Propagation of Extrinsic Fluctuations in Biochemical Birth–Death Processes

Abstract

Biochemical reactions are often subject to a complex fluctuating environment, which means that the corresponding reaction rates may themselves be time-varying and stochastic. If the environmental noise is common to a population of downstream processes, then the resulting rate fluctuations will induce statistical correlations between them. In this paper we investigate how such correlations depend on the form of environmental noise by considering a simple birth–death process with dynamical disorder in the birth rate. In particular, we derive expressions for the second-order statistics of two birth–death processes evolving in the same noisy environment. We find that these statistics not only depend on the second-order statistics of the environment, but the full generator of the process describing it, thus providing useful information about the environment. We illustrate our theory by considering applications to stochastic gene transcription and cell sensing.

Appendix: Continuous Environmental Noise

One of the useful features of the method developed in our paper is that a similar analysis can be applied to the case of the environmental variable $\xi (t)$ being a continuous stochastic process $X(t)\in \varOmega \subset {{\mathbb {R}}}$, satisfying the SDE

$$\begin{aligned} \mathrm{d}X=F(X)\mathrm{d}t+\sqrt{2D}\mathrm{d}W(t), \end{aligned}$$

(58)

with W(t) a Wiener process:

$$\begin{aligned} \langle W(t)\rangle =0,\quad \langle W(t)W(t')\rangle =\min (t,t'). \end{aligned}$$

As a further simplification, we will assume that X(t) is a stationary process with mean ${\overline{X}}$ and covariance

$$\begin{aligned} \varSigma (\tau ):=\langle (X(t)-{\overline{X}})(X(t+\tau )-{\overline{X}})\rangle = \varSigma _0 \mathrm {e}^{-\tau /\tau _c}, \end{aligned}$$

(59)

where $\tau _c$ is the correlation time. For example, if X(t) evolves according to an Ornstein-Uhlnebeck process for which $F(X)=-\eta X$ in Eq. (58), then ${\overline{X}}=0$, $\tau _c=1/\eta$ and $\varSigma _0=D/\eta$ (Gardiner 2009). The discrete probability distribution $q_j(t)$ evolving according to the master Eq. (15) is replaced by the probability density $q(x,t)dx={{\mathbb {P}}}[x<X(t)<x+dx]$, which satisfies the Fokker–Planck equation

$$\begin{aligned} \frac{\partial q}{\partial t}={{\mathbb {L}}}_x q, \end{aligned}$$

(60)

where $L_x$ is the generator of the forward Fokker–Planck equation for the SDE (58), which is given by the differential operator

$$\begin{aligned} {{\mathbb {L}}}_x=-\frac{\partial }{\partial x}F(x)+D\frac{\partial ^2}{\partial x^2}, \end{aligned}$$

(61)

which is supplemented by appropriate boundary conditions on $\partial \varOmega$. The analysis now proceeds along similar lines to Sects. 3 and 4, except that the matrix generator $\mathbf{Q}$ is replaced by the differential operator ${{\mathbb {L}}}_x$.

The first step is to consider a single BD process and introduce the joint probability density

$$\begin{aligned}&{{\mathcal {P}}}({{\mathbf {p}}},x,t)d{{\mathbf {p}}}dx \\&\quad ={{\mathbb {P}}}[{{\mathbf {p}}}<\mathbf{P}(t)<{{\mathbf {p}}}+d{{\mathbf {p}}}, x<X(t)<x+dx], \end{aligned}$$

which evolves according to the CK equation (see also Eq. 24)

$$\begin{aligned} \frac{\partial {{\mathcal {P}}}}{\partial t}=-\sum _{m\ge 0}\frac{\partial }{\partial p_m}V_m({{\mathbf {p}}},x){\mathcal P}({{\mathbf {p}}},x,t)+{{\mathbb {L}}}_{x}{{\mathcal {P}}}({{\mathbf {p}}},x,t), \end{aligned}$$

(62)

where $V_m$ is given by Eq. (21). Introduce the first-order moments of ${{\mathcal {P}}}$,

$$\begin{aligned} P_{m}(x,t)=\int p_m{{\mathcal {P}}}({{\mathbf {p}}},x,t)d{{\mathbf {p}}}, \end{aligned}$$

Multiplying both sides of (62) by $p_m$ and then integrating with respect to ${{\mathbf {p}}}$ gives the effective differential CK equation

$$\begin{aligned} \frac{\partial P_{m}}{\partial t}&=x [\alpha _0-\alpha _1(m-1)] P_{m-1}+\beta (m+1)P_{m+1} \nonumber \\&\qquad -[x(\alpha _0-\alpha _1 m)+\beta m]P_{m} +{\mathbb L}_{x}P_{m}. \end{aligned}$$

(63)

We can now derive an equation for the first-order moments

$$\begin{aligned} z(x,t)=\sum _{m\ge 0}^NmP_{m}(x,t). \end{aligned}$$

Multiplying both sides of (63) by m and summing over m yields

$$\begin{aligned} \frac{\partial z}{\partial t}=x(\alpha _0 \rho (x)-\alpha _1 z(x,t))-\beta z(x,t)+{{\mathbb {L}}}_xz(x,t), \end{aligned}$$

(64)

where $\rho (x)$ is the stationary density of the stochastic process X(t). Taking the limit $t\rightarrow \infty$ then yields the time-independent equation

$$\begin{aligned}{}[{{\mathbb {L}}}_x-\alpha _1x -\beta ]z^{\infty }(x)=-x\alpha _0 \rho (x). \end{aligned}$$

(65)

This can be inverted to give the integral solution

$$\begin{aligned} z^{\infty }(x)=\alpha _0\int _{\varOmega }G_1(x,x') x'\rho (x')\mathrm{d}x', \end{aligned}$$

(66)

where $G_k(x,x')$ is the Green’s function defined according to the equation

$$\begin{aligned}{}[{{\mathbb {L}}}_{x}-k(\alpha _1x +\beta )]G_k(x,x')=-\delta (x-x'), \end{aligned}$$

(67)

together with boundary conditions on $\partial \varOmega$.

