Linking Stochastic Dynamics to Population Distribution: An Analytical Framework of Gene Expression

We present an analytical framework describing the steady-state distribution of protein concentration in live cells, considering that protein production occurs in random bursts with an exponentially distributed number of molecules. We extend this framework for cases of transcription autoregulation and noise propagation in a simple genetic network. This model allows for the extraction of kinetic parameters of gene expression from steady-state distributions of protein concentration in a cell population, which are available from single cell data obtained by flow cytometry or fluorescence microscopy.

Figure 1

Analytical solution to stochasticity in gene expression. (a) Kinetic scheme for describing noise in gene expression. (b) Distribution of protein concentration in a cell population. Simulation parameters are biologically relevant for bacteria: ${\gamma }_1=0.01{\mathrm{s}}^{-1}$, ${\gamma }_2=4\times {10}^{-4}{\mathrm{s}}^{-1}$, $V_0=1.7\mathrm{fl}$ A: $a=0.5$, $b=30\ln 2$; B: $a=5$, $b=30\ln 2$; C: $a=50$, $b=3\ln 2$; D: $a=50$, $b=30\ln 2$. Simulations include random (binomial) partitioning of protein molecules during cell division and exponential growth in cell volume. Gamma distribution [lines, Eq. (5)] matches stochastic stimulation results (dots) well over a large range of parameter values. (c),(d) Simulation and analytic results for $a<1$ and $a>1$, corresponding to curves A, B of (b), respectively. In the first case, $p(x)$ peaks at $x=0$, regardless of the value of $b$. In the second case, $p(x)$ peaks at $x>0$. Insets show typical simulated trajectories.

Figure 2

Autoregulation. (a) Dots show simulated distribution of protein concentration without (curve A) and with (B) negative feedback. Lines are analytical solutions [Eq. (9)]. Parameters: $a=10$, $b=20$, $k=70\mathrm{n}M$, $\epsilon =0.05$, $H=1$. (b) Same as (a), for positive feedback. Inset: kinetic scheme. X: the autoregulated transcription factor, m: corresponding mRNA. Parameters: as in (a) except for : $H=-1$ (C, cyan) and $H=-4$, $\epsilon =0.2$ (D, magenta). In this case, $a\epsilon >1$ resulting in low peak at $x>0$. (c) Mean of $x$ as a function of $k$ calculated from Eq. (9), for positive autoregulation (black curve). Parameters are as in [(b), curve C)]. Circles: values calculated from stochastic simulations. Representative distributions $p(x)$ are shown in insets. (d) Standard deviation of $x$ vs $k$ [(parameters as in (c)]. Inset: the Fano factor, ${\sigma }^2/⟨x⟩$, peaking at the transition region.

Figure 3

Joint distribution of a repressor, $R$, and a regulated gene, $x$. Contour plots of steady-state distributions obtained by analytic solution [Eq. (11), top row] and by numerical stochastic simulation (bottom row) for a case when fluctuation in the repressor is slow. Distributions are shown for 2 average values of repressor concentration. The black dashed line represents the deterministic steady-state solution for $x(R)$. Averages of the distributions are marked (X). The distribution $p(x)$ is shown on the right (blue line), compared to the case of fast repressor fluctuations [dashed-red line, Eq. (10): negligible extrinsic noise]. Parameters: ${\gamma }_{1x}$, ${\gamma }_{2x}$, $V$: as in Fig. 1b; $k=20\mathrm{n}M$, $k_{1x}=0.01{\mathrm{s}}^{-1}$, $k_{2x}=0.4{\mathrm{s}}^{-1}$, ($a_x=25$, $b_x=40$); ${\gamma }_{1R}=0.02{\mathrm{s}}^{-1}$, ${\gamma }_{2R}={\gamma }_{2x}$, $k_{2R}=k_{2x}$, ($b_R=20$); left: $k_{1R}=4\times {10}^{-5}{\mathrm{s}}^{-1}$ ($a_R=0.1$), right: $k_{1R}=4\times {10}^{-3}{\mathrm{s}}^{-1}$ ($a_R=10$).