Consequences of mRNA Transport on Stochastic Variability in Protein Levels

Abstract

Homogeneous cell populations can exhibit considerable cell-to-cell variability in protein levels arising from the stochastic nature of the gene-expression process. In particular, transcriptional bursting of mRNAs from the promoter has been implicated as a major source of stochasticity in the expression of many genes. In eukaryotes, transcribed pre-mRNAs have to be exported outside the nucleus and in many cases, export rates can be slow and comparable to mRNA turnover rates. We investigate whether such export processes can be effective mechanisms in buffering protein levels from transcriptional bursting of pre-mRNAs in the nucleus. For a stochastic gene-expression model with both transcriptional bursting and export, we derive an exact solution of the steady-state probability-generating function for both the nuclear and the cytoplasmic mRNA levels. These formulas reveal that decreasing export rates can dramatically reduce variability in cytoplasmic mRNA levels. However, our results also show that decreasing export rates enhance mRNA autocorrelation times, which function to increase heterogeneity in protein levels. Our overall analysis concludes that under physiologically relevant parameter regimes, a pre-mRNA export step can decrease steady-state variability at the mRNA level but not at the protein level. Finally, we reinforce previous observations that saturation in the pre-mRNA transport machinery can be an important mechanism in suppressing protein variability from underlying transcriptional bursts.

Introduction

The inherent probabilistic nature of biochemical reactions can lead to large stochastic fluctuations in RNA and protein copy numbers over time in individual cells (1–6). Cell-to-cell variability in protein levels generated by these fluctuations is often referred to as “gene-expression noise”. Increasing evidence suggests that expression noise profoundly affects biological function and phenotype. For example, diverse diseased states have been attributed to an elevated expression noise (7–9). Not surprisingly, genes actively use different regulatory mechanisms to reduce stochastic fluctuations in protein levels (10–20). Many genes also exploit protein level variability to drive probabilistic cell-fate decisions and generate phenotypic heterogeneity across a clonal cell population (21–25). Given these functional roles of expression noise, it is important to understand how different cellular processes shape stochastic variability in protein levels.

Random fluctuations between different transcriptional states of the promoter (i.e., promoter switching) has been implicated as a major source of noise in the expression of many genes. An important consequence of promoter switching is transcriptional bursting, where multiple mRNAs are created per promoter-firing event (26–33). In eukaryotes, transcribed pre-mRNAs have to be processed and exported outside the nucleus before they can become functionally active mRNAs that encode protein molecules. In vivo tracking of nuclear pre-mRNAs show export times ranging from a few minutes to up to an hour (34). Considering that many mRNAs have short half-lives (35) comparable to these export times suggests that export processes can significantly alter mRNA dynamics. We investigate whether slow pre-mRNA export from the nucleus can be an effective mechanism in buffering protein levels from bursts of transcriptional activity at the promoter.

Retention of pre-mRNAs in the nucleus after transcription creates a stochastic delay in the gene-expression process. Because the exact distribution of this delay is unknown, one can consider two limiting cases: deterministic and exponentially distributed delay. If the delay is deterministic (i.e., each pre-mRNA spends a fixed amount of time in the nucleus), then it will not affect steady-state variability in protein levels. We here consider the opposite scenario where pre-mRNA processing and export happens in an exponentially distributed time interval. This delay is incorporated in the gene-expression model by representing pre-mRNA export as a first-order reaction characterized by an export rate.

For a stochastic model with transcriptional bursting and export, we derive an exact analytical solution for the corresponding Chemical Master Equation. This solution provides insights into the shape of the mRNA distribution for different export rates and shows that pre-mRNA export can dramatically reduce the extent of fluctuations in mRNA population counts. Interestingly, pre-mRNA export also enhances the duration of mRNA level fluctuations (i.e., autocorrelation times), which increase protein noise levels. Taking both the above effects into account shows that in physiologically relevant parameter regimes, protein noise level is invariant of the export rate. Thus, stochasticity variability arising from transcriptional bursting is effectively attenuated by export processes at the mRNA level but resurrected at the protein level through enhanced mRNA autocorrelation times.

Gene-Expression Model with mRNA Transport: An Exact Solution

We begin by introducing a model for capturing the stochastic dynamics of nuclear pre-mRNA (M_n) and cytoplasmic mRNA (M_c) expression levels (see Fig. 1). The model can be represented by the following set of reactions:

$$\begin{array}{c}Gene\stackrel{k_m}{\to }Gene+B\times M_n,\\ M_n\stackrel{{\gamma }_e}{\to }{M}_c,M_c\stackrel{{\gamma }_c}{\to }\varnothing .\end{array}$$ (1)

The first reaction corresponds to pre-mRNA transcription in bursts from the promoter, with each round of transcription creating B mRNA transcripts. Remaining reactions in Eq. 1 represent pre-mRNA export from the nucleus and cytoplasmic mRNA degradation. Because pre-mRNAs tend to be stable inside the nucleus (36), we ignore pre-mRNA nuclear degradation. In Eq. 1, B represents the transcriptional burst size, k_m is the burst frequency, γ_e is the export rate, and γ_c is the rate constant for cytoplasmic mRNA degradation. We assume the transcription burst size B to be a random variable with distribution

$$\begin{array}{c}Probability\left\{B=z\right\}={\alpha }_z,z\in \left\{0,1,\dots \right\},\end{array}$$ (2)

and mean 〈B〉. Here and in the sequel we use 〈.〉 to denote the expected value. Our gene-expression model corresponds to the stochastic formulation of the above biochemical reactions, where each reaction is a probabilistic event and occurs at exponentially distributed time intervals (37,38). Moreover, whenever a particular reaction fires, the number of molecules of the pre-mRNA and/or cytoplasmic mRNA are reset based on the stoichiometry of the reaction.

