Stochastic Modeling of Gene Regulation by Noncoding Small RNAs in the Strong Interaction Limit

Abstract

Noncoding small RNAs (sRNAs) are known to play a key role in regulating diverse cellular processes, and their dysregulation is linked to various diseases such as cancer. Such diseases are also marked by phenotypic heterogeneity, which is often driven by the intrinsic stochasticity of gene expression. Correspondingly, there is significant interest in developing quantitative models focusing on the interplay between stochastic gene expression and regulation by sRNAs. We consider the canonical model of regulation of stochastic gene expression by sRNAs, wherein interaction between constitutively expressed sRNAs and mRNAs leads to stoichiometric mutual degradation. The exact solution of this model is analytically intractable given the nonlinear interaction term between sRNAs and mRNAs, and theoretical approaches typically invoke the mean-field approximation. However, mean-field results are inaccurate in the limit of strong interactions and low abundances; thus, alternative theoretical approaches are needed. In this work, we obtain analytical results for the canonical model of regulation of stochastic gene expression by sRNAs in the strong interaction limit. We derive analytical results for the steady-state generating function of the joint distribution of mRNAs and sRNAs in the limit of strong interactions and use the results derived to obtain analytical expressions characterizing the corresponding protein steady-state distribution. The results obtained can serve as building blocks for the analysis of genetic circuits involving sRNAs and provide new insights into the role of sRNAs in regulating stochastic gene expression in the limit of strong interactions.

Introduction

Intrinsic randomness in the process of gene expression can play an important role in stochastic cell fate decisions (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12). Cellular control of such intrinsic randomness (noise) depends on the regulation of gene expression, which occurs at various stages. Although considerable research has focused on regulation of gene expression by transcription factors, there has been growing interest in understanding gene regulation at the post-transcriptional level by noncoding small RNAs (sRNAs). Noncoding sRNAs are known to play key roles in regulating diverse cellular processes (13, 14, 15, 16, 17, 18) such as stress response (19), virulence gene expression (20, 21, 22, 23), and canalization during development (24, 25), to name a few. Furthermore, dysregulation of and by sRNAs is linked to various diseases, including cancer (26, 27, 28). Correspondingly, there is significant interest in quantitatively modeling the effect of post-transcriptional regulation on noise in gene expression, i.e., in analyzing stochastic models of regulation by sRNAs.

A central feature of post-transcriptional regulation by sRNAs is that the interaction between sRNAs and mRNAs can lead to stoichiometric mutual degradation. This is the key element of a widely studied canonical stochastic model of stoichiometric regulation by sRNAs (29). Interestingly, despite its simplicity, the nonlinearity of the interaction terms between the sRNAs and mRNAs makes the model analytically intractable. Specifically, it has been challenging to obtain exact analytical expressions for even basic quantities of interest such as the mean mRNA/protein levels, let alone the corresponding steady-state distributions.

Previous approaches have been largely based on the mean-field approximation (29) and have predicted interesting results such as a threshold behavior in target gene expression (30, 31), which has been observed experimentally. The mean-field approximation is clearly valid in the absence of any interaction between sRNAs and the target mRNAs. By extension, it is expected that the mean-field results provide a reasonable approximation in the weak interaction limit. However, it is reasonable to assume that effective post-transcriptional regulation requires strong interactions between mRNAs and sRNAs as indicated in previous work (31, 32), with the sRNA-induced mutual degradation rate being in the range of 10–100-fold higher than the natural mRNA degradation rate. Furthermore, there are multiple examples (33, 34, 35) of synthetic genetic circuits involving sRNA-based regulation wherein the strength of mRNA-sRNA interactions can be tuned (35). In these applications, naturally occurring sRNAs can be used to rationally design synthetic RNAs with an mRNA-sRNA interaction strength that correlates with mRNA-sRNA binding energy, which in turn can be tuned by changing the binding sequence. Thus, in synthetic biology applications, the strong interaction limit can be realized by tuning the mRNA-sRNA interaction strength. However, in this strong interaction limit, the evolution of sRNAs and mRNAs is strongly correlated, and mean-field approaches (which neglect correlation) will not give accurate results. The aim of this work is to carry out the analysis in this important regime of strong interactions between sRNAs and mRNAs. In this limit, we derive analytic results characterizing the joint probability distributions of mRNAs and sRNAs. Furthermore, the analytical results derived at the level of mRNAs are then applied to study the statistics of the corresponding protein steady-state distribution.

