Limitations of the two-stage model

We based our approach on one of the simplest classic stochastic models of protein production, the two-stage model [7–9, 14]. This model describes the production and degradation of mRNAs and proteins of one particular type. In contrast to a three-stage model [9], it considers a constitutive gene, as it does not integrate a regulatory stage at the transcription initiation level, and thus represents the expression of a constitutive gene. The additional noise observed in Taniguchi et al. uniformly impacts highly expressed proteins, and therefore seems to be the result of cell-scale phenomena. Even if gene-specific mechanisms such as gene regulation can have a large impact on protein variance (see [15, 16] for instance), we rather are looking at mechanisms at the scale of the cell to explain the feature observed in Taniguchi et al.

The two-stage model represents the evolution of the random variables M and P, respectively representing the number of mRNAs and proteins associated with this gene inside a single cell (see Fig 1). It is important to note that the quantities considered in this model are integers. In Fig 1, M and P respectively denote the number of mRNAs and proteins inside the cell, but their concentrations are not explicit in the model, it does not incorporate the notion of cell volume.

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Fig 1. Classic two-stage model of one constitutive gene.

M and P represent the respective numbers of mRNAs and proteins in a given cell. The parameter λ₁ is the molecular-specific rate at which mRNAs are produced, λ₂M is the rate at which proteins are created, σ₁ indicates the molecular-specific mRNA degradation rate and σ₂ is the rate at which protein leaves the cell (by the effect of volume growth and division).

https://doi.org/10.1371/journal.pone.0226016.g001

Both the transcription and translation mechanisms are represented in the model, as well as mRNA degradation. There is also a protein decay mechanism that can either represent the degradation of the protein or the dilution effect induced by the growth of cell volume, since the proteolysis often occurs on a timescale much longer than the cell cycle (see [17]). All the events, like the production and degradation of mRNAs and proteins, are assumed to occur at exponentially distributed random times, the rates of which depend on the current state of the system.

The simplicity of the two-stage model is such that the identification of closed-form formulas for the mean and variance of mRNAs and proteins is possible (see [9]); moreover as it only represents mechanisms that are specific to protein production, the predicted variance can be naturally regarded as representative of the intrinsic noise, as has been done by Taniguchi et al. But the model lacks several features that are necessary for making quantitative comparisons with experimental results, and neglects several mechanisms that might have an impact on protein variance.

Impact of random partitioning

The first intermediate model focuses on the effect of random partitioning. Fluctuations in the availability of RNA-polymerases and ribosomes, as well as the gene replication is therefore not considered. As the global sharing of ribosomes and RNA-polymerases are not represented, it results in a model where there are no interactions between the productions of different proteins; each production can be considered separately. It is a gene-centered model and one can focus on the production of one particular type of protein; M and P being respectively the number of mRNAs and proteins of this type.

Fig 2 depicts this intermediate model. The parameter λ₁ of transcription (resp. λ₂ for the translation) implicitly includes some aspects specific to the gene (the promoter–polymerase affinity, for instance) and the effective constant concentration of free RNA-polymerases (resp. free ribosomes). As the gene is in constant concentration, the rate of mRNA creation increases alongside the volume; it is therefore equal to λ₁V(t), with V(t) the volume of the cell at time t. Hence, the rate of mRNA production per volume unit remains constant.

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Fig 2. Intermediate model with volume growth and partition at division.

(A): The model considers transcription, translation, volume growth, and partitioning at division. (B): Model of production for one type of protein. The parameters λ₁, σ₁ and λ₂ are specific to the type of protein. The gene concentration is considered as constant, giving a rate of transcription proportional to cell volume. (C): The evolution of the normalized protein concentration during the cell cycle for three proteins. The central thick line represents the average production during the cell cycle and the colored area the standard deviation. (D) Main: For each type of protein, the variance in the case of random partition divided by the variance in the case of exact partition as a function of the Fano Factor (see main text and Section 1.5 of S1 Appendix). The proportion of variance attributed to the effect of the random partitioning globally follows the prediction of the simplified model in blue dashed line (see Section 1.5 of S1 Appendix). Section Inset: Comparison between the two partitions for the protein OmpC. The thick line is the average protein concentration during the cell cycle and the colored areas correspond to the standard deviation in the two cases.

https://doi.org/10.1371/journal.pone.0226016.g002

For this intermediate model, a deterministic volume growth is considered. Based on experimental results (see [29]), we represent the growth of the bacteria volume as exponential. For this model, if t is the time spent since the last division, the volume is given by with V₀ being the typical size of a cell at birth and τ_D the duration of the cell cycle. The explicit description of the volume leads to consider the concentration of mRNA M(t)/V(t) and proteins P(t)/V(t) at any time t of the cell cycle.

