Thibault A Etienne

Muriel Cocaign-Bousquet

Delphine Ropers

Competitive effects in bacterial mRNA decay

Keywords: Michaelis-Menten kinetics, Quasi-steady-state assumption, Substrate competition, mRNA turnover, Regulation of gene expression

In living organisms, the same enzyme catalyses the degradation of thousands of different mRNAs, but the possible influence of competing substrates has been largely ignored so far. We develop a simple mechanistic model of the coupled degradation of all cell mRNAs using the total quasi-steady-state approximation of the Michaelis-Menten framework. Numerical simulations of the model using carefully chosen parameters and analyses of rate sensitivity coefficients show how substrate competition alters mRNA decay. The model predictions reproduce and explain a number of experimental observations on mRNA decay following transcription arrest, such as delays before the onset of degradation, the occurrence of variable degradation profiles with increased non linearities and the negative correlation between mRNA halflife and concentration. The competition acts at different levels, through the initial concentration of cell mRNAs and by modifying the enzyme affinity for its targets. The consequence is a global slow down of mRNA decay due to enzyme titration and the amplification of its apparent affinity. Competition happens to stabilize weakly affine mRNAs and to destabilize the most affine ones. We believe that this mechanistic model is an interesting alternative to the exponential models commonly used for the determination of mRNA half-lives. It allows analysing regulatory mechanisms of mRNA degradation and its predictions are directly comparable to experimental data.

Introduction

Intracellular levels of bacterial mRNAs result from a balance between transcription and degradation. It has long been believed that mRNA decay plays a minor role in the regulation of bacterial gene expression, most likely because bacterial transcription has been early shown the target of so many regulations. mRNA degradation has gained renewed attention when studies have evidenced tight modulations of mRNA half-life by environmental conditions, through the action of small RNAs and RNA-binding proteins (see e.g. [START_REF] Bobrovskyy | Small RNAs regulate primary and secondary metabolism in Gram-negative bacteria[END_REF][START_REF] Mohanty | Regulation of mRNA decay in bacteria[END_REF]. Short mRNA half-lives allow both a rapid remodelling of gene expression when cells undergo environmental perturbations and a recycling of the ribonucleotide pool [START_REF] Carpousis | The RNA degradosome of Escherichia coli : an mRNAdegrading machine assembled on RNase E[END_REF]. In addition, a complicated multi-protein machinery called degradosome is responsible for both mRNA instability and RNA processing (see [START_REF] Marcaida | The RNA degradosome: life in the fast lane of adaptive molecular evolution[END_REF]Lykke-Andersen et al., 2009, for reviews).

Experimental approaches for analysing mRNA decay are based on transcriptional arrest by antibiotics addition (e.g. rifampicin). The kinetics of mRNA degradation is monitored by measuring mRNA levels along time.

With the advent of high-throughput technologies, this is nowadays achieved at the whole cell level using microarrays or RNA sequencing. Fitting the degradation profiles to an exponential model provides half-life values for cell mRNAs. They typically vary between 3 and 8 minutes in the model bacterium Escherichia coli, much shorter than the duration of the cell cycle [START_REF] Bernstein | Global analysis of Escherichia coli RNA degradosome function using DNA microarrays[END_REF][START_REF] Chen | Genome-wide study of mRNA degradation and transcript elongation in Escherichia coli[END_REF][START_REF] Esquerré | Dual role of transcription and transcript stability in the regulation of gene expression in Escherichia coli cells cultured on glucose at different growth rates[END_REF]. By comparison, ribosomal and transfer RNAs are stable and function over several generations. In some extreme cases in the stationary phase, half-lives as high as 50-100 minutes are measured in E. coli [START_REF] Dressaire | PNPase is involved in the coordination of mRNA degradation and expression in stationary phase cells of Escherichia coli[END_REF] and up to 90 minutes in the budding yeast [START_REF] Wang | Precision and functional specificity in mRNA decay[END_REF].

These exponential models of mRNA decay fall short of providing mechanistic interpretations to the degradation kinetics. They are based on the assumption that mRNA decay follows a first-order reaction, that is, the reaction rate depends linearly on the mRNA concentration [START_REF] Bernstein | Global analysis of mRNA decay and abundance in Escherichia coli at singlegene resolution using two-color fluorescent DNA microarrays[END_REF][START_REF] Chen | Genome-wide study of mRNA degradation and transcript elongation in Escherichia coli[END_REF]. They do not explicitly include regulations of mRNA decay, as this would require complexifying both the model and the procedure of parameter estimation. Notably, one largely ignored regulatory mechanism is the possible implication of substrate competition, while experimental studies point at its possible involvement [START_REF] Esquerré | Genome-wide investigation of mRNA lifetime determinants in Escherichia coli cells cultured at different growth rates[END_REF][START_REF] Nouaille | The stability of an mRNA is influenced by its concentration: a potential physical mechanism to regulate gene expression[END_REF].

Substrate competition is encountered in many biochemical processes, from signal transduction to gene expression or transport in living organisms (e.g., [START_REF] Cookson | Queueing up for enzymatic processing: correlated signaling through coupled degradation[END_REF][START_REF] De Vos | How molecular competition influences fluxes in gene expression networks[END_REF][START_REF] Kiel | Challenges ahead in signal transduction: MAPK as an example[END_REF].

It occurs when several substrates compete for binding to the same active site of an enzyme, which possibly leads to complex dynamical behaviours. For instance, the substrates of the MAP kinases (MAPK), which regulate various cell fate decisions (proliferation, differentiation, and apoptosis), compete for binding to MAPK in early Drosophila embryos [START_REF] Kim | MAPK substrate competition integrates patterning signals in the Drosophila embryo[END_REF][START_REF] Kim | Gene regulation by MAPK substrate competition[END_REF][START_REF] Legewie | Competing docking interactions can bring about bistability in the mapk cascade[END_REF]. Their local concentration and their relative binding affinities for the MAPK is predicted to control both MAPK activity and the choice of substrate. This enables the same MAPK pathway to operate in different modes and elicit distinct cell type responses.

In the case of mRNA degradation, the same machinery is responsible for the fate of messenger RNAs. In E. coli for instance, the main degradation pathway is the internal cleavage of mRNAs by RNase E, an enzyme also involved in RNA processing [START_REF] Clarke | Direct entry by RNase E is a major pathway for the degradation and processing of RNA in Escherichia coli[END_REF]. In the alternative, so-called 5'-end-dependent, pathway, mRNA cleavage by RNase E occurs after the prior recognition and removal of two of the three phosphate groups at the 5' end of the transcript by the pyrophosphohydrolase RppH. The fragments of mRNAs are then rapidly degraded by exoribonucleases. RNase E is abundant in E. coli cells [START_REF] Mackie | RNase E: at the interface of bacterial RNA processing and decay[END_REF][START_REF] Valgepea | Escherichia coli achieves faster growth by increasing catalytic and translation rates of proteins[END_REF], at least in excess with respect to individual substrates, but the total substrate concentration largely exceeds that of the enzyme. As a consequence, pools of RNAse E are limited in cells. It is thus expected that differences of substrate affinities and concentrations play a determinant role in the degradation kinetics of mR-NAs. Experimental observations seem to agree with this hypothesis, at least for mRNA initial concentrations, for which a strong negative correlation was found with the mRNA lifetime [START_REF] Esquerré | Genome-wide investigation of mRNA lifetime determinants in Escherichia coli cells cultured at different growth rates[END_REF][START_REF] Nouaille | The stability of an mRNA is influenced by its concentration: a potential physical mechanism to regulate gene expression[END_REF].

In this manuscript, we study the possible role of substrate competition on mRNA degradation kinetics. We need for this a mechanistic modelling of mRNA degradation describing the binding of RNase E to competing substrates and the subsequent titration of the enzyme. The model should make a trade-off between complexity and biological realism, and have physiologically interpretable parameters. In addition model predictions should be directly comparable to experimental data. Common models of mRNA degradation kinetics do not meet this requirement. We thus developed a mechanistic model of mRNA decay in this study and analysed the effect of mRNA competition on the degradation kinetics.

In what follows, we briefly survey mRNA degradation (Section 2). We show in Section 3 how to assimilate mRNA degradation by the degradosome to a macro-reaction catalysed by the limiting enzyme RNase E. The kinetics of the reaction is described by means of a Michaelis-Menten equation. To take into account the fact that RNase E is in limited concentration with respect to all mRNA concentrations, we used the tQSSA form of the Michaelis-Menten equation for describing the degradation kinetics. By extending previous works by [START_REF] Pedersen | The total quasi-steady-state approximation for fully competitive enzyme reactions[END_REF] and [START_REF] Tang | A total quasi-steady-state formulation of substrate uptake kinetics in complex networks and an example application to microbial litter decomposition[END_REF], we developed an approximated version of the tQSSA form which includes competition between all cell mRNAs. We used this approximation in Section 4 to study the contribution of competition to mRNA decay by means of numerical simulations, based on carefully chosen parameter values and distributions. Our results show that the introduction of competition leads to variability, delays and increased non linearities in the degradation curves, similarly to experimental observations. Such diversity and non linearities are difficult if not impossible to reproduce in the absence of competition. By means of rate response coefficients in Section 5 and 6, we show how competition globally affects degradation by slowing it down and how it differentially affects the fate of mRNAs. We discuss these results in Section 7.

2. A short survey of mRNA degradation: mechanism, monitoring and kinetic modelling 2.1. The mechanism of mRNA degradation and its regulation In many bacteria, RNase E (EC 3.1.26.12) is a central player of mRNA degradation and RNA processing. It initiates the processing of small RNAs and of a majority of pre-tRNAs, participates to the maturation of ribosomal RNAs in addition to degrading mRNAs. The tetrameric enzyme forms a multi-protein complex called the RNA degradosome, together with the RNA helicase B (RhlB), the glycolytic enzyme enolase (Eno), and the phosphorolytic exonuclease polynucleotide phosphorylase (PNPase) [START_REF] Carpousis | The RNA degradosome of Escherichia coli : an mRNAdegrading machine assembled on RNase E[END_REF]. It is mostly anchored to the cytoplasmic inner membrane, where it forms highly mobile foci diffusing all over the membrane [START_REF] Strahl | Membrane recognition and dynamics of the RNA degradosome[END_REF].

These foci disappear upon inhibition of transcription and the subsequent disappearance of mRNAs. mRNA degradation results from the combined action of endo-and exoribonucleases (Fig. 1). As an endoribonuclease, RNase E catalyses the first internal cleavage, followed by a rapid degradation of the RNA fragments by exoribonucleases. This initial cleavage is considered to be the limiting step in mRNA degradation [START_REF] Kushner | mRNA decay in Escherichia coli comes of age[END_REF]. RNase E has also an indirect effect on mRNA decay, through the processing of small regulatory RNAs that regulate the stability of their target mRNAs [START_REF] Chao | In vivo cleavage map illuminates the central role of RNase E in coding and non-coding RNA pathways[END_REF]. This is for instance the case of CsrB and C, which control the activity of the post-transcriptional regulator CsrA. Their stabilization or destabilization alters the half-life of many CsrA-regulated genes [START_REF] Suzuki | Identification of a novel regulatory protein (CsrD) that targets the global regulatory RNAs CsrB and CsrC for degradation by RNase E[END_REF].