Figure 1.

Schematic of the gene expression model where pre-mRNAs are made in transcriptional bursts. The pre-mRNAs are transported outside the nucleus to become functional mRNAs that encode protein molecules.

For the above model, the probability P(m_n, m_c, t) of having m_n copies of pre-mRNA in the nucleus and m_c copies of mRNA in the cytoplasm at time t satisfies the chemical master equation (CME),

$$\frac{dP\left(m_n,m_c,t\right)}{dt}=k_m\left(\sum _{z=0}^{m_n}{\alpha }_{z}P\left(m_n-z,m_c,t\right)-P\left(m_n,m_c,t\right)\right)+{\gamma }_e\left(\left(m_n+1\right)P\left(m_n+1,m_c-1,t\right)-m_{n}P\left(m_n,m_c,t\right)\right)+{\gamma }_c\left(\left(m_c+1\right)P\left(m_n,m_c+1,t\right)-m_{c}P\left(m_n,m_c,t\right)\right),$$ (3)

subject to an initial condition

$$P\left(m_n,m_c,0\right)=P^{\left(0\right)}\left(m_n,m_c\right),$$ (4)

which, being a probability distribution, satisfies the normalization condition

$$\sum _{m_n=0}^{\mathrm{\infty }}\sum _{m_c=0}^{\mathrm{\infty }}{P}^{\left(0\right)}\left(m_n,m_c\right)=1.$$ (5)

Our aim in this section will be to study the solution P(m_n, m_c, t) to Eqs. 3 and 4, in particular its large-time behavior.

We introduce the probability-generating function (PGF) of the probability distribution of mRNA levels, and that of the burst distribution B,

$$\begin{array}{c}G\left(x,y,t\right)=\sum _{m_n=0}^{\mathrm{\infty }}\sum _{m_c=0}^{\mathrm{\infty }}{x}^{m_n}{y}^{m_c}P\left(m_n,m_c,t\right),\\ F\left(x\right)=\sum _{z=0}^{\mathrm{\infty }}{\alpha }_z{x}^z.\end{array}$$ (6)

Multiplying Eqs. 3 and 4 by the factor $x^{m_n}{y}^{m_c}$ and summing over all m_n and m_c, we find that the generating function satisfies a first-order partial differential equation (PDE),

$$\frac{\partial G}{\partial t}=k_m\left(F\left(x\right)-1\right)G+{\gamma }_e\left(y-x\right)\frac{\partial G}{\partial x}+{\gamma }_c\left(1-y\right)\frac{\partial G}{\partial y},$$ (7)

subject to

$$G\left(x,y,0\right)=G^{\left(0\right)}\left(x,y\right)=\sum _{m_n=0}^{\mathrm{\infty }}\sum _{m_c=0}^{\mathrm{\infty }}{x}^{m_n}{y}^{m_c}{P}^{\left(0\right)}\left(m_n,m_c\right).$$ (8)

Here and elsewhere we write zero in the superscript to indicate an initial condition that is imposed at time t = 0. The normalization condition in Eq. 5 implies that

$$G^{\left(0\right)}\left(1,1\right)=1.$$ (9)

We shall be primarily interested in the marginal distribution of cytoplasmic mRNA counts. The marginals are defined by

$$\begin{array}{c}{P}_n\left(m_n,t\right)=\sum _{m_c=0}^{\mathrm{\infty }}P\left(m_n,m_c,t\right),\\ P_c\left(m_c,t\right)=\sum _{m_n=0}^{\mathrm{\infty }}P\left(m_n,m_c,t\right).\end{array}$$ (10)

The generating functions of the marginal distributions are

$$\begin{array}{c}{G}_n\left(x,t\right)=\sum _{m_n=0}^{\mathrm{\infty }}{x}^{m_n}{P}_n\left(m_n,t\right),\\ G_c\left(y,t\right)=\sum _{m_c=0}^{\mathrm{\infty }}{y}^{m_c}{P}_c\left(m_c,t\right).\end{array}$$ (11)

The marginal and joint generating functions are related by

$$\begin{array}{c}{G}_n\left(x,t\right)=G\left(x,1,t\right),\\ G_c\left(y,t\right)=G\left(1,y,t\right),\end{array}$$ (12)

enabling us to determine the marginals from the joint generating function.

It is often convenient to use the PGF in a slightly different guise. The factorial-cumulant generating function (FCGF) corresponding to our joint distribution (which we refer to as the joint FCGF below) is defined by (see Johnson et al. (39)):

$$\phi \left(u,\upsilon ,t\right)=\text{ln}G\left(1+u,1+\upsilon ,t\right).$$ (13)

The FCGFs for the marginal distributions (marginal FCGFs) of nuclear and cytoplasmic mRNA are defined by

$$\begin{array}{c}{\phi }_n\left(u,t\right)=\text{ln}{G}_n\left(1+u,t\right),\\ {\phi }_c\left(\upsilon ,t\right)=\text{ln}{G}_c\left(1+\upsilon ,t\right).\end{array}$$ (14)

Using the relation between the joint and marginal generating functions in Eq. 12, we find that

$$\begin{array}{c}{\phi }_n\left(u,t\right)=\phi \left(u,0,t\right),\\ {\phi }_c\left(\upsilon ,t\right)=\phi \left(0,\upsilon ,t\right),\end{array}$$ (15)

which enable us to determine the marginal FCGFs from the joint generating functions.