Note that in this model and the subsequent ones, we interpret the division mechanism as a “sizer” model: the division occurs when the cell reaches the volume 2V₀ [29–32]. Even if in the present model, the exact dependence of the volume to the cell cycle time can bring us to either interpret it as a “timer” (occurring at time τ_D) or a “sizer” (occurring when reaching volume 2V₀); the “sizer” interpretation will be preferred as it is the one that ensures cell size stability and has been shown to be a good first approximation to explain experimental cell size distributions [32, 33]. The last intermediate model will later explicitly use a “sizer” mechanism.

In this first model, we study the effect of partitioning on protein variance. Two mechanisms of segregating compounds at division are compared, either an exact or a random partitioning.

• for the exact partitioning, the number of proteins and mRNAs at division are equally allocated between the two daughter cells. Clearly, this mechanism does not have an impact on the variance of the mRNA and protein concentrations;
• for the random partitioning at division, each mRNA and protein has an equal chance to be in either one of the two daughter cells (so that with probability 1/2, they are in the next cell of interest). An additional variability, due to this random allocation, should be therefore added in this case.

As depicted in Materials and Methods, we first perform a theoretical analysis of this model in order to predict its mean concentrations of mRNA and protein averaged over the cell cycle: respectively and as they are defined in Eq (6) of the Materials and Methods section. In this model, the mean concentration 〈M(t)/V(t)〉 and 〈P(t)/V(t)〉 during the cell cycle remain constant. We then have that for any time t of the cell cycle: (1) Proofs of these formulas can be found in Sections 1.1.1 and 1.1.2 of S1 Appendix. The mRNA degradation times and the time of cell cycles are directly measured in [6], thus setting the parameters σ₁ and τ_D. With that, these previous formulas make it possible, for every gene considered in [6], to set the parameters λ₁ and λ₂ in order to have an average production that corresponds to those experimentally measured. This gives a series of parameters corresponding to a representative sample of real bacterial genes (more details in Section 1.2 of S1 Appendix). As described in Section 1.3 of S1 Appendix, simulations are performed using an algorithm derived from Gillespie method in order to determine the evolution of protein concentration across the cell cycle and determine its variance averaged over the cell cycle —as it is defined in Eq (7) of Materials and Methods section. We then check that the average protein and mRNA concentrations correspond to those experimentally measured (see S1 Fig).

Fig 2C shows results of exact simulations (using Gillepsie-related algorithm): the figures show the profile of protein concentration during the cell cycle for three representative genes (adk, fabH and yjiE) which are respectively highly, moderately and lowly expressed. As predicted theoretically, the mean concentration 〈P(t)/V(t)〉 does not change across the cell cycle (it is due to the fact that the gene concentration remains constant).

Fig 2D shows the proportion of variance that is added with the introduction of random partitioning. It appears that for all genes, their protein variance indeed increases with random partition, up to be doubled for some genes (like for the protein OmpC whose profile is shown in the inset, where the average production remains constant in both cases, but the random partition increases protein variance at the beginning of the cell cycle.).

The x-axis of Fig 2D is somewhat unusual as it is the protein Fano Factor, defined as . It is used because the proportion of variance added by the random partition shows a remarkable clear dependence to the Fano Factor: proteins with a low Fano Factor are particularly more impacted. Note that rare proteins also tend to have low Fano factor (see S1(B) Fig), the global tendency remains the same with having the average production as an x-axis, even if this dependence is less strong (see S1(C) Fig).

In order to theoretically explain this clear dependence on the Fano factor, another simplified model, that focuses only on the partition effect without considering volume growth, has been analyzed (details in Section 1.5 of S1 Appendix). Its predictions are shown in blue dotted line of the main Fig 2D. It globally predicts the proportion of noise that can be attributed to the random partitioning. It confirms that this effect is only significant for proteins with very low Fano factor.

We also want to decompose the protein concentration variance in the same way as it is done experimentally using the dual reporter technique [1] where are observed correlated and non-correlated variances between the expression of two proteins with identical promoters: the part of the variance that is not correlated between the two protein expressions is interpreted as solely due to the variability of the production mechanism itself; while the correlated variance is due to the common environment in the cell of the two genes (such as the cell volume, the number of available RNA polymerases, etc.) that influences equally the two productions. Hilfinger and Paulson [34] showed that the theoretical counterpart of such decomposition is the environmental state decomposition of the variance (see details in Materials and methods). With the environmental state decomposition, the protein variance can be decomposed into the two terms and that would respectively represent the intrinsic (uncorrelated variance in the dual reporter technique) and extrinsic contribution (correlated variance in the dual reporter technique) to protein variance. A general formula for this decomposition can be found in the Materials and Methods, but in the current model, as the only external environment considered is the cell cycle (represented by the cell volume V), the docomposition of the variance of each protein is the sum of (2) (3) (see Section 1.4 of S1 Appendix for more details).