The gene coding RNase E is essential, presumably due to its role in RNA processing. Knocking down RNase E activity in vivo or decreasing its cellular level result in a slowdown in the turnover of bulk mRNAs [START_REF] Bernstein | Global analysis of Escherichia coli RNA degradosome function using DNA microarrays[END_REF][START_REF] Sousa | Autoregulation allows Escherichia coli RNase E to adjust continuously its synthesis to that of its substrates[END_REF][START_REF] Stead | Analysis of Escherichia coli RNase E and RNase III activity in vivo using tiling microarrays[END_REF]. Microarray analyses have

shown RNase E to initiate the decay of 60% of E. coli mRNAs [START_REF] Bernstein | Global analysis of Escherichia coli RNA degradosome function using DNA microarrays[END_REF][START_REF] Stead | Analysis of Escherichia coli RNase E and RNase III activity in vivo using tiling microarrays[END_REF]. Recently RNA-seq experiments have mapped thousands of RNase E cleavage sites in both E. coli and Salmonella transcriptomes [START_REF] Chao | In vivo cleavage map illuminates the central role of RNase E in coding and non-coding RNA pathways[END_REF][START_REF] Clarke | Direct entry by RNase E is a major pathway for the degradation and processing of RNA in Escherichia coli[END_REF]. Other endoribonucleases have been identified, such as RNase III and the paralog of RNase E in E. coli, RNase G, but they have a much more limited set of substrates [START_REF] Carpousis | The RNA degradosome of Escherichia coli : an mRNAdegrading machine assembled on RNase E[END_REF][START_REF] Kushner | mRNA decay in Escherichia coli comes of age[END_REF].

RNase E prefers binding to single-stranded RNA regions that are typically A-U rich, with a strong preference for U at position +2 with respect to the scissile phosphate [START_REF] Chao | In vivo cleavage map illuminates the central role of RNase E in coding and non-coding RNA pathways[END_REF][START_REF] Clarke | Direct entry by RNase E is a major pathway for the degradation and processing of RNA in Escherichia coli[END_REF][START_REF] Huang | Poly (A)-and poly (U)-specific RNA 3' tail shortening by E. coli ribonuclease E[END_REF][START_REF] Mcdowall | A+U content rather than a particular nucleotide order determines the specificity of RNase E cleavage[END_REF][START_REF] Mcdowall | Site-specific RNase E cleavage of oligonucleotides and inhibition by stem-loops[END_REF]. Two determinants of substrate recognition are the 5'-monophosphate termini -known to activate degradation through a higher enzyme affinity [START_REF] Kime | Rapid cleavage of RNA by RNase E in the absence of 5' monophosphate stimulation[END_REF][START_REF] Mackie | Ribonuclease E is a 5'-end-dependent endonuclease[END_REF] -and the presence of a stem-loop upstream of the cleavage site in most mRNAs cleaved by RNase E [START_REF] Del Campo | Secondary structure across the bacterial transcriptome reveals versatile roles in mRNA regulation and function[END_REF]. The current model proposes two modes of substrate recognition, which do not need to be mutually exclusive [START_REF] Bandyra | Substrate recognition and autoinhibition in the central ribonuclease RNase E[END_REF]. In the 5'-sensing mode, RNase E binds to the 5' monophosphate and the first bases of the 5' termini, causing an appropriate alignment at the active site of the single-stranded region containing the cleavage site. The binding can be strengthened or weakened by regulatory proteins or RNAs. k + and k -are the rate constants for the association and dissociation of RNase E to the mRNA, respectively. k cat is the catalytic constant of the degradation reaction.

In this case, degradation by RNase E requires a preliminary processing by RppH of the 5'-triphosphate termini to 5'-monophosphate [START_REF] Celesnik | Initiation of RNA decay in Escherichia coli by 5' pyrophosphate removal[END_REF][START_REF] Deana | The bacterial enzyme RppH triggers messenger RNA degradation by 5' pyrophosphate removal[END_REF][START_REF] Garrey | Roles of the 5'-phosphate sensor domain in RNase E[END_REF]. In the 5' bypass mode, which seems to be the most frequent one [START_REF] Clarke | Direct entry by RNase E is a major pathway for the degradation and processing of RNA in Escherichia coli[END_REF][START_REF] Richards | Influence of translation on RppHdependent mRNA degradation in Escherichia coli[END_REF], the direct binding of RNase E to a structured region of the mRNA allows to align the single-strand region at the active site independently of the nature of the 5' end [START_REF] Bandyra | Substrate recognition and autoinhibition in the central ribonuclease RNase E[END_REF].

The binding of RNase E to individual bacterial mRNAs is also regulated by trans-acting factors (Fig. 1) [START_REF] Richards | Obstacles to scanning by RNase E govern bacterial mRNA lifetimes by hindering access to distal cleavage sites[END_REF]. The base pairing of a small RNA with a mRNA can lead for instance to translational activation or repression in response to stresses. The later case is often associated to an increased destabilization of the messenger RNA. In many cases the process involves the protein chaperone Hfq, which facilitates the base pairing between the sRNA and the mRNA (see for review [START_REF] Santiago-Frangos | Hfq chaperone brings speed dating to bacterial sRNA[END_REF]. Ribosomes bound to the ribosome-binding site protect mRNAs from degradation by RNase E (for review, [START_REF] Deana | Lost in translation: the influence of ribosomes on bacterial mRNA decay[END_REF].

Changes in the level and activity of RNase E also modulates the degradation rate. An auto-regulation mechanism has been shown sensing the intracellular level of RNase E (Jain and Belasco, 1995a). However, under normal circumstances there appears to be excess RNase E in the cell, because enzyme levels have to be significantly reduced to see any phenotypic effect (Jain and Belasco, 1995b). The regulatory proteins RraA and RraB were also shown to

repress RNase E activity in vivo when over-expressed [START_REF] Lee | RraA: a protein inhibitor of RNase E activity that globally modulates RNA abundance in E. coli[END_REF], but their physiological role remains to be determined.

Monitoring mRNA degradation kinetics

Current experimental methods monitor mRNA decay in Prokaryotes and Eukaryotes through three different approaches: general inhibition of transcription by drugs (e.g. rifampycin) or mutation, transcriptional control by regulatable promoters and pulse-chase labelling [START_REF] Lugowski | Determining mRNA half-lives on a transcriptome-wide scale[END_REF].

Transcription is inhibited in the two first cases, either that of all cellular genes or of a single gene, following which the mRNA level starts to decay.

Pulse-chase labelling relies on the cell incorporation of labelled nucleotides into newly synthesized RNAs. The addition of natural nucleotides wash out the labelled nucleotides. mRNA decay is then monitored by following the decline of labelled mRNAs.

Transcription inhibition is often used in Prokaryotes (see [START_REF] Bernstein | Global analysis of mRNA decay and abundance in Escherichia coli at singlegene resolution using two-color fluorescent DNA microarrays[END_REF][START_REF] Esquerré | Dual role of transcription and transcript stability in the regulation of gene expression in Escherichia coli cells cultured on glucose at different growth rates[END_REF][START_REF] Dressaire | PNPase is involved in the coordination of mRNA degradation and expression in stationary phase cells of Escherichia coli[END_REF][START_REF] Hundt | Global analysis of mRNA decay in Halobacterium salinarum NRC-1 at single-gene resolution using DNA microarrays[END_REF] a few examples), coupled nowadays to high throughput technologies for the quantification over time of residual mRNA concentrations. Large scale data obtained with micro-arrays are by far the most abundant in the literature (for review [START_REF] Laguerre | Largescale measurement of mRNA degradation in Escherichia coli : To delay or not to delay[END_REF]. At the time of transcription inhibition and in the following minutes, the culture is sampled, mRNAs are extracted, purified, and used as matrix for the synthesis of complementary cDNAs. The latter are then quantified by hybridization to DNA chips. As an illustration, mRNA degradation profiles obtained in E. coli with this approach are shown in Appendix A.1. Alternatively, qRT-PCR approaches allow obtaining medium scale data through the amplification of selected cDNAs. RNA sequencing approaches starts also being used [START_REF] Chen | Genome-wide study of mRNA degradation and transcript elongation in Escherichia coli[END_REF][START_REF] Laguerre | Largescale measurement of mRNA degradation in Escherichia coli : To delay or not to delay[END_REF][START_REF] Moffitt | Spatial organization shapes the turnover of a bacterial transcriptome[END_REF][START_REF] Potts | Global role of the bacterial post-transcriptional regulator csra revealed by integrated transcriptomics[END_REF]. They consist in the depletion of ribosomal RNAs and the retro-transcription of mRNAs into cD-NAs, before the latter are sequenced.

Depending on the method chosen for the quantification of residual mR-NAs, different types of data normalization can be applied [START_REF] Laguerre | Largescale measurement of mRNA degradation in Escherichia coli : To delay or not to delay[END_REF]. The objective is to compensate for technical variations between samples at different time points. The mRNA levels can be further multiplied by RNA extraction yields to obtain mRNA concentrations in arbitrary units per gram dry weight. Mathematical models are subsequently used to estimate mRNA half-lives from these normalized data, as we shall see in the following section. Other modelling efforts include mRNA degradation beside other processes, some models being used to analyse regulatory mechanisms of mRNA degradation kinetics.

Kinetic modelling of mRNA degradation in literature

Various mathematical models of mRNA degradation kinetics have been proposed in the literature, the most common and simplest one being the exponential model. This mathematical description simplifies the mechanism of mRNA degradation to a first-order reaction modelled by the following linear ordinary differential equation (ODE):

d dt m i (t) = -k deg i m i (t), m i (0) = m i0 (1)
where m i (t) is the concentration at time t of the messenger RNA of gene i.

k deg is the degradation constant, from which mRNA half-life can be determined:

t 1/2 = log(2)/k deg .
This model is valid once transcription has been stopped at time 0, which means that the transcription rate does no longer counterbalance the degradation rate of mRNAs.

This mathematical modelling implies that the degradation kinetics depends linearly on the initial concentration of mRNA i, according to the following analytical solution of the ODE system:

m i (t) = m i0 × e (-k deg i ×t) . (2)
The exponential model in Eq. 2 is commonly used to estimate mRNA half-life by linear regression on a semilog plot of the concentration levels (for review, Laguerre et al 2018). The approach has been used for decades to estimate individual mRNA half-lives. It has been extended more recently to genome-wide data (e.g. [START_REF] Bernstein | Global analysis of mRNA decay and abundance in Escherichia coli at singlegene resolution using two-color fluorescent DNA microarrays[END_REF][START_REF] Bernstein | Global analysis of Escherichia coli RNA degradosome function using DNA microarrays[END_REF][START_REF] Esquerré | Dual role of transcription and transcript stability in the regulation of gene expression in Escherichia coli cells cultured on glucose at different growth rates[END_REF][START_REF] Potts | Global role of the bacterial post-transcriptional regulator csra revealed by integrated transcriptomics[END_REF]. Other related models are now emerging. They extend the exponential model with a delay before the onset of the decay [START_REF] Chen | Genome-wide study of mRNA degradation and transcript elongation in Escherichia coli[END_REF][START_REF] Dar | Extensive reshaping of bacterial operons by programmed mRNA decay[END_REF][START_REF] Moffitt | Spatial organization shapes the turnover of a bacterial transcriptome[END_REF] and a residual mRNA concentration at the end of the degradation process [START_REF] Moffitt | Spatial organization shapes the turnover of a bacterial transcriptome[END_REF]. The delay is generally attributed to the time required by RNA polymerase to complete transcription initiated prior to the drug addition.