Changing variables in Eqs. 7 and 8 according to x = 1 + u, y = 1 + υ, and G = exp(φ), we find that the joint FCGF satisfies a partial differential equation

$$\frac{\partial \phi }{\partial t}=k_m\left(M\left(u\right)-1\right)+{\gamma }_e\left(\upsilon -u\right)\frac{\partial \phi }{\partial u}-{\gamma }_c\upsilon \frac{\partial \phi }{\partial \upsilon },$$ (16)

where M(u) = F(1 + u) is the factorial-moment generating function associated with the burst distribution. The initial condition for Eq. 16 is

$$\phi \left(u,\upsilon ,0\right)={\phi }^{\left(0\right)}\left(u,\upsilon \right)=\text{ln}{G}^{\left(0\right)}\left(1+u,1+\upsilon \right).$$ (17)

The normalization condition in Eq. 9 for the PGF implies one for the FCGF,

$${\phi }^{\left(0\right)}\left(0,0\right)=0.$$ (18)

Below, we aim to determine the solution to Eqs. 16 and 17 using the method of characteristics (40).

The characteristics of Eq. 16 emanating from a given point t, u, υ satisfy

$$\begin{array}{cc}\frac{d\tilde{t}}{ds}=-1,& \tilde{t}\left(0\right)=t,\\ \frac{d\tilde{u}}{ds}={\gamma }_e\left(\tilde{\upsilon }-\tilde{u}\right),& \tilde{u}\left(0\right)=u,\\ \frac{d\tilde{\upsilon }}{ds}=-{\gamma }_c\tilde{\upsilon },& \tilde{\upsilon }\left(0\right)=\upsilon ,\end{array}$$ (19)

which give

$$\tilde{t}\left(s\right)=t-s,\tilde{\upsilon }\left(s\right)=\upsilon e^{-{\gamma }_{c}s},$$ (20)

and

$$\tilde{u}\left(s\right)=\{\begin{array}{ll}\frac{\upsilon {\gamma }_e}{{\gamma }_e-{\gamma }_c}{e}^{-{\gamma }_{c}s}+\left(u-\frac{\upsilon {\gamma }_e}{{\gamma }_e-{\gamma }_c}\right)e^{-{\gamma }_{e}s}\hfill & \text{if}{\gamma }_e\ne {\gamma }_c,\hfill \\ e^{-{\gamma }_{e}s}\left(u+{\gamma }_e\upsilon s\right)\hfill & \text{if}{\gamma }_e={\gamma }_c.\hfill \end{array}$$ (21)

By the chain rule and Eq. 16

$$-\frac{d}{ds}\phi \left(\tilde{u}\left(s\right),\tilde{\upsilon }\left(s\right),\tilde{t}\left(s\right)\right)=k_m\left(M\left(\tilde{u}\left(s\right)\right)-1\right),$$ (22)

from which, by integrating from 0 to t,

$$\phi \left(u,\upsilon ,t\right)=k_m\underset{0}{\overset{t}{\int }}\left(M\left(\tilde{u}\left(s\right)\right)-1\right)ds+{\phi }^{\left(0\right)}\left(\tilde{u}\left(t\right),\tilde{\upsilon }\left(t\right)\right).$$ (23)

Because by Eqs. 18 and 21, we have

$$\underset{t\to \mathrm{\infty }}{\text{lim}}{\phi }^{\left(0\right)}\left(\tilde{u}\left(t\right),\tilde{\upsilon }\left(t\right)\right)={\phi }^{\left(0\right)}\left(0,0\right)=0,$$ (24)

we find that the large-time (i.e., stationary) FCGF satisfies

$$\overline{\phi }\left(u,\upsilon \right)=\underset{t\to \mathrm{\infty }}{\text{lim}}\phi \left(u,\upsilon ,t\right)=k_m\underset{0}{\overset{\mathrm{\infty }}{\int }}\left(M\left(\tilde{u}\left(s\right)\right)-1\right)ds,$$ (25)

which, together with Eq. 21, give an integral representation of the joint FCGF. By Eq. 15, marginal FCGFs can be obtained by setting either u = 0 or υ = 0 in Eq. 21. The PGFs (both joint and marginal) can be determined from Eq. 25 by returning to the original variables, x, y, and G, by means of Eq. 13.

Below we illustrate some of the possible uses of the formula in Eq. 25 in studying the distribution of cytoplasmic mRNA for the specific physiologically important case of geometric burst sizes.

Geometrically distributed transcriptional burst sizes

In this subsection we study Eq. 1 in the special case of transcriptional bursts following the geometric distribution. For this case,

$$\begin{array}{l}Probability\left\{B=z\right\}={\alpha }_z=\frac{1}{1+b}{\left(\frac{b}{1+b}\right)}^z,\\ z=0,1,\dots ,\end{array}$$ (26)

where b = 〈B〉 is the mean transcriptional burst size. If the export mechanism in Eq. 1 is fast, the nuclear intermediate gets eliminated, and Eq. 1 reduces to a well-studied case in which the steady-state probability $\overline{P_c}\left(m_c\right)$ of having m_c mRNA molecules in the cytoplasm is given by the negative binomial distribution (41):

$$\overline{P_c}\left(m_c\right)=\frac{b^{m_c}}{{\left(1+b\right)}^{m_c+k_m/{\gamma }_c}}\frac{\Gamma \left(m_c+k_m/{\gamma }_c\right)}{m_c!\Gamma \left(k_m/{\gamma }_c\right)}\text{if}{\gamma }_e\gg {\gamma }_c.$$ (27)

Throughout the article, we use a bar over a variable to denote that the variable is taken at steady state. Note that

$$\underset{m_c\to \mathrm{\infty }}{\text{lim}}\frac{\overline{P_c}\left(m_c+1\right)}{\overline{P_c}\left(m_c\right)}=\frac{b}{1+b},$$ (28)

implying that for large values of mean burst size b we expect a relatively slow (i.e., with a quotient close to one) geometric decay of the tail of the distribution.

