continuous process with left limits at any positive real number.

If H is a locally compact metric space, we denote by ${{\mathcal {C}}}_c(H)$ the set of continuous functions with compact support on H. It is endowed with the topology of the uniform norm.

3 Sub-critical case

In this section we investigate the asymptotic properties of the auxiliary process of Sect. 2.4. It is assumed throughout this section that $c_m{>}1$ holds, the total number of possible sites for transcription of mRNAs is much larger than the total number of polymerases.

Definition 2

For $g{\in }{{\mathcal {C}}}_{c}\left( {\mathbb {R}}_+{\times }{\mathbb {N}}\right)$

$$\begin{aligned} \left\langle \mu _N,g\right\rangle {\mathop {=}\limits ^{\text { def.}}} \int _0^{+\infty } g\left( u,{F}^0_N(u)\right) \mathop {}\mathopen {}\textrm{d}u. \end{aligned}$$

(12)

We start with a technical result on a birth and death process.

Lemma 3

For $\kappa _0{>}0$ and $\kappa _1{>}0$, let (Y(t)) be a birth and death process on ${\mathbb {N}}$ whose Q-matrix is given by

$$\begin{aligned} q(x,x{+}1)=\kappa _0 \text { and } q(x,x{-}1)=\kappa _1 x,\quad x{\in }{\mathbb {N}}, \end{aligned}$$

• (a) if $Y(0){=}N$ and

  $$\begin{aligned} H_Y^N{\mathop {=}\limits ^{\text { def.}}}\inf \{t{>}0: Y(t){=}0\}, \end{aligned}$$

  then $(H_Y^N/\ln N)$ is converging in probability to a constant.

• (b) if $Y(0){=}0$, then for any $\delta {>}0$, the convergence in distribution of processes

  $$\begin{aligned} \lim _{N\rightarrow +\infty }\left( \frac{Y\left( N^{\delta }t\right) }{\ln (N)^2}\right) =(0) \end{aligned}$$

  holds.

The process (Y(t)) can be thought as a kind of discrete Ornstein-Uhlenbeck process on ${\mathbb {N}}$. In a queueing context, this is the process of the number of jobs of an $M/M/\infty$ queue. See Chapter 6 of Robert (2003) for example. Its invariant distribution is Poisson with parameter $\kappa _0/\kappa _1$.

Proof

The first assertion comes directly from Proposition 6.8 of Robert (2003). If $Y(0){=}0$ and, for $p{\ge }1$,

$$\begin{aligned} T_p=\inf \{t{>}0:Y(t)>p\}, \end{aligned}$$

Proposition 6.10 Robert (2003) gives that, if $\rho {{\mathop {=}\limits ^{\text { def.}}}}\kappa _0/\kappa _1$, the sequence

$$\begin{aligned} \left( \rho ^p\frac{T_p}{p!}\right) \end{aligned}$$

is converging in distribution to an exponential distribution. In particular for any $K{>}0$,

$$\begin{aligned} \lim _{N\rightarrow +\infty }{\mathbb {P}}\left( T_{\ln (N)^2}<KN^\delta \right) =0, \end{aligned}$$

since, by Stirling’s Formula,

$$\begin{aligned} \lim _{N\rightarrow +\infty } \frac{(\ln (N)^2)!}{\rho ^{\ln (N)^2}N^\delta }={+}\infty . \end{aligned}$$

The lemma is proved. $\square$

We begin with a lemma showing that the initial state of $({X}^0_N(t))$ can be taken with few free polymerases.

Lemma 4

If $({F}^0_N(0),{S}^0_N(0),{Z}^0_N(0)){=}(f_N,s_N,z_N)$, such that

$$\begin{aligned} \lim \frac{1}{N}(f_N,s_N,z_N)=(f_0,s_0,z_0){\in }{\mathbb {R}}_+^3, \end{aligned}$$

with $f_0{+}s_0{<}1$, and, if

$$\begin{aligned} {\tau }^0_N{\mathop {=}\limits ^{\text { def.}}}\inf \left\{ t:{F}^0_N(t)=0\right\} , \end{aligned}$$

then the sequence $({\tau }^0_N/(\ln N)^2)$ is converging in probability to 0.

The Condition $f_0{+}s_0{<}1$ is to take into account the fact that ${F}^0_N(0){+}{S}^0_N(0){\le }N$.

Proof

Because of the assumption $c_m{>}1$, there exists $\varepsilon _0{>}0$ such that, for N sufficiently large, $C_m^N{-}N{>}\varepsilon _0N$. The relations (7) for the transition rates gives that ${+}1$ jumps of $(F_N^0(t))$ occur at rate less than $N(\beta _m{+}\eta )$ since $S_N(t)$ is always less than N, and for ${-}1$ jumps of $F_N^0(t)$ occur at a rate greater than $\varepsilon _0N F^0_N(t)$. One can therefore construct a coupling $({F}^0_N(t),Y(t))$ such that $Y(0){=}{F}^0_N(0)$ and the relation

$$\begin{aligned} {F}^0_N(t)\le Y(Nt),\qquad \forall t{\ge }0, \end{aligned}$$

(13)

holds almost surely for all $t{\ge }0$, where (Y(t)) is a process as defined in Lemma 3 with $\kappa _0{=}\beta _m$ and $\kappa _1{=}\alpha _m \varepsilon _0$. We conclude the proof by using Lemma 3. $\square$

We can now state the main result of this section. It shows that for the asymptotic system, when N is large, all polymerases are eventually in the transcription phase of mRNAs, i.e. the fraction of sequestered polymerases is close to 0. A sketch of the proof is given in Section A.1 of the Appendix.

Proposition 5

(Starting from a Congested State) If the initial state $({F}^0_N(0),{S}^0_N(0), {Z}^0_N(0)){=}(0,s_N,z_N)$ is

$$\begin{aligned} \lim \frac{1}{N}(s_N,z_N)=(s_0,z_0)\in {\mathbb {R}}_+^2, \end{aligned}$$

and such that $s_0{+}z_0{<}1$, then, for the convergence in distribution of processes

$$\begin{aligned} \lim _{N\rightarrow +\infty } \left( \frac{{S}^0_N(t)}{N},\frac{{Z}^0_N(t)}{N}\right) =(s(t),z(t)), \end{aligned}$$

where (s(t), z(t)) is the unique solution of the system of ODEs,

$$\begin{aligned} {\dot{s}}(t)={-}\eta s(t){+}\lambda z(t) \frac{\beta _m{-}\left( \beta _m{-}\eta \right) s(t)}{\alpha _m(c_m{-}1{+}s(t))+\lambda z(t)}, \quad {\dot{s}}(t){+}{\dot{z}}(t)={-}\delta _6 z(t), \end{aligned}$$

with $(s(0),z(0))=(s_0,z_0)$. The function (s(t), z(t)) is converging to (0, 0) at infinity.

To study the asymptotic behavior of the model in the exponential regime, we investigate the occupation measure associated to free polymerases when the initial state is “small”. In this case, contrary to the last proposition, the processes $(S^0_N(t),Z^0_N(t))$ should be “slow”, i.e. their transition rate are of the order of O(1), only $(F^0_N(t))$ is “fast”.

Proposition 6

(Fixed Initial Point) If the initial state is such that $({F}^0_N(0),{S}^0_N(0), {Z}^0_N(0)){=}(f_0,s_0,z_0){\in }{\mathbb {N}}^3$, then, for the convergence in distribution

$$\begin{aligned} \lim _{N\rightarrow +\infty } \left\langle \mu _N,g\right\rangle = \int _0^{+\infty } {\mathbb {E}}\left( g\left( u,{{\mathcal {N}}}_1\left( 0,\rho _m\right) \right) \right) \mathop {}\mathopen {}\textrm{d}u, \end{aligned}$$

for any $g{\in }{{\mathcal {C}}}_{c}\left( {\mathbb {R}}_+{\times }{\mathbb {N}}\right)$, where $\rho _m{=}{\beta _m}/{(\alpha _m(c_m-1))}$, $\mu _N$ is the occupation measure defined by Relation (12), and ${{\mathcal {N}}}_1$ is a Poisson process with rate 1.

The sequence of processes $(S^0_N(t),Z^0_N(t))$ converges in distribution for the Skorohod topology to a jump process (Y(t)) on ${\mathbb {N}}^2$ whose transition rates are given by

$$\begin{aligned} (s,z)\longrightarrow (s,z){+} {\left\{ \begin{array}{ll} (1,{-}1)&{} \lambda \rho _m z,\\ ({-}1,1)&{} \eta s, \end{array}\right. } \quad (s,z)\longrightarrow (s,z) {\left\{ \begin{array}{ll} (0,1)&{} \beta _6,\\ (0,{-}1)&{} \delta _6z. \end{array}\right. } \end{aligned}$$

See Section A.2 of the appendix.

4 Super-critical case

In this section we study the auxiliary process under the condition $c_m{<}1$, so that $C_m^N{<}N$ for N sufficiently large. In this case the places for transcription of mRNAs are likely to be saturated quickly. Consequently, there should remain many free polymerases and the sequestration mechanism has to play a role.

If there are no 6S RNAs initially, since the creation of 6S RNAs is constant, the sequestration of a significant fraction of the polymerases will occur after a duration of time at least of the order of N. In this case, when a 6S RNA is created, it is right away paired with a free polymerase and will paired again and again that after the successive steps of sequestration/desequestration, as long as the number of free polymerases is sufficiently “large”. The sequestration occurs always before a possible degradation of the 6S RNAs takes place. The precise result is in fact more subtle than that. It will be shown that, on the fast time scale $t{\mapsto }N t$, the sequestration of polymerases increases but, due to the degradation of 6S RNAs there will remain a positive fraction of free polymerases.

The goal of this section is of proving an averaging principle for the process $(F_N^0(t),Z_N^0(t))$. A coupling and a technical lemma are presented in Sect. 4.1, tightness properties of occupations measures are proved in Sect. 4.2, finally the main convergence results are proved in Sect. 4.3.

Definition 7

For $N{>}0$ and $t{\ge }0$, we define

$$\begin{aligned} G^0_N(t){\mathop {=}\limits ^{\text { def.}}}C_m^N{-}\left( N{-}F^0_N(t){-}S^0_N(t)\right) , \end{aligned}$$

the number of “empty” places for transcription of mRNAs at time t.