The work is organized as follows. The canonical model for post-transcriptional gene regulation is presented in Model along with a description of the mean-field analysis. In Beyond Mean Field: Analytical Results for the Joint Probability Distribution, we derive an exact expression for the joint probability distribution of mRNAs and sRNAs in the limit of infinitely strong interactions. These results are then extended to derive analytical results for the joint probability distribution, which are valid in the strong interaction regime. The results derived at mRNA level are then used to study the steady-state statistics of protein copy numbers in Protein Statistics, and conclusions are presented in the Conclusions.

Materials and Methods

Model

To gain a quantitative understanding of post-transcriptional control of stochastic gene expression by sRNAs, we consider stoichiometric regulation in the canonical model (29) shown in Fig. 1. In this model, both mRNAs and sRNAs are expressed constitutively with rates k_m and k_s, respectively. Furthermore, mRNAs (sRNAs) can undergo individual degradation with rates μ_m (μ_s). Apart from individual degradation, the interaction between sRNAs and mRNAs leads to stoichiometric mutual degradation, wherein both the molecular species are jointly degraded. The interaction between a single mRNA and a single sRNA leading to joint degradation occurs with rate γ. We note that the mode of regulation considered in the model is noncatalytic, whereas many sRNAs are also known to act catalytically. We focus initially on the distributions at the mRNA/sRNA level; the corresponding protein steady-state distributions will be analyzed in subsequent sections.

Figure 1.

Schematic representation of the canonical gene-expression model with post-transcriptional regulation by noncoding small RNAs (sRNAs). The production of mRNAs (sRNAs) occurs with rate k_m (k_s), and they degrade naturally with rate μ_m (μ_s). Stoichiometric mutual degradation due to mRNA-sRNA interactions occurs with rate γ. To see this figure in color, go online.

At any time t, the state of the system is characterized by the number of mRNAs (m) and number of sRNAs (s). Let P(m,s,t) denote the joint probability distribution of having m mRNAs and s sRNAs at time t. Then, the evolution of this joint probability is given by the following master equation:

$$\frac{\partial P\left(m,s,t\right)}{\partial t}=k_{m}P\left(m-1,s,t\right)+{\mu }_m\left(m+1\right)P\left(m+1,s,t\right)+k_{s}P\left(m,s-1,t\right)+{\mu }_s\left(s+1\right)P\left(m,s+1,t\right)+\gamma \left(s+1\right)\left(m+1\right)P\left(m+1,s+1,t\right)-\left(k_m+{\mu }_{m}m+k_s+{\mu }_{s}s+\gamma ms\right)P\left(m,s,t\right).$$ (1)

The presence of the nonlinear term corresponding to interactions between mRNAs and sRNAs makes the model analytically intractable, and it has not been possible, so far, to obtain exact expressions for P(m,s,t) in the steady-state limit. In fact, even obtaining exact expressions for mean mRNA and sRNA levels remains a challenge. To illustrate the difficulties in doing so, consider the following: multiplying Eq. 1 by m(s) and summing over all possible values of m and s, we obtain the following evolution equation for the mean number of mRNAs $\left(\langle m\rangle \right)$ and mean number of sRNAs $\left(\langle s\rangle \right)$,

$$\begin{array}{l}\frac{\partial \langle m\rangle }{\partial t}=k_m-{\mu }_m\langle m\rangle -\gamma \langle ms\rangle ,\\ \frac{\partial \langle s\rangle }{\partial t}=k_s-{\mu }_s\langle s\rangle -\gamma \langle ms\rangle .\end{array}$$ (2)

As indicated in the above, the evolution of their mean values, $\langle m\rangle$ and $\langle s\rangle$, is coupled to the higher-order moment $\langle ms\rangle$, which in turn is coupled to higher moments, eventually forming an infinite hierarchy of equations. By applying suitable moment closure techniques (36), approximate expressions for moments can be obtained. One commonly used approximation is to replace $\langle ms\rangle$ by $\langle m\rangle \langle s\rangle$, the so-called mean-field approximation, which neglects correlations between the molecular species. This correlation is usually quantified by the correlation function

$$C_{ms}=\frac{\langle ms\rangle -\langle m\rangle \langle s\rangle }{\langle m\rangle \langle s\rangle }.$$

The correlation function can have values between −1 and 1; positive values correspond to positive correlations, whereas negative values correspond to negative correlations.