On the other hand, if the export mechanism in Eq. 1 is slow, one expects that, in the long run, large quantities of mRNA accumulate in the nucleus, which then start to trickle down, one molecule at a time, to the cytoplasm. In other words, slow export effectively cancels out the transcription bursts, and the resulting cytoplasmic mRNA distribution can be expected to be a Poisson with mean k_mb/γ_c:

$$\overline{P_c}\left(m_c\right)=e^{-k_{m}b/{\gamma }_c}\frac{{\left(k_{m}b/{\gamma }_c\right)}^{m_c}}{m_c!}\text{if}{\gamma }_e\ll {\gamma }_c.$$ (29)

Notably,

$$\underset{m_c\to \mathrm{\infty }}{\text{lim}}\frac{\overline{P_c}\left(m_c+1\right)}{\overline{P_c}\left(m_c\right)}=0,$$ (30)

which means that the tail of the distribution decays faster than geometrically in the limit of very slow export.

Below we study the distribution of cytoplasmic mRNA level for intermediate values of the export rate. In addition, we use the theoretical results presented above to find an efficient method for numerical evaluation of the exact distribution and use this method to visualize the transition between the negative binomial and Poisson scenarios.

The factorial-moment generating function of the geometric distribution in Eq. 26 is given by

$$M\left(u\right)=\frac{1}{1-bu}.$$ (31)

Therefore, the factorial-cumulant generating function in Eq. 25 of the stationary joint distribution of nuclear and cytoplasmic mRNA levels reduces to

$$\overline{\phi }\left(u,\upsilon \right)=k_m\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{b\tilde{u}\left(s\right)ds}{1-b\tilde{u}\left(s\right)},$$ (32)

where $\tilde{u}\left(s\right)$ is given by Eq. 21. The marginal FCGF of cytoplasmic mRNA level is obtained by setting u = 0 in Eq. 21, i.e.,

$${\overline{\phi }}_c\left(\upsilon \right)=\overline{\phi }(0,\upsilon )=\{\begin{array}{ll}{k}_m\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\frac{b\upsilon {\gamma }_e}{{\gamma }_e-{\gamma }_c}\left(e^{-{\gamma }_{c}s}-e^{-{\gamma }_{e}s}\right)ds}{1-\frac{b\upsilon {\gamma }_e}{{\gamma }_e-{\gamma }_c}\left(e^{-{\gamma }_{c}s}-e^{-{\gamma }_{e}s}\right)}\hfill & \text{if}{\gamma }_e\ne {\gamma }_c,\hfill \\ k_m\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{b{\gamma }_e\upsilon s{e}^{-{\gamma }_{e}s}ds}{1-b{\gamma }_e\upsilon s{e}^{-{\gamma }_{e}s}}\hfill & \text{if}{\gamma }_e={\gamma }_c.\hfill \end{array}$$ (33)

Note that the FCGF ${\overline{\phi }}_c\left(v\right)$ is defined only if the denominator of the integrand is positive for all s > 0. For this to happen, the variable υ must satisfy the following restrictions:

$$\begin{array}{ll}\upsilon <\frac{{\left({\gamma }_e/{\gamma }_c\right)}^{{\gamma }_c/{\gamma }_e{\gamma }_c}}{b}\hfill & \text{if}{\gamma }_e\ne {\gamma }_c,\hfill \\ \upsilon <\frac{e}{b}\hfill & \text{if}{\gamma }_e={\gamma }_c.\hfill \end{array}$$ (34)

Thus, the stationary marginal PGF of cytoplasmic mRNA,

$${\overline{G}}_c\left(y\right)=exp\left({\overline{\phi }}_c\left(y-1\right)\right),$$ (35)

is defined only for y, satisfying

$$\begin{array}{ll}y<1+\frac{{\left({\gamma }_e/{\gamma }_c\right)}^{{\gamma }_c/{\gamma }_e{\gamma }_c}}{b}\hfill & \text{if}{\gamma }_e\ne {\gamma }_c,\hfill \\ y<1+\frac{e}{b}\hfill & \text{if}{\gamma }_e={\gamma }_c.\hfill \end{array}$$ (36)

The PGF ${\overline{G}}_c\left(y\right)$ was originally defined as the power series whose coefficients are the marginal stationary probabilities $\overline{P_c}\left(m_c\right)$ of having m mRNA molecules in the cytoplasm. The upper bound from Eq. 36 for the domain of the PGF thus determines the radius of convergence R of the power series. Hence,

$$\underset{m_c\to \mathrm{\infty }}{\text{lim}}\frac{\overline{P_c}\left(m_c+1\right)}{\overline{P_c}\left(m_c\right)}=\frac{1}{R}=\{\begin{array}{ll}{\left(1+\frac{{\left({\gamma }_e/{\gamma }_c\right)}^{{\gamma }_c/{\gamma }_e{\gamma }_c}}{b}\right)}^{-1}\hfill & \text{if}{\gamma }_e\ne {\gamma }_c,\hfill \\ {\left(1+\frac{e}{b}\right)}^{-1}\hfill & \text{if}{\gamma }_e={\gamma }_c.\hfill \end{array}$$ (37)

The quotient of Eq. 37 is close to one if b is large and the ratio γ_e/γ_c is moderate to large, suggesting a slow geometric decay of the distribution’s tail for such parameter values. On the other hand, the quotient tends to zero if the export is slow, which is consistent with the Poisson limit, whose tail decays faster than geometrically.