Now consider two identical BD processes $Z_1(t)$ and $Z_2(t)$ as in Sect. 4. Introduce the joint probability density

$$\begin{aligned}&{{\mathcal {P}}}({{\mathbf {p}}}^1,{{\mathbf {p}}}^2,x,t)d{{\mathbf {p}}}^1d{{\mathbf {p}}}^2dx \\&\quad ={{\mathbb {P}}}[({{\mathbf {p}}}^1,{{\mathbf {p}}}^2)<( \mathbf{P}^1(t),\mathbf{P}^2(t))<({{\mathbf {p}}}^1+d{{\mathbf {p}}}^1,{{\mathbf {p}}}^2+d{{\mathbf {p}}}^2), x<X(t)<x+dx], \end{aligned}$$

The differential CK Eq. (37) becomes

$$\begin{aligned} \frac{\partial {{\mathcal {P}}}}{\partial t}=-\sum _{r=1,2}\sum _{m\ge 0}\frac{\partial }{\partial p_m^r}V_m({{\mathbf {p}}}^r,x){\mathcal P}({{\mathbf {p}}}^1,{{\mathbf {p}}}^2,x,t)+{{\mathbb {L}}}_{x}{{\mathcal {P}}}({{\mathbf {p}}}^1,{{\mathbf {p}}}^2,x,t), \end{aligned}$$

(68)

where $V_m$ is given by Eq. (21). Define the second-order cross-moments of the density ${{\mathcal {P}}}_n$ according to

$$\begin{aligned} C_{m_1m_2}(x,t)= \int p^1_{m_1}p^2_{m_2}{\mathcal P}({{\mathbf {p}}}^1,{{\mathbf {p}}}^2,x,t)d{{\mathbf {p}}}^1d{{\mathbf {p}}}^2 \end{aligned}$$

(69)

These satisfy the CK equation

$$\begin{aligned} \frac{\partial C_{m_1m_2}}{\partial t}&=x[\alpha _0-\alpha _1 (m_1-1)] C_{m_1-1,m_2}(x,t)+\beta (m_1+1)C_{m_1+1,m_2}(x,t) \nonumber \\&\quad +x[\alpha _0-\alpha _1(m_2-1)]C_{m_1,m_2-1}(x,t)+\beta (m_2+1)C_{m_1,m_2+1}(x,t)\nonumber \\&\quad -[x(2\alpha _0 -\alpha _1 (m_1+m_2)) +\beta (m_1+m_2)]C_{m_1m_2}(x,t)\nonumber \\&\quad +{{\mathbb {L}}}_xC_{m_1m_2}(x,t). \end{aligned}$$

(70)

We can now derive equations for the second-order moments

$$\begin{aligned} {\mu }(x,t)=\sum _{m_1\ge 0}\sum _{m_2\ge 0}m_1m_2C_{m_1m_2}(x,t), \end{aligned}$$

(71)

by multiplying both sides of (70) by $m_1m_2$ and summing over $m_1$ and $m_2$. This yields

$$\begin{aligned} \frac{\partial {\mu }}{\partial t}=2x (\alpha _0 z(x,t)-\alpha _1\mu (x,t)) -2\beta {\mu }(x,t)+{\mathbb L}_x{\mu }(x,t). \end{aligned}$$

(72)

Taking the limit $t\rightarrow \infty$, we obtain the time-independent equation

$$\begin{aligned}{}[{{\mathbb {L}}}_x-2(\alpha _1x +\beta ) ]\mu ^{\infty }(x)=-2x\alpha _0z^{\infty }(x). \end{aligned}$$

(73)

This can be inverted to give the integral solution

$$\begin{aligned} \mu ^{\infty }(x)=2\alpha _0\int _{\varOmega }G_2(x,y) yz^{\infty }(y)\mathrm{d}y . \end{aligned}$$

(74)

Finally, substituting for $z^{\infty }$ using Eq. (66), we obtain the integral solution

$$\begin{aligned} \mu _{\infty }:= \int _{\varOmega }\mu ^{\infty }(x)\mathrm{d}x=2\alpha _0^2\int _{\varOmega }\left[ \int _{\varOmega } \int _{\varOmega } G_2(x,y)yG_1(y,x') x '\rho (x')\mathrm{d}x'\mathrm{d}y \right] dx.\nonumber \\ \end{aligned}$$

(75)

In the special case $\alpha _1=0$,

$$\begin{aligned} \mu _{\infty }= \frac{\alpha _0^2}{\beta }\int _{\varOmega }\int _{\varOmega } x G(x,x')x'\rho (x')\mathrm{d}x'\mathrm{d}x, \end{aligned}$$

(76)

where

$$\begin{aligned}{}[{{\mathbb {L}}}_{x}-\beta ]G(x,x')=-\delta (x-x'). \end{aligned}$$

(77)

Equations (75) and (76) are the analogs of Eqs. (44) and (45).