First-order models provide valuable information on reaction rates through the determination of mRNA half-lives. However, they fell short of describing the biological mechanisms at work. In the literature, mechanistic models including mRNA degradation are generally broader in scope. With a relative simple description of the degradation mechanism, they relate several cellular processes such as the synthesis and degradation of mRNAs and proteins, the DNA replication and the cell growth... (e.g. [START_REF] Tadmor | A coarse-grained biophysical model of E. coli and its application to perturbation of the rRNA operon copy number[END_REF][START_REF] Thomas | Sources, propagation and consequences of stochasticity in cellular growth[END_REF][START_REF] Weiße | Mechanistic links between cellular trade-offs, gene expression, and growth[END_REF]. In [START_REF] Thomas | Sources, propagation and consequences of stochasticity in cellular growth[END_REF] for instance, mRNA degradation is identified as one source of growth heterogeneity at slow grow rates, when ribosomes are in too low numbers to protect mRNA from degradation by endonucleases. This leads to variations in biomass production and division times. Some modelling efforts are more focused on the mechanism of mRNA degradation and its regulation, such as the 5' binding and sliding of endonucleases on mRNAs and the mRNA protection by ribosomes in bacteria [START_REF] Carrier | Mechanistic modeling of prokaryotic mRNA decay[END_REF][START_REF] Deneke | Effect of ribosome shielding on mRNA stability[END_REF], or else the modes of action of miRNAs on protein translation and mRNA degradation in Eukaryotes [START_REF] Morozova | Kinetic signatures of microRNA modes of action[END_REF]. For instance, [START_REF] Carrier | Mechanistic modeling of prokaryotic mRNA decay[END_REF] and [START_REF] Deneke | Effect of ribosome shielding on mRNA stability[END_REF] have developed models of this type using a stochastic framework. However, the simulation of these models comes with a computational cost, which makes it inappropriate to analyse the coupled degradation of all cell mRNAs.

Knowing that large scale data are mostly available for mRNA degradation nowadays, these mechanistic modelling studies tell us that a compromise has to be reached in terms of model complexity and biological realism. The ideal model should i) be realistic from a biological point of view and useful to analyse regulatory mechanisms of mRNA degradation including competition; ii) alleviate the computational cost of numerical simulations; iii) include parameters that can be interpreted from a biological point of view; and iv) predict degradation kinetics that can be directly confronted to experimental data. We develop in the following section a kinetic model of the coupled degradation of all cell mRNAs, which meets these constraints as much as possible. We justify how we simplified the complexity of mRNA degradation to a macro-reaction catalysed by the limiting enzyme RNase E. We modelled this reaction and its regulations with the total quasi-steady-state approximation of the Michaelis-Menten equation. We incorporate substrate competition among the regulatory mechanisms. This required extending earlier modelling works on substrate competition to the case when multiple substrates compete for the same enzyme and when individual substrates are in excess of enzyme, but not the total substrate concentration.

Model development

Simplified description of mRNA degradation

As explained in Section 2.3, developing a model that can be confronted to experimental data constrains the level of description and the nature of the mechanisms to consider. We described in Section 2.2 how high throughput approaches are used to quantify residual mRNA concentrations after transcriptional arrest. These approaches quantify full length mRNAs through a good coverage of mRNA sequences. For instance, qRT-PCR and microarray experiments make use of oligonucleotides distributed along mRNAs (see [START_REF] Esquerré | Dual role of transcription and transcript stability in the regulation of gene expression in Escherichia coli cells cultured on glucose at different growth rates[END_REF][START_REF] Nouaille | The stability of an mRNA is influenced by its concentration: a potential physical mechanism to regulate gene expression[END_REF] for example), and read counts for each gene are determined in RNAseq experiments. In other words, messenger RNAs that have been cut once or several times by endo-and exo-nucleases are not tractable in these experiments. Especially since the rapidity of degradation by exonucleases makes it unlikely that partially degraded mRNAs could be confused with full length mRNAs. According to Fig. 1, this implies that our model does not need to describe reaction steps posterior to the initial cleavage by the endoribonuclease. In most cases, the endoribonuclease is RNase E and we will refer to that enzyme hereafter, although in some cases other endoribonucleases like RNase G are known to specifically degrade some target mRNAs. Our model remains valid for these alternative enzymes as well.

Experiments monitor the disappearance of full length mRNAs following the binding of RNase E to these mRNAs and their internal cleavage. As summarized in Section 2.1, this is the limiting step in mRNA degradation. The initial cleavage by RNase E is followed by the complete degradation of mRNAs into ribonucleotides, which implies that the reaction is irreversible and we do not expect end-product inhibition. So essentially, the simplest mechanistic description of mRNA degradation would simplify the reaction steps in Fig. 1 to the binding of free RNase E to a given mRNA i and its subsequent cleavage according to the general biochemical equation:

RNase E + mRNA i k + i k - i RNase E : mRNA i kcat → RNase E + ∅ .
The rate constants k + i and k - i denote respectively the rate of association or dissociation of mRNA i with the enzyme. Sequence and structure specificities may affect the rate at which RNase E binds to or unbinds from mRNAs. k cat is the catalytic constant of the reaction. The nature of the 5' extremity of some mRNAs is known to affect this constant when the 5' sensing mode is the mechanism of substrate recognition. In this case, RNase E can bind to mRNAs with 5' triphosphate termini but do not cleave them due to steric hindrance [START_REF] Mackie | Ribonuclease E is a 5'-end-dependent endonuclease[END_REF][START_REF] Callaghan | Structure of Escherichia coli RNase E catalytic domain and implications for RNA turnover[END_REF]. These mRNAs need a pre-processing by RppH to convert the 5' triphosphate into a 5' monophosphate termini. The conversion of the 5' end by RppH seems to concern only a small fraction of each transcript [START_REF] Luciano | A novel RNA phosphorylation state enables 5' end-dependent degradation in Escherichia coli[END_REF][START_REF] Richards | Influence of translation on RppHdependent mRNA degradation in Escherichia coli[END_REF], but the fact has not been established for all cell mR-NAs. Given that the direct entry mechanism is the most frequently used and that it functions independently of the 5' end, we will thus consider for simplicity that the catalytic constant is the same for all mRNAs. According to this view, the recognition of the 5' end is expected to act on the enzyme affinity only, in agreement with [START_REF] Kime | Rapid cleavage of RNA by RNase E in the absence of 5' monophosphate stimulation[END_REF].

The binding of RNase E to its target mRNA can be modulated, for instance by small RNAs, regulatory factors such as the proteins Hfq and CsrA, translating ribosomes or else, as we shall see later, by competition between mRNAs.

In the following section, we will show how to model the biochemical equation and its regulation. This equation is reminiscent of enzymatic reactions driven by Michaelis-Menten kinetics. However, we will see that the classical Michaelis-Menten expression based on the standard quasi-steady-state approximation does not always hold for this particular system along the degradation kinetics and for all mRNAs.

Kinetic modelling using quasi-steady-state approximations

Measurements of residual mRNA concentrations are performed experimentally on populations of sampled cells. We will thus use a deterministic modelling framework to describe the degradation kinetics. The model will predict averaged in-tracellular mRNA concentrations over the population of cells that can be related directly to experimental data.

In biochemistry, Michaelis-Menten-Henri kinetics is classically used to model enzymatic reactions (for review, Cornish-Bowden, 2013). The expression is derived from mass-action law and additional simplifying hypotheses introduced by [START_REF] Briggs | A note on the kinetics of enzyme action[END_REF], based on a standard quasi-steady-state approximation (sQSSA). It relates the reaction rate to the substrate concentration. This gives the following formula with our system:

v i = - d dt m i (t) = V m × m i (t) Km i + m i (t) , (3)
where m i (t) is the total concentration of mRNA i, assumed to be equal to the free substrate concentration:

m i (t) m i,f ree (t). V m = k cat E 0 is the maxi-
mum velocity achieved by the system at saturating concentration of mRNA i and E 0 is the total enzyme concentration. The value of the Michaelis constant

Km i = (k cat + k - i )/k +
i is defined as the substrate concentration at half maximum velocity. Kinetic parameters and concentrations are in standard units: molar units for Km values and intracellular concentrations, and min -1 for the k cat .

The sQSSA form of Michaelis-Menten kinetics in Eq. 3 reduces not only the model complexity but also allows relating model predictions with experimental observations. This canonical equation is commonly used in biochemical models, without regard to the validity of the underlying assumptions. Under the sQSSA, however, the substrate concentration must be greater than the enzyme concentration and the following relation should hold in our conditions:

Km i + m T i E 0 ,
where m T i is the total concentration of mRNA i corresponding to the concentrations of the free form of mRNA i, its form complexed to RNase E and the degradation product. This condition is verified in most in vitro enzymatic assays, where the enzyme concentration is chosen much lower than the substrate concentration.

However, the sQSSA condition can break down in in vivo conditions (e.g. [START_REF] Albe | Cellular concentrations of enzymes and their substrates[END_REF][START_REF] Sols | Concentrations of metabolites and binding sites. implications in metabolic regulation[END_REF]. Intra-cellular enzyme concentrations are usually higher or at least of the same magnitude as their substrates. Consequently, a significant fraction of substrates can be bound as complexes [START_REF] Bennett | Absolute metabolite concentrations and implied enzyme active site occupancy in Escherichia coli[END_REF].

What happens to mRNA degradation? The difficulty to verify sQSSA validity lies in the fact that Km i values are unknown. Available values in the literature were obtained in in vitro assays using synthetic RNAs and resulted in variable values whose validity in in vivo conditions is questionable [START_REF] Jeske | BRENDA in 2019: a European ELIXIR core data resource[END_REF][START_REF] Jiang | Catalytic activation of multimeric RNase E and RNase G by 5'-monophosphorylated RNA[END_REF][START_REF] Kim | Kinetic analysis of precursor M1 RNA molecules for exploring substrate specificity of the N-terminal catalytic half of RNase E[END_REF][START_REF] Kime | Identifying and characterizing substrates of the RNase E/G family of enzymes[END_REF][START_REF] Kime | Rapid cleavage of RNA by RNase E in the absence of 5' monophosphate stimulation[END_REF]. We expect that Km values in vivo vary between mRNAs due to sequence and structure specificities.

Regulations by small RNAs or RNA-binding proteins may affect as well the affinity of the enzyme for its substrates. Quantitative data are available for the concentrations of RNase E and mRNAs though. For example, [START_REF] Valgepea | Escherichia coli achieves faster growth by increasing catalytic and translation rates of proteins[END_REF] have quantified these concentrations in E. coli cultures growing at various rates in steady state. It appears that the median mRNA concentration at 0.48h -1 (0.5 nM) is way smaller than the concentration of the tetrameric form of RNase E ( 411nM). This enzyme excess with respect to individual mRNAs is no longer observed at the mRNA population level, since the total concentration of mRNAs is much larger (5927 nM).