Before illustrating the result from Eq. 37 for a specific parameter set, let us briefly describe a numerical method based on generating functions for evaluation of the cytoplasmic mRNA distribution, which we shall use in the example that follows. Equations 33 and 35 enable us to evaluate the PGF of the cytoplasmic mRNA distribution within its domain of convergence. In particular, we can evaluate the PGF for y from the complex unit circle, which in turn enables us to determine the underlying probability distribution using a discrete Fourier transform method (for details, see, e.g., Bokes et al. (42)). Using the efficient fast Fourier transform algorithm (43) to implement the discrete Fourier transform, we obtain a valuable numerical recipe that complements the customary methods of evaluating the distribution, such as the finite state projection (44) or repeated Gillespie simulations (45). The Fourier method appears to be effective in providing an accurate description of the tail of the distribution, which is convenient for the purpose of illustrating the validity of Eq. 37.

In Fig. 2, we show in the left panel the stationary distribution $\overline{P_c}\left(m_c\right)$ of cytoplasmic mRNA, obtained using the Fourier method. All reaction rates are normalized by the mRNA degradation rate constant (i.e., γ_c = 1). We consider the transcription frequency k_m = 3, a geometrically distributed burst size with mean equal to b = 8, and a selection of choices for the export rate constant. The first bar chart is concerned with the situation of γ_e = ∞, for which the distribution reduces to the negative binomial. Given the relatively large mean burst size (b = 8), this distribution is widely spread and has a fat tail. As the export rate constant γ_e decreases in the successive bar charts, we observe a transformation in the overall shape of the distribution, until it reduces for γ_e = 0⁺ to the Poisson limit. In agreement with previous observations (45), we note that the cytoplasmic mRNA variability is reduced as the export rate decreases.

Figure 2.

Stationary distribution of cytoplasmic mRNA count $\left(\overline{P_c}\left(m_c\right)\right)$ for k_m = 3, b = 8, γ_c = 1 and various choices of the export rate constant γ_e(left-hand panel). All rates are normalized by the cytoplasmic mRNA degradation rate γ_c. Behavior of the tail of the mRNA probability distribution for different export rates (right-hand panel).

To get a precise picture of the behavior of the tail of the exact distribution, we show in the right panel of Fig. 2 the ratio $\overline{P_c}\left(m_c+1\right)/\overline{P_c}\left(m_c\right)$ against m_c (solid line with dot marks). As m_c tends to infinity, the ratio tends to a limit (dashed line) given by Eq. 37. This limiting quotient is close to one (unless γ_e gets too small), suggesting that the tail of the cytoplasmic mRNA distribution is heavy.

Export Processes Enhance mRNA Autocorrelation Times

Results in the previous section have shown that decreasing the export rate results in lower variability in cytoplasmic mRNA levels (Fig. 2). We here analyze how steady-state mRNA autocorrelation times change with varying export rate. To determine the autocorrelation of the stochastic process m_c(t) we use the fact that

$$\langle m_c(t+s)m_c\left(s\right)\rangle =\langle m_c\left(s\right)\langle m_c(t+s)\left|m_c\right(s),m_n(s)\rangle \rangle ,$$ (38)

where 〈m_c(t + s)|m_c(s)〉, and the value 〈m_n(s)〉 is the expected number of cytoplasmic mRNA transcripts at time t + s, given m_c(s) and m_n(s). We first compute this conditional expectation using the linear system of differential equations that describe the time evolution of the mean population counts.

It is relatively straightforward to derive differential equations that describe the time evolution of the different statistical moments of the population counts (46,47). From Eq. 3, the pre-mRNA and cytoplasmic mRNA moment dynamics is given by

$$\begin{array}{l}\frac{d\langle m_n\rangle }{dt}=k_m\langle B\rangle -{\gamma }_e\langle m_n\rangle ,\\ \frac{d\langle m_c\rangle }{dt}={\gamma }_e\langle m_n\rangle -{\gamma }_c\langle m_c\rangle ,\end{array}$$ (39a)

$$\frac{d\langle m_n^2\rangle }{dt}=k_m\langle B^2\rangle +{\gamma }_e\langle m_n\rangle +2k_m\langle B\rangle \langle m_n\rangle -2{\gamma }_e\langle m_n^2\rangle ,$$ (39b)

$$\frac{d\langle m_c^2\rangle }{dt}={\gamma }_e\langle m_n\rangle +{\gamma }_c\langle m_c\rangle +2{\gamma }_e\langle m_n{m}_c\rangle -2{\gamma }_c\langle m_c^2\rangle ,$$ (39c)

$$\frac{d\langle m_n{m}_c\rangle }{dt}={\gamma }_e\langle m_n^2\rangle +k_m\langle B\rangle \langle m_c\rangle -{\gamma }_e\langle m_n\rangle -\langle m_n{m}_c\rangle \left({\gamma }_e+{\gamma }_c\right),$$ (39d)

and yields the steady-state population averages of

$$\begin{array}{l}\overline{\langle m_n\rangle }=\frac{k_m\langle B\rangle }{{\gamma }_e},\\ \overline{\langle m_c\rangle }=\frac{k_m\langle B\rangle }{{\gamma }_c},\end{array}$$ (40)