The mean-field approximation can be used to find a closed-form expression for the mean levels of mRNAs and sRNAs. For example, using Eq. 2, mean mRNA copy numbers in the steady state are given by the following (29):

$$\langle m\rangle =\frac{1}{2\gamma {\mu }_m}\left[\gamma \left(k_m-k_s\right)-{\mu }_m{\mu }_s+\sqrt{{\left(\gamma \left(k_m-k_s\right)-{\mu }_m{\mu }_s\right)}^2+4\gamma k_m{\mu }_m{\mu }_s}\right].$$ (3)

However, in the limit of strong interactions, we expect the actual evolution of the system to show significant deviations from the corresponding mean-field results. To analyze this, we carried out stochastic simulations using the Gillespie algorithm (37). The initial state was chosen such that the abundances of all molecular species are set to zero (i.e., m = s = p = 0). For all simulations, we set the unit of time such that μ_m = 1, and system evolution was carried out up to 5000 time units. The results obtained are based on averaging over 50,000 realizations of the process. As shown by the results of our simulations for the model in (Fig. 2), the correlation function C_ms varies gradually from 0 to −1, i.e., from no correlation to perfect anticorrelation as we vary the interaction parameter γ. To understand this variation, we note that, in one of the extreme limits, $\gamma \to 0$, the evolution of mRNAs and sRNAs is independent, and thus we have $\langle ms\rangle =\langle m\rangle \langle s\rangle$, which leads to C_ms = 0. In the other extreme limit, $\gamma \to \infty$, the mutual degradation rate dominates all other reaction rates, and mRNA-sRNA degradation can be assumed to occur instantaneously. In this limit, mutual coexistence of mRNA and sRNA molecules cannot occur, which leads to $\langle ms\rangle \to 0$ and consequently $C_{ms}\to -1$. The results clearly demonstrate the breakdown of the mean-field approximation in the strong interaction limit. Furthermore, apart from obtaining the mean values of m and s, there is growing interest in studying higher-order statistics such as the variance of the corresponding distributions. In previous work (38), a variational approach was developed that provides accurate results for the moments; however, closed-form analytical results are currently lacking. In the following, we address this issue in the limit of strong sRNA-mRNA interactions.

Figure 2.

Deviations from results obtained using the mean-field approximation. Simulation results for the steady-state correlation function C_ms between mRNAs and sRNAs are plotted as a function of the interaction strength γ for k_m = 1, μ_m = 1, k_s = 0.5, μ_s = 0.1.

Results and Discussion

Beyond mean field: Analytical results for the joint probability distribution

The standard approach to obtain the joint probability distributions is to introduce the corresponding probability generating function, $G\left(z_1,z_2,t\right)$, given by

$$G\left(z_1,z_2,t\right)=\sum _{m=0}^{\infty }\sum _{s=0}^{\infty }{z}_1^m{z}_2^{s}P\left(m,s,t\right)\text{.}$$ (4)

Using this in Eq. 1 reduces the master equation, which corresponds to an infinite number of coupled differential equations for different values of m and s, to a single partial differential equation for the joint probability generating function. The resulting equation, in the steady-state limit, is given by

$$\left[k_m\left(z_1-1\right)+k_s\left(z_2-1\right)\right]G+{\mu }_m\left(1-z_1\right)\frac{\partial G}{\partial z_1}+{\mu }_s\left(1-z_2\right)\frac{\partial G}{\partial z_2}+\gamma \left(1-z_1{z}_2\right)\frac{{\partial }^{2}G}{\partial z_1\partial z_2}=0.$$ (5)

Solving Eq. 5 for an arbitrary value of interaction strength γ is challenging and is currently an open problem in the field. However, in the simple case of $\gamma =0$ the evolution of mRNAs and sRNAs is independent, leading to the solution (see section A, Supporting Materials and Methods)

$$G_0\left(z_1,z_2\right)=\text{exp}\left(\frac{k_m}{{\mu }_m}\left(z_1-1\right)\right)\text{exp}\left(\frac{k_s}{{\mu }_s}\left(z_2-1\right)\right)\text{,}$$ (6)

which is simply the product of generating functions corresponding to two Poisson distributions. It has been proposed, based on some assumptions and approximations, that the joint distribution of mRNAs and sRNAs can always be expressed as a product form for all γ values (39). However, when the interaction strength is not negligible, it is straightforward to see that deviations from a simple product-form solution are unavoidable for the exact analytical form of the distribution. In the limit $\gamma \to 0$, we can use a perturbative approach to obtain an explicit expression for the generating function (section A, Supporting Materials and Methods). Keeping terms up to linear order in γ leads to the following expression:

$$G=G_0\left(z_1,z_2\right)\left[1+\gamma \frac{k_m{k}_s}{{\mu }_m{\mu }_s}\left(\frac{1-z_1}{{\mu }_m}+\frac{1-z_2}{{\mu }_s}-\frac{\left(1-z_1\right)\left(1-z_2\right)}{{\mu }_m+{\mu }_s}\right)\right]\text{,}$$ (7)

which indicates deviations from the product-form solution. Thus, the product-form approach cannot be used to characterize the exact joint distribution and alternative approaches are needed.