The scaled process is defined by

$$\begin{aligned} \left( {\overline{X}}^0_N(t)\right) =\left( {\overline{F}}^0_N(t),G^0_N(Nt),Z^0_N(Nt)\right) \text { with } \left( {\overline{F}}^0_N(t)\right) {{\mathop {=}\limits ^{\text { def.}}}}\left( \frac{F^0_N(N t)}{N}\right) .\nonumber \\ \end{aligned}$$

(14)

If g is non-negative Borelian function on ${\mathbb {R}}_+^2{\times }{\mathbb {N}}^2$, we define the occupation measure

$$\begin{aligned} \left\langle \overline{\Lambda }^0_N,g\right\rangle {\mathop {=}\limits ^{\text { def.}}} \int _{{\mathbb {R}}_+}g\left( s,{\overline{X}}^0_N(s)\right) \mathop {}\mathopen {}\textrm{d}s. \end{aligned}$$

(15)

Note that the, a priori, slow process $({\overline{F}}^0_N(t))$ is also included in the definition of the occupation measure $\overline{\Lambda }^0_N$. The reason is that the proof of the tightness of $({\overline{F}}^0_N(t))$ (for the topology of the uniform norm on càdlàg functions) is not clear. Due to the fast time scale, the proof that the martingale component of $({\overline{F}}^0_N(t))$ vanishes does not seem to be straightforward.

The following initial conditions will be assumed,

$$\begin{aligned} \lim _{N\rightarrow +\infty } \frac{F^0_N(0)}{N}{=}{\overline{f}}_0{\in }(0,1{-}c_m),\, G^0_N(0){=}m_0,\, \text { and } Z^0_N(0){=}z_0, \end{aligned}$$

(16)

with $m_0$ $z_0{\in }{\mathbb {N}}$. Consequently, a fraction ${\overline{f}}_0$ of the polymerases are initially free and there are $z_0$ 6S RNAs and $C_m^N{-}m_0$ polymerases in the transcription phase of mRNAs and the number of sequestered polymerases $S^0_N(0)$ is such that

$$\begin{aligned} \lim _{N\rightarrow +\infty } \frac{S^0_N(0)}{N}{=}1{-}{\overline{f}}_0{-}c_m. \end{aligned}$$

As it will be seen in Sect. 4.1 there is no loss of generality to consider these initial conditions.

Before proving the convergence of the sequence of processes $({\overline{X}}_N^0(t))$, we analyze the convergence of a “stopped” version of it. In several technical arguments we will need that the fraction of free polymerases is not too small. A second step is of showing that, essentially, the stopped process does not differ from the original process.

Definition 8

For $a{>}0$, the stopping time $\tau _N(a)$ is defined by

$$\begin{aligned} \tau _N(a){\mathop {=}\limits ^{\text { def.}}}\inf \left\{ t{>}0: F^0_N(N t) \le a N\right\} , \end{aligned}$$

(17)

and

• (a) if (W(t)) is a càdlàg process, we denote $(W_N^a(t)){=}(W(N(t{\wedge }\tau _N(a))))$;

• (b) The “stopped” occupation measure $\overline{\Lambda }^{0,a}_N$ is defined by, if g is non-negative Borelian function on ${\mathbb {R}}_+^2{\times }{\mathbb {N}}^2$,

  $$\begin{aligned} \left\langle \overline{\Lambda }_N^{0,a},g\right\rangle {\mathop {=}\limits ^{\text { def.}}} \int _{0}^{\tau _N(a)}g\left( s,{\overline{X}}^0_N(s)\right) \mathop {}\mathopen {}\textrm{d}s. \end{aligned}$$

With a slight abuse, the notation $({\overline{F}}_N^a(t)){=}(F^0_N(N(t{\wedge }\tau _N(a)))/N)$ will be used in the following.

4.1 Technical Lemmas

The two processes $(G^0_N(t)$ and $(Z^0_N(t))$ are in fact in a neighborhood of 0 quickly. They will be the fast processes (on the timescale $t{\mapsto }Nt$) of our averaging principle. Using Relations (7), in state $F^0_N{=}f$, $G^0_N{=}g$ and $Z^0_N{=}z$, f, g, $z{\in }{\mathbb {N}}$, the jump rates of the process $(G^0_N(t)$ and $(Z^0_N(t))$ are respectively

$$\begin{aligned} {\left\{ \begin{array}{ll} {+}1,&{} \beta _m(C_m^N{-}g),\\ {-}1,&{} \alpha _m f g, \end{array}\right. } \quad \text { and } \quad {\left\{ \begin{array}{ll} {+}1,&{} \left( \beta _6{+}\eta (N{-}f-C_m^N+g)\right) ,\\ {-}1,&{} \left( \lambda f{+}\delta _6\right) z. \end{array}\right. } \end{aligned}$$

If $\eta _0{>}\eta$ and $\eta _1{>}\beta _mc_m$, and N sufficiently large, up to time $\tau _N(a)$, a simple coupling shows that there exist independent processes $(Y_G(t))$ and $(Y_Z(t))$ such that

$$\begin{aligned} G^0_N(Nt)\le Y_G(N^2t) \text { and } Z^0_N(Nt)\le Y_Z(N^2t), \end{aligned}$$

(18)

holds for all $t{\in }(0,\tau _N(a))$. The process $(Y_G(t))$, resp. the process $(Y_Z(t))$, is as in Lemma 3 with $\kappa _{i,G}{=}\eta _1$ and $\kappa _{o,G}{=}\alpha _m a$ (resp. $\kappa _{i,Z}{=}\eta _0$ and $\kappa _{o,Z}{=}\lambda a$), and $Y_G(0){=}G^0_N(0)$, resp. $Y_Z(0){=}Z^0_N(0)$. It is not difficult, using again Lemma 3, as in Sect. 3, that the hitting time of (0, 0) by $(Y_G(t),Y_Z(t))$ is of the order of $\ln N$ so that Condition (16) for the initial state can be assumed.

Lemma 9

Under Conditions (16) and $c_m{<}1$, and if $a{\in }(0,{\overline{f}}_0)$, then

$$\begin{aligned} \lim _{N\rightarrow +\infty } {\mathbb {P}}\left( \tau _N(a){<} t_0^a\right) =0, \end{aligned}$$

with $t_0^a{=}({\overline{f}}_0{-}a)/\beta _6$, and the relation

$$\begin{aligned} \lim _{N\rightarrow +\infty } \left( \frac{G^{0,a}_N(Nt)}{\ln (N)^2},\frac{Z^{0,a}_N(Nt)}{(\ln (N))^2}\right) =(0,0). \end{aligned}$$

holds for the convergence in distribution of processes.

In the following, we will use the notation $t_0^a$, where $a{\in }(0,{\overline{f}}_0)$ is fixed.

Proof

The first relation is clear since, for $x{\le }1$ and $t{>}0$, on the event $\{{\overline{F}}^0_N(t){<}x\}$ there are at least $F^0_N(0){-}\lfloor Nx\rfloor {-}z_0$ new 6S RNAs which have been created up to time t. The rest of the proof follows from the coupling with $(Y_G(t),Y_Z(t))$, Relation (18), and Lemma 3. $\square$

4.2 Tightness properties

Throughout this section, conditions (16) and $c_m{<}1$ are assumed to hold.

Proposition 10

The sequence of measure-valued processes $(\overline{\Lambda }_N^{0,a})$ on the state space $[0,t_0^a){\times }{\mathbb {R}}_+{\times }{\mathbb {N}}^2$ is tight for the convergence in distribution and any limiting point $\overline{\Lambda }_\infty ^{0,a}$ can be expressed as,

$$\begin{aligned} \left\langle \overline{\Lambda }_\infty ^{0,a},f\right\rangle =\int _{[0,t_0^a){\times }{\mathbb {R}}_+{\times }{\mathbb {N}}^2} f\left( s,x,p\right) \pi _s^a(\mathop {}\mathopen {}\textrm{d}x,\mathop {}\mathopen {}\textrm{d}p)\mathop {}\mathopen {}\textrm{d}s, \end{aligned}$$

(19)

for any function f with compact support on $[0,t_0^a){\times }{\mathbb {R}}_+{\times }{\mathbb {N}}^2$, where $t_0^a{=}({\overline{f}}_0{-}a)/\beta _6$ and $(\pi _s^a)$ is an optional process with values in the space of probability distributions on ${\mathbb {R}}_+{\times }{\mathbb {N}}^2$.

For an introduction to the convergence in distribution of measure-valued processes, see (Dawson 1993). The optional property is just used to have convenient measurability properties to define random variables as integrals with respect to $(\pi _s^a,s{>}0)$. See Section VI.4 of Rogers and Diffusions (2000).

Proof

Note that, for $K{>}0$ and $t{<}t_0^a$, since

$$\begin{aligned} \int _0^t\mathbbm {1}_{\left\{ Z^0_N(Ns)\ge K\right\} }\mathop {}\mathopen {}\textrm{d}s= \int _0^t\mathbbm {1}_{\left\{ Z^{0,a}_N(s)\ge K\right\} }\mathop {}\mathopen {}\textrm{d}s \end{aligned}$$

holds on the event $\{\tau _N(a){\ge }t_0^a\}$, then

$$\begin{aligned}{} & {} {\mathbb {E}}\left( \overline{\Lambda }^{0,a}_N([0,t_0^a]{\times }[0,1]{\times }{\mathbb {N}}{\times }[K,{+}\infty ])\right) \\{} & {} \quad \le {\mathbb {E}}\left( \mathbbm {1}_{\left\{ \tau _N(a)>t_0^a\right\} }\int _0^{t_0^a}\mathbbm {1}_{\left\{ Z^{0,a}_N(s)\ge K\right\} }\mathop {}\mathopen {}\textrm{d}s\right) {+}t_0^a{\mathbb {P}}\left( \tau _N(a){<}t_0^a\right) , \end{aligned}$$

and, with Relation (18) and Lemma 9, we have

$$\begin{aligned}{} & {} {\mathbb {E}}\left( \mathbbm {1}_{\left\{ \tau _N(a)>t_0^a\right\} }\int _0^{t_0^a}\mathbbm {1}_{\left\{ Z^{0,a}_N(s)\ge K\right\} }\mathop {}\mathopen {}\textrm{d}s\right) \\{} & {} \qquad \le \int _0^{t_0^a}{\mathbb {P}}(Y_Z(N^2s)\ge K)\mathop {}\mathopen {}\textrm{d}s=\frac{1}{N^2}\int _0^{N^2t_0^a}{\mathbb {P}}(Y_Z(s)\ge K)\mathop {}\mathopen {}\textrm{d}s, \end{aligned}$$

since the Markov process $(Y_Z(t))$ converges in distribution to a Poisson distribution with parameter $\kappa _{i,Z}/\kappa _{o,Z}$, the ergodic theorem for Markov processes and Lemma 9 give therefore the inequality

$$\begin{aligned} \limsup _{N\rightarrow +\infty } {\mathbb {E}}\left( \overline{\Lambda }^{0,a}_N([0,t_0^a]{\times }[0,1]{\times }{\mathbb {N}}{\times }[K,{+}\infty ])\right) \le t_0^a {\mathbb {P}}({{\mathcal {N}}}_1(0,\eta /(\lambda a)){\ge }K), \end{aligned}$$

where ${{\mathcal {N}}}_1$ is a Poisson process on ${\mathbb {R}}_+$ with rate 1. One can choose K sufficiently large such that ${\mathbb {E}}\left( \overline{\Lambda }^{0,a}_N([0,t_0^a]{\times }[0,1]{\times }{\mathbb {N}}{\times }[K,{+}\infty ])\right)$ is arbitrarily small uniformly in N. Similarly, by replacing $(Z^0_N,Y_Z)$ by $(G^0_N,Y_G)$ the same property can be proved for ${\mathbb {E}}\left( \overline{\Lambda }^{0,a}_N([0,t_0^a]{\times }[0,1]{\times }[K,{+}\infty ]{\times }{\mathbb {N}}\right)$ for K and N sufficiently large. For any $\varepsilon {>}0$, there exists some $K_0$ such that

$$\begin{aligned} \sup _{N} {\mathbb {E}}\left( \overline{\Lambda }^{0,a}_N([0,t_0^a]{\times }[0,1]{\times }[0,K_0]^2\right) \ge (1{-}\varepsilon )t_0^a. \end{aligned}$$

