Fluctuating-rate model with multiple gene states
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Mathematical Biology
Fluctuating-rate model with multiple gene states Jingwei Li1 · Hao Ge2
· Yunxin Zhang3
Received: 20 February 2019 / Revised: 28 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Multiple phenotypic states of single cells often co-exist in the presence of positive feedbacks. Stochastic gene-state switchings and low copy numbers of proteins in single cells cause considerable fluctuations. The chemical master equation (CME) is a powerful tool that describes the dynamics of single cells, but it may be overly complicated. Among many simplified models, a fluctuating-rate (FR) model has been proposed recently to approximate the full CME model in the realistic intermediate region of gene-state switchings. However, only the scenario with two gene states has been carefully analysed. In this paper, we generalise the FR model to the case with multiple gene states, in which the mathematical derivation becomes more complicated. The leading order of fluctuations around each phenotypic state, as well as the transition rates between phenotypic states, in the intermediate gene-state switching region is characterized by the rate function of the stationary distribution of the FR model in the Freidlin–Wentzell-type large deviation principle (LDP). Under certain reasonable assumptions, we show that the derivative of the rate function is equal to the unique nontrivial solution of a dominant generalised eigenvalue problem, leading to a new numerical algorithm for obtaining the LDP rate function directly. Furthermore, we prove the Lyapunov property of the rate function for the corresponding deterministic mean-field dynamics. Finally, through a tristable example, we show that the local fluctuations (the asymptotic variance of the stationary distribution at each phenotypic state) in the intermediate and rapid regions of gene-state switchings are different. Finally, a tri-stable example is constructed to illustrate the validity of our theory. Keywords Fluctuating-rate model · Nonequilibrium landscape function · Transition rates · Dominant generalised eigenvalue Mathematics Subject Classification 60F10 · 47A75 · 15B48 Abbreviations CME Chemical master equation FR Fluctuating-rate
Extended author information available on the last page of the article
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J. Li et al.
DMFD LDP NLF GE DGE
Deterministic mean-field dynamics Large deviation principle Nonequilibrium landscape function Generalised eigenvalue Dominant generalised eigenvalue
1 Introduction Genes switch among different states due to the regulation of transcription factors and synthesise proteins at a state-dependent rate. This paper considers self-regulatory genes with positive feedback regulations, in which the transcription factors are synthesised by the regulated gene itself and reversely help the genes switch to a state with a relatively large synthesis rate. This may lead to a copy-number distribution with multiple modals, which, from a biological point of view, correspond to multiple phenotypic states of a living cell c
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