An alternative to sQSSA is the total quasi-steady-state approximation (tQSSA) [START_REF] Borghans | Extending the quasi-steady state approximation by changing variables[END_REF][START_REF] Tzafriri | Michaelis-Menten kinetics at high enzyme concentrations[END_REF]. It replaces the concentration of free substrate with its total concentration. Unmistakably, the total substrate cannot be depleted by the formation of a complex with the enzyme. As a consequence of this simple modification, tQSSA is valid for almost any substrate and enzyme concentrations. Its domain of validity is also larger than a third approximation, the reverse quasi-steady state approximation (rQSSA), which assumes that the substrate concentration is in quasi-steady-state with respect to that of the enzymesubstrate complex [START_REF] Schnell | Enzyme kinetics at high enzyme concentration[END_REF][START_REF] Segel | The quasi-steady-state assumption: a case study in perturbation[END_REF][START_REF] Tzafriri | The total quasi-steady-state approximation is valid for reversible enzyme kinetics[END_REF]). The rQSSA is valid for high enzyme concentrations and would be appropriate to describe the degradation of individual mRNAs but not for a population of mRNAs. For these reasons of larger domain of validity, we will thus use the total QSSA of the Michaelis-Menten equation. Moreover, this in line with the experimental procedure, which most likely quantifies the total pool of mRNAs due to their deproteination following cell lysis. In the next section, we describe the modelling of the degradation of individual mRNAs using total QSSA. We will extend further the model in Section 3.4 with mRNA competition.

Kinetic modelling of the enzymatic degradation of individual mRNAs

We start here with the simplest case of mRNA degradation, where mRNAs are isolated and do not compete for binding to RNase E. This allows us to develop a simple model for the degradation of isolated mRNAs, which we will refer to as 'isolated system' in the rest of the article. We denote m i,f ree (t) the free concentration of mRNA i. c i (t) is the concentration of complex RN ase E : mRN A i . We have the following mass conservation relation for the total concentration of mRNA i:

m i (t) = m i,f ree (t) + c i (t) . (4)
Employing the total substrate concentration instead of the free concentration results in the so-called single reaction tQSSA [START_REF] Borghans | Extending the quasi-steady state approximation by changing variables[END_REF][START_REF] Tzafriri | Michaelis-Menten kinetics at high enzyme concentrations[END_REF]. This equation is virtually valid for any substrate and enzyme concentrations:

d dt m i (t) = -k cat × E 0 +Km i +m i (t) 2 (5) - √ (E 0 +Km i +m i (t)) 2 -4×E 0 ×m i (t) 2 , m i (0) = m i0 .
From the somewhat complex expression in (5), [START_REF] Tzafriri | Michaelis-Menten kinetics at high enzyme concentrations[END_REF] developed an approximation called first order tQSSA. It resembles the sQSSA form of Michaelis-Menten kinetics:

d dt m i (t) = - kcat × E 0 × m i (t) Km i + E 0 + m i (t) (6)
The equation ( 6) is valid if either of the following criteria is satisfied. Let K i be the individual constant of Van Slyke defined by

K i = k cat /k + i : E 0 + Km i m i0 and K i Km i or : (7) E 0 m i0 and E 0 Km i ≈ K i (8)
As for the sQSSA form of the Michaelis-Menten, model parameters are expressed in the same international standard units. Note that the comparison of model predictions generated by these models with experimental data may require preliminary conversions or normalizations. Indeed, some experimental protocols used for mRNA quantification generate data in arbitrary units. For instance, microarrays quantifying residual mRNAs generate concentrations in arbitrary units per gram dry weight (see Esquerré et al., 2013, for instance), and Ct values for qRT-PCR data (e.g. [START_REF] Nouaille | The stability of an mRNA is influenced by its concentration: a potential physical mechanism to regulate gene expression[END_REF].

Model extension with competition between mRNAs

The tQSSA derived in the previous section is well adapted for the description of the degradation of isolated mRNAs. However, the reality is that cells include a large number of mRNAs that share the same degradation enzyme, not to mention the small RNAs, pre-tRNAs and pre-rRNAs that are also processed by RNase E.

Can these multiple substrates titrate the enzyme?

To the best of our knowledge, there is no direct experimental evidence for competition, through measurements of the intracellular concentration of free or bound RNase E for instance. However, several studies support indirectly this hypothesis.

E. coli cells transcribe mRNAs from about 4000 genes, resulting in a much higher intracellular mRNA concentration with respect to the enzyme (Section 3.1). Their capacity to titrate RNase E depends on the enzyme kinetic properties. An efficient enzyme would process the excess of mRNA without being significantly titrated.

However, available data for RNase E suggest that the enzyme is far from being a kinetically perfect enzyme (see Table B.1 for details). The rate of turnover of the mRNA-RNase E complex is rather small (11 min -1 on average), while the low Km value (0.4µM) indicates a rapid binding of mRNAs to the enzyme. This suggests that the enzyme cannot process the excess of mRNA efficiently and that significant enzyme titration can be expected. A similar regulatory mechanism has been proposed for another ribonuclease with a much smaller set of substrates in E. coli, RNase P. This enzyme is involved in the maturation of the 5' end of eighty tRNA precursors [START_REF] Yandek | Alternative substrate kinetics of Escherichia coli ribonuclease P determination of relative rate constants by internal competition[END_REF].

Among the multiple RNase E substrates, those at higher concentration have generally a better chance to bind to the poorly available enzyme and be degraded.

This has been experimentally observed by means of reporter genes in E. coli [START_REF] Nouaille | The stability of an mRNA is influenced by its concentration: a potential physical mechanism to regulate gene expression[END_REF]. Increasing the concentration of given mRNAs reduced their stability, which is an indirect evidence supporting the existence of competition in RNase E-mediated degradation.

Along these lines, mRNAs would thus compete for binding to the active site of RNase E (Fig. 2), with the consequence that the rate of formation and the concentration of the complex between a mRNA i and RNase E might influence other mRNAs j. We thus extend the model in Section 3.3 to include competition between mRNAs. We will refer to this system as 'competitive' in the rest of the paper.

The competition between mRNAs requires to track the free enzyme concentration. This was done in [START_REF] Pedersen | The total quasi-steady-state approximation for fully competitive enzyme reactions[END_REF], in which the authors studied the case of two competing substrates. [START_REF] Tang | A total quasi-steady-state formulation of substrate uptake kinetics in complex networks and an example application to microbial litter decomposition[END_REF] extended this work to

k cat RNase E + mRNA 1 RNase E : mRNA 1 k 1 k 1 - ∅ 1 + k cat + mRNA 2 RNase E : mRNA 2 k 2 k 2 - ∅ 2 + k cat + mRNA 3 RNase E : mRNA 3 k 3 k 3 - ∅ 3 + k cat + mRNA n RNase E : mRNA n k n k n - ∅ n + . . . + + + + RNase E
Figure 2: mRNA degradation by RNase E in conditions of limiting enzyme concentration. ∅ i denotes the degradation product of mRNA i.

complex systems with n substrates, m enzymes and different types of inhibitory mechanisms. The authors also studied the special case of a system with n substrates and one enzyme (Eq. 20 in [START_REF] Tang | A total quasi-steady-state formulation of substrate uptake kinetics in complex networks and an example application to microbial litter decomposition[END_REF]). We applied this derivation to the case of n mRNAs competing for binding to RNase E. Using mass-action law, we wrote the following ODE system, with i = 1, • • • , n mRNAs (see Appendix C for details):

d dt m i (t) = -k cat × c i (t), m i (0) = m i0 d dt c i (t) = k + i × [(E 0 - n j=1 c j (t)) (9) ×(m i (t) -c i (t)) -Km i × c i (t)], c i (0) = 0.
In their respective works, [START_REF] Pedersen | The total quasi-steady-state approximation for fully competitive enzyme reactions[END_REF] and [START_REF] Tang | A total quasi-steady-state formulation of substrate uptake kinetics in complex networks and an example application to microbial litter decomposition[END_REF] applied approximations to the mass-action law system. They used the quasi-steadystate approximation dc i (t) dt ∼ 0 and either the total two Padé approximation [START_REF] Pedersen | The total quasi-steady-state approximation for fully competitive enzyme reactions[END_REF] or perturbation theory [START_REF] Tang | A total quasi-steady-state formulation of substrate uptake kinetics in complex networks and an example application to microbial litter decomposition[END_REF] to eventually obtain similar expressions. The application of the global approximation in [START_REF] Tang | A total quasi-steady-state formulation of substrate uptake kinetics in complex networks and an example application to microbial litter decomposition[END_REF] to our system of competition allowed us to obtain a simplified formula, much easier to interpret, which can be related to other Michaelis-Menten expressions (see Appendix C for details):

d dt m i (t) = - k cat × E 0 × m i (t) Km i × 1 + j =i m j (t) Km j + E 0 + m i (t) (10) with j = 1 • • • n.
The expression differs from the model of the isolated system in Eq. 6 with the introduction of a competition term (1 + j =i (m j (t)/Km j )) multiplied to the Km value, similarly to competitive inhibition systems. We will refer to the resulting constant

Km app i (t) = Km i ×   1 + j =i m j (t) Km j  
as "apparent Km" in the rest of the article. The competition term increases the value of the apparent Km value, indicative of a reduced RNase E affinity for its target mRNAs in competitive systems compared to isolated systems.

Tang and Riley (2013) studied the range of enzyme and substrate concentrations for which their approximation is valid. [START_REF] Pedersen | The total quasi-steady-state approximation for fully competitive enzyme reactions[END_REF] give also these criteria of validity, as well as the range of Km values for the system with two competing substrates. We could easily extend these criteria to the case of a system with multiple competing substrates. With K = kcat min{k 1,i } , any of the following conditions imply the validity of the approximation:

E 0 Km app i (0) + m i0 and K Km app i (0) + m i0 Km i n i=1 m i0 and K Km app i (0) Km i n i=1 m i0 and E 0 K Km app i (0) E 0 Km app i (0) + m i0 and E 0 K
Using numerical simulations of in vivo reactions, [START_REF] Pedersen | The total quasi-steady-state approximation for fully competitive enzyme reactions[END_REF] showed that their approximation is valid for a large range of parameter values. It fits correctly the original mass-action law model. In [START_REF] Tang | A total quasi-steady-state formulation of substrate uptake kinetics in complex networks and an example application to microbial litter decomposition[END_REF], the generalisation of the approximation to multiple substrates and/or enzymes prove to outperform the Michaelis-Menten expression. We will use in what follows this approximated version to investigate competitive effects in mRNA degradation kinetics.

Numerical simulation of mRNA degradation kinetics in isolated and competitive systems

At this stage, we have developed two alternative models of mRNA degradation kinetics in bacteria. Both rely on approximations of tQSSA forms of the Michaelis-Menten expression with extended criteria of validity compared to the classical sQSSA form. One model describes the reaction kinetics in situations where mRNAs are isolated and degraded independently of other mRNAs (Eq. 6).

The second model includes the effect of competition between substrates, where the formation of complexes between RNase E and mRNAs possibly affects the formation of other complexes (Eq. 10). Whether this is a true phenomenon in the control of mRNA turnover or not has not been established. Two recent studies suggest that this might be the case, since mRNAs with higher concentrations are destabilized in the bacteria E. coli and Lactococcus lactis [START_REF] Esquerré | Genome-wide investigation of mRNA lifetime determinants in Escherichia coli cells cultured at different growth rates[END_REF][START_REF] Nouaille | The stability of an mRNA is influenced by its concentration: a potential physical mechanism to regulate gene expression[END_REF]. It has been proposed that higher mRNA concentrations increase the probability that RNase E encounters its substrates in these two bacterial strains.