Lemma 1.3 of Kurtz (1992) shows that the sequence $(\overline{\Lambda }^{0,a}_N)$ is tight, and Lemma 1.4 of the same reference gives the representation (19). $\square$

Proposition 10 has established tightness properties $(\overline{\Lambda }_N^{0,a})$. The following simple lemma extends this result in terms of the convergence of stochastic processes. It will be used repeatedly, in particular to identify the possible limits of $(\overline{\Lambda }_N^{0,a})$. See (Dawson 1993) for example.

Lemma 11

If $(\overline{\Lambda }_{N_k}^{0,a})$ is a subsequence converging to $\overline{\Lambda }^{0,a}_\infty$ satisfying Relation (19), then for any $g{\in }{{\mathcal {C}}}_c({\mathbb {R}}_+{\times }{\mathbb {N}}^2)$, for the convergence in distribution of processes associated to the uniform norm,

$$\begin{aligned} \lim _{k\rightarrow +\infty } \left( \int _0^t g\left( {\overline{X}}^0_{N_k}(s)\right) \mathop {}\mathopen {}\textrm{d}s\right) = \left( \int _0^t\int _{{\mathbb {R}}_+{\times }{\mathbb {N}}^2} g\left( x,p\right) \pi _s^a(\mathop {}\mathopen {}\textrm{d}x,\mathop {}\mathopen {}\textrm{d}p)\mathop {}\mathopen {}\textrm{d}s \right) . \end{aligned}$$

Proof

The tightness of the sequence of stochastic processes is obtained by the use of the criterion of the modulus of continuity. See Theorem 7.3 of Billingsley (1999). The identification of the limit is a straightforward consequence of the convergence of $(\overline{\Lambda }_{N_k}^{0,a})$ $\square$

If we divide by $N^2$ Relation (40) of the appendix, we get that, on the event $\{\tau _N(a){>}t\}$, the relation

$$\begin{aligned} \frac{1}{N^2}&f\left( {\overline{X}}^0_N(t)\right) =\frac{1}{N^2}f\left( {\overline{X}}^0_N(0)\right) {+}\frac{M_{f,N}(t)}{N^2}\nonumber \\&+\lambda \int _0^{t} \nabla _{-\frac{e_1}{N}-e_3}(f)\left( {\overline{X}}^0_N(s)\right) {\overline{F}}^0_N(Ns)Z^0_N(Ns)\mathop {}\mathopen {}\textrm{d}s\nonumber \\&{+}\eta \int _0^{t}\nabla _{\frac{e_1}{N}{+}e_3}(f)\left( {\overline{X}}^0_N(s)\right) \left( 1{-}\frac{C_m^N}{N}{+}\frac{G^0_N(Ns)}{N}{-}{\overline{F}}^0_N(Ns)\right) \mathop {}\mathopen {}\textrm{d}s\nonumber \\&+\alpha _m \int _0^{t} \nabla _{-\frac{e_1}{N}-e_2}(f)\left( {\overline{X}}^0_N(s)\right) G^0_N(Ns){\overline{F}}^0_N(Ns)\mathop {}\mathopen {}\textrm{d}s\nonumber \\&+\beta _m \int _0^{t} \nabla _{\frac{e_1}{N}{+}e_2}(f)\left( {\overline{X}}^0_N(s)\right) \left( \frac{C_m^N}{N}{-}\frac{G^0_N(Ns)}{N}\right) \mathop {}\mathopen {}\textrm{d}s\nonumber \\&+\frac{\beta _6}{N} \int _0^{t} \nabla _{e_3}(f)\left( {\overline{X}}^0_N(s)\right) \mathop {}\mathopen {}\textrm{d}s+\frac{\delta _6}{N} \int _0^{t} \nabla _{-e_3}(f)\left( {\overline{X}}^0_N(s)\right) Z^0_N(Ns)\mathop {}\mathopen {}\textrm{d}s \end{aligned}$$

(20)

holds. The process (M f, N(t)) is a martingale whose previsible increasing process is given by Relation (41). For $i{\in }\{1,2,3\}$, $e_i$ is the ith unit vector of ${\mathbb {R}}^3$ and, if g is a function on ${\mathbb {R}}_+{\times }{\mathbb {N}}^2$, for x, $a{\in }{\mathbb {R}}_+{\times }{\mathbb {N}}^2$, $\nabla _a(g)(x){=}g(x{+}a){-}g(x)$.

Lemma 12

If f is a continuous bounded function on ${\mathbb {R}}_+{\times }{\mathbb {N}}$, then the martingale $(M_{f,N}(t), t<t_0^a)$ of Relation (20) converges in distribution to 0.

Proof

We take care of one of the six terms of $(\left\langle M_{f,N}/N^2\right\rangle (t))$ of Relation (41), the arguments are similar for the others, even easier.

$$\begin{aligned} A_{1,N}(t){\mathop {=}\limits ^{\text { def.}}}\frac{\lambda }{N^2} \int _0^{t} \left[ \nabla _{-\frac{e_1}{N}-e_3}(f)\left( {\overline{X}}^0_N(s)\right) \right] ^2{\overline{F}}^0_N(s)Z^0_N(Ns)\mathop {}\mathopen {}\textrm{d}s \end{aligned}$$

We note that for $t{\ge } 0$, $0{\le }{Z}^0_N(t){\le }N{+}{{\mathcal {P}}}_5((0,\beta _6){\times }(0,t])$. Consequently, Doob’s Inequality shows the convergence of $(M_{f,N}(t)/N^2)$ to 0. The lemma is proved.

$\square$

Proposition 13

If $\overline{\Lambda }_\infty ^{0,a}$ is a limiting point of $\overline{\Lambda }_n^{0,a}$ with the representation (19) of Proposition 10, then, if $\pi ^{1,a}_t{=}\pi _t^{0,a}(\cdot ,{\mathbb {N}}^2)$, for any $t{<}t_0^a$ and any continuous function g on ${\mathbb {R}}_+{\times }{\mathbb {N}}^2$ we have

$$\begin{aligned} \int _0^t \int g(x,p)\pi _s^a(\mathop {}\mathopen {}\textrm{d}x,\mathop {}\mathopen {}\textrm{d}p)\mathop {}\mathopen {}\textrm{d}s= & {} \int _0^t \int _{{\mathbb {R}}_+} {\mathbb {E}}\left[ g\left( x,{{\mathcal {N}}}_1\left( \left[ 0,\rho _m\frac{c_m}{x}\right] \right) ,\right. \right. \nonumber \\{} & {} \left. \left. {{\mathcal {N}}}_2\left( \left[ 0,\rho _1\frac{1{-}c_m{-}x}{x}\right] \right) \right) \right] \pi _s^{1,a}(\mathop {}\mathopen {}\textrm{d}x)\mathop {}\mathopen {}\textrm{d}s,\qquad \end{aligned}$$

(21)

where ${{\mathcal {N}}}_1$ and ${{\mathcal {N}}}_2$ are two independent Poisson processes on ${\mathbb {R}}_+$ with rate 1 and

$$\begin{aligned} \rho _1{=}\frac{\eta }{\lambda } \text { and }\rho _m{=}\frac{\beta _m}{\alpha _m}. \end{aligned}$$

(22)

Relation (21) states that, for almost all $t{<}t_0^a$, $\pi _t$ conditioned on the first coordinate x is a product of two Poisson distributions with respective parameters $\rho _mc_m/x$ and $\rho _1(1{-}c_m{-}x)/x$.

Proof

Let $(\overline{\Lambda }^0_{N_k})$ be a subsequence of $(\overline{\Lambda }^0_N)$ converging to some $\overline{\Lambda }^0_\infty$ of the form (19). By letting k go to infinity in Relation (20), with Lemmas 9, 11 and 12, we obtain that there exists an event ${{\mathcal {E}}}_1$ with ${\mathbb {P}}({{\mathcal {E}}}_1){=}1$ on which the relation

$$\begin{aligned}{} & {} \int _0^{t}\int _{{\mathbb {R}}_+{\times }{\mathbb {N}}^2} \left( \eta \left( 1{-}c_m{-}x\right) \nabla _{e_3}(f)(x,p){+} \lambda xp_2\nabla _{-e_3}(f)(x,p) \right) \pi _s^{a}(\mathop {}\mathopen {}\textrm{d}x,\mathop {}\mathopen {}\textrm{d}p)\mathop {}\mathopen {}\textrm{d}s\\{} & {} \quad + \int _0^{t}\int _{{\mathbb {R}}_+{\times }{\mathbb {N}}^2} \left( \beta _mc_m \nabla _{e_2}(f)(x,p){+}\alpha _m\nabla _{-e_2}(f)(x,p) p_1x\right) \pi _s^{a}(\mathop {}\mathopen {}\textrm{d}x,\mathop {}\mathopen {}\textrm{d}p)\mathop {}\mathopen {}\textrm{d}s=0, \end{aligned}$$

< holds for all $t{\le }T$ and for all functions $f{\in }{{\mathcal {C}}}_c({\mathbb {R}}_+{\times }{\mathbb {N}}^2)$, by using the separability property of this space of functions for the uniform norm. See Corollary 11.2.5 of Richard (2018) for example. If $f(x,p){=}f_1(x)f_2(p)$, this relation can be rewritten as

$$\begin{aligned} \int _0^t\int _{{\mathbb {R}}_+{\times }{\mathbb {N}}^2} f_1(x)\Omega [x](f_2)(p)\pi _s^{a}(\mathop {}\mathopen {}\textrm{d}x,\mathop {}\mathopen {}\textrm{d}p)\mathop {}\mathopen {}\textrm{d}s=0 \end{aligned}$$

where, for $h:{{\mathbb {N}}^2}{\rightarrow }{\mathbb {R}}_+$ and $p{=}(p_1,p_2){\in }{\mathbb {N}}^2$,

$$\begin{aligned} \Omega [x](h)(p){} & {} =\beta _mc_m \nabla _{e_1}(h)(p){+}\alpha _m p_1x\nabla _{-e_1}(h)(p) \\{} & {} \quad {+}\eta (1{-}c_m{-}x)\nabla _{e_2}(h)(p) {+}\lambda xp_2 \nabla _{-e_2}(h)(p). \end{aligned}$$

$\Omega [x]$ is the jump matrix of two independent birth and death processes $(Y_1(t))$ and $(Y_2(t))$ as in Lemma 3 with parameters $\kappa _0{=}\beta _mc_m$, $\kappa _1{=}\alpha _m x$ for $(Y_1(t))$ and $\kappa _0{=}\eta (1{-}c_m{-}x)$, $\kappa _1{=}\lambda x$ for $(Y_2(t))$.