In this intermediate model, even with random partitioning, such a decomposition shows surprisingly no external contribution for every protein (since the protein concentration P(t)/V(t) remains constant across the cell cycle). It is therefore remarkable that this decomposition only captures a part of what is generally accepted as the extrinsic noise. We verified this decomposition by a simulation that reproduces the dual reporter technique: we considered the expression of two identical promoters, with the same parameters, in the same cell. The covariance of the two protein concentrations measured in the simulation is much smaller than the variance of each protein concentration (see Section 1.4 in S1 Appendix and S1(D) Fig).

Impact of gene replication

Taking back the model previously described with volume growth and random partitioning at division, we introduce the notion of gene replication. As in the case of the slowing growing bacteria of [6], we consider only one DNA replication per cell cycle. The gene is represented for now on as an entity that is replicated at some instant τ_R of the cell cycle, hence doubling the transcription rate (τ_R is determined depending on the position of the gene on the chromosome, see Materials and methods): before a time τ_R after the cell birth, the rate of transcription will be λ₁, after the time τ_R, the rate of transcription is doubled (see Fig 3B).

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Fig 3. Intermediate model with cell cycle and gene replication.

(A): Features of the intermediate model. The model now takes into account replication. (B): The production of one particular type of protein. The number of mRNAs and proteins are respectively M and P; the difference with the previous model is the introduction of replication at a time τ_R after the cell birth where the transcription rate is doubled. (C): The evolution of the normalized protein concentration during the cell cycle for three proteins. The central thick line and the colored areas represent the same quantities as in Fig 2C. (D) Main: for each type of protein, protein variance of the previous model (gene in constant concentration and random partitioning) divided by the one in this model. The replication paradoxically tends to slightly diminish the fluctuations of the concentration. Insight: on the left, the concentration through the cell cycle for the protein Adk (a close-up of the one presented in Fig 3C); on the right, profile of a modified version of Adk with parameters chosen in order to minimize the variance (see main text).

https://doi.org/10.1371/journal.pone.0226016.g003

The mean of mRNAs concentration can be determined at any moment of the cell cycle. At any time t in the cell cycle (with 0 ≤ t < τ_D), it is given by, (4) with 1_{x≥y} being the Heaviside function (1_{x≥y} = 1 if x ≥ y and 0 otherwise). One can refer to Section 2.1.1 of S1 Appendix for the proof.

Similarly, the mean of protein concentration can also be explicitly determined. Before the replication, if t indicates the time after the birth of the cell (i.e. 0 ≤ t < τ_R), we can determine the mean of P(t)/V(t) as a function of the mean of M(0) and P(0). Similarly, after the replication, the mean of P(t + τ_R)/V(t + τ_R) (with t such as 0 ≤ t < τ_D − τ_R) is known as a function of the mean of M(τ_R) and P(τ_R). They are given by the formula, (5) with τ = 0 and η = 1 for the case before the replication and τ = τ_R and η = 2 for the case after the replication. At steady state we are able to determine explicit values for 〈P(0)〉 and 〈P(τ_R〉 as a function of the parameters (see Section 2.1.2 of S1 Appendix).

As for the previous model, Eqs (4) and (5) make it possible, for every gene experimentally measured in [6], to give a set of parameters λ₁, σ₁ and λ₂ that corresponds to it (Section 2.2 of S1 Appendix and S2(A) Fig). To determine the variance of protein concentration, simulations can be performed, but it is noticeable that we also managed to have formulas for mRNA and protein variances. These formulas greatly simplify the analysis of this intermediate model (see Sections 2.1.1 and 2.1.2 of S1 Appendix).

Fig 3C presents the normalized profiles for three different proteins with different replication times in the cell cycle: their normalized average concentration (thick line) and their normalized standard deviation (colored area) during the cell cycle are shown. Globally, these profiles present little changes compared to the ones of the previous model (Fig 2C). In the left figure of the inset of Fig 3D is shown the profile of the protein Adk (a closeup of the first profile of Fig 3C). It appears that the mean concentration at any given time t of the cell cycle 〈P(t)/V(t)〉 (the thick line of the profile) is not constant during the cell cycle, as it was the case in the model of Fig 2. The curve of 〈P(t)/V(t)〉 fluctuates around 2% of the global average protein concentration .