Although our two models are relatively simple at first glance, the introduction of competition between mRNAs couples the ordinary differential equations. This makes the system behaviour difficult to understand intuitively, in particular when studying degradation of all cell mRNAs. We will rather rely on numerical simulations for our purpose. This requires setting parameters to reasonable physiological values in order to predict degradation kinetics. Since literature data on the subject are generally scarce, we will use data obtained in the bacterium E. coli for which more knowledge and quantitative information are available. However, the model structure is sufficiently general and the mechanisms considered, well conserved among bacteria, for the model to be valid in other biological backgrounds.

Model parameters and initial conditions

Our objective with the models is to generate progress curves comparable to those obtained experimentally. The experimental curves vary a lot, from exponential to more linear shapes (see Appendix A.1 as an example). Their fitting by an exponential function allows determining mRNA half-lives ranging from one up to a dozen of minutes. This variability in mRNA stability reflects the diversity of mRNA sequence and structure determinants that modulate the enzyme affinity for its targets. It also results from regulatory mechanisms, such as regulations by small RNAs or mRNA protection by ribosomes to name but a few, which also impact the binding of RNase E to mRNAs. According to our modelling hypotheses described in Section 3.1, this results in an expected large variability of Km values because the k cat value is held constant between mRNAs.

To account for this variability, we constructed a parameter distribution for Km values based on diverse biological information. First, we chose a log-normal distribution to guarantee positive Km values (Fig. 3):

Km i ∼ logN (µ Km , σ 2 Km ),
where the mean µ Km is set to the empirical median value of 13 different Km values estimated from experimental data [START_REF] Jeske | BRENDA in 2019: a European ELIXIR core data resource[END_REF][START_REF] Jiang | Catalytic activation of multimeric RNase E and RNase G by 5'-monophosphorylated RNA[END_REF][START_REF] Kim | Kinetic analysis of precursor M1 RNA molecules for exploring substrate specificity of the N-terminal catalytic half of RNase E[END_REF][START_REF] Kime | Identifying and characterizing substrates of the RNase E/G family of enzymes[END_REF][START_REF] Kime | Rapid cleavage of RNA by RNase E in the absence of 5' monophosphate stimulation[END_REF]) (see also Table B.1 for some values). The dispersion of these data did not reflect the expected variability of Km values in vivo. We thus chose the standard deviation σ Km such that it mirrors the variability of mRNA concentrations measured experimentally: σ Km = σ m0 , similarly to what is found generally for many enzymes and their substrates (e.g. [START_REF] Bennett | Absolute metabolite concentrations and implied enzyme active site occupancy in Escherichia coli[END_REF].

The chosen k cat and Km i values result in a mean catalytic efficiency constant of 6•10 5 s -1 M -1 , in the average of values determined experimentally for RNase E (Table B.1). The mRNA concentrations at time 0, together with the RNase E concentration, correspond to transcriptomics and proteomics data in [START_REF] Valgepea | Escherichia coli achieves faster growth by increasing catalytic and translation rates of proteins[END_REF], which we discussed in Section 3. decay. This is altogether a reasonable assumption, given the similar fraction of mRNAs shown or predicted to be regulated at the degradation level [START_REF] Keseler | The EcoCyc database: reflecting new knowledge about escherichia coli K-12[END_REF][START_REF] Wright | CopraRNA and IntaRNA: predicting small RNA targets, networks and interaction domains[END_REF]. The distribution of initial mRNA concentrations is also characterized by a longer tail, where few mRNAs have high concentrations. Most of them are present at low intracellular levels. For the catalytic constant, we used the mean value estimated from experimental data obtained with various mRNAs [START_REF] Jeske | BRENDA in 2019: a European ELIXIR core data resource[END_REF]. Overall, the distributions and parameter values set verify the criteria of validity for the tQSSA described in Sections 3.3 and 3.4. Remember that in these data, the individual mRNA concentrations are well below the RNase E concentration, while the total mRNA concentration is in large excess (Fig. 3).

Competition between mRNAs slows down the degradation kinetics and increases the variability of progress curves

With the parameter values and concentrations described above, we simulated the kinetic models of the isolated and competitive systems. The transcriptomics data used for the initial conditions includes information for 4312 mRNAs. We thus ran as many simulations. We took every time a Km value from the log-normal distribution as well as the fixed catalytic constant and enzyme concentration. The same values of parameters and initial conditions were used to simulate the models with and without competition. All simulations were performed with the function ode15s of Matlab (Mathworks).

Remember that the simulated models are approximated forms of the tQSSA, valid under the criteria described in Sections 3.3 and 3.4. We could not exclude that some combinations of parameters and initial concentrations may violate these criteria. Beforehand we thus compared the predictions generated by the various tQSSA models to make sure that approximations of poorest quality did not impact significantly the degradation profiles (see Appendix D.1). However, the strong resemblance of the model predictions prompted us to use the approximated versions in the rest of the study. the Km value. However, the variability of the mRNA concentrations is such that it hides the effect of differences of RNase E affinity, represented by curves from red (high Km values) to blue (low Km). These differences are more visible in Panels (c) and (d), where the same curves normalized to the initial mRNA concentrations are shown. They differ by their Km value only. A striking difference between the isolated and competitive systems is the rapidity of the decay in the absence of competition. After two minutes, all mRNAs are at least half degraded in the isolated system (Panel (d)), whereas it takes almost five minutes to the competitive system to reach the same state (Panel (c)). This slower rate of degradation is consistent with measurements of mRNA half-lives in E. coli cells [START_REF] Bernstein | Global analysis of mRNA decay and abundance in Escherichia coli at singlegene resolution using two-color fluorescent DNA microarrays[END_REF][START_REF] Esquerré | Dual role of transcription and transcript stability in the regulation of gene expression in Escherichia coli cells cultured on glucose at different growth rates[END_REF]. For instance in [START_REF] Esquerré | Dual role of transcription and transcript stability in the regulation of gene expression in Escherichia coli cells cultured on glucose at different growth rates[END_REF], mRNAs have median half-lives ranging from 2.9 to 4.2 minutes in E. coli cells at various doubling rates. This general slowdown of mRNA degradation results from the reduced RNase E affinity in a competitive system compared to an isolated one (Section 3.4). This can be easily seen in The competitive effects result in more variable and more non-linear degradation profiles. Hence, the competition affects how degradation is launched. It starts immediately in isolated systems while it is delayed in competitive ones (Fig. 4(a,b)).

This delay is slightly more pronounced with the approximated tQSSA, but still present in the absence of approximation (see Fig. D.1 for comparison). It is therefore a feature of competing mRNAs. The occurrence of delays in mRNA degradation kinetics is a known phenomenon. The primary reason is the common use of rifampicin as a drug to block the initiation of transcription (e.g. [START_REF] Chen | Genome-wide study of mRNA degradation and transcript elongation in Escherichia coli[END_REF].

Elongating RNA polymerase is insensitive to the drug and pursues transcription.

During a transient time, the transcription rate counterbalances the degradation rate. It is thus expected that longer transcripts have longer delays before mRNA levels start decreasing [START_REF] Chen | Genome-wide study of mRNA degradation and transcript elongation in Escherichia coli[END_REF][START_REF] Esquerré | Genome-wide investigation of mRNA lifetime determinants in Escherichia coli cells cultured at different growth rates[END_REF][START_REF] Moffitt | Spatial organization shapes the turnover of a bacterial transcriptome[END_REF].

Are there other determinants to the delay before degradation?

In our study we have not modelled the delay explicitly, as least not the time required for the completion of transcription. The predicted delay results from the titration of RNase E by competing mRNAs. As shown in Panel (e), the intracellular pool of free RNase E is strongly depleted in the first minutes of the degradation kinetics in the competitive system. While the enzyme level is indeed much higher than those of individual mRNAs, the total mRNA concentration is bigger and the formation of complexes between the enzyme and the mRNAs limits the enzyme availability. This implies that sole mRNAs at high initial concentration or with low Km values are more likely to be degraded by this small enzymatic pool. The intracellular pool of RNase E raises to higher levels when mRNAs are degraded to a large extent. At this time of the degradation kinetics, RNase E is in excess and all mRNAs can be degraded. This is what happens during the whole kinetics for mRNAs in the isolated system, since the enzyme concentration is never limiting (Panel (f)). The mRNAs can all bind to RNase E and be degraded soon afterwards.

Another striking difference is the variability of the degradation profiles driven by competing mRNAs (compare Panels (a) and (b) or (c) and (d)). Even though we set the variance of the Km distribution to quite a large value, it is clearly not enough to produce a large diversity of curves in the isolated system. With the same parameters and initial concentrations, the introduction of the competition term gives rise to curves with variable shapes. This is especially true for the curves in dark blue in Panel (d), characterized by low Km values. Addition of the competition term spread them out, as can be seen in Panel (c). Cyan to red curves, characterized by a high Km, are not scattered but simply shifted. The corresponding mRNAs seem merely to be stabilized.

Where do these variability and increased non linearity come from? As can be seen in Panels (a) and (b) for both isolated and competitive systems, many mRNAs have a low initial concentration. Many of them have also higher Km values, as shown in Panels (c) and (d) (lines coloured from cyan to red). In these conditions, the mRNA concentration becomes negligible with respect to the Km and/or the RNase concentration: m i0 Km i + E 0 . Therefore, the kinetics in Eq. 6 can be approximated by the following system for isolated mRNAs:

d dt m i (t) ≈ k cat × E 0 Km i + E 0 × m i (t) . (11)
For mRNAs in competition, it becomes possible to approximate the degradation kinetics in Eq. 10 by:

d dt m i (t) ≈ k cat × E 0 Km i 1 + j =i m j (t) Km j + E 0 × m i (t) . (12)
In both expressions, the degradation rate varies linearly with the mRNA concentration. Note as a side comment that the solution of these ODEs is an exponential function comparable to the classical exponential model used to determine mRNA half-life (see Eq. 2). It is thus appropriate to fit the degradation profiles of these particular mRNAs (higher Km and/or lower initial concentration) with an exponential model. The decrease of enzyme affinity resulting from the increased apparent Km value in the competitive system (Fig. E.1(a)) globally shifts the curves to the right compared to the isolated system (look for instance at the cyan to red curves in Panel (c) versus (d)). All mRNAs with higher Km and/or lower initial concentrations will react in the same manner. While they tend to be stable in the isolated system, competition stabilizes them even more.

The situation is different for mRNAs characterized by higher initial concentrations and/or lower Km values. In these conditions, the approximation in Eqs. 11 and 12 does not hold any more. RNase E is closer to saturation at the beginning of the degradation kinetics (see Fig. 4), the substrates start to compete and their degradation rates depend on each other. The degradation rate is no longer linear in the mRNA concentration and the non linearity is amplified by the multiplication of the Km value with the competitive term. We thus expect that these mRNAs react differently to the competition. This is visible in In summary, competition between mRNAs acts globally on mRNA decay by decreasing RNase E affinity for its substrates and by retarding the onset of degradation. It also acts on each mRNA individually. To better understand how the own characteristics of mRNAs in terms of enzyme affinity and initial concentrations lead to variable responses to competition, we determined rate sensitivity coefficients for the two models, as we shall see in the following section.

Determination of rate sensitivity coefficients disentangles the effect of substrate concentration and enzyme affinity in competitive systems

In our models, mRNAs start to decay following the transcription arrest at time zero. The initial velocity of the degradation reaction depends on the saturation of RNase E. Any change in the initial concentrations of mRNAs and the enzyme affinity is expected to affect the degree of saturation. This is exactly what bacteria experience in nature. They constantly adjust their growth rate to changing environmental conditions. This involves a wide reorganization of gene expression, which includes, among others, changes in mRNA abundance and half-life resulting notably from post-transcriptional regulations (see [START_REF] Bernstein | Global analysis of Escherichia coli RNA degradosome function using DNA microarrays[END_REF]Esquerré et al., 2014, for instance). The latter affect the enzyme affinity for its substrates in our modelling framework. The consequence is that mRNA stabilities should react differently to these changes according to our simulation results in Section 4.2.