where the symbol $\overline{\langle .\rangle }$ denotes the steady-state expected value. Recall from Eq. 2 that 〈B〉 represents the average transcriptional burst size, i.e., the mean number of mRNA transcripts made from the promoter in one round of transcription. From Eq. 39, we obtain the following steady-state variances and covariance,

$$Var\left(m_n\right)=\overline{\langle m_n^2\rangle }-{\overline{\langle m_n\rangle }}^2=\overline{\langle m_n\rangle }\left(1+B_e\right),$$ (41a)

$$Var\left(m_c\right)=\overline{\langle m_c^2\rangle }-{\overline{\langle m_c\rangle }}^2=\overline{\langle m_c\rangle }\left(1+\frac{B_e{\gamma }_e}{{\gamma }_e+{\gamma }_c}\right),$$ (41b)

$$Co\upsilon \left(m_n,m_c\right)=\overline{\langle m_n{m}_c\rangle }-\overline{\langle m_n\rangle }\overline{\langle m_c\rangle }=\frac{B_e{\gamma }_e\overline{\langle m_n\rangle }}{{\gamma }_e+{\gamma }_c},$$ (41c)

where

$$B_e=\frac{\langle B^2\rangle -\langle B\rangle }{2\langle B\rangle }.$$ (42)

Note that consistent with Fig. 2, decreasing the export rate in Eq. 41b leads to lower variability in cytoplasmic mRNA expression level. Solving Eq. 39a results in the following conditional expectation:

$$\langle m_c\left(t+s\right)|m_c\left(s\right),m_n\left(s\right)\rangle =\overline{\langle m_c\rangle }+\left(m_c\left(s\right)-\overline{\langle m_c\rangle }\right)e^{-{\gamma }_{c}t}+{\gamma }_e\left(m_n\left(s\right)-\overline{\langle m_n\rangle }\right)\frac{e^{-{\gamma }_{e}t}-e^{-{\gamma }_{c}t}}{{\gamma }_c-{\gamma }_e}.$$ (43)

Substituting Eq. 43 into Eq. 38, we obtain

$$\langle m_c\left(t+s\right)m_c\left(s\right)\rangle ={\overline{\langle m_c\rangle }}^2+Var\left(m_c\right)e^{-{\gamma }_{c}t}+{\gamma }_{e}Co\upsilon \left(m_n,m_c\right)\frac{e^{-{\gamma }_{e}t}-e^{-{\gamma }_{c}t}}{{\gamma }_c-{\gamma }_e},$$ (44)

which, using Eq. 42, yields the autocorrelation function

$$\begin{array}{c}R\left(t\right)=\frac{m_c\left(t+s\right)m_c\left(s\right)-{\overline{\langle m_c\rangle }}^2}{Var\left(m_c\right)}\\ =e^{-{\gamma }_{c}t}+{\gamma }_e\frac{Co\upsilon \left(m_n,m_c\right)}{Var\left(m_c\right)}\frac{e^{-{\gamma }_{e}t}-e^{-{\gamma }_{c}t}}{{\gamma }_c-{\gamma }_e}\\ =e^{-{\gamma }_{c}t}+\frac{{\gamma }_e{\gamma }_c{B}_e}{\left(1+B_e\right){\gamma }_e+{\gamma }_c}\frac{e^{-{\gamma }_{e}t}-e^{-{\gamma }_{c}t}}{{\gamma }_c-{\gamma }_e}.\end{array}$$ (45)

Note that the autocorrelation function reduces to R(t) = exp(−γ_ct) in three cases: 1), no transcriptional bursting (i.e., B = 1 with probability one, which implies B_e = 0); 2), export is rapid compared to the mRNA half-life (γ_e >> γ_c); and 3), export is slow compared to the mRNA half-life (γ_e << γ_c). However, when there is transcriptional bursting, nuclear pre-mRNA transport can dramatically increase cytoplasmic mRNA autocorrelation times at intermediate timescales (γ_e ≈ γ_c; see Fig. 3).

Figure 3.

Cytoplasmic mRNA autocorrelation functions for different export rates. mRNA half-life is assumed to be 1 h and transcriptional burst size is geometrically distributed with an average burst size of 10 transcripts.

Quantifying Stochastic Fluctuations in Protein Levels

The results above show that increasing the pre-mRNA export half-life decreases cell-to-cell variability in cytoplasmic mRNA levels, which should lead to lower variability in protein levels. However, increasing pre-mRNA export half-life also increases mRNA autocorrelation times that function to enhance variability in protein levels by making it harder for protein molecules to average-out fluctuations in underlying mRNA population counts (13,48,49). We investigate how both these factors contribute to the variability in protein levels.

Protein production and degradation can be represented by the chemical reactions

$$\begin{array}{c}{M}_c\stackrel{k_p}{\to }{M}_c+P,\\ P\stackrel{{\gamma }_p}{\to }\varnothing ,\end{array}$$ (46)

where k_p is the protein translation per mRNA and γ_p represents the protein degradation rate. We denote by p(t) the number of protein molecules at time t. In the stochastic formulation of biochemical reactions in Eqs. 1 and 46, the probability P(m_n, m_c, p, t) of observing m_n molecules of nuclear pre-mRNA, m_c molecules of cytoplasmic mRNA, and p molecules of the protein at time t evolves according to the following CME:

$$\frac{dP\left(m_n,m_c,p,t\right)}{dt}=\sum _{z=0}^{m_n}{k}_m{\alpha }_{z}P\left(m_n-z,m_c,p,t\right),$$ (47)

$$+{\gamma }_e\left(m_n+1\right)P\left(m_n+1,m_c-1,p,t\right)+{\gamma }_c\left(m_c+1\right)P\left(m_n,m_c+1,p,t\right)+k_p{m}_{c}P\left(m_n,m_c,p-1,t\right)+{\gamma }_p\left(p+1\right)P\left(m_n,m_c,p+1,t\right),$$ (48)

$$-P\left(m_n,m_c,p,t\right)\left(k_m+{\gamma }_e{m}_n+{\gamma }_c{m}_c+k_p{m}_c+{\gamma }_{p}p\right).$$ (49)