Strong interaction limit

As reported in various studies (31, 32), we now consider the case of strong mRNA-sRNA interactions i.e., large γ/μ_m. Initially, we consider the limiting case $\gamma /{\mu }_m\to \infty$. Unless otherwise stated, we will be working with units such that μ_m = 1. In these units, the strong interaction limit corresponds to having $\gamma \to \infty$, whereas all the other rate constants remain finite. In this limit, mutual degradation of mRNAs and sRNAs can be taken to occur instantaneously. This leads to an effective reduction in the allowed state space, because coexistence of sRNAs and mRNAs is not allowed. Accordingly, the allowed states of the system (m,s) can be decomposed into three subgroups as follows:

$$\left\{\left(m,s\right)\right\}=\{\begin{array}{cc}\left(\mathrm{0,0}\right);& \\ \left\{\left(m,0\right)\right\},m=\mathrm{1,2}\dots ;& \\ \left\{\left(0,s\right)\right\},s=\mathrm{1,2}\dots & \end{array}$$

Transitions between these states have been illustrated in Fig. 3 a. The simplification provided by the $\gamma /{\mu }_m\to \infty$ limit is that the effective reduction in state space allows us to use the principle of detailed balance to obtain the steady-state solution. To see this, let us denote the steady-state probability current between two states i and j by $J_{i\to j}=P\left(i\right)W_{i\to j}-P\left(j\right)W_{j\to i}$, with $W_{i\to j}$ representing the rate of transition from state i to j. Now, the state space considered in Fig. 3 a does not have any cycles, and correspondingly, the steady-state is characterized by a single constant probability current J. Furthermore, because P(m,0)(P(0,s)) tends to 0 for large enough m(s), we must have J = 0. Thus, all probability currents vanish in the steady state, which is a statement of the condition of detailed balance. Using the detailed balance condition for transitions between allowed states, the steady-state probability generating function $G\left(z_1,z_2\right)$ in the limiting case of $\gamma \to \infty$ can be written explicitly as follows (see section B, Supporting Materials and Methods):

$$G\left(z_1,z_2\right)=\frac{{}_{1}F_1\left(\mathrm{1,1}+\frac{k_s}{{\mu }_m},\frac{k_m}{{\mu }_m}{z}_1\right)+{}_{1}F_1\left(\mathrm{1,1}+\frac{k_m}{{\mu }_s},\frac{k_s}{{\mu }_s}{z}_2\right)-1}{{}_{1}F_1\left(\mathrm{1,1}+\frac{k_s}{{\mu }_m},\frac{k_m}{{\mu }_m}\right)+{}_{1}F_1\left(\mathrm{1,1}+\frac{k_m}{{\mu }_s},\frac{k_s}{{\mu }_s}\right)-1}\text{.}$$ (8)

Figure 3.

(a) State space and transitions in the limit of infinitely strong mRNA-sRNA interactions. States with nonzero numbers of both sRNAs and mRNAs are not considered (in the steady-state limit), given that mutual degradation of mRNAs and sRNAs occurs instantaneously. (b) The transitions between allowed states $\left\{\left(m,0\right),\left(m,1\right),\left(0,s\right),\left(1,s\right)\right\}$ for large γ values are shown. To see this figure in color, go online.

It is interesting to note that the above generating function $G\left(z_1,z_2\right)$, which is exact for $\gamma \to \infty$, has an additive structure, unlike the product form solution for γ = 0. Furthermore, the expression derived for the generating function can be used to obtain various moments associated with mRNAs or sRNA copy numbers. For example, explicit expressions for the first two moments for mRNAs are given by the following (section B, Supporting Materials and Methods):

$$\langle m\rangle =\frac{\left(\frac{k_m}{{\mu }_m+k_s}\right){}_{1}F_1\left(\mathrm{2,2}+\frac{k_s}{{\mu }_m},\frac{k_m}{{\mu }_m}\right)}{{}_{1}F_1\left(\mathrm{1,1}+\frac{k_s}{{\mu }_m},\frac{k_m}{{\mu }_m}\right)+{}_{1}F_1\left(\mathrm{1,1}+\frac{k_m}{{\mu }_s},\frac{k_s}{{\mu }_s}\right)-1}\text{,}$$ (9)