The determination of scaled rate sensitivity coefficients allow assessing how changes of initial concentrations and RNase E affinity impact the degradation rates. For each model, we derived such a coefficient for the initial velocity, by determining the scaled change of the initial degradation rate of each mRNA i upon a variation of mRNA concentrations at time 0 or of Km value. These coefficients are the equivalent of the elasticities and control coefficients determined at steady state in Metabolic Control Analysis [START_REF] Heinrich | The Regulation of Cellular Systems[END_REF][START_REF] Ingalls | Sensitivity analysis of stoichiometric networks: an extension of metabolic control analysis to non-steady state trajectories[END_REF]. Their analytical expressions are shown in Table 1.

We first examined how changing the initial concentration (m i0 ) or Km value (Km i ) of a messenger RNA i affects the initial velocity (v i ) of its degradation reaction. The sign of the analytical expressions of the response coefficients in Table 1 shows that the direction of the change is the same in both isolated and competitive systems: increasing the initial concentration destabilizes the mRNA (R v i m i ≥ 0), while increasing the Km value (and thus decreasing the enzyme affinity) stabilizes it (R v i Km i ≤ 0). However, the amplitude of the change may differ between both systems. We quantified it, by evaluating the analytical expressions of the response coefficients for the various pairs of mRNA concentration and Km used for the numerical simulations in Section 4.2. The linear interpolations of the coefficients are reported as contour plots in Fig. 5. We also displayed as points the pairs of concentration and Km value used to generate the degradation profiles in Section 4.2.

The response coefficients with respect to the change in mRNA concentrations are close to unity for both models (see Panels (a) and (b)). This means that increasing the initial concentration by 10 % results in a similar ∼ 10% increase in the rate of degradation and conversely, a decrease in the time at which mRNAs are half degraded. The coefficients tend to be even higher for the competitive system. The initial velocity of mRNA degradation is thus essentially determined by the initial messenger concentration and this effect is reinforced in competitive systems. This

Model Coefficients

Isolated system (Eq. 6)

Effect of changes of m i0 and Km i on v i :

R v i m i = ∂v i ∂m i0 m i0 v i = Km i + E 0 Km i + E 0 + m i0 R v i Km i = ∂v i ∂Km i Km i v i = -Km i Km i + E 0 + m i0
Effect of changes of m j0 and Km j on v i :

R v i m j = ∂v i ∂m j0 m j0 v i = 0 R v i Km j = ∂v i ∂Km j Km j v i = 0
Competitive system (Eq. 10)

Effect of changes of m i0 and Km i on v i :

R v i m i = ∂v i ∂m i0 m i0 v i = Km i × 1 + j =i m j0 Km j + E 0 Km i × 1 + j =i m j0 Km j + E 0 + m i0 R v i Km i = ∂v i ∂Km i Km i v i = -Km i × 1 + j =i m j0 Km j Km i × 1 + j =i m j0 Km j + E 0 + m i0
Effect of changes of m j0 and Km j on v i :

R v i m j = ∂v i ∂m j0 m j0 v i = -m j0 × Km i Km j Km i × 1 + j =i m j0 Km j + E 0 + m i0 R v i Km j = ∂v i ∂Km j Km j v i = -R v i m j
agrees well with experimental observations, where mRNA concentration has been found negatively correlated with mRNA half time in E. coli and L. lactis bacteria [START_REF] Esquerré | Genome-wide investigation of mRNA lifetime determinants in Escherichia coli cells cultured at different growth rates[END_REF][START_REF] Nouaille | The stability of an mRNA is influenced by its concentration: a potential physical mechanism to regulate gene expression[END_REF]. To reach this conclusion, the authors of these studies analysed large data sets previously obtained in these bacteria growing in environmental conditions leading to changes in mRNA concentrations and half-lives. Increasing specifically the concentrations of four mRNAs by means of inducible promoters elicited again the same negative relationship between mRNA concentration and stability [START_REF] Nouaille | The stability of an mRNA is influenced by its concentration: a potential physical mechanism to regulate gene expression[END_REF]. Similarly to our observations, the correlation is more pronounced at low to middle mRNA concentrations, but less at high mRNA concentration (R v i m i coefficients are a bit smaller).

We then studied how the isolated and competitive systems react to changes in enzyme affinity. The corresponding response coefficients are shown in Panels (c) and (d) of Fig. 5. For the pairs of initial concentration and Km value used for the numerical simulations in Fig. 4, changing the Km value by 10% leads, on average, to a 3% decrease of the initial velocity in the isolated system (R v i Km i ∼ -0.3) and to a 9% decrease in the competitive system (R v i Km i ∼ -0.9). This effect does not depend on the initial mRNA concentration. It is stronger at higher Km, but the range of Km values for which the system response is highly negative is much larger in the competitive system than in the isolated one. It encompasses the majority of the pairs of initial concentration and Km value used in the numerical simulations.

It is interesting to realize that initial concentrations impact the degradation kinetics at two different levels in competitive systems. First of all, as described above, they strongly influence mRNA stability due to the strong, positive response of the degradation rate to changes of concentration. Secondly, they have an antagonistic effect on the degradation rate in the competitive system, due to the strong negative response to Km changes. This response to Km changes is low in the isolated system, which implies that the enzyme affinity hardly compensates for the effect of the initial concentration (R v i Km i ∼ -0.3 and R v i m i ∼ 1 on average for the pairs of Km and initial concentration used in the numerical simulations). The degradation kinetics of an isolated system is thus almost exclusively driven by the initial mRNA concentration. Now, in the competitive system, the response coefficients are of opposite sign and similar amplitude (R v i Km i ∼ -0.9 and R v i m i ∼ 1 on average). The combination of these two antagonistic effects explain the non linearities observed in the predicted degradation curves in Fig. 4(a,c), in particular the occurrence of delayed degradation when mRNAs compete between them. Indeed, as long as the total mRNA concentration is high at the beginning of the kinetics, it augments the apparent Km value to sufficiently high values to compensate for the effect of the initial mRNA concentrations. RNase E is totally saturated and the initial velocity is almost null. It is only when the total mRNA concentration decreases significantly that the weight of the competitive term in Eq. 10 fades away. RNase E is no longer saturated and the competitive system resembles the isolated one, with less sensitivity to changes of enzyme affinity.

With these analyses, we were able to disentangle the contribution of Km and mRNA concentration to the degradation kinetics in competitive systems. One question remains open though. Since the degradation kinetics of all mRNAs are coupled due to the competition, are they impacted by the cell composition in mRNAs? The competition term in Eq. 10 includes the affinity of cell mRNAs j different from the mRNA i of interest, as well as their concentration. Given that there are 4312 different mRNAs in our biological system, the intuition would say that the change of concentration or Km of a single mRNA should not affect the degradation kinetics of another one. But how true is it? We further study this question in the following section.

Effect of coupling on mRNA degradation kinetics

We analysed a new set of rate sensitivity coefficients to analyse how the changes of concentration or Km value of another mRNA j affect the degradation kinetics of a given mRNA i. The analytical expressions for these new coefficients are shown in Table 1. The isolated system is characterized by the absence of coupling between mRNAs. Consequently, the response coefficient to changes of Km value or initial concentration of mRNA j is zero (R v i m j = R v i Km j = 0). In the competitive system, the analytical expression depends on the concentration of the other mRNA j and the relative affinities of mRNAs j and i. The sign of the response coefficients R v i m j and R v i Km j is opposed to the previous coefficients R v i m i and R v i Km i . It shows that increasing the initial concentration of mRNA j stabilizes mRNA i (R v i m j ≤ 0), while increasing the Km value of mRNA j (and thus decreasing the enzyme affinity) destabilizes it (R v i Km j ≥ 0). As previously, this is a matter of enzyme titration. If a given mRNA j titrates more RNase E due to a high initial concentration or low Km value, less RNase E will be available for mRNA i, which is stabilized.

We R v i m j and the dark blue area in Panel (c) for R v i Km j ). However, when it does impact the initial velocity, it does so for all mRNAs i (see horizontal lines coloured from But interestingly, they are not negligible. The competitive effect is the strongest when the mRNA j has a high concentration or a low Km value. Increasing its concentration by 10% in this case reduces by 0.2% the initial velocity of mRNA i degradation (R v i m j ∼ -0.02; Panel (e)). The effect is exactly the opposite when increasing the Km by 10%. This stimulates the initial degradation rate of the average mRNA i by 0.2% (R v i Km j ∼ 0.02; Panel (f)) and reduces the time at which it is half degraded. Hence, a highly competitive mRNA, which titrates more RNase E due to a high initial concentration or a low Km value, can affect the degradation rate of all cell mRNAs, with an average reduction of 0.2% of the velocity.

The competing effects that mRNAs exert on each other appear limited at first glance. However, these effects do exist whenever a single competing mRNA is considered and could be even stronger if several competing mRNAs are subject to changes. Indeed mRNA abundances and half-lives are known to massively adjust to environmental conditions (see [START_REF] Bernstein | Global analysis of Escherichia coli RNA degradosome function using DNA microarrays[END_REF][START_REF] Schneider | A DNA architectural protein couples cellular physiology and DNA topology in Escherichia coli[END_REF]

Discussion

We have developed in this study a mechanistic model of the coupled degradation of all cell mRNAs. The model represents a compromise in terms of complexity and biological realism. Its predictions can be compared directly to degradation profiles obtained experimentally. We hence assimilated the degradation mechanism by the degradosome machinery to a macro-reaction catalysed by RNase E.

Literature data motivated the use of the tQSSA form of the Michaelis-Menten equation for describing the degradation kinetics. We further approximated the tQSSA model by extending previous work by [START_REF] Pedersen | The total quasi-steady-state approximation for fully competitive enzyme reactions[END_REF] and [START_REF] Tang | A total quasi-steady-state formulation of substrate uptake kinetics in complex networks and an example application to microbial litter decomposition[END_REF]. The resulting simplified model allows analysing the effect of regulatory mechanisms on mRNA decay, which are difficult to test experimentally.

The possible implication of mRNA competition is one of them. While there is no direct experimental proof for significant titration of RNase E, a body of evidences -on its limited concentration relative that of mRNAs and its mean catalytic efficiency -suggest that competition between mRNAs could well impact the mRNA degradation kinetics (Section 3.4). We analysed the contribution of competition to mRNA decay by means of numerical simulations and rate sensitivity coefficients, based on carefully chosen parameter values and distributions. To the best of our knowledge, this is the first study focusing on the role of competition in mRNA turnover. A similar study was conducted on the maturation of transfer RNAs by RNase P in conditions of limiting substrates [START_REF] Yandek | Alternative substrate kinetics of Escherichia coli ribonuclease P determination of relative rate constants by internal competition[END_REF], but the number of alternative substrates was low compared to our system. In addition the authors of the study opted for the classical sQSSA form of the Michaelis-Menten framework, which would not be suitable in our case, given its limits of validity.