Consequently, for almost all $t{\le }T$, the relation

$$\begin{aligned} \int _{{\mathbb {R}}_+{\times }{\mathbb {N}}^2} f_1(x)\Omega [x](f_2)(p)\pi _t^a\mathop {}\mathopen {}\textrm{d}x,\mathop {}\mathopen {}\textrm{d}p)=0 \end{aligned}$$

holds. Hence, if $\widetilde{\pi }^a_t(\cdot |x)$ is the conditional probability on ${\mathbb {N}}^2$ of $\pi ^a_t(\mathop {}\mathopen {}\textrm{d}x,\mathop {}\mathopen {}\textrm{d}p)$ given x, we have

$$\begin{aligned} \int _{{\mathbb {R}}_+} f_1(x)\int _{{\mathbb {N}}^2}\Omega [x](f_2)(p)\widetilde{\pi }^a_t(\mathop {}\mathopen {}\textrm{d}p|x)\pi ^{1,a}(\mathop {}\mathopen {}\textrm{d}x)=0, \end{aligned}$$

we deduce that the relation

$$\begin{aligned} \int _{{\mathbb {N}}^2}\Omega [x](f_2)(p)\widetilde{\pi }^a_t(\mathop {}\mathopen {}\textrm{d}p|x)=0 \end{aligned}$$

holds $\pi _t^{1,a}(\mathop {}\mathopen {}\textrm{d}x)$ almost surely, for all functions $f_2$ with finite support on ${\mathbb {N}}^2$. Consequently, $\pi ^{1,a}(\mathop {}\mathopen {}\textrm{d}x)$ almost surely, $\widetilde{\pi }^a_t(\mathop {}\mathopen {}\textrm{d}p|x)$ is the invariant distribution associated to the Q-matrix $\Omega [x]$. The proposition is proved. $\square$

We fix $(N_k)$ an increasing sequence such the sequence $(\overline{\Lambda }_{N_k}^{0,a})$ is converging in distribution to the law of $\overline{\Lambda }_\infty ^{0,a}$ with a representation given by Relations (19) and (21).

4.3 Averaging principle

We define, for $t{\ge }0$,

$$\begin{aligned} {\widetilde{Z}}^0_N(t)=S^0_N(t){+}Z^0_N(t), \end{aligned}$$

${\widetilde{Z}}^0_N(t)$ is in fact the total number of 6S RNAs (free or paired) of the system at time t. Using the SDEs (9) and (10), we have

$$\begin{aligned} \frac{{\widetilde{Z}}^0_N(Nt)}{N}=M_{Z,N}(t){+}\frac{{\widetilde{Z}}^0_N(0)}{N}{+}\beta _6 t{-}\delta _6\int _0^tZ^0_N(Ns)\mathop {}\mathopen {}\textrm{d}s, \end{aligned}$$

(23)

where $(M_{Z,N}(t))$ is a local martingale whose previsible increasing process is given by

$$\begin{aligned} \left( \left\langle M_{Z,N}\right\rangle (t)\right) =\left( \frac{1}{N}\left( \beta _6 t{+}\delta _6\int _0^tZ^0_N(Ns)\mathop {}\mathopen {}\textrm{d}s\right) \right) . \end{aligned}$$

(24)

Proposition 14

Under Conditions (16) and $c_m{<}1$, for the convergence in distribution of processes

$$\begin{aligned} \lim _{k\rightarrow +\infty } \left( \int _0^t Z_{N_k}(N_ks)\mathop {}\mathopen {}\textrm{d}s,t<t_0^a\right) =\left( \rho _1\int _0^t\int _{{\mathbb {R}}_+}\frac{1{-}c_m{-}x}{x}\pi _s^{1,a}(\mathop {}\mathopen {}\textrm{d}x)\mathop {}\mathopen {}\textrm{d}s, t< t_0^a\right) , \end{aligned}$$

with $t_0^a{=}({\overline{f}}_0{-}a)/\beta _6$ and $\rho _1{=}\eta /\lambda$. Furthermore, $(M_{Z,N}(t),t{<}t_0^a)$ is converging to 0.

Proof

The convergence of the sequence of stochastic processes $(M_{Z,N}(t),t{<}t_0^a)$ to 0 is a consequence of Relations (23) and (24), and of Doob’s Inequality. For $0{\le }s{\le }t$, the coupling (18) and Cauchy–Schwartz’ Inequality give

$$\begin{aligned} {\mathbb {E}}\left( \left( \mathbbm {1}_{\left\{ \tau _N(a){>}t\right\} }\int _s^t Z^0_{N_k}(Ns)\mathop {}\mathopen {}\textrm{d}s \right) ^2\right)\le & {} (t-s){\mathbb {E}}\left( \mathbbm {1}_{\left\{ \tau _N(a){>}t\right\} }\int _s^t Z^0_{N_k}(s)^2\mathop {}\mathopen {}\textrm{d}s \right) \\\le & {} (t{-}s){\mathbb {E}}\left( \int _s^t Y_Z(Ns)^2\mathop {}\mathopen {}\textrm{d}s \right) \\ {}\le & {} (t{-}s)^{2}\sup _{u{\ge }0}{\mathbb {E}}\left( Y_Z(u)^2\right) . \end{aligned}$$

We now use the Kolmogorov-Čentsov’s criterion, see Theorem 2.8 and Problem 4.11, page 64 of Karatzas and Shreve (1998) and Lemma 9 to show that the sequence of stochastic processes

$$\begin{aligned} \left( \int _0^t Z_{N_k}^0(Ns)\mathop {}\mathopen {}\textrm{d}s, t<t_0^a\right) \end{aligned}$$

is tight for the convergence in distribution.

Lemma 11 and Relation (21) give the convergence in distribution of processes

$$\begin{aligned}{} & {} \lim _{k\rightarrow +\infty } \left( \int _0^t Z^0_{N_k}(N_ks){\wedge }K \mathop {}\mathopen {}\textrm{d}s,t<t_0^a\right) \\{} & {} \quad =\left( \int _0^t\int _{{\mathbb {R}}_+}{\mathbb {E}}\left( {{\mathcal {N}}}_1\left( 0,\rho _1\frac{1{-}c_m{-}x}{x}\right) {\wedge }K\right) \,\pi _s^{1,a}(\mathop {}\mathopen {}\textrm{d}x)\mathop {}\mathopen {}\textrm{d}s,t<t_0^a\right) . \end{aligned}$$

By using again Relation (18), we have

$$\begin{aligned} {\mathbb {E}}\left( \int _0^{t_0^a} Z_{N_k}(N_ks)\mathbbm {1}_{\left\{ Z^0_{N_k}(N_ks)\ge K\right\} } \mathop {}\mathopen {}\textrm{d}s\right) \le {\mathbb {E}}\left( \int _0^{t_0^a} Y_Z({N_k}^2s)\mathbbm {1}_{\left\{ Y_Z(N_k^2s)\ge K\right\} } \mathop {}\mathopen {}\textrm{d}s\right) \end{aligned}$$

and the convergence in distribution of processes of $(Y_Z(t))$, as t goes to infinity, to $Y_Z(\infty )$ a random variable with a Poisson distribution with parameter $\rho _Z{=}\kappa _{i,Z}/\kappa _{o,Z}$ give

$$\begin{aligned} \lim _{k\rightarrow +\infty } {\mathbb {E}}\left( \int _0^{t_0^a} Y_Z(N_k^2s)\mathbbm {1}_{\left\{ Y_Z(N_k^2s)\ge K\right\} } \mathop {}\mathopen {}\textrm{d}s\right) =t_0^a{\mathbb {E}}\left( Y_Z(\infty )\mathbbm {1}_{\left\{ Y_Z(\infty ){\ge }K\right\} }\right) . \end{aligned}$$

It is then easy to obtain the first convergence by letting K go to infinity.

The proposition is proved. $\square$

Relation (23) therefore shows that, on the time interval $I_a{=}[0,t_0^a)$, the sequence of processes

$$\begin{aligned} \left( \frac{{\widetilde{Z}}^0_{N_k}(N_kt)}{N_k}\right) \end{aligned}$$

is converging in distribution. Since

$$\begin{aligned} \left( \frac{{\widetilde{Z}}^0_{N_k}(N_kt)}{N_k}\right) =\left( 1{-}\frac{F_{N_k}^0(N_kt)}{N_k}{-}\frac{C_m^{N_k}}{N_k}{+}\frac{G_{N_k}^0(N_kt)}{N_k}{+}\frac{Z_{N_k}^0(N_kt)}{N_k}\right) \end{aligned}$$

with Lemma 9, we therefore obtain that the sequence of processes $(F_{N_k}(N_kt)/N_k)$ is converging in distribution to some process $({\overline{f}}(t))$ on $I_a$. In particular, for $t{<}t_0^a$ and $g{\in }{{\mathcal {C}}}_c({\mathbb {R}}_+)$, we have

$$\begin{aligned} \int _0^t \int g(x)\pi _s^1(\mathop {}\mathopen {}\textrm{d}x)\mathop {}\mathopen {}\textrm{d}s =\int _0^t g\left( {\overline{f}}(s)\right) \mathop {}\mathopen {}\textrm{d}s, \end{aligned}$$

hence, $\pi _s^1$ is in fact the Dirac measure at ${\overline{f}}(s)$ for $s{<}t_0^a$.