$$\langle m^2\rangle =\left(\frac{\left(\frac{k_m}{{\mu }_m+k_s}\right)}{{}_{1}F_1\left(\mathrm{1,1}+\frac{k_s}{{\mu }_m},\frac{k_m}{{\mu }_m}\right)+{}_{1}F_1\left(\mathrm{1,1}+\frac{k_m}{{\mu }_s},\frac{k_s}{{\mu }_s}\right)-1}\right)\left[\frac{2k_m}{2{\mu }_m+k_s}{}_{1}F_1\left(\mathrm{3,3}+\frac{k_s}{{\mu }_m},\frac{k_m}{{\mu }_m}\right)+{}_{1}F_1\left(\mathrm{2,2}+\frac{k_s}{{\mu }_m},\frac{k_m}{{\mu }_m}\right)\right].$$ (10)

To highlight the significance of the results derived, consider the case μ_m = μ_s. In this case, the mean field result predicts $\langle m\rangle =0$ for k_s > k_m. However, the mean-field prediction is not in good agreement with the exact result, Eq. 9 (see Fig. 4). Also, as can be seen, the result obtained for $\gamma \to \infty$ matches reasonably well with the simulation results, even for γ/μ_m = 20. Interestingly, the difference between mean field and exact result is maximum at k_s = k_m, the point that marks the transition in the mean field behavior for large γ (29).

Figure 4.

Steady-state mean mRNA levels plotted as function of the sRNA synthesis rate k_s. Mean-field predictions for $\gamma \to \infty$ and for $\gamma =20$ are shown as dotted and dashed lines, respectively. Corresponding analytical result for the case $\gamma \to \infty$ have been shown by a solid line. The points correspond to the simulations results for γ = 20. Other parameters are k_m = 2.5, μ_m = μ_s = 1. In the inset, the difference between the mean-field result and analytical result $\mathit{\Delta }\langle m\rangle$ is plotted.

Extensions for large γ

The results obtained in the preceding section are strictly valid in the limiting case of infinitely strong interaction between mRNAs and sRNAs. The accessible states in this limit correspond to mutual exclusion of mRNAs and sRNAs, i.e., P(m,s) = 0, if m ≥ 1 and s ≥ 1. However, to go beyond this simplifying limit for large γ, we need to include states allowing sRNAs and mRNAs to coexist. For large γ, the simplest extension is to include the states: $\left\{\left(m,1\right)\right\}$, $\left\{\left(1,s\right)\right\}$, with m ≥ 1, s ≥ 1, in addition to the states $\left\{\left(m,0\right),\left(0,s\right),\left(\mathrm{0,0}\right)\right\}$, m ≥ 1, s ≥ 1 (Fig. 3 b). However, unlike the preceding analysis for infinitely strong interaction, the topology of state space now permits cycles, and the condition of detailed balance does not hold in this case. In particular, it is clearly violated for transitions to and from states $\left\{\left(m,1\right)\right\}$, $\left\{\left(1,s\right)\right\}$, with m ≥ 1, s ≥ 1. Nevertheless, it is straightforward to extend the previous analysis using the condition that, in the steady state, there is no net accumulation of probability current at any node. Correspondingly, we obtain the following expression for the probability generating function for the joint distribution of mRNAs and sRNAs (see section C, Supporting Materials and Methods):

$$G\left(z_1,z_2\right)=P\left(\mathrm{0,0}\right)[{}_{1}F_1\left(\mathrm{1,1}+\frac{k_s}{{\mu }_m},\frac{k_m}{{\mu }_m}{z}_1\right)+{}_{1}F_1\left(\mathrm{1,1}+\frac{k_m}{{\mu }_s},\frac{k_s}{{\mu }_s}{z}_2\right)-1+\underset{\mathrm{\epsilon }\to 0}{\text{lim}}\frac{k_m}{\gamma }{z}_1\frac{1}{\mathrm{\epsilon }}\left({}_{1}F_1\left(\mathrm{\epsilon },1+\frac{k_m}{{\mu }_s},\frac{k_s}{{\mu }_s}{z}_2\right)-1\right)+\underset{\mathrm{\epsilon }\to 0}{\text{lim}}\frac{k_s}{\gamma }{z}_2\frac{1}{\mathrm{\epsilon }}\left({}_{1}F_1\left(\mathit{\epsilon },1+\frac{k_s}{{\mu }_m},\frac{k_m}{{\mu }_m}{z}_1\right)-1\right)],$$ (11)