Our results provide a plausible explanation to the observed delays and variable degradation profiles in experimental studies. Under the effect of competition, the profiles vary from seemingly exponential to more linear profiles. Such diversity is difficult to reproduce in the absence of competition, while it is observed experimentally (e.g., see Fig. The lag effect caused by mRNA competition is a striking result. Residual transcription is generally associated with delayed degradation, in particular for long genes or genes towards the 3' end of operons [START_REF] Chen | Genome-wide study of mRNA degradation and transcript elongation in Escherichia coli[END_REF]. Could mRNA competition be an additional mechanism? In the absence of direct evidence of this phenomenon, through measurements of the free or bound concentration of RNase E for instance, we cannot answer formally the question. However, an analysis of available dynamical transcriptomics data in E. coli indicates that residual transcription alone does not suffice to explain the duration of the delay before degradation kicks in (see Appendix G for details). We estimated the delay before degradation for 3140 mRNAs of this data set [START_REF] Esquerré | Dual role of transcription and transcript stability in the regulation of gene expression in Escherichia coli cells cultured on glucose at different growth rates[END_REF] The competition explains also the negative correlation observed between mRNA half-life and concentration [START_REF] Esquerré | Genome-wide investigation of mRNA lifetime determinants in Escherichia coli cells cultured at different growth rates[END_REF][START_REF] Nouaille | The stability of an mRNA is influenced by its concentration: a potential physical mechanism to regulate gene expression[END_REF]. Our model has the advantage that it provides a mechanistic interpretation to the correlation, through the coupling of the degradation kinetics of each mRNA. This results from the fact that the competition acts at two different levels. First of all, the degradation rate and thus the time at which the mRNA is half degraded is extremely sensitive to mRNA concentration (Fig. 5(a)). This explains why the correlation is the highest in the experimental data, in particular for mRNAs with low to intermediate concentrations [START_REF] Nouaille | The stability of an mRNA is influenced by its concentration: a potential physical mechanism to regulate gene expression[END_REF]. At higher mRNA levels, the degradation rate becomes less sensitive to the concentrations while it can be still strongly sensitive to the Km values (Fig. 5(a,b)). This is where the correlation between half-life and concentration is smaller: the mRNA half-lives no longer vary with high mRNA concentrations.

Our simulation results and the response coefficients show that mRNA compe-tition amplifies the effect of the Km value in the degradation kinetics. mRNAs that have intermediate to low affinities compete for binding to RNase E. The less affine mRNAs will have less chance to be degraded: the competition term in their apparent Km value in Eq. 10 increases due to the presence of competing mRNAs with higher affinities (the ratio j =i m j (t)

Km i takes higher values). This results in a higher apparent Km reflecting a stabilisation of these mRNAs ( ). The consequence is that these unstable mRNAs in the isolated system are destabilized by competing mRNAs in the competitive system.

By affecting differently the mRNAs, the competition brings about a diversity of effects that is responsible for the larger variability of degradation profiles in a competitive system with respect to an isolated one. Surprisingly, a single mRNA can affect the degradation profiles of the other cell mRNAs, provided it is sufficiently concentrated or affine for RNase E. All these observations are interesting because it implies that cells can efficiently control the stability of their entire population of mRNAs through the sole use of a competitive mechanism and a limiting enzyme concentration. This suggests that mRNA competition allows cells to adjust degradation to transcription. Similar coupling effects are observed in Eukaryotes and Prokaryotes like E. coli. Buffering mechanisms against changes in mRNA stability or transcription ensure that intracellular levels of mRNAs are maintained for instance, although the molecular mechanisms have not been fully elucidated [START_REF] Dori-Bachash | Coupled evolution of transcription and mRNA degradation[END_REF], 2012;[START_REF] Esquerré | Dual role of transcription and transcript stability in the regulation of gene expression in Escherichia coli cells cultured on glucose at different growth rates[END_REF][START_REF] Sun | Comparative dynamic transcriptome analysis (cDTA) reveals mutual feedback between mRNA synthesis and degradation[END_REF][START_REF] Shalem | Transcriptome kinetics is governed by a genome-wide coupling of mRNA production and degradation: a role for RNA Pol II[END_REF]; see Hartenian and Glaunsinger 2019 for review). In our simulation results, the positive coupling between transcription and degradation results from the higher probability that mRNAs with increased concentration will be degraded.

However, we also observe a counter-acting effect that negatively correlates transcription and degradation rates. The increase of mRNA concentration through transcription increases RNase E apparent affinity and slows down degradation, although the amplitude of the effect varies between mRNAs.

The occurrence of non linearities resulting from enzyme titration is a well studied phenomenon in biochemistry. It has been observed in many biological systems, ranging from the MAP kinases to transcription and translation (e.g. [START_REF] Buchler | Protein sequestration generates a flexible ultrasensitive response in a genetic network[END_REF][START_REF] De Vos | How molecular competition influences fluxes in gene expression networks[END_REF][START_REF] Huang | Ultrasensitivity in the mitogen-activated protein kinase cascade[END_REF][START_REF] Mehra | An algorithmic framework for genome-wide modeling and analysis of translation networks[END_REF]. Quite often, this results in ultrasensitive responses. We have not really observed such phenomenon in our simulation study. The response coefficients that we determined from the pairs of Km value and initial concentrations used for the numerical simulations are generally between -1 and 1. Sole responses above one in absolute value reflect an ultrasensitive behaviour. Values slightly below -1 were sometimes obtained in specific conditions, when the Km value of more affine mRNAs i was varied. This does not mean than no ultra-sensitivity occurs in mRNA degradation since the amplitude of the response coefficients strongly depends on the chosen parametrization.

While we were able to retrieve quite precise values for the enzyme concentration, the mRNA concentrations and the catalytic constant in the literature, this prove to be more difficult for the Km value. Available data for the enzyme affinity were obtained with a limited number of substrates in (often) in vitro assays. While this gave a possible average affinity of RNase E for its substrates, it did not reflect the variety of cell mRNAs. The latter were expected having different affinities, depending on their structural and sequence determinants, as well as their possible regulations by small RNAs and other regulatory proteins. This is the reason why we chose a distribution of Km as large as the distribution of measured initial concentrations to take into account as much as possible these different situations.

This said, using different Km distributions would only tune the amplitude of the competition effect. Indeed there will be always an antagonism between the contributions of initial concentrations and enzyme affinities to the competition effect.

Depending on their relative distributions, one may become prevailing. In our simulations, the choice of distributions with similar variances lead to contributions of initial concentrations and Km of similar strength in many cases.

Estimating the Km value from experimental data, and possibly also the catalytic constant, could help refine our analysis on the role of mRNA competition in mRNA decay. This is beyond the scope of this study and will be done elsewhere (T.A. Etienne, L. Girbal, M. Cocaign-Bousquet, D. Ropers, in preparation). Using a total quasi-steady-state approximation rather than the classical quasi-steady-state approximation is known to improve the estimation of parameter values by introducing the enzyme concentration in the equation [START_REF] Choi | Beyond the Michaelis-Menten equation: Accurate and efficient estimation of enzyme kinetic parameters[END_REF]. In addition to helping with the identifiability problems related to the use of Michaelis-Menten kinetics, estimating the parameter values of such competition model allows to determine the true value of the Km and not an apparent constant that includes a competition term.

The model developed is sufficiently general to be applied to other organisms and regulatory mechanisms of mRNA degradation. For instance, the apparent Km value in our equations may be extended to include regulations by small RNAs or the competition by stable RNAs, which are also known to be matured by RNase E. The fact that RNase E is active both in the cytoplasm and bound to the membrane as part of the degradosome allows to interpret the Km in a different manner. It also reflects the diffusion of substrates to the membrane and allows to study the effect of substrate localization in the degradation kinetics [START_REF] Tang | A total quasi-steady-state formulation of substrate uptake kinetics in complex networks and an example application to microbial litter decomposition[END_REF][START_REF] Tzafriri | Reaction diffusion model of the enzymatic erosion of insoluble fibrillar matrices[END_REF]. Overall, we believe that our mechanistic model of mRNA degradation and its regulation by competition and other mechanisms is an interesting alternative to the classically used exponential models. Note that the total concentration of RNase E, but not the free concentration, is a variable in Eqs. 5 and 9. We thus calculated a posteriori the concentration of free RNase E from the total RNase E and complex concentrations. Since the model at time zero is not at steady state, we could not determine a steady-state value for the initial complex concentration, which would have given a more realistic value for the initial concentration of free RNase E. Instead, we used the same initial condition for the complex concentration as in the work of [START_REF] Pedersen | The total quasi-steady-state approximation for fully competitive enzyme reactions[END_REF], with ci(0) = 0. This effect is transitory only, since the free RNase E concentration predicted with Eq. 9 becomes quickly similar to that obtained with the approximated tQSSA model in Fig. 4(a). position of RNA polymerase when rifampicin inhibits the holoenzyme. Such data is not available, which makes it impossible to determine the residual transcription time precisely. However, we can set an upper limit to this time, by determining the time needed to transcribe entire mRNAs. If this transcription time is smaller than the delay before degradation, this means than residual transcription cannot be the sole determinant of the delay. Results are more difficult to interpret when the maximal transcription time is larger than the delay before degradation. They can reflect co-transcriptional degradation as in [START_REF] Chen | Genome-wide study of mRNA degradation and transcript elongation in Escherichia coli[END_REF] for instance, but also include false negative: mRNAs with low residual transcription time, but total gene transcription time larger than the delay, should be counted as mRNAs for which residual transcription is not the sole determinant of the delay. To determine the time needed for transcription, we multiplied the elongation rate of transcription determined at the genome scale by Chen and co-authors (25 nts/s on average), with the length of each mRNA. When a gene is located within an operon, we used the promoter operon to determine a maximal size for that mRNA. We focused on that, for a vast majority of genes, transcription alone does not explain the whole duration of the delay with the transcription rate considered. This does not prove the role of competition experimentally, but does not exclude that this phenomenon could play a role in the retardation of degradation. Note we cannot rule out that heterogeneity in the polymerisation rate in transcription or in degradation may affect the estimated transcription times and delays before degradation. However, our analysis is based on population data for which information on heterogeneity is not available.

Figure 1 :

 1 Figure1: RNA degradation in Prokaryotes. Endoribonucleolytic cleavage by RNase E generates fragments that are further converted to nucleotides by oligoribonucleases. RNase E binds to a A-U rich sequence of the messenger RNA, often downstream of a stem loop. The binding can be strengthened or weakened by regulatory proteins or RNAs. k + and k -are the rate constants for the association and dissociation of RNase E to the mRNA, respectively. k cat is the catalytic constant of the degradation reaction.

  2. The mRNA concentrations form a log-normal distribution illustrated on a logarithmic scale in Panel (a) of Fig. 3. The corresponding distribution for Km values is shown in Panel (b). On the logarithmic scale, the distribution is normal, while it is characterized by a longer tail towards higher Km values on a linear scale. It reflects the proportion of mRNAs potentially affected by specific regulations antagonizing RNase E (less than 10% with the chosen variance). As well, an even smaller fraction of mRNAs have low Km values, reflecting possible regulatory mechanisms activating mRNA

FrequencyFigure 3 :

 3 Figure 3: Log-normal distributions of initial concentrations (a) and Km values (b). The x-axes are on a logarithmic scale. The red and green lines denote the total concentrations of RNase E and mRNA, respectively.