The main Fig 3D shows the effect of replication on the variance: it represents the ratio of protein variance between the previous model (with genes in constant concentration and random partitioning) and the current model with gene replication. For all the genes, the variances predicted show little difference from the previous intermediate model. The ratio is even surprisingly slightly above one for many genes, indicating that for these genes the replication tends to reduce the variance.

As for the previous intermediate model, we can use the environmental state decomposition to separate the part of variance (defined in Eq (2)) specific to the gene expression and (defined in Eq (3)) attributed to cell cycle fluctuations. As the mean concentration, 〈P(t)/V(t)〉 is no longer constant during the cell cycle is no longer null. Yet it appears that only represents a very small part of the global variance (for 99% of the genes, it represents less than 2%). For this intermediate model, the extrinsic contribution of DNA replication computed with this decomposition is small.

Using the analytical formula of protein variance (Section 2.1.2 of S1 Appendix) and by performing variations on certain parameters, we can analyze the effect of several aspects on protein variance in this model. By considering a given protein (protein Adk), we subsequently modified different parameters while making sure that the average protein concentration remain unchanged, to see how each of these changes impact the protein variance:

• Changing the position of the gene on the chromosome (specifically changing τ_R and slightly adapting the gene activation rate λ₁ to keep the same average mRNA concentration)
• Changing the mRNA lifetime (by increasing the gene activity λ₁ and increasing the mRNA degradation rate σ₁ so that it keeps the same average mRNA concentration)
• Changing the mRNA number (by increasing the gene activity λ₁ while decreasing the mRNA activity λ₂ so that it keeps the same average protein concentration)

Results are shown in S2 Fig(C) of S1 Appendix. Changing the gene position have almost no impact on global protein variance . The effect of mRNA lifetime is more noticeable as a shorter mRNA lifetime can diminish protein variance at most about 40%. The mRNA number seems to have the most important effect on protein production: for the same average protein concentration, having more mRNAs greatly diminish protein variance; such effect has been experimentally observed [2, 35]. This can be interpreted as lower bursting effect in protein production: as it is known that mRNAs in few copies with large activity display a protein production with large bursts, conversely a large number of mRNAs less active leads to a more stable protein production.

The right insight of S2 Fig(C) of S1 Appendix shows an example of such a protein with reduced variance: this protein is based on Adk, the protein average production is the same but there are ten times more mRNAs, with a ten times shorter lifetime. The variance is indeed reduced but with the cost of the production of additional mRNAs. Yet, even in this case, the protein expression is not strongly cycle-dependent (see inset of S2 Fig(D) of S1 Appendix); in particular, this profile is not precise enough to be used as a “trigger” for periodic cell events (such as DNA replication initiation, or partition at division): the evolution of the protein concentration across the cell cycle is not precise enough to robustly distinguish different phases of the cell cycle.

Impact of the sharing of RNA-polymerases and ribosomes

We now consider the complete model, which includes a limited amount of RNA-polymerases and ribosomes: the model is explained in detail in the Materials and Methods. Now, RNA-polymerases and ribosomes are explicitly represented in the model and each of these macromolecules is considered either allocated (i.e. sequestered on a gene if it is an RNA-polymerase, or on an mRNA if it is a ribosome), or free (i.e. either moving freely in the cytoplasm or, in the case of RNA-polymerases, potentially non-specifically sliding on the DNA).

The previous intermediate models were “gene-centered”: each class of proteins was considered independently from each other. The common sharing of RNA-polymerases and ribosomes is an additional key feature that leads us to investigate a multi-protein model where all the genes of the bacteria are considered altogether. For each type of protein i, we denote by G_i(t), M_i(t) and P_i(t) respectively the number of gene copies, of messengers and of proteins at time t in the cell cycle. For each gene i, E_Y,i(t) is the number of RNA-polymerases sequestered on the i-th gene for transcription and E_R,i(t) is the number of ribosomes sequestered on an mRNA of type i for translation. The non-allocated RNA-polymerases and ribosomes are respectively denoted by F_Y(t) and F_R(t). In a first step, we have considered that the gene pool of [6] (841 genes with their mRNA and protein expression measured) would represent the whole genome. We will later see that the addition of new genes does not change significantly our results.

New ribosomes and RNA-polymerases are added to the system as cell volume increases: in a first step, these macromolecules are regularly added such as their total concentration in the cell remains constant during the cell cycle; we will later consider a more realistic way to represent RNA-polymerase and ribosome production.