Relation (23) gives that, on the time interval $I_a$ and under the initial conditions (16), then the sequence of processes $({\overline{F}}^0_{N_k}(t))$ of Relation (14) is converging in distribution to $({\overline{f}}(t))$ such that

$$\begin{aligned} 1{-}{\overline{f}}(t){} & {} = 1{-}{\overline{f}}_0{+}\beta _6 t {-} \delta _6\rho _1\int _0^t\int _{{\mathbb {R}}_+}\frac{1{-}c_m{-}x}{x}\pi _s^{1,a}(\mathop {}\mathopen {}\textrm{d}x)\mathop {}\mathopen {}\textrm{d}s\nonumber \\{} & {} = 1{-}{\overline{f}}_0{+}\beta _6 t {-} \delta _6\rho _1\int _0^t\frac{1{-}c_m{-}{\overline{f}}(s)}{{\overline{f}}(s)}\mathop {}\mathopen {}\textrm{d}s \end{aligned}$$

(25)

with Proposition 14 and Notation (22).

Hence, by uniqueness of the solution of the integral equation,

$$\begin{aligned} {\overline{f}}(t) = {\overline{f}}_0{-}\delta _6(\rho _6{+}\rho _1)t{+} \delta _6\rho _1(1{-}c_m)\int _0^t\frac{1}{{\overline{f}}(s)}\mathop {}\mathopen {}\textrm{d}s \end{aligned}$$

holds on $I_a$, for the convergence in distribution, we thus have

$$\begin{aligned} \lim _{N\rightarrow +\infty } \left( {\overline{F}}_{N}(t), t{\in }I_a\right) =\left( {\overline{f}}(t), t{\in }I_a\right) . \end{aligned}$$

The equilibrium point of $({\overline{f}}(t))$ is ${\overline{f}}_\infty {=}{\rho _1}(1{-}c_m)/{(\rho _6{+}\rho _1)}$, if ${\overline{f}}_0{<}{\overline{f}}_\infty$, then ${\overline{f}}(t){\ge }{\overline{f}}_0$ for all $t{\ge }0$, and otherwise ${\overline{f}}(t){\ge }{\overline{f}}_{\infty }$. By induction, this implies that the convergence in distribution of $({\overline{F}}_{N}(t))$ can be extended on time intervals $(0,nt_0^a)$, for all $n{\ge }1$ and, consequently, on ${\mathbb {R}}_+$. We summarize our results.

Theorem 15

(Law of Large Numbers) If

$$\begin{aligned} \lim _{N\rightarrow +\infty } \frac{F^0_N(0)}{N}={\overline{f}}_0{\in }(0,1{-}c_m), \end{aligned}$$

and $(G_N(0),Z_N(0){=}(m_0,z_0)$, then, for the convergence in distribution of processes,

$$\begin{aligned} \lim _{N\rightarrow +\infty } \left( \frac{F^0_N(Nt)}{N}\right) =({\overline{f}}(t)), \end{aligned}$$

where $({\overline{f}}(t))$ is the solution of the ODE

$$\begin{aligned} \frac{\mathop {}\mathopen {}\textrm{d}{\overline{f}}}{\mathop {}\mathopen {}\textrm{d}t} (t)= {-}\delta _6\left( \rho _6 {+}\rho _1\right) {+}\delta _6\rho _1(1{-}c_m)\frac{1}{{\overline{f}}(t)}, \end{aligned}$$

(26)

where $\rho _6$ and $\rho _1$ are defined by Relation (22).

We summarize the results obtained for the convergence in distribution of the occupation measures $(\overline{\Lambda }^0_N)$. This is an extension of Proposition 13.

Corollary 16

Under the conditions of Theorem 15, the sequence of empirical distributions $(\overline{\Lambda }^0_N)$ converge in distribution to the measure $\overline{\Lambda }^0$ such that

$$\begin{aligned} \left\langle \overline{\Lambda }^0,g\right\rangle {=}\int _{{\mathbb {R}}_+}\hspace{-2mm} {\mathbb {E}}\left( g\left( s,{\overline{f}}(s), {{\mathcal {N}}}_1\left( \left[ 0,\rho _m\frac{c_m}{{\overline{f}}(s)}\right] \right) , {{\mathcal {N}}}_2\left( \left[ 0,\rho _1\frac{1{-}c_m{-}{\overline{f}}(s)}{{\overline{f}}(s)}\right] \right) \right) \right) \mathop {}\mathopen {}\textrm{d}s, \end{aligned}$$

for any continuous function g on ${\mathbb {R}}_+^2{\times }{\mathbb {N}}^2$, where ${{\mathcal {N}}}_1$ and ${{\mathcal {N}}}_2$ are two independent Poisson processes on ${\mathbb {R}}_+$ with rate 1 and $({\overline{f}}(t))$ is the solution of Relation (26) with ${\overline{f}}(0){=}{\overline{f}}_0$.

5 Exponential phase

Throughout this section, Conditions (3) and of exponential phase of Definition 1 hold. Heuristically, if there are sufficiently many polymerases, there will be an accumulation of them in the elongation phase of rRNAs and, therefore, the output rate of all types of rRNAs is maximal. The goal of this section is to prove precise results for this assertion.

Under this condition, for any $1{\le }j{\le }J$, the initiation rate $\alpha _{r,j}$ of rRNA of type j, is larger than $\beta _{r,j}$, the rate at which an rRNA of type j grows.

5.1 A coupling

We introduce a coupling to study the occupancy of the places for transcription of rRNAs. The idea is quite simple: for $1{\le }j{\le }J$, as long as $R_j^N(t)$ is strictly less than $C_{r,j}^N$, when $U_j^N(t){=}1$, a new polymerase is added for transcription after an exponential with parameter $\alpha _{r,j}$ and, if at that time $F_N(t)$ is positive, then the variable $U_j^N(t)$ remains at 1. See the part of transcription of rRNAs in the Q-Matrix of our process in Sect. 2.2.

Otherwise, if $F_N(t){=}0$, there is a total of at least $A_N{{\mathop {=}\limits ^{\text { def.}}}}N{-}C_{r,1}^N{-}\cdots {-} C_{r,J}^N{-}J$ polymerases either in transcription of mRNAs or sequestered. If $\delta {=}\min (\eta ,\beta _m)$, the duration of time after which there will be a free polymerase which can be accommodated by the jth promoter of rRNAs, with probability at least 1/J, is stochastically bounded by an exponential random variable with parameter $2\delta A_N$. Hence, if $F_N(t){=}0$ and $U_j^N(t){=}0$, then $U_j(t)$ returns to 1 after a duration whose distribution is stochastically bounded by an exponential random variable with parameter $2\delta A_N/J$.

We choose $N_0$ sufficiently large, so that

$$\begin{aligned} \frac{1}{\alpha _{r,j}}{+}\frac{J}{\delta N_0} < \frac{1}{\beta _{r,j}},\qquad \forall 1{\le }j{\le }J \text { with }\delta {=}\min (\eta ,\beta _m). \end{aligned}$$

(27)

We are interested in the behavior of $(Q_j^N(t)){{\mathop {=}\limits ^{\text { def.}}}}(C_{r,j}^N(t){-}R_j^N(t)), 1{\le }j{\le }J$, which measures the congestion of the transcription of the rRNAs. The above coupling shows that if $N{\ge }N_0$, it can be stochastically bounded by independent queueing processes $({\overline{Q}}_j(t))$, $1{\le }j{\le }J$ characterized as follows: for $1{\le }j{\le }J$,

• The arrivals of customers is a Poisson process with rate $\beta _{r,j}$. If $R_j^N(t){\ne }0$, the rate of jumps of size ${+}1$ of $Q_j^N(t)$ is $\beta _{r,j}$.

• The service of a customer is expressed as the sum of two independent exponential random variables with respective parameters $\alpha _{r,j}$ and ${\delta N_0}/{J}$.

To summarize, the instant jumps ${+}1$ of $(Q_j(t))$ are a subset of the ${+}1$ jumps of $({\overline{Q}}_j(t))$ and the time between two jumps ${-}1$ is greater for $({\overline{Q}}_j(t))$.

The process $({\overline{Q}}_j(t))$ is the process of the number of customers of an M/G/1 queue, see Chapter 2 of Robert (2003). It has a Markovian representation as $(I_j(t),{\overline{Q}}_j(t))$ where $I_j(t){\in }\{1,2\}$, $I_j(t){=}1$ indicates that the customer being served is in phase $\alpha _{r,j}$ and $I_j(t){=}2$ when it is in the phase ${\delta N_0}/{J}$.

Under Condition (27), $({\overline{Q}}(t)){=}((I_j(t),{\overline{Q}}_j(t)), 1{\le }j{\le }J)$ is a positive recurrent Markov process, since the coordinates are independent positive recurrent Markov processes. In particular if ${\overline{Q}}(0){\in }(\{0,1\}{\times }{\mathbb {N}})^J$, then

$$\begin{aligned} \inf \left\{ t{>}0: {\overline{Q}}(t){=}((1,0),j{=}1,\ldots ,J)\right\} \end{aligned}$$

is almost surely finite and integrable and for any $\varepsilon {>}0$ and $T{>}0$, there exists K such that

$$\begin{aligned} {\mathbb {P}}\left( \sup _{0\le t\le T}\max _{1\le j\le J} {\overline{Q}}_j(t) \ge K\right) \le \varepsilon . \end{aligned}$$

Furthermore if

$$\begin{aligned} \overline{\tau }_j^N=\inf \{t{>}0: {\overline{Q}}_j(t){=}0\}, \text { with } {\overline{Q}}_j(0){=}C_{r,j}^N, \end{aligned}$$

then it is not difficult to show, with the classical law of large numbers, that, if $i{\in }\{0,1\}$,

$$\begin{aligned} \lim _{K\rightarrow +\infty } \frac{{\mathbb {E}}_{(i,K)}(\overline{\tau }^N_j)}{N}= c_{r,j}\bigg /\left( \frac{1}{{1}/{\alpha _{r,j}}+{J}/{(\delta N_0)}}{-}\beta _{r,j}\right) . \end{aligned}$$

We have thus proved the following proposition which shows that in the exponential phase, the transcription of rRNAs is essentially congested.