From this CME, we obtain

$$\frac{d\langle p\rangle }{dt}=k_p\langle m_c\rangle -{\gamma }_p\langle p\rangle ,$$ (50a)

$$\frac{d\langle p^2\rangle }{dt}=k_p\langle m_c\rangle +{\gamma }_p\langle p\rangle +2k_p\langle m_{c}p\rangle -2{\gamma }_p\langle p^2\rangle ,$$ (50b)

$$\frac{d\langle m_{n}p\rangle }{dt}=k_p\langle m_n{m}_c\rangle +k_m\langle B\rangle \langle p\rangle -{\gamma }_e\langle m_{n}p\rangle -{\gamma }_p\langle m_{n}p\rangle ,$$ (50c)

$$\frac{d\langle m_{c}p\rangle }{dt}=k_p\langle m_c^2\rangle +{\gamma }_e\langle m_{n}p\rangle -{\gamma }_c\langle m_{c}p\rangle -{\gamma }_p\langle m_{c}p\rangle ,$$ (50d)

which, together with Eq. 39, describe the time-evolution of all the first- and second-order moments of m_n(t), m_c(t), and p(t). Steady-state analysis of the above moment equations yields the following expression for the steady-state coefficient of variation of protein levels:

$$\frac{{\sigma }_p^2}{{\overline{\langle p\rangle }}^2}=\frac{1}{\overline{\langle p\rangle }}+\frac{{\gamma }_p}{{\gamma }_p+{\gamma }_c}\frac{1}{\overline{\langle m_c\rangle }}\left(1+\frac{B_e{\gamma }_e}{{\gamma }_e+{\gamma }_c}\left(1+\frac{{\gamma }_c}{{\gamma }_p+{\gamma }_e}\right)\right).$$ (51)

In the limit γ_e >> γ_p, the protein noise level reduces to

$$\frac{{\sigma }_p^2}{{\overline{\langle p\rangle }}^2}\approx \frac{1}{\overline{\langle p\rangle }}+\frac{{\gamma }_p}{{\gamma }_p+{\gamma }_c}\frac{1}{\overline{\langle m_c\rangle }}\left(1+B_e\right),$$ (52)

and is insensitive to changes in γ_e (up to a level so that γ_p << γ_e remains satisfied). In general, protein half-lives are much longer than pre-mRNA export half-lives (γ_e >> γ_p) and in this physiologically relevant parameter regime, variability in protein level is invariant of γ_e. Thus, in contrast to stochastic variability at the mRNA level that is effectively attenuated by export processes, nuclear pre-mRNA export cannot buffer protein levels from transcriptional bursting (right panel in Fig. 4). For variations in γ_e to affect protein variability, the protein will have to be highly unstable or the export processes need to occur sufficiently slowly such that the export rate is comparable to the protein turnover rate (left panel in Fig. 4).

Figure 4.

Cell-to-cell variability in protein/mRNA levels measured by the coefficient-of-variation squared as a function of pre-mRNA export rate for different mRNA and protein stabilities. Transcriptional burst size is assumed to be geometrically distributed with an average burst size of 100 transcripts, $\overline{\langle m_c\rangle }=500$, $\overline{\langle p\rangle }=10,000$. Cell-to-cell variability is normalized by its corresponding value in the limit γ_e → ∞.

Discussion

In this article, we studied a model for stochastic dynamics of eukaryotic gene expression that incorporates a number of key biological mechanisms, namely burstlike transcription of messenger RNA in the nucleus, its subsequent transport out to the cytoplasm, and eventual translation into functional protein products (Fig. 1). For a minimal Markovian model of these mechanisms, we analyzed the corresponding chemical master equation by a range of available mathematical methods. We aimed to improve our understanding of how the stochastic fluctuations spurred in bursts of transcription are transmitted to downstream elements—i.e., cytoplasmic mRNA and protein.

Here we used the generating-function technique to solve the chemical master equation for the cytoplasmic mRNA dynamics. The solution was found in two steps: first, in transforming the master equation, we obtained a partial differential equation for the factorial-cumulant generating function of the sought-after cytoplasmic mRNA distribution; second, the partial differential equation was solved using the method of characteristics. The resulting formula (Eq. 25) for the generating function represents the exact solution to the chemical master equation. The presented method is similar to that used by Bokes et al. (42) to characterize the steady-state solution to the master equation of a two-stage model for prokaryotic gene expression; to our knowledge, the obtained solution is a new contribution to a growing collection of stochastic gene expression models for which an exact solution is known (31,42,50–55).

For the physiologically important case of geometrically distributed burst size, it was noted that, on the one hand, for very fast export rates, the cytoplasmic mRNA levels have negative binomial distribution; on the other hand, in the case of very slow export, we can expect a Poisson distribution. Because the negative binomial and Poisson distributions are qualitatively different from one another—the former typically has a much heavier tail than the latter—we aimed to investigate the properties of the cytoplasmic mRNA distribution for intermediate export rates. Using the exact solution we determined the asymptotic behavior of the tail of the distribution, noting that a geometric decay of the tail, similar to that of the negative binomial distribution, is typical for a wide range of export rates. Our results complement the previous analysis of cytoplasmic mRNA variance (45), confirming the important observation that transport attenuates the noise in mRNA levels (compare to Fig. 2); what we regard as the novelty of our approach is that we provide a more detailed account (including the exact solution) of the distribution for intermediate export rates.