The previous intermediate models represented the production of a specific type of protein immersed into a “background environment” where the cell grows and divides, the current model includes simultaneously all the genes altogether. In this model, as we are on the scale of the whole cell, we would like to model the impact of global protein production on the cell growth. We therefore can no longer consider that the production of each type of protein has no effect on the global performance of the cell. The volume V(t) depends now on the global production of proteins, and it is not an independent and deterministic feature anymore. As the density of cell components tends to be constant in real-life experiment [36] and proteins represent more than half of the dry mass of the cell [37], the model considers the volume as proportional to the current total mass of proteins in the cell. The mass of each protein is given by the length of its peptide chain (see Section 3.2.1 of S1 Appendix for an exhaustive description of the model). Like in the previous model, division occurs when the cell reaches the volume 2V₀ (making it a “sizer” model).

The processes of mRNA and protein productions are both separated in two parts: the binding and initiation on one side, and the elongation and termination on the other side. The rate at which an RNA-polymerase is sequestered on a gene of type i at time t depends on the copy number of the i-th gene G_i(t), the free RNA-polymerase concentration F_Y(t)/V(t) and a parameter λ_1,i specific to the gene that takes into account the RNA-polymerase–promoter affinity. The elongation rate of each mRNA only depends on the average transcription speed and the length of the gene. The mechanism for translation is similar.

As this model is more complex than the previous ones, the complete analytical description of mRNA and protein dynamics seems to be out of reach. To address this problem, we try to predict the average behavior of this model using a system of ordinary differential equations (ODEs). We used the predictions of these equations to fix the parameters. An a posteriori validation has been made to check that this system of ODEs well predicts the average behavior of the stochastic model (see Section 3.2.1 of S1 Appendix for more details on the procedure).

In Fig 4E is shown an example of the cell volume evolution in a simulation. It appears that the volume seems to grow exponentially during the cell cycle. The number of available ribosomes and RNA-polymerases also changes rapidly, of the order of the second for the RNA-polymerases, and of the order of one tenth of a second for the ribosomes (see S3 Fig(C) and S3 Fig(D) of S1 Appendix).

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Fig 4. Complete model.

(A): The model now considers the sharing of RNA-polymerases and ribosomes between the different productions. (B): The model of production of all proteins. The i-th gene is associated with one particular type of mRNA (whose number is M_i) and proteins (whose number is P_i). The number of free RNA-polymerases (resp. ribosomes) is F_Y (resp. F_R), the number of those sequestered on the i-th gene is E_Y,i (resp. E_R,i). (C): The evolution of the normalized protein concentration during the cell cycle for three proteins. The central thick line and the colored areas represent the same quantities as in Fig 2C. (D) Main: variance in the previous model with replication divided by the variance in this model with sharing of RNA-polymerases and ribosomes. Insight: the concentration through the cell cycle for the protein FabH (a close-up of the one presented in Fig 4C). (E) Main: A simulation sample that shows that cell volume grows exponentially from around V₀ up to around 2V₀. Insight: The growth rates of the simulation are centered around the expected growth rate log 2/τ_D.

https://doi.org/10.1371/journal.pone.0226016.g004

Fig 4D compares the results of the simulations with the previous intermediate model with gene replication and random partitioning. It shows that, for 90% of the genes, the interactions between protein productions only represent at most 10% of variability. In inset is shown the example of the protein FabH profile during the cell cycle, showing that sharing of RNA-polymerases and ribosomes introduces little change.

We also analyze the model using the environmental state decomposition. Two genes of the dual reporter technique would undergo the same volume growth with the same evolution of free RNA-polymerases and ribosomes; as a consequence, the common cellular environment on which the decomposition is operated now includes the concentration of free RNA-polymerases and ribosomes (in addition to the cell cycle). Details can be found in Section 3.7 of S1 Appendix. It appears that for all the genes, the extrinsic contribution of the variance represents only a very small portion of (for 99% of the cells, the ratio represents less than 1%).

We compared these simulation results with a simplified theoretical model: Section 3.6 of S1 Appendix presents a multi-protein model, that is inspired by the one described in [26]. Even if it is a multi-protein model as it represents the expression of a large number of genes altogether, it is a simpler model than the one presented here as it considers separately transcription and translation, and it does not consider neither volume growth, partitioning at division, nor DNA replication. We show that the predicted distributions of free RNA-polymerases (and ribosomes in the adapted model) fits well the one observed in our simulations (see S4 Fig). As we will see in the discussion, this good correspondence between the models would suggest that the mean-field mathematical properties proven for the simplified model could be applied to our complete model.

Model and parameter sensitivity

The complete model supposes a series of modeling and parameter choices that might legitimately influence protein production. We have analyzed several of these aspects and have shown that they do not appear to significantly change the results previously presented.