Theorem 17

(Saturation of Transcription of rRNAs) If Conditions (3) and (4) hold and if $F_N(0){=}N$, $Z_N(0){=}0$ and $(U_N(0),R_N(0)){=}(0,0)$, i.e. all polymerases are initially free, then the variable $\tau ^e_N$ defined by

$$\begin{aligned} \tau ^e_N{\mathop {=}\limits ^{\text { def.}}} \inf \{t{>}0: R_{N,j}(t){=}C_{r,j}^N, \forall 1{\le }j{\le }J\}, \end{aligned}$$

(28)

is almost surely finite and

$$\begin{aligned} \sup _N\frac{{\mathbb {E}}(\tau ^e_N)}{N}<{+}\infty . \end{aligned}$$

For any $\varepsilon {>}0$ and $T{>}0$, there exists K such that

$$\begin{aligned} {\mathbb {P}}_{((1,0))}\left( \sup _{0\le t\le T}\max _{1\le j\le J} C_{r,j}^N{-}R_j^N(t) \ge K \right) \le \varepsilon \end{aligned}$$

(29)

The variable $\tau ^e_N$ is the first time when all places for transcription of rRNAs are occupied, i.e. the first instant when this part of the system is saturated. Our proposition gives an upper bound linear in N for the average value of this random variable when Condition (4) holds.

Now we investigate the asymptotic behavior of the remaining part of the system after time $\tau ^e_N$. We introduce

$$\begin{aligned} \left\langle \Lambda ^F_N,g\right\rangle {\mathop {=}\limits ^{\text { def.}}} \int g\left( s,F_N(s))\right) \mathop {}\mathopen {}\textrm{d}s \text { and } \left\langle \Lambda ^{0,F}_N,g\right\rangle {\mathop {=}\limits ^{\text { def.}}} \int g\left( s,F^0_{A_N}(s))\right) \mathop {}\mathopen {}\textrm{d}s, \end{aligned}$$

if g is a continuous function with compact support on ${\mathbb {R}}_+{\times }{\mathbb {N}}$, where $A_N$ defined by Relation (6) is the number of polymerases available when transcription of rRNA is saturated. The process $(F^0_{A_N}(t))$ is the solution of Relation (8) whose initial condition is the same as the process $(F_N(t),S_N(t),Z_N(t))$.

Lemma 18

(Coupling with the Auxiliary Process) If $(F_N(0),S_N(0),Z_N(0)){=}(f,s,z){\in }{\mathbb {N}}^3$ and if $(U_N(0),R_N(0)){=}((1,C_{r,j}^N))$, then for any $g{\in }{{\mathcal {C}}}_c({\mathbb {R}}_+{\times }{\mathbb {N}})$,

$$\begin{aligned} \lim _{N\rightarrow +\infty } \left| {\mathbb {E}}\left( \left\langle \Lambda ^{0,F}_{A_N},g\right\rangle \right) {-}{\mathbb {E}}\left( \left\langle \Lambda ^{F}_N,g\right\rangle \right) \right| =0. \end{aligned}$$

Proof

From Relation (29), we know that for K sufficiently large, the probability of the event

$$\begin{aligned} {{\mathcal {E}}}_K{\mathop {=}\limits ^{\text { def.}}} \left\{ \sup _{t\le T} A_N-\left( N-\sum _{1}^JU_j^N(t){+}R_j^N(t)\right) \le K\right\} \end{aligned}$$

is close to 1.

Given our initial state, at time 0 there are $A_N$ polymerases either sequestered, free or in transcription of an mRNA. On the event ${{\mathcal {E}}}_K$, on the time interval [0, T], there may be at most K additional polymerases. Since they enter this part of the system as free, at rate at least $C_m^N{-}(N{-}C_{r,1}^N\cdots {-}C_{r,J}^N)$, they go into transcription of an mRNA. Note that, almost surely, any of these K polymerases may return a finite number of times as free on [0, T]. Hence, with high probability, their contribution to the integral defining the occupation measure is arbitrarily small as N gets large, and so is their impact on the random variable $(F_N(t),S_N(t),Z_N(t))$. $\square$

We can now state convergence results for the number of free and sequestered polymerases. It is a direct consequence of the arguments of the proof of the last lemma and Proposition 5. It shows that in this case, basically, the number of free polymerases has a Poisson distribution and the process of the number of sequestered polymerases and free 6S RNAs is a positive recurrent Markov process on ${\mathbb {N}}^2$.

Theorem 19

(Free/Sequestered Polymerases and 6S RNAs) Under Conditions (3) and (4) and if $(F_N(0),S_N(0),Z_N(0)){=}(f,s,z){\in }{\mathbb {N}}^3$ and $(U_N(0),R_N(0)){=}((1,C_{r,j}^N))$, then, for the convergence in distribution,

$$\begin{aligned} \lim _{N\rightarrow +\infty } \int g\left( s,F_N(s))\right) \mathop {}\mathopen {}\textrm{d}s = \int _0^{+\infty } {\mathbb {E}}\left( g\left( u,{{\mathcal {N}}}_1\left( 0,\rho _m\right) \right) \right) \mathop {}\mathopen {}\textrm{d}u, \end{aligned}$$

for any $g{\in }{{\mathcal {C}}}_{c}\left( {\mathbb {R}}_+{\times }{\mathbb {N}}\right)$, where ${{\mathcal {N}}}_1$ is a Poisson process with rate 1 and

$$\begin{aligned} \rho _m{=}\frac{\beta _m(1{-}c_r)}{\alpha _m(c_m{+}c_r{-}1)}, \quad c_r{{\mathop {=}\limits ^{\text { def.}}}}\sum _{j=1}^J c_{r,j}. \end{aligned}$$

Furthermore, the sequence of processes $(S_N(t),Z_N(t))$ converges in distribution for the Skorohod topology to a jump process (S(t), Z(t)) on ${\mathbb {N}}^2$ whose transition rates are given by

$$\begin{aligned} (s,z)\longrightarrow (s,z){+} {\left\{ \begin{array}{ll} (1,-1)&{} \lambda \rho _m z,\\ (-1,1)&{} \eta s, \end{array}\right. } \quad {\left\{ \begin{array}{ll} (0,1)&{} \beta _6,\\ (0,-1)&{} \delta _6z. \end{array}\right. } \end{aligned}$$

Note that the process (S(t), Z(t)) is a positive recurrent Markov process. Indeed, if, for $a{>}0$,

$$\begin{aligned} H_a(s,z){\mathop {=}\limits ^{\text { def.}}} a s{+}z, \end{aligned}$$

then it is easily seen that $H_a$ is a Lyapunov function for this Markov process if $a{\in }{\mathbb {R}}_+$ is chosen so that

$$\begin{aligned} 1< a < 1{+}\frac{\delta _6}{\lambda \rho _m}, \end{aligned}$$

see Proposition 8.14 of Robert (2003).

6 Stationary phase

Conditions (3) and of stationary phase of Definition 1 now hold. For any type j of rRNA, the initiation rate $\alpha _{r,j}$ is less than its production rate.

6.1 A coupling

As in Sect. 5 we introduce a simple coupling to study the occupancy of the slots for transcription of rRNAs. Since a polymerase enters in elongation phase of an rRNA of type $j{\in }\{1,\ldots ,J\}$ at rate at most $\alpha _{r,j}$, it is easy to construct a coupling with J independent M/M/1 processes $(Q_j(t))$ with respective input rate $\alpha _{r,j}$ and service rate $\beta _{r,j}$, so that the relations

$$\begin{aligned} R_j^N(t)\le Q_j^N(t), \quad \forall t{\ge }0, 1{\le }j{\le }J, \end{aligned}$$

hold. See Chapter 5 of Robert (2003) for example. The following proposition is a direct consequence of this coupling and the fact that, for the convergence in distribution, the hitting time of p starting from a fixed initial state is exponential with respect to p, for p large. See Proposition 5.16 of Robert (2003)

Proposition 20

Under Conditions (3) and (5), and if $F_N(0){=}N$, $Z_N(0){=}0$ and $(U_N(0),R_N(0)){=}(0,0)$, all polymerases are initially free, then the variable $\tau _N$ defined by

$$\begin{aligned} \tau ^s_N{\mathop {=}\limits ^{\text { def.}}} \inf \{t{>}0: R_{N,j}(t)=0, \forall 1{\le }j{\le }J\}, \end{aligned}$$

(30)

is almost surely finite and

$$\begin{aligned} \sup _N\frac{{\mathbb {E}}(\tau ^s_N)}{N}<{+}\infty . \end{aligned}$$

Lemma 21

Under Condition (5) then, for any $K{>}0$,

$$\begin{aligned} \lim _{N\rightarrow +\infty } {\mathbb {P}}_{(u,r)}\left( \sup _{t\le NT}\frac{R_j^N(t)}{(\ln (N))^2}>K\right) =0. \end{aligned}$$

Proof

This is a simple consequence of the independence of the $(Q_j(t))$ and of Proposition 5.11 of Robert (2003). $\square$

The above result shows that few polymerases are in transcription of an rRNA, hence the results of Sect. 4 on the auxiliary process can be used, in particular Theorem 15.

Theorem 22

(Asymptotic Behavior in Stationary Phase) Under Conditions (3) and (5), and the initial state such that

$$\begin{aligned} \lim _{N\rightarrow +\infty }\left( \frac{F_N(0)}{N},\frac{S_N(0)}{N}\right) =\left( {\overline{f}},{\overline{s}}\right) \in [0,1]^2, \text { with } {\overline{f}}{+}{\overline{s}}{=}1-c_m, \end{aligned}$$

and $(U_N(0),R_N(0)){=}(u,r){\in }(\{0,1\}{\times }{\mathbb {N}})^J$ then, for the convergence the sequence of processes

$$\begin{aligned} \lim _{N\rightarrow +\infty } \left( \frac{F_N(t)}{N},\frac{S_N(t)}{N})\right) =\left( {\overline{f}}(t),1{-}c_m{-}{\overline{f}}(t)\right) , \end{aligned}$$

where $({\overline{f}}(t))$ is the solution of the ODE

$$\begin{aligned} \frac{\mathop {}\mathopen {}\textrm{d}{\overline{f}}}{\mathop {}\mathopen {}\textrm{d}t} (t)= {-}\delta _6\left( \rho _6 {+}\rho _1\right) {+}\delta _6\rho _1(1{-}c_m)\frac{1}{{\overline{f}}(t)}, \end{aligned}$$

(31)

with $\rho _1{=}{\eta }/{\lambda }$ and $\rho _6{=}{\beta _6}/{\delta _6}$.

If $g{\in }{{\mathcal {C}}}_c({\mathbb {R}}_+{\times }{\mathbb {N}})$ then, for the convergence in distribution,

$$\begin{aligned} \lim _{N\rightarrow +\infty }\left( \int _{{\mathbb {R}}_+} g(t,Z_N(t))\mathop {}\mathopen {}\textrm{d}t\right) = \int _{{\mathbb {R}}_+} {\mathbb {E}}\left[ g\left( t, {{\mathcal {N}}}_1\left( \left[ 0,\rho _1\frac{1{-}c_m{-}{\overline{f}}(t)}{{\overline{f}}(t)}\right] \right) \right) \right] \mathop {}\mathopen {}\textrm{d}t, \end{aligned}$$

where ${{\mathcal {N}}}_1$ is a Poisson processes on ${\mathbb {R}}_+$ with rate 1.