where P(0,0) is obtained from the normalization condition G(1,1) = 1. The above expression can be considered as a correction to the limiting case of infinite γ. We expect the above results to hold for large γ and to improve the predictions based on results from Eq. 8, which is valid strictly for $\gamma \to \infty$. In Fig. 5, we have compared the results for the first two steady-state moments of mRNAs from this analysis with corresponding simulation results. As can be seen, our analytical prediction works well for large γ values. We have further explored this agreement for an expanded range of parameters and observe that the relative error between the simulation results and our analytical predictions is within 10% for the range of parameters explored (see Fig. S1). This agreement can be systematically improved by increasing the interaction parameter γ. At the same time, for a given interaction strength, we also observe deviations of our analytic results from the simulation results as the mean mRNA levels are reduced.

Figure 5.

Steady-state moments for mRNAs as a function of mRNas-sRNA interaction strength γ, with other parameters as k_m = k_s = 1.5, μ_m = μ_s = 1. The points are simulation results, and lines correspond to analytical predictions based on Eq. 11.

Protein statistics

After deriving analytical expressions for the steady-state joint probability distribution of mRNAs-sRNAs, we next turn to extend our analysis to the corresponding steady-state protein statistics. Obtaining analytical results for protein distributions is a key ingredient for quantification of noise in gene expression.

To begin, let us consider the master equation that describes the time evolution for the joint probability distribution of number of mRNAs (m), sRNAs (s) and proteins (p). The time evolution for the joint distribution P(m,s,p,t) at any time t is given by

$$\frac{\partial P\left(m,s,p,t\right)}{\partial t}=k_{m}P\left(m-1,s,p,t\right)+{\mu }_m\left(m+1\right)P\left(m+1,s,p,t\right)+k_{s}P\left(m,s-1,p,t\right)+{\mu }_s\left(s+1\right)P\left(m,s+1,p,t\right)+k_{p}mP\left(m,s,p-1,t\right)+{\mu }_p\left(p+1\right)P\left(m,s,p+1,t\right)+\gamma \left(s+1\right)\left(m+1\right)P\left(m+1,s+1,p,t\right)-\left(k_m+{\mu }_{m}m+k_s+{\mu }_{s}s+k_{p}m+{\mu }_{p}p+\gamma ms\right)P\left(m,s,p,t\right).$$ (12)

Protein moments

Let us now focus on the evolution equations for protein moments. For example, by multiplying Eq. 12 by p and summing over all possible values of m,s,p, the evolution equation for the mean protein levels can be derived, which in the steady state gives

$$\langle p\rangle =\frac{k_p}{{\mu }_p}\langle m\rangle \text{.}$$ (13)

The above expression for the mean protein levels is valid for all interaction strengths γ. So, if we know the mean mRNA level $\langle m\rangle$ exactly, we can obtain an exact expression for the corresponding mean protein levels. This implies that the results derived in previous sections for mean mRNA levels can be used to obtain analytical results for mean protein levels. In particular, in the limit $\gamma \to \infty$, we can use the exact expression for the mean mRNA levels, Eq. 9, to obtain the mean steady-state protein levels. Explicitly, mean protein levels are given by

$$\langle p\rangle =\frac{k_p}{{\mu }_p}\left[\frac{\left(\frac{k_m}{{\mu }_m+k_s}\right){}_{1}F_1\left(\mathrm{2,2}+\frac{k_s}{{\mu }_m},\frac{k_m}{{\mu }_m}\right)}{{}_{1}F_1\left(\mathrm{1,1}+\frac{k_s}{{\mu }_m},\frac{k_m}{{\mu }_m}\right)+{}_{1}F_1\left(\mathrm{1,1}+\frac{k_m}{{\mu }_s},\frac{k_s}{{\mu }_s}\right)-1}\right].$$ (14)

Furthermore, using Eq. 11 in Eq. 13, we can extend the result to obtain an expression for mean protein levels that is valid for large γ. In Fig. 7, we have compared the simulation results for mean protein levels with our analytical predictions that are valid for large γ. As indicated in the figure, the analytical predictions work very well; interestingly, the agreement is good even for $\gamma /{\mu }_m\to 1$. We have further validated our analytical predictions for a wider range of parameters (see Fig. S2). As protein moments depend on corresponding mRNA moments, the deviations observed in the case of mRNA are also reflected at protein level.

Figure 7.

Mean and variance in protein copy number are shown in (a) and (b), respectively. Points are simulation results for γ = 1 (triangles) and 10 (circles). The solid line is the analytic prediction based on large γ results and taking γ = 20 results. Other parameters are k_m = 1, μ_m = 1, k_s = 0.5, μ_s = 0.1, μ_p = 0.5.