The formulae for stationary cytoplasmic mRNA variance derived in Xiong et al. (45), as well the exact steady-state probability distributions, provide a static description of cytoplasmic mRNA stochasticity. Nevertheless, fluctuations are a dynamic process, and the timescale on which these fluctuations operate is of equal importance for determining the extent to which these fluctuations are transmitted downstream: clearly, short-lived fluctuations have a lesser impact than long-term ones. With this in mind, we determined the autocorrelation function of cytoplasmic mRNA level, finding that slower export rates imply longer autocorrelation times (compare to Fig. 3). Combining this result with the observations made in Xiong et al. (45) we conclude that slower transport implies lesser, but more permanent fluctuations, and it is the balance between these two effects that ultimately determines the extent to which mRNA fluctuations are passed downstream to the protein stage.

Indeed, it turns out that in physiologically relevant situations these counteracting effects cancel each other out, and the protein noise remains relatively unaffected by the speed of mRNA transport (see right panel of Fig. 4). For example, consider a gene with high levels of transcriptional bursting, a 1 h mRNA half-life, and a 20 h protein half-life. Increasing the pre-mRNA export time from 1 min to 1 h reduces stochastic variability in mRNA levels by 50% (Fig. 4). However, this effect is balanced by an approximately twofold increase in mRNA autocorrelation time (see Fig. 2; note that mRNA autocorrelation time t_m is defined as R(t_m) = 0.5, where R(t) is given by Eq. 45), resulting in a reduction of protein noise level by only 2%. Our analysis shows that for pre-mRNA export to significantly attenuate the effects of transcriptional bursting at the protein level, the export rate will have to be comparable to the protein turnover rate. Given recent genomewide measurements of protein half-lives in mammalian cells (see Fig. 2 in Schwanhäusser et al. (35)) and export times ranging from a few minutes to 1 h (34), γ_e ≈ γ_p seems highly unlikely.

The impact of the transport step on the noise in cytoplasmic mRNA and protein levels can alternatively be visualized by considering the transient response of the system to a single transcriptional burst (Fig. 5). This is simply obtained as a solution to the system of differential equations

$$\begin{array}{c}\frac{d\langle m_n\rangle }{dt}=-{\gamma }_e\langle m_n\rangle ,\\ \frac{d\langle m_c\rangle }{dt}={\gamma }_e\langle m_n\rangle -{\gamma }_c\langle m_c\rangle ,\\ \frac{d\langle p\rangle }{dt}=k_p\langle m_c\rangle -{\gamma }_p\langle p\rangle ,\end{array}$$ (53)

satisfying the initial conditions 〈m_n(0)〉 = b, 〈m_c(0)〉 = 0, and 〈p(0)〉 = 0, where b is a given size of the burst. Provided that γ_e >> γ_c >> γ_p (Fig. 5, left column), the transcriptional burst spurs a successive peak in the cytoplasmic mRNA level, and a slower burst in protein levels. If γ_e is decreased to values comparable to γ_c, the peak in cytoplasmic mRNA level is significantly reduced; however, the protein burst reaches almost the same height as before (Fig. 5, right column). Thus, for a slower rate of pre-mRNA export, the impact of single transcriptional burst is attenuated at the cytoplasmic mRNA level but not at the protein level. This dynamics illustrates a connection between our results on stochastic variability and the qualitative properties of the transient response of the transport/translation machinery to a transcriptional burst.

Figure 5.

Transmission of a burst of transcriptional activity through the transport and translation channels in the case of fast export (left column) and slow export (right column). The impact of a single transcriptional burst is attenuated at the mRNA level for slow export, but the peak of the protein burst remains the same for both fast and slow export.

In analyzing the effects of pre-mRNA export on stochastic gene-expression we completely ignored the presence of any direct/indirect negative feedback loops. These feedbacks can be manifested through saturation of the export machinery, in particular, the nuclear pore complexes (NPC) that are involved in exporting pre-mRNAs across the nuclear envelope. As shown previously (45), saturation of the export machinery can be effective in attenuating expression variability arising from transcriptional bursting. For example, if NPCs operate at saturation then mRNA production rate is independent of nuclear pre-mRNA levels, and mRNA/protein levels are completely buffered from underlying transcriptional bursts. However, experimental visualization of NPC-RNA interactions show no NPC pile-up, suggesting that the export machinery is operating far from saturation (34). Clearly, further experimental and theoretical work is required to identify regulatory mechanisms at the pre-mRNA export level that may function to filter out stochasticity in gene-expression.

In conclusion, we analyzed a Markovian model for eukaryotic expression based on mechanisms depicted in Fig. 1 by a range of mathematical techniques, namely by generating functions, autocorrelation analysis, and variance decomposition. Mathematical analysis provided us, as it often does, with intricate algebraic formulae: an integral representation of the generating function (Eq. 25), a formula for cytoplasmic autocorrelation (Eq. 45), and the protein variance equation (Eq. 51). The challenge lay not as much in derivation of these formulae, but rather in their interpretation with respect to the underlying biological phenomena. Our results indicate that the protein noise levels are robust with respect to change in the mRNA export rate; it is left to further investigation, whether theoretical or experimental, to determine the full implications of robustness of this kind in gene regulation. We also believe that the methodology applied in this article will be helpful time and again in analyzing other models for stochastic gene expression.