In particular, the asymptotic fraction of free polymerases is

$$\begin{aligned} \frac{\rho _1}{\rho _6{+}\rho _1}(1{-}c_m), \end{aligned}$$

and, in this state, the number of free 6S RNAs has a Poisson distribution with parameter $\rho _6$.

Appendices

Appendix A: Sub-critical Case

It is assumed throughout this section that $c_m{>}1$ holds. We give a sketch of the proof of the averaging principle at the basis of the proof of Proposition 5 for the sake of completeness. The analogue of this result in the super-critical case in Sect. 4 is quite different and more challenging. The corresponding tightness property is less clear in this case, in particular the definition of occupation measures has to include the slow processes. The arguments of the proofs of Sect. 4 can be used in the same way. As it will be seen, it is easy to show that the sequences of “slow” processes $({S}^0_N(t)/N)$ and $({Z}^0_N(t)/N)$ are tight.

Recall that $\mu _N$ is the occupation measure defined by Relation (12). For $K{>}0$, with the same notations as in the proof of Lemma 4, Relation (13) gives the inequality

$$\begin{aligned} {\mathbb {E}}\left( \left\langle \mu _N,[0,t]{\times }[0,K]\right\rangle \right) \ge \int _0^t {\mathbb {P}}(Y(Ns)\le K)\mathop {}\mathopen {}\textrm{d}s =\frac{1}{N}\int _0^{Nt}{\mathbb {P}}(Y(s)\le K)\mathop {}\mathopen {}\textrm{d}s. \end{aligned}$$

Since (Y(t)) is converging in distribution to a Poisson distribution with parameter a/b, for any $\varepsilon {>}0$ and $t{>}0$, there exists $K_0$ and $N_0$ such that if $K{\ge }K_0$ and $N{\ge }N_0$, then ${\mathbb {E}}\left( \left\langle \mu _N,[0,t]{\times }[0,K]\right\rangle \right) {>}(1{-}\varepsilon )t$. Lemma 1.3 and 1.4 of Kurtz (1992) show that the sequence $(\mu _N)$ of random measures is tight and any limiting point $\mu _\infty$ can be expressed as

$$\begin{aligned} \left\langle \mu _\infty ,g\right\rangle = \int _{{\mathbb {R}}_+{\times }{\mathbb {N}}} g\left( u,x\right) \pi _u(\mathop {}\mathopen {}\textrm{d}x)\mathop {}\mathopen {}\textrm{d}u \end{aligned}$$

where $(\pi _u)$ is a previsible process with values in the state space of probability distributions on ${\mathbb {N}}$.

1.1 A.1 Proof of Proposition 5

By integrating Relations (9) and (10), we obtain the identities, for $t{\ge }0$,

$$\begin{aligned} {S}^0_N(t)&={S}^0_N(0){+}M_6^N(t){-}\eta \int _0^t {S}^0_N(s)\mathop {}\mathopen {}\textrm{d}s{+}\lambda \int _0^t {F}^0_N(s){Z}^0_N(s)\mathop {}\mathopen {}\textrm{d}s, \end{aligned}$$

(32)

$$\begin{aligned} {Z}^0_N(t)&={Z}^0_N(0){+}M_Z^N(t){+}\beta _6 t{-}\delta _6\int _0^t {Z}^0_N(s)\mathop {}\mathopen {}\textrm{d}s \nonumber \\&\quad {+}\eta \int _0^t {S}^0_N(s)\mathop {}\mathopen {}\textrm{d}s{-}\lambda \int _0^t {F}^0_N(s){Z}^0_N(s)\mathop {}\mathopen {}\textrm{d}s, \end{aligned}$$

(33)

where $(M_6^N(t))$ and $(M_Z^N(t))$ are martingales whose previsible increasing processes are given by

$$\begin{aligned} \left\langle M_6^N\right\rangle (t)&=\eta \int _0^t {S}^0_N(s)\mathop {}\mathopen {}\textrm{d}s{+}\lambda \int _0^t {F}^0_N(s){Z}^0_N(s)\mathop {}\mathopen {}\textrm{d}s, \end{aligned}$$

(34)

$$\begin{aligned} \left\langle M_Z^N\right\rangle (t)&=\beta _6 t{+}\delta _6\int _0^t {Z}^0_N(s)\mathop {}\mathopen {}\textrm{d}s {+}\eta \int _0^t {S}^0_N(s)\mathop {}\mathopen {}\textrm{d}s{+}\lambda \int _0^t {F}^0_N(s){Z}^0_N(s)\mathop {}\mathopen {}\textrm{d}s. \end{aligned}$$

(35)

Relations (34) and (35), Relation (13), and Doob’s Inequality show that, for convergence in distribution, then

$$\begin{aligned} \lim _{N\rightarrow +\infty } \left( \frac{M^N_6(t)}{N}\right) =\lim _{N\rightarrow +\infty } \left( \frac{M^N_Z(t)}{N}\right) =0. \end{aligned}$$

We note that, for $t{\ge }0$, ${S}^0_N(t){\in }[0,N]$ and $0{\le }{Z}^0_N(t){\le }N{+}{{\mathcal {P}}}_5((0,\beta _6){\times }(0,t])$ by Relation (10). Relations (32) and (33), and the criterion of the modulus of continuity, see (Billingsley 1999), give that the sequence of processes $\left( {{S}^0_N(t)}/{N},{{Z}^0_N(t)}/{N}\right)$ is tight for the convergence in distribution associated to the uniform norm on compact sets of ${\mathbb {R}}_+$.

We can therefore take a subsequence of $\left( \mu _N,\left( {{S}^0_N(t)}/{N}\right) ,\left( {{Z}^0_N(t)}/{N}\right) \right)$ with indices $(N_k)$ converging in distribution to $(\mu _\infty ,(s(t)),(z(t)))$, where (s(t)) and (z(t)) are continuous processes.

If $f{\in }{{\mathcal {C}}}_{c}\left( {\mathbb {N}}{\times }{\mathbb {R}}_+^2\right)$, Relation (8) gives the identity

$$\begin{aligned}{} & {} f\left( {F}^0_{N_k}(t),\frac{{S}^0_{N_k}(t)}{{N_k}},\frac{{Z}^0_{N_k}(t)}{{N_k}}\right) = f\left( f_{N_k},s_{N_k},z_{N_k}\right) +M_f^{N_k}(t)\\{} & {} \quad +\beta _m\int _0^t\nabla _{e_1}(f)\left( {X}^0_{N_k}(s)\right) \left( {N_k}{-}{F}^0_{N_k}(s){-}{S}^0_{N_k}(s)\right) \mathop {}\mathopen {}\textrm{d}s\\{} & {} \quad + \alpha _m\int _0^t\nabla _{-e_1}(f)\left( {X}^0_{N_k}(s)\right) \left( C_m^{N_k}{-}{N_k}{+}{F}^0_{N_k}(s){+}{S}^0_{N_k}(s)\right) {F}^0_{N_k}(s)\mathop {}\mathopen {}\textrm{d}s\\{} & {} \quad +\lambda \int _0^t\nabla _{-e_1+\frac{e_2}{{N_k}}-\frac{e_3}{{N_k}}}(f)\left( {X}^0_{N_k}(s)\right) {F}^0_{N_k}(s){Z}^0_{N_k}(s)\mathop {}\mathopen {}\textrm{d}s\\{} & {} \quad +\eta \int _0^t\nabla _{e_1-\frac{e_2}{{N_k}}+\frac{e_3}{{N_k}}}(f)\left( {X}^0_{N_k}(s)\right) {S}^0_{N_k}(s)\mathop {}\mathopen {}\textrm{d}s\\{} & {} \quad +\beta _6\int _0^t\nabla _{\frac{e_3}{{N_k}}}(f)\left( {X}^0_{N_k}(s)\right) \mathop {}\mathopen {}\textrm{d}s +\delta _6\int _0^t\nabla _{-\frac{e_3}{{N_k}}}(f)\left( {X}^0_{N_k}(s)\right) {Z}^0_{N_k}(s)\mathop {}\mathopen {}\textrm{d}s, \end{aligned}$$

with the notation $\nabla _a(f)(x){=}f(x{+}a){-}f(x)$, for a and $x{\in }{\mathbb {N}}{\times }{\mathbb {R}}_+^2$.

With the same arguments as for the martingales $(M_S^{N_k}(t))$ and $(M_Z^{N_k}(t))$, the process $(M_f^{N_k}(t))$ is converging in distribution to 0. By dividing by ${N_k}$ the last relation, and by letting k go to infinity, we get

$$\begin{aligned}{} & {} \int _0^t\nabla _{e_1}(f)(x,s(u),z(u))\left( \beta _m{-}\left( \beta _m-\eta \right) s(u)\right) \pi _u(\mathop {}\mathopen {}\textrm{d}x) \mathop {}\mathopen {}\textrm{d}u\\{} & {} \quad +\int _0^t\nabla _{-e_1}(f)(x,s(u),z(u))\left( \alpha _m(c_m{-}1{+}s(u)){+}\lambda z(u)\right) x\pi _u(\mathop {}\mathopen {}\textrm{d}x)\mathop {}\mathopen {}\textrm{d}u=0, \end{aligned}$$

and therefore

$$\begin{aligned} \int _0^t \int _{{\mathbb {N}}} \Omega _{s(u),z(u)}(g)(x)\pi _u(\mathop {}\mathopen {}\textrm{d}x) \mathop {}\mathopen {}\textrm{d}u =0, \end{aligned}$$

(36)

with, for s, $z{\ge }0$, $s{+}z{<}1$ and $x{\in }{\mathbb {N}}$,

$$\begin{aligned} \Omega _{s,z}(g)(x){} & {} =\left( \beta _m{-}\left( \beta _m{-}\eta \right) s\right) (g(x{+}1){-}g(x))\\{} & {} \quad +\left( \alpha _m(c_m{-}1{+}s){+}\lambda z\right) (g(x{-}1){-}g(x)), \end{aligned}$$

$\Omega _{s,z}$ is the infinitesimal generator of the Markov process (Y(t)) of Lemma 3 with $a{=}a(s,z){=}\left( \beta _m{-}\left( \beta _m{-}\eta \right) s\right)$ and $b{=}b(s,z){=}\alpha _m(c_m{-}1{+}s){+}\lambda z)$. From Relation (36) and with the same methods as in Sect. 4, we obtain that, almost surely,

$$\begin{aligned} \int _0^t \int _{\mathbb {N}}g(x)\pi _u(\mathop {}\mathopen {}\textrm{d}x)\mathop {}\mathopen {}\textrm{d}u =\int _0^t \int _{\mathbb {N}}g(x)\pi _u(\mathop {}\mathopen {}\textrm{d}x)\mathop {}\mathopen {}\textrm{d}u = \int _0^t {\mathbb {E}}\left( g\left( P_{u}\right) \right) \mathop {}\mathopen {}\textrm{d}u \end{aligned}$$

holds for all $t{>}0$ and all functions g with finite support on ${\mathbb {N}}$, where $P_{u}$ is a Poisson random variable with parameter a(s(u), z(u))/b(s(u), z(u)), $u{\ge }0$.