Apart from mean values, it is also of interest to study higher moments such as the variance in protein copy numbers, $\langle p^2\rangle -{\langle p\rangle }^2$. To find an expression for protein variance, we first obtain the second moment by multiplying Eq. 12 by p² and then summing over all values of m,s,p. This leads to

$$\frac{\partial \langle p^2\rangle }{\partial t}=k_p\left(\langle m\rangle +2\langle mp\rangle \right)+{\mu }_p\left(\langle p\rangle -2\langle p^2\rangle \right)\text{,}$$ (15)

which in the steady state and using Eq. 13 leads to the following expression for the second moment:

$$\langle p^2\rangle =\frac{k_p}{{\mu }_p}\left(\langle m\rangle +\langle mp\rangle \right)\text{.}$$ (16)

The above expression is exact for all values of the interaction strength γ. However, it is nontrivial to solve for $\langle p^2\rangle$ because of the dependence on $\langle mp\rangle$. Using Eq. 12, we obtain the following expression for $\langle mp\rangle$:

$$\langle mp\rangle =\frac{1}{{\mu }_m+{\mu }_p}\left(k_m\langle p\rangle +k_p\langle m^2\rangle -\gamma \langle msp\rangle \right)\text{.}$$ (17)

Note that the second-order moment depends on the third-order moment $\langle msp\rangle$, a consequence of the nonlinearity in the model.

In the absence of post-transcriptional regulation (γ = 0), the mRNA steady state is given by a Poisson distribution with generating function $G\left(z_1\right)=\text{exp}\left[\left(k_m/{\mu }_m\right)\left(z_1-1\right)\right)$, which can be used to find $\langle m\rangle =k_m/{\mu }_m$, $\langle m^2\rangle =\left(k_m/{\mu }_m\right)\left(1+k_m/{\mu }_m\right)$. Using these moments in Eqs. 16 and 17, we can find an explicit expression for $\langle p^2\rangle$ and hence the protein variance. Beyond this simple limit, finding $\langle p^2\rangle$ becomes challenging. However, for large γ, wherein mutual degradation of mRNAs and sRNAs dominates, we can use the approximation

$$C\equiv \langle msp\rangle -\langle ms\rangle \langle p\rangle =0.$$ (18)

This approximation has been verified by our simulation results, Fig. 6 (also, see Fig. S3, which provides support for validity of the approximation as different model parameters are varied). In this limit, we can thus replace $\langle msp\rangle$ by $\langle ms\rangle \langle p\rangle$, and then, using Eqs. 2, 10, and 13 in Eq. 17, we find $\langle mp\rangle$. Plugging the resulting expression for $\langle mp\rangle$ in Eq. 16 and using Eq. 9 for $\langle m\rangle$, we obtain the following expression for $\langle p^2\rangle$:

$$\langle p^2\rangle =\frac{k_p}{{\mu }_p}\left(\langle m\rangle +\frac{1}{{\mu }_m+{\mu }_p}\left(\frac{{\mu }_m{k}_p}{{\mu }_p}{\langle m\rangle }^2+k_p\langle m^2\rangle \right)\right)\text{.}$$ (19)

Figure 6.

Steady-state correlation function $C=\langle msp\rangle -\langle ms\rangle \langle p\rangle$ as a function of mRNA-sRNA interaction strength γ for three different sets of parameters. Circles correspond to k_m = 0.5, μ_m = 1, k_s = 1, μ_s = 0.1, k_p = 2, μ_p = 0.1; triangles correspond to k_m = 1.5, μ_m = 1, k_s = 1.5, μ_s = 1, k_p = 0.1, μ_p = 0.5; and squares are for k_m = 0.5, μ_m = 1, k_s = 1, μ_s = 0.1, k_p = 0.1, μ_p = 0.5. Lines are drawn as a guide to the eye.

The above expression is in terms of moments associated with mRNAs, which can be found using Eqs. 9 and 10. For large γ, we can use Eq. 18 as a reasonable approximation, and then, using Eq. 11, we can find the moments associated with mRNAs and hence the variance for proteins. In Fig. 7, we have compared our analytical predictions for protein variance based on large γ with the simulation results, and they are in excellent agreement, even for $\gamma /{\mu }_m\to 1$ (also, see Fig. S2).