Fig. 4 Figure 4 :

 44 Fig. 4 describes the simulation results obtained with the approximated tQSSA models. Predicted mRNA concentrations are shown for the competitive system in Panel (a) and the isolated system in Panel (b) (see Fig. D.2 for a log scale representation of these results). The curves differ by the initial mRNA concentration and

  Fig. E.1, where the distribution at time 0 of the apparent Km values (including the competition term) is shifted to a higher median value compared to the distribution of Km values.

  Fig. E.1(b), where normalized values of the Km and apparent Km for each mRNA are compared. We observe the global rescaling of the apparent Km values at time 0 in the competitive system.However, for a non negligible part of the mRNAs, the trend of the rescaling is different. mRNAs with higher Km values in the isolated system can have lower apparent Km values in the competitive system. It is only later in the kinetics, when mRNAs are degraded in large parts, that the curves for the isolated and competitive systems are more alike. In these conditions, the mRNA concentration becomes negligible. The degradation kinetics become again linear in the mRNA concentration and can be approximated by the simplified expressions in Eqs. 11 and 12.

  evaluated the response coefficients by determining how a change of 10% of the initial concentration or Km of each of the 4312 mRNAs j affects the initial velocity of each of the 4311 other mRNAs i and j itself. The results are shown as heatmaps in Fig. F.1. In most cases, the mRNA j marginally affects the other mRNAs (response coefficients close to zero, see the yellow area in Panel (b) for

Figure 5 :

 5 Figure 5: Scaled rate sensitivity coefficients for the initial velocity of mRNA i degradation. The points correspond to the pairs of initial concentration and Km used in the numerical simulations in Fig. 4. For each pair, the contour plots display by means of linear interpolation the corresponding response coefficients with respect to (a) the initial concentrations (R vi mi ) or (c) the Km (R vi Kmi ), in a competitive system or in the isolated system ((b) and (d), respectively). Colour bars indicate the value of the elasticities. Panels (e) and (f) display the median response coefficients to changes of mRNA j concentration (R vi mj ) or Km (R vi Kmj ), respectively. The x-and y-axes are on a logarithmic scale.

  Fig.3(a)) and most likely also a relatively weak affinity (Fig.3(b)), such that they are not saturated by RNase E. Consequently, it is well possible that a majority of

  A.1). The explanation lies in the titration of RNase E at the beginning of the kinetics and as long as the total mRNA concentration remains sufficiently high, which delays the onset of degradation. During this period of time, mRNAs compete between them. The consequence is that the degradation rate varies along time, from zero during the delay to maximal when free RNase E is available, and back to zero towards the end of the kinetics. Such variation is visible in experimental data (e.g., see Fig. A.1).

  , as well as the time needed to transcribe each of them. In the absence of information on the position of RNA polymerase at the time of rifampicin addition, determining the time needed for transcribing an entire mRNA gives an upper bound to the residual transcription time. Among the 2454 mRNAs that are not immediately degraded after rifampicin addition, 51% of them have a delay before degradation larger than the time needed for transcription. The delay is even twice larger for 21% of them (Fig.G.2). While this quick analysis clearly underestimates the number of mRNAs for which the transcription elongation takes a shorter time than the delay before degradation, it indicates that residual transcription is not the sole determinant for the delay in the data set studied. Competition between mRNAs could well be another one.

  Fig. E.1(b)). Hence, the more stable mRNAs are stabilized by competition. On the contrary, the most affine mRNAs are saturated by RNase E, in both isolated and competitive systems. They are among the less sensitive to the competition and can bind to the low enzyme levels. The ratio j =i m j (t) Km i in the competition term is among the smallest throughout the population of mRNAs, because the competing mRNAs have higher affinities. These mRNAs will have the smallest apparent Km values among the population of mRNAs (Fig. E.1(b)

  Figure D.1: Numerical simulation of mRNA degradation kinetics in isolated and competitive systems using Eqs. 5 and 9. Predicted profiles for the competitive (a) and isolated (b) systems. The profiles are normalized to their respective initial concentrations for the competitive (c) and isolated (d) systems. Predicted free RNase E concentrations for (e) the competitive system and (f) the isolated one. 4312 curves are displayed in this case, due to the lack of coupling between mRNAs. The colour bars on the right side represent the normalized gradient of Km values, on a scale from zero (the minimal Km value) to one (maximal value). Note that the total concentration of RNase E, but not the free concentration, is a variable in Eqs. 5 and 9. We thus calculated a posteriori the concentration of free RNase E from the total RNase E and complex concentrations. Since the model at time zero is not at steady state, we could not determine a steady-state value for the initial complex concentration, which would have given a more realistic value for the initial concentration of free RNase E. Instead, we used the same initial condition for the complex concentration as in the work of[START_REF] Pedersen | The total quasi-steady-state approximation for fully competitive enzyme reactions[END_REF], with ci(0) = 0. This effect is transitory only, since the free RNase E concentration predicted with Eq. 9 becomes quickly similar to that obtained with the approximated tQSSA model in Fig.4(a).

Figure

  Figure D.2: Semi-logarithmic representation of simulation results shown in Figs. 4(a,b) (Panels (a) and (b)) and D.1(a,b) (Panels (c) and (d)). The predicted profiles for the competitive (a) and isolated (b) systems were obtained with the approximated tQSSA models (Eqs. 9 and 5, respectively). The predicted profiles obtained with the original models (Eqs. 10 and 6) are shown in Panels (c) and (d) respectively.

Figure F. 1 :

 1 Figure F.1: Scaled rate sensitivity coefficients for the initial velocity of mRNA i degradation. (a) Schematic drawing of the coupling between the degradation of mRNA i and the other cell mRNAs j. We determined, for each of the 4312 E. coli mRNAs i, the effect of a 10% change in (b) the initial concentration and (c) the Km value of mRNA j on the initial degradation rate of each of the 4312 mRNAs i. Colour bars indicate the value of the response coefficients.

Figure G. 1 :

 1 Figure G.1: The distance from the 5' end of the operon nuo scales proportionally to the duration of the delay before start of RNA degradation. (a) Organization of the nuo operon. Numbers indicate genome position in base. (b) RNA concentration over time of exponential phase MG1655 cells growing at 0.4 h -1 on a semilogarithmic scale. Vertical lines indicate the duration of the delay. (c) Delay before degradation increases with the distance of the gene transcript to the 5' end of the operon.

  the 2454 out of 3868 mRNAs with a delay before the onset of degradation (the remaining 1414 are immediately degraded). As shown in Fig. G.2, 1255 out of 2454 mRNAs have a delay before degradation larger than the maximal theoretical time needed for transcription (points in blue). 512 of them have a delay before degradation twice as large as the transcription time. Overall these results show

Table 1 :

 1 Scaled rate sensitivity coefficients of the initial velocity v i with respect to changes in initial mRNA concentrations and Km values.

Table B .

 B 1: Catalytic efficiency of RNase E. Kinetic parameters k cat and Km were estimated from the same assays.

	References: (1) -Jiang and Belasco (2004); (2) -Kim et al. (2004)
	Ref. Experimental	Substrates	k cat (s -1 )	Km (M)	k cat /Km
		conditions					(M -1 .s -1 )
	(1)	pH 7.5, 25°C, recom-	5' hydroxylated fluoro-	0.015	3.3 •10 -7	45455
		binant N-terminal do-	genic oligonucleotide
		main
	(1)	pH 7.5, 25°C, recom-	5'	monophospho-	0.014	2.3 •10 -7	60870
		binant N-terminal do-	rylated	fluorogenic
		main	oligonucleotide
	(2)	pH 7.5, 30°C	AAUUU-containing	0.192	7.3 •10 -7	263,014
			RNA oligonucleotide
	(2)	pH 7.5, 30°C	CAUUU-containing	0.150	4.6 •10 -7	326,087
			RNA oligonucleotide
	(2)	pH 7.5, 30°C	GAUUU-containing	0.765	1.2 •10 -6	616,935
			RNA oligonucleotide
	(2)	pH 7.5, 30°C	GUUUU-containing	0.121	3.7 •10 -7	327,027
			RNA oligonucleotide
	(2)	pH 7.5, 30°C	UAUUU-containing	0.209	8.1 •10 -7	258,025
			RNA oligonucleotide
	(2)	pH 7.5, 30°C	UUUUU-containing	0.023	3.2 •10 -6	71,875
			RNA oligonucleotide
			Mean		0.186 (11.2 min -1 ) 5.6•10 -7	246,161
			Median	0.136 (8.1 min -1 )	4.1•10 -7	260,519
			Standard deviation	0.247	3.4 •10 -7	191,168

Acknowledgements

This work was supported by the Agence Nationale de la Recherche under project RibEco (ANR-18-CE43-0010), as well as by the Institut National de la Recherche Agronomique and Inria, from which T.A.E received a doctoral fellowship. The authors thank Hidde de Jong and Marco Mauri (Ibis, Inria) for discussions, and Laurence Girbal (TBI, CNRS) for discussions as well as comments on a previous version of the manuscript. The authors also thank an anonymous reviewer for helpful comments.

Appendix A. Example of observed degradation kinetics

We illustrate the type of degradation profiles obtained experimentally with data obtained for mRNAs in E. coli cells growing at 0.4 h -1 in continuous cultures.

We display in Fig. A.1 21 degradation profiles out of the 4254 profiles obtained in [START_REF] Esquerré | Dual role of transcription and transcript stability in the regulation of gene expression in Escherichia coli cells cultured on glucose at different growth rates[END_REF]. [START_REF] Esquerré | Dual role of transcription and transcript stability in the regulation of gene expression in Escherichia coli cells cultured on glucose at different growth rates[END_REF]. After delays of various duration, most mRNAs have a degradation kinetics resembling an exponential decay (blue curves), while other mRNAs can be slowly decaying, like the murein lipoprotein mRNA lpp (black curve).

Appendix C. Development of the tQSSA model with competition

We first describe below the development of the mass-action law model in Equation 9 and its subsequent approximations into tQSSA and first-order tQSSA models.

With i = 1 • • • , n, we can write the following mass-action law system:

Using the conservation laws

we can reduce the previous system:

We apply the total quasi-steady-state approximation

and obtain:

with Km i the Michaelis constant:

We apply the quasi-steady-state assumption : dc i (t) dt ∼ 0 and obtain:

and

Two alternative methods have been proposed in the literature to solve the non-linear system (C.6) for c i . Tang andRiley (2013, 2017) use perturbation theory applied to sQSSA to derive an approximated solution for the tQSSA system. [START_REF] Pedersen | The total quasi-steady-state approximation for fully competitive enzyme reactions[END_REF] develop their approximation for a competing system with two substrates by means of the total two Padé approximation. Applied to our system, these approaches allow obtaining the following formulation for the approximated concentration of complex c i :

Appendix F. Heatmaps of rate sensitivity coefficients

In this section, we assess the effect of changing the initial concentration or Km value of a competing mRNA j on the initial velocity of the degradation of a given mRNA i. To that aim, we determine rate sensitivity coefficients for the effect of each of the 4312 mRNAs j on the degradation kinetics of each of the 4312 mRNAs i (they include the mRNA j itself). We obtain a square matrix of 4312×4312 coefficients, represented as heatmap in Fig. (c)). However, this is not a general rule. For instance, we noticed that many monocistronic mRNAs of low to medium size have also a delay.

We then assessed the contribution of residual transcription to the total duration of the delay before degradation. In principle, this would require to know the On log scales and for each mRNA with a non negligible delay, the duration of transcription is plotted versus the delay before degradation. mRNAs with a delay higher than the time needed for transcription are displayed in blue and red, otherwise.