Hence, with similar arguments as in Sect. 4, for $T{\ge }0$ such that $s(t){+}z(t){<}1$ holds for all $t{\le }T$, we obtain that the identities

$$\begin{aligned}&s(t)=s_0{-}\eta \int _0^t s(u)\mathop {}\mathopen {}\textrm{d}u{+}\lambda \int _0^t z(u) \frac{\beta _m{-}\left( \beta _m-\eta \right) s(u)}{\alpha _m(c_m{-}1{+}s(u)){+}\lambda z(u)}\mathop {}\mathopen {}\textrm{d}u,\end{aligned}$$

(37)

$$\begin{aligned}&s(t){+}z(t)=s_0{+}z_0{-}\delta _6\int _0^t z(u)\mathop {}\mathopen {}\textrm{d}u, \end{aligned}$$

(38)

hold almost surely, for $t{\le }T$. From Relation (38) we obtain that $(s(t){+}z(t))$ is a non-increasing function, hence $0{\le }s(t){+}z(t){\le }s_0{+}z_0{<}1$, for $t{\ge }0$, the above system has therefore a unique solution defined on ${\mathbb {R}}_+$. Since the function $(s(t){+}z(t))$ is converging at infinity, Equation (38) shows that (z(t)) is an integrable function on ${\mathbb {R}}_+$, hence the second integral of the right-hand side of Relation (37) is converging as t goes to infinity. Since (s(t)) is bounded, the other integral of (37) also converges at infinity. We conclude that (s(t)) has a limit at infinity and therefore also (z(t)). By dividing Relation (38) by t and let t go to infinity, we obtain that the limit of (z(t)) is 0. With a similar method in Relation (37), we get that both limits are zero. Proposition 5 is proved.

1.2 A.2 Proof of Proposition 6

The first assertion on the convergence of the occupation is obtained in the same way but with $(s,z){=}(0,0)$, hence for $u{\ge }0$, $s(u){=}z(u){=}0$, and the operator is

$$\begin{aligned} \Omega _{(s(u),z(u))}(g)(x)=\beta _m (g(x{+}1){-}g(x)){+}\alpha _m(c_m{-}1)(g(x{-}1){-}g(x)). \end{aligned}$$

Therefore $P_{u}$ is a Poisson random variable with parameter $\rho _m$.

Let, for $k{\ge }1$, $t_k^N$ be the kth jump of $(Y_N(t)){{\mathop {=}\limits ^{\text { def.}}}}(S_N(t),Z_N(t))$ when the initial state is (s, z), there are four random variables $A_i$, $i{\in }\{1,2,3,4\}$, to trigger a change of state of $(Y_N(t))$,

• (a) $A_1^N$ is a random variable such that, for $t{\ge }0$,

  $$\begin{aligned} {\mathbb {P}}\left( A_1^N{\ge }t\mid (F_N^0(s))\right) =\exp \left( {-}\lambda z\int _0^tF_N^0(s)\mathop {}\mathopen {}\textrm{d}s\right) ; \end{aligned}$$

  (39)

• (b) $A_2$, $A_3$, $A_4$ are independent exponential random variables with respective parameters $\eta s$, $\beta _6$ and $\delta _6 z$,

and, conditionally on $(F_N^0(t))$, the random variables $A_1^N$, $A_i$, $i{\in }\{2,3,4\}$ are independent.

Relation (39) and the convergence of the sequence $(\mu _N)$ of occupation measure of $(F_N^0(t))$ given that $A_1^N$ is converging in distribution to an exponential distribution with parameter $\lambda z\rho _m$.

For $t{\ge }0$, we have

$$\begin{aligned} {\mathbb {P}}_{(s,z)}\left( Y_N\left( t_1^N\right) =(s{+}1,z{-}1), t_1^N\ge t \right)&={\mathbb {P}}\left( A_1^N\ge t, A_1^N\le A_2{\wedge }A_3{\wedge }A_4\right) \\&={\mathbb {E}}\left( \mathbbm {1}_{\left\{ A_1^N{\ge }t\right\} }\exp \left( {-}(\eta s{+}\beta _6{+}\delta _6 z)A_1^N\right) \right) , \end{aligned}$$

hence,

$$\begin{aligned}{} & {} \lim _{N\rightarrow +\infty } {\mathbb {P}}_{(s,z)}\left( Y_N\left( t_1^N\right) =(s{+}1,z{-}1), t_1^N{\ge }t\right) \\{} & {} \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \ =\frac{\lambda \rho _m z}{(\lambda \rho _m z{+}\eta s{+}\beta _6{+}\delta _6z)} e^{{-}(\lambda \rho _m z{+}\beta _6{+}\delta _6 z{+}\eta s)t}, \end{aligned}$$

and this last quantity is ${\mathbb {P}}_{(s,z)}(Y(t_1)=(s{+}1,z{-}1), t_1\ge t)$, where (Y(t)) is the jump process defined in Proposition 6 and $(t_i)$ is the non-decreasing sequence of its instants of jumps. A similar convergence result is obtained in the same manner for the other possibilities for the first jump of $(Y_N(t))$. By induction, one can show that for $k{\ge }1$ and any sequence $(a_i){\in }{\mathbb {N}}^2$,

$$\begin{aligned} \lim _{N\rightarrow +\infty } {\mathbb {P}}\left( Y_N(t_i^N)=a_i, 1{\le }i{\le }k\right) ={\mathbb {P}}(Y(t_i)=a_i, 1{\le }i{\le }k). \end{aligned}$$

We conclude the proof of the convergence by using directly the very definition of the Skorohod topology. See (Billingsley 1999).

Super-critical case

The assumption $c_m{<}1$ holds throughout this section. Technical results used in Sect. 4 are presented here.

Recall that $({\overline{X}}^0_N(t)){=}({\overline{F}}^0_N(t),G^0_N(Nt),Z^0_N(Nt))$, with

$$\begin{aligned} G^0_N(t){\mathop {=}\limits ^{\text { def.}}}C_m^N{-}\left( N{-}F^0_N(t){-}S^0_N(t)\right) . \end{aligned}$$

If f be a non-negative Borelian function on ${\mathbb {R}}_+{\times }{\mathbb {N}}^2$, the SDEs (8), (9), and (10) give directly the relations

$$\begin{aligned} f&\left( {\overline{X}}^0_N(t)\right) =f\left( {\overline{X}}^0_N(0)\right) {+}M_{f,N}(t)\nonumber \\&+\lambda N\int _0^{t} \nabla _{-\frac{e_1}{N}-e_3}(f)\left( {\overline{X}}^0_N(s)\right) F^0_N(Ns)Z^0_N(Ns)\mathop {}\mathopen {}\textrm{d}s\nonumber \\&{+}\eta N\int _0^{t}\nabla _{\frac{e_1}{N}{+}e_3}(f)\left( {\overline{X}}^0_N(s)\right) \left( N{-}C_m^N{+}G^0_N(Ns){-}F^0_N(Ns)\right) \mathop {}\mathopen {}\textrm{d}s\nonumber \\&+\alpha _m N\int _0^{t} \nabla _{-\frac{e_1}{N}-e_2}(f)\left( {\overline{X}}^0_N(s)\right) G^0_N(Ns)F^0_N(Ns)\mathop {}\mathopen {}\textrm{d}s\nonumber \\&+\beta _m N \int _0^{t} \nabla _{\frac{e_1}{N}{+}e_2}(f)\left( {\overline{X}}^0_N(s)\right) \left( C_m^N{-}G^0_N(Ns)\right) \mathop {}\mathopen {}\textrm{d}s\nonumber \\&+\beta _6 N\int _0^{t} \nabla _{e_3}(f)\left( {\overline{X}}^0_N(s)\right) \mathop {}\mathopen {}\textrm{d}s+\delta _6 N \int _0^{t} \nabla _{-e_3}(f)\left( {\overline{X}}^0_N(s)\right) Z^0_N(Ns)\mathop {}\mathopen {}\textrm{d}s, \end{aligned}$$

(40)

where, for $i{\in }\{1,2,3\}$, $e_i$ is the ith unit vector of ${\mathbb {R}}^3$, and $(M_{f,N}(t))$ is a local martingale and its previsible increasing process is given by

$$\begin{aligned} \left\langle M_{f,N}\right\rangle (t)&=\lambda N \int _0^{t} \nabla _{-\frac{e_1}{N}-e_3}(f)\left( {\overline{X}}^0_N(s)\right) ^2F^0_N(Ns)Z^0_N(Ns)\mathop {}\mathopen {}\textrm{d}s\nonumber \\&+\eta N \int _0^{t} \nabla _{\frac{e_1}{N}+e_3}(f)\left( {\overline{X}}^0_N(s)\right) ^2 (N{-}C_m^N{+}G^0_N(Ns){-}F^0_N(Ns))\mathop {}\mathopen {}\textrm{d}s\nonumber \\&+\alpha _m N\int _0^{t} \nabla _{-\frac{e_1}{N}{-}{e_2}}(f)\left( {\overline{X}}^0_N(s)\right) ^2G^0_N(Ns)F^0_N(Ns)\mathop {}\mathopen {}\textrm{d}s\nonumber \\&+\beta _m N \int _0^{t} \nabla _{\frac{e_1}{N}{+}{e_2}}(f)\left( {\overline{X}}^0_N(s)\right) ^2\left( C_m^N{-}G^0_N(Ns)\right) \mathop {}\mathopen {}\textrm{d}s\nonumber \\&+\beta _6 N\int _0^{t} \nabla _{e_3}(f)\left( {\overline{X}}^0_N(s)\right) ^2\mathop {}\mathopen {}\textrm{d}s\nonumber \\&+\delta _6 N\int _0^{t}\nabla _{-e_3}(f)\left( {\overline{X}}^0_N(s)\right) ^2Z^0_N(Ns)\mathop {}\mathopen {}\textrm{d}s. \end{aligned}$$

(41)