Protein distribution

The explicit expressions derived for protein mean and variance can be used to obtain an approximate analytical expression for the protein steady-state distribution. In the absence of sRNAs, if we consider the so-called burst limit (i.e., when the degradation rate of mRNAs is much greater than the degradation rate of proteins), it is well-known that the protein steady-state distribution is a negative binomial distribution (40). Correspondingly, the steady-state protein distribution P(p) is given by

$$P\left(p\right)=\frac{\left(p+r-1\right)!}{p!\left(r-1\right)!}{q}^p{\left(1-q\right)}^r\text{,}$$ (20)

where r and q are the two parameters of the negative binomial distribution, determined by the mean and variance of protein copy numbers,

$$q=1-\frac{\langle p\rangle }{{\sigma }_p^2}\text{and}r=\frac{1-q}{q}\langle p\rangle .$$

To obtain an approximate analytical representation for the protein distribution in the presence of interaction with sRNAs, we consider a reduced representation of the original model, wherein a single mRNA degrades individually with rate μ_m and with an additional fluctuating rate γs due to interaction. The interaction with sRNAs leads to an increased rate of mRNA degradation, thus the burst limit approximation will continue to be valid. If we further assume that the burst distribution can be well-approximated by a geometric distribution, then we obtain that protein steady-state distribution can be approximated by a negative binomial distribution with renormalized parameters. Using the derived expressions for the mean and variance in the case of large γ, we can obtain the parameters q and r for the negative binomial and hence the corresponding protein steady-state distribution. As illustrated in the figure, for large γ, this approximate negative binomial distribution fits the simulation results very well (Fig. 8).

Figure 8.

Approximate protein steady-state distributions. Points are simulation results for γ = 1 (triangles) and 10 (circles). The line shows the analytical prediction based on large γ and taking γ = 20. Other parameters are k_m = 1, μ_m = 1, k_s = 0.5, μ_s = 0.1, μ_p = 0.1, k_p = 1.

Conclusions

In this work, we analyzed the canonical stochastic model of post-transcriptional regulation by sRNAs via stoichiometric degradation. The model studied plays a fundamental role in quantitative understanding of the effects of post-transcriptional gene regulation in cell-fate decisions. An exact solution of the model for arbitrary interaction strength remains an open problem in the field. Correspondingly, most studies have used results based on mean-field approximation to analyze system response, which can lead to incorrect predictions in the strong interaction limit (31, 32). In this limit, we have derived an analytical expression for the steady-state joint probability distributions of mRNAs and sRNAs. The resulting exact expression can be used to shed light on the statistics of mRNA distributions under post-transcriptional regulation by sRNAs.

The results obtained at the level of mRNAs are used to study the statistics associated with protein steady-state distribution. We have derived analytic expressions for the mean and variance of proteins in the strong interaction regime. Furthermore, we observed that in the burst limit, i.e., when the mRNA degradation rate is much greater than the protein degradation rate, an approximate negative binomial distribution agrees very well with results from simulations for the protein steady-state distribution. The analytic results derived in this work provide an important step toward a quantitative understanding of post-transcriptional regulation of stochastic gene expression. The limit of strong mRNA-sRNA interactions studied in this work can also be realized in synthetic genetic circuits by tuning the strength of mRNA-sRNA interactions. In these cases, analytical results are currently lacking in the literature, and our derived expressions for the moments can be used to guide experimental efforts to achieve desired mean levels (or noise) of gene products. In particular, they can be used the quantify the impact of changes in combinations of parameters in tuning the mean and noise in gene expression. On the flip side, the results derived can be used to make inferences about underlying parameters based on observations of mean and noise of gene expression. Furthermore, the results obtained can serve as building blocks for the analysis of more complicated genetic circuits involving sRNAs, and the approach developed can potentially be generalized to analyze a more general class of stochastic models of gene regulation in the strong interaction limit. Finally, it is interesting to note that, although our results are based on the assumption that mRNAs or sRNAs are produced constitutively, future work should include the possibility of their production in bursts (8, 41, 42, 43, 44, 45, 46). The simplification that occurs when we consider constitutive gene expression, i.e., using the principle of detailed balance, cannot be used directly for the case of gene expression in bursts. Previous work (18) focusing on bursty gene expression has derived expressions for the protein-steady state distribution for a limited range of parameters with the constraint that the mRNA degradation rate is much larger than the protein degradation rate. It would thus be of interest to develop approaches, potentially combining current and previous works, to address models with extensions to accommodate biologically observed features such as catalytic degradation and bursty gene expression.

Author Contributions

N.K., K.Z., and R.V.K. designed the research. N.K. and R.V.K. performed the research. N.K. performed the simulations. N.K., K.Z., and R.V.K. wrote the manuscript.

Editor: Karin Musier-Forsyth.