Additional genes.

As previously said, to perform our first simulations, only 841 genes from which the average mRNA and protein concentration have been measured in Tanuguchi et al. are considered. To have a pool of proteins that might represent a global gene expression in E. coli, we studied the case of a simulation with a set of genes that represent about 2000 genes, more in accordance to the expected number of genes expressed in a growing condition. To propose realistic parameters for these fictional genes, we sample them according to different empirical distributions observed in the empirical data, and also by taking into account the possible correlations observed (the correlation that exists between the average mRNA and protein concentration for instance). See Fig 5A and Section 3.6.1 of S1 Appendix. No changes in protein concentration variance is observed.

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Fig 5. Model and parameter sensitivity.

(A): Creation of additional genes by sampling of mRNA and protein concentrations in accordance to their correlation (see Section 3.6.1 of S1 Appendix for details on the procedure). (B): A variation on the complete model where the RNA-polymerases and ribosomes are produced by the cell. Above: clear correlation between the total number of ribosomes and the estimation of volume growth. Below: comparison between the two versions of the model (with or without production of RNA-polymerases and ribosomes); the distribution of protein number distribution of FabH protein for cells of different volumes in both cases.

https://doi.org/10.1371/journal.pone.0226016.g005

Interpretation of the model predictions

The experimental data of Taniguchi et al. gave us the opportunity to systematically and quantitatively inspect the impact on protein variance of many cell mechanisms that are not often considered in stochastic models of protein production. The broad variety of genes experimentally measured in Taniguchi et al. has been a good opportunity for us to realistically test our models for a wide number of different genes, with different mRNA and protein concentrations, different mRNA lifetimes or gene position on the chromosome or gene length.

From this analysis, it appears that among the different features included in the model, the random partitioning has the most significant effect on the variance of protein concentration, especially for the less expressed proteins. We recover here one of the conclusions made in [27]. The gene replication induces little difference (it even tends, to a small extent, to reduce the variance in some cases); the important fluctuations of free RNA-polymerases and ribosomes have little impact on protein production, which does not fit the hypothesis made in [6]. It is confirmed by the environmental state decomposition, which separates the intrinsic and extrinsic contribution to protein variance (in an analog way as it is done with the dual reporter technique): the extrinsic contributions represent at most a few percents of the total variance.

We interpret the surprising little impact of sharing of RNA-polymerases and ribosomes on the proteins variance by noticing the similarities of our model with the one described in [26]. Indeed, as previously explained, the global behavior of free RNA-polymerases and ribosomes can be predicted by a simplified model derive from [26], where the RNA-polymerases and ribosomes are shared among the different productions. The main result of [26] is a mean-field theorem: as the number of genes increases, the production process of different types of proteins can be seen as independent production processes. The reason is that the dynamic of free RNA-polymerases and ribosomes is much faster than the production of mRNAs and proteins of one particular type. The rate at which an mRNA or protein is elongated only depends on the “local steady state” concentrations of free RNA-polymerases and ribosomes (a similar phenomenon can be found in [39]). Our model seems to display such mechanism: with a global sharing of RNA-polymerases and ribosomes by a large amount of protein productions, the dynamic of free RNA-polymerases and ribosomes is faster than the production of each mRNA and protein of each type. As a consequence, it is not surprising to see that this multi-protein model, which takes into account the production of all proteins displays little difference with the intermediate “gene-centered” model due to a mean-field effect.

Comparison with experimental measures

In the end, we can compare the results of our models with the experiments. Firstly, one can remark that the profile of the mean production 〈P(t)/V(t)〉 (the plain line in Fig 3D is representative of all cells) during the cell cycle corresponds to the one observed experimentally in [22]. Furthermore, the maximum deviation of the average 〈P(t)/V(t)〉 around the global average protein (red dashed line in Fig 3D) is between 2% and 4% for all the proteins of our models, and Walker et al. measure such fluctuations also around 2% of the global average for genes at different positions on the chromosome (see Figure 1.d and Figure S6.b of [22]).

