Noise dissipation in gene regulatory networks via second order statistics of networks of infinite server queues

Abstract

RNA and protein concentrations within cells constantly fluctuate. Some molecular species typically have very low copy numbers, so stochastic changes in their abundances can dramatically alter cellular concentration levels. Such noise can be harmful through constrained functionality or reduced efficiency. Gene regulatory networks have evolved to be robust in the face of noise. We obtain exact analytical expressions for noise dissipation in an idealised stochastic model of a gene regulatory network. We show that noise decays exponentially fast. The decay rate for RNA molecular counts is given by the integral of the tail of the cumulative distribution function of the degradation time. For proteins, it is given by the slowest rate-limiting step of RNA degradation or proteolytic breakdown. This is intuitive because memory of the chemical composition of the system is manifested through molecular persistence. The results are obtained by analysing a non-standard tandem of infinite server queues, in which the number of customers present in one queue modulates the arrival rate into the next.

Appendix

In this section we reproduce a number of definitions from Last and Penrose (2017) for the convenience of the reader.

Let $\left( \mathbb {X},{\mathcal {X}}\right)$ be a measurable space. Let the space of all measures $\mu$ on $\mathbb {X}$ with the property that $\mu (A)\in \mathbb {N}_0$ for any $A\in {\mathcal {X}}$, be denoted by ${{\textbf {N}}}_{<\infty }(\mathbb {X})$, or just ${{\textbf {N}}}_{<\infty }$ for short. Now write ${{\textbf {N}}}(\mathbb {X})$ (or just ${{\textbf {N}}}$ for short) for the space of all measures that can be written as a countable sum of elements of ${{\textbf {N}}}_{<\infty }$. We define the $\sigma$-algebra ${\mathcal {N}}$ on ${{\textbf {N}}}$ by

$$\begin{aligned} {\mathcal {N}}:=\sigma \left( \left\{ \mu \in {{\textbf {N}}}:\mu (A)=k\right\} :A\in {\mathcal {X}},k\in \mathbb {N}_0\right) . \end{aligned}$$

Definition 10

(Last and Penrose (2017) Definition 2.1) Given a probability space $\left( {\varOmega },{\mathcal {F}},{{\mathbb {P}}}\right)$, a point process on $\mathbb {X}$ is a measurable mapping $\eta :{\varOmega }\rightarrow {{\textbf {N}}}$.

Definition 11

(Last and Penrose (2017) Definition 2.5) The intensity measure of a point process $\eta$ on $\mathbb {X}$, is the measure $\lambda$ defined by $\lambda (A):=\mathbb {E}[\eta (A)]$, where $A\in {\mathcal {X}}$.

Note that $\lambda (A)$ is well-defined, though possibly infinite because $\eta (A)$ is a non-negative random variable.

Definition 12

(Last and Penrose (2017) Page 10) We say that a measure $\nu$ on $\mathbb {X}$ is s-finite if it is a countable sum of finite measures.

Definition 13

(Last and Penrose (2017) Definition 3.1) Let $\lambda$ be an s-finite measure on $\mathbb {X}$. A Poisson process with intensity measure $\lambda$ is a point process $\eta$ on $\mathbb {X}$ with the following two properties:

• 1. For every $B\in {\mathcal {X}}$, the distribution of $\eta (B)$ is Poisson with parameter $\lambda (B)$.

• 2. For every $m\in \mathbb {N}$ and all pairwise disjoint sets $B_1,\ldots ,B_m\in {\mathcal {X}}$ the random variables $\eta (B_1),\ldots ,\eta (B_m)$ are mutually independent.

Denote by ${{\textbf {M}}}(\mathbb {X})={{\textbf {M}}}$, the set of s-finite measures on $\mathbb {X}$. We denote by ${\mathcal {M}}(\mathbb {X})={\mathcal {M}}$ the $\sigma$-algebra generated by $\left\{ \mu \in {{\textbf {M}}}:\mu (B)\le t\right\} ,B\in {\mathcal {X}},t\in \mathbb {R}_+$. This $\sigma$-algebra is the smallest possible to make the mappings $\mu \mapsto \mu (B)$ measurable for all B in ${\mathcal {X}}$.

In what follows there is a fixed probability space $({\varOmega },{\mathcal {F}},{{\mathbb {P}}})$ on which all random elements are defined.

Definition 14

(Last and Penrose (2017) Definition 13.1) A random measure on $\mathbb {X}$ is a random element $\xi$ of the space $({{\textbf {M}}},{\mathcal {M}})$, that is, a measurable mapping $\xi :{\varOmega }\rightarrow {{\textbf {M}}}$.

Now let $\lambda \in {{\textbf {M}}}(\mathbb {X})$ and write ${\varPi }_{\lambda }$ to denote the distribution of a Poisson point process with intensity measure $\lambda$. Theorem 3.6 of Last and Penrose (2017) guarantees that such a process exists.

Definition 15

(Last and Penrose (2017) Definition 13.5) Let $\xi$ be a random measure on $\mathbb {X}$. A point process $\eta$ on $\mathbb {X}$ is called a Cox process directed by $\xi$ if

$$\begin{aligned} {{\mathbb {P}}}(\eta \in A|\xi )={\varPi }_{\xi }(A),\quad {{\mathbb {P}}}\text {-a.s.}, A\in {\mathcal {N}}. \end{aligned}$$

Then $\xi$ is called a directing random measure of $\eta$.