Analytical results for non-Markovian models of bursty gene expression

Zihao Wang, Zhenquan Zhang, and Tianshou Zhou

Phys. Rev. E 101, 052406 – Published 22 May 2020

Abstract

Modeling stochastic gene expression has long relied on Markovian hypothesis. In recent years, however, this hypothesis is challenged by the increasing availability of time-resolved data. Correspondingly, there is considerable interest in understanding how non-Markovian reaction kinetics of gene expression impact protein variations across a population of genetically identical cells. Here, we analyze a stochastic model of gene expression with arbitrary waiting-time distributions, which includes existing gene models as its special cases. We find that stationary probabilistic behavior of this non-Markovian system is exactly the same as that of an equivalent Markovian system with the same substrates. Based on this fact, we derive analytical results, which provide insight into the roles of feedback regulation and molecular memory in controlling the protein noise and properties of the steady states, which are inaccessible via existing methodology. Our results also provide quantitative insight into diverse cellular processes involving stochastic sources of gene expression and molecular memory.

Received 11 April 2019
Revised 5 May 2019
Accepted 24 March 2020

DOI:https://doi.org/10.1103/PhysRevE.101.052406

Physics Subject Headings (PhySH)

Chemical kinetics, dynamics & catalysis Fluctuations Fluctuations & noise Gene expression

Physics of Living SystemsStatistical Physics & ThermodynamicsInterdisciplinary PhysicsNetworks

Authors & Affiliations

Zihao Wang, Zhenquan Zhang , and Tianshou Zhou ^*

Guangdong Province Key Laboratory of Computational Science School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China

^*Corresponding author: mcszhtsh@mail.sysu.edu.cn

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Vol. 101, Iss. 5 — May 2020

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Figure 1
This paper proves that the stationary behavior of a non-Markovian gene expression system with general waiting-time distributions $ψ_{i} (t; n)$ (a) is exactly the same as that of an equivalent Markovian gene expression system with exponential waiting-time distributions $K_{i} (n) e^{- K_{i} (n) t}$ (b), where $1 \leq i \leq 4$ , $X$ stands for protein and $n$ for its molecular number, and $K_{i} (n)$ represents the effective transition rate (see the main text). (c) It is demonstrated that dynamic behavior of a non-Markovian gene expression system described by $D N A \overset{(μ^{k} / Γ (k)) t^{k - 1} e^{- μ t}}{\to} D N A + B \cdot X$ and $X \overset{δ e^{- δ t}}{\to} \emptyset$ can be well approximated by that of an equivalent Markovian gene expression system described by $D N A \overset{K_{1} (n)}{\to} D N A + B \cdot X$ and $X \overset{K_{2} (n) = n δ e^{- n δ t}}{\to} \emptyset$ , where $B$ represents burst size that is assumed to follow a geometric distribution: $p r o b {B = i} = b^{i} / {(1 + b)}^{i + 1}$ , and $Γ (\cdot)$ is the common Gamma function. In (c), $D$ represents the Kullback-Leibler divergence between two distributions in the sense of Laplace transform. Parameter values are set as $k = 2$ , $μ = 10$ , $δ = 1,$ and $b = 2$ .
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Figure 2
Convergence of series $\sum_{n = 1}^{\infty} a_{n} / n!$ and $\sum_{n = 1}^{\infty} b_{n} / n!$ , where bursts are not considered. (a), (b) ${\tilde{K}}_{1} (n) = \tilde{α}$ , ${\tilde{K}}_{2} (n) = \tilde{β}$ , and ${\tilde{K}}_{3} (n) = \tilde{μ}$ , where $\tilde{α} = 1$ , $\tilde{β} = 10$ , $\tilde{μ} = 10$ for $α + β > 2 δ$ ; $\tilde{α} = 1$ , $\tilde{β} = 1$ , $\tilde{μ} = 10$ for $\tilde{α} + \tilde{β} = 2$ ; and $\tilde{α} = 0.1$ , $\tilde{β} = 0 . 5$ , $\tilde{μ} = 10$ for $α + β < 2 δ$ . (c), (d) ${\tilde{K}}_{1} (n) = \tilde{α} + \tilde{f} \frac{n^{h}}{K^{h} + n^{h}}$ , ${\tilde{K}}_{2} (n) = \tilde{β} + \tilde{g} \frac{n^{h}}{K^{h} + n^{h}}$ , ${\tilde{K}}_{3} (n) = \tilde{μ}$ , where $\tilde{α} = 1$ , $\tilde{β} = 10$ , $\tilde{μ} = 10$ , $\tilde{f} = 1.2$ , $\tilde{g} = 1$ , $h_{1} = h_{2} = 2$ , $K_{1} = K_{2} = \sqrt{10}$ for $\tilde{α} + \tilde{f} + \tilde{β} + \tilde{g} > 2$ ; $\tilde{α} = 0 . 5$ , $\tilde{β} = 0 . 5$ , $\tilde{μ} = 10$ , $\tilde{f} = 0 . 5$ , $\tilde{g} = 0 . 5$ , $h_{1} = h_{2} = 2$ , $K_{1} = K_{2} = \sqrt{10}$ for $\tilde{α} + \tilde{f} + \tilde{β} + \tilde{g} = 2$ ; and $\tilde{α} = 0.1$ , $\tilde{β} = 0.1$ , $\tilde{μ} = 10$ , $\tilde{f} = 0.2$ , $\tilde{g} = 0.2$ , $h_{1} = h_{2} = 2$ , $K_{1} = K_{2} = \sqrt{10}$ for $\tilde{α} + \tilde{f} + \tilde{β} + \tilde{g} < 2$ . The tilde bar represents the normalization by $δ$ , e.g., ${\tilde{K}}_{1} (n) = K_{1} (n) / δ$ . Note that the barred parameters are actually the same as the corresponding unbarred parameters since $δ = 1$ has been set.
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Figure 3
Effects of molecular memory on protein expression. This is the case where burst is not considered, where solid lines represent the results obtained by theoretical prediction, whereas circles represent the results obtained by the Gillespie algorithm [34]. Parameter values are set as: $β = 5$ , $b = 1$ (i.e., burst is not considered), and $δ = 1$ . Other parameters are determined by relations: $α = 5 I_{1}$ , $μ = 10 I_{3}$ , $f_{1} = 0.2 I_{1}$ , $f_{3} = 0.2 I_{3}$ . Here we set $λ_{1} (n) = α + n f_{1}$ and $λ_{3} (n) = μ + n f_{3}$ .
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Figure 4
(a) Molecular memory can induce bimodal protein expressions: Case 1, where empty circles represent the results obtained by a numerical method [36] whereas the solid curves represent the results obtained by theoretical prediction. (b) Dependence of the peaks of stationary distribution on memory index $I_{1}$ , where blue symbols correspond to the left peak (i.e., the peak at $x = 0$ ) whereas orange symbols to the right peak [(i.e., the peak at $x = b μ$ )]. In (b), the vertical dashed line represents the boundary of unimodality and bimodality, and the solid curves conduct numerical results (since analytical results cannot be given for some $I_{1}$ ). In (a) and (b), we set $λ_{1} (n) = α$ , $α = 2 . 2$ , $β = 0 . 8$ , $μ = 2$ , $b = 2$ , $δ = 1$ .
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