Secondly, we compared the global mRNA and protein fluctuations predicted in our models with those measured in Taniguchi et al. Fig 6A and 6B shows respectively, for every gene, the protein CV of mRNAs and proteins (resp. defined as and ) against their respective average concentration (resp. and ); it is compared in both cases with the same results obtained experimentally (respectively corresponding to measurements shown in Figures 2.D and 2.B of [6]). For the mRNAs (Fig 6A), the noise globally scales inversely the average mRNA concentration. Experimental measurements in [6]—made using the FISH technique for 137 of the highest expressed mRNAs—show a similar tendency (the CVs are normalized on the figure because the uncertainty in the cell volume at birth can introduce a shift, but the shape of the normalized CV of both experimental and simulated CV remains exactly the same regardless of this effect). For the protein CV (Fig 6B), it appears that the noise approximately scales inversely the average protein concentration like in the first “intrinsic noise” regime of [6]. But unlike in the [6] experiment, there is no lower plateau for highly expressed proteins: for the highest produced proteins, the CV should be in the order of 10³ fold higher than the one predicted. It confirms that the features considered here cannot correctly explain the noise observed experimentally.

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Fig 6. Comparison with experimental dataset of Taniguchi et al.

(A): The normalized CV of mRNA concentration (defined as ) for each gene predicted by the complete model (in yellow, the corresponding experimental measurements). (B): The CV of protein concentration for each gene predicted by the complete model (in yellow, the corresponding experimental measurements).

https://doi.org/10.1371/journal.pone.0226016.g006

We can propose different interpretations to explain the discrepancy between the predictions of the models and the experimental measures. For the biological processes not included in our models and that can have an impact on the variability, one can first mention the gene regulation as in our models, all the genes are considered as constitutive. The introduction of a gene regulation can indeed induce a large variability in protein concentration [9, 14]. Nonetheless, the “extrinsic noise plateau” observed in Taniguchi et al. only concerns the proteins and not the mRNAs (compare the yellow areas of Fig 6A and 6B). As a consequence, one can expect that the mechanism explaining the extrinsic noise plateau takes place at the translation level and not at the transcription. Moreover for highly expressed proteins, the protein CV is independent of the protein expression; it is therefore not gene-dependent as it would be the case for gene regulation. Finally, we have considered a simple model with gene regulation (like the three-stage model of [9]), and determined the regulations parameters in order to predict protein variance observed in [6]; we came with an activation/deactivation timescale has to be very high (several times the doubling time) in order to reproduce the variance experimentally observed, which is way above the typical biologically expected parameters.

One can also mention other possible mechanisms not represented in our models such as the fluctuations of availability of amino-acids or free RNA nucleotides in the medium, thus inducing additional fluctuations in the translation speed. Even if one can see here a clear analogy with the fluctuations in RNA-polymerase and ribosome availability (which also impact the transcription and translation speeds), the different timescales of the dynamics of amino-acids or free RNA nucleotides might result in a different effect. One can also challenge the hypothesis time intervals between events are modeled by exponentially distributed random variables: for instance, elongation times would be better represented as having Erlang distribution, that is the sum of independent exponential random variables. Yet, some results incline to say that it has a limited impact [13]. Also, in this model, the binding and initiation (either of RNA-polymerases or ribosomes) are considered as a single event. A more precise representation would be to describe them as two different processes (Reference [40] gives for instance a median transcriptional initiation time of 15t which is of the same order of magnitude as the elongation time).

One can also consider that, as this effect mainly affects proteins with the highest fluorescence, it is possible that some saturation induces a bias in the estimation of variance of highly produced proteins. To our knowledge, exhaustive measures of [6] for mean and variance of protein and mRNA concentrations have not been replicated at the same scale, so we have not been able to confront our results to other measures.

The complete model

In this subsection is presented in detail the complete model of Fig 4 that includes the sharing of RNA-polymerases and Ribosomes (the other models being simplifications of this complete model). It represents, for any gene, both the number of mRNA and protein molecules associated with the gene inside a given cell. But, contrary to the two-stage model (Fig 1) it also explicitly represents the volume V(t) that is changing across the time t due to cell growth, so that, if M(t) and P(t) represent the respective number of mRNAs and proteins of a given gene at any time t, one can now explicitly represent their concentration by Furthermore, contrary to the two-stage model, all genes in the bacteria are represented altogether here (in order to represent the global sharing of RNA-polymerases and ribosomes in the different productions of proteins) and the division (to represent a partitioning of components at septation). When dividing, the model focuses on only one of the two daughter cells in order to follow one lineage of cells.

Analysis of each intermediate models

Each of the intermediate models is systematically analyzed with the same method, see Fig 7A. The average behavior of the model is analytically predicted (either with exact formulas for the first two intermediate models, or approximately for the last complete model); it makes it possible, for each of the 841 genes of Taniguchi et al. (for which both the average protein and mRNA production have been measured), to determine the set of parameters of the model. An overview of the parameters hence determined is shown in Fig 7C. Simulations are then performed—using methods derived from Gillespie algorithm—with these parameters over 10000 cell cycles: it makes it possible to check the accuracy of the average concentration and to predict the variance of the concentration of proteins. For each model, we then have the variance of protein concentration predicted for the wide range of genes measured in [6].

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Fig 7. Experimental data and modeling principles.

(A): The complete model. (B): Scheme of analysis to determine the variance of protein concentration for every gene predicted in each intermediate model. (C): [6] measures of mRNAs and proteins for 841 genes. (D): Box-plots presenting the parameters deduced from experiments to corresponds to the model of Fig 2; for the other models, these rates are in the same order of magnitude.

https://doi.org/10.1371/journal.pone.0226016.g007

Means and variances

Throughout this paper we use the notation 〈X(t)〉 and Var [X(t)] for the mean and variance of a random variable X(t) at a time t of the cell cycle. We introduced the concentrations for mRNAs and proteins. If M(t) and P(t) denote the random variables representing the number of mRNAs and proteins of a given type at time t and V(t) is the volume at this instant, the corresponding concentrations are M(t)/V(t) and P(t)/V(t). One of the goals of this work is to study the properties of the mean, 〈P(t)/V(t)〉, and variance, Var [P(t)/V(t)], of these concentrations. These quantities correspond to the mean and the variance of a population of synchronized cells of volume V(t). The measurements of [6] consider a cell population in exponential growth; by consequence, we also have to define the notions of global mean and variance for a heterogeneous population (see [33, 41, 42] for the population distribution in exponential growth). By denoting by u the age distribution of the cell population, we can define the global average and variance averaged over the population, (6) (7) We observe that the choice of the age distribution u (either uniform or corresponding to an exponentially growing population) does not seem to have much impact; see Section 2.2.3 of S1 Appendix for more details. We therefore simply considered u as uniform in the Results section.

Intrinsic and extrinsic effects

As for the previous studies of [1, 12], we want to decompose protein variance that can be attributed to the intrinsic expression from the one due to extrinsic mechanisms. It appears that this decomposition can be computed using two different ways:

• First, there is the method used in Taniguchi et al. As the intrinsic noise is usually attributed to the protein production mechanism alone, a model that represents only transcription and translation, as the classic two-stage model, are usually considered as predicting the intrinsic noise.
  Yet, in our case, the consideration of a more realistic mechanism for protein disappearance (through segregation at division rather than regular decay) prevent us from using directly the two-stage models as a quantitative representation of the intrinsic variance. Our very first intermediate model, where the number of proteins and mRNAs are exactly halved at division, is considered as our baseline model. This baseline model is very close to the two-stage model in that sense that it contains no other features than those intrinsically linked to protein production. Therefore, we use protein variance predicted by this baseline model as intrinsic protein variance.
  From this baseline model, any additional variance added by the introduction to the model of external mechanisms (random partition at division, gene replication, etc.) would be considered as extrinsic. For every type of protein, we will look how the global variance changes with the subsequent introduction of the external mechanisms.
• Secondly, it appears that the previous method of extrinsic noise estimation is not exactly the same from the first attempt made by [1] using the dual reporter technique. [34] showed that decomposition using the dual reporter technique can be interpreted as an estimator of the environmental state decomposition (also called the law of total variance). It decomposes protein variance between the effects specifically due to the stochastic nature of the instants of birth and death of mRNAs and proteins (intrinsic noise) and the external effect of the biological environment (extrinsic noise). If Z represents the state of the cell, the number of RNA-polymerases, the volume, etc…, then the protein concentration P/V can be decomposed such as, (8) where: where indicates the integration over the variable Z in a population of cells of age t. (Note that in the case of the two first intermediate models, the environment Z is simply the deterministic volume and simplified expressions for and in these cases can be found in Eqs (2) and (3)).
  The variable Z represents the common environment in which the two similar genes of the dual reporter technique evolve; yet mathematically, it is dependent on the model that we consider (for each model, it is what is considered as being part of the “environment” of the gene). We have therefore explicitly described for each intermediate models what it represents in this context (the second intermediate model with gene replication shows an illustrative example of this decomposition). Once the variable Z fixed, the decomposition is explicit and separates the total variance in two parts: that corresponds to the intrinsic contributions and corresponds to the variance induced by external contributions represented by the environment Z (volume fluctuation, concentrations of free RNA-polymerases and ribosomes, etc.) The term is indeed the variance that can be expected from a model without any external fluctuation. A model like the two-stage model does not consider any change in the environment of protein production, so the term of the decomposition would be null.

In the intermediate models, we quantify the external contributions by these two means, either by looking at the increase of the global variance , or by performing the environmental state decomposition and looking the portion of external variance predicted. We have seen that these two ways to quantify the external contributions of protein variance are not strictly equivalent.