Phase reduction of stochastic biochemical oscillators

Document Type

Article

Publication Date

1-1-2020

Abstract

A common method for analyzing the effects of molecular noise in chemical reaction networks is to approximate the underlying chemical master equation by a Fokker-Planck equation and to study the statistics of the associated chemical Langevin equation. This so-called system-size expansion involves performing a perturbation expansion with respect to a small dimensionless parameter ε = Ω 1, where Ω characterizes the system size. For example, Ω could be the mean number of proteins produced by a gene regulatory network. In the deterministic limit Ω → ∞, the chemical reaction network evolves according to a system of ordinary differential equations based on classical mass action kinetics. In this paper we develop a phase reduction method for chemical reaction networks that support a stable limit cycle in the deterministic limit. We present a variational principle for the phase reduction, yielding an exact analytic expression for the resulting phase dynamics. We demonstrate that this decomposition is accurate over timescales that are exponential in the system size Ω. This contrasts with the phase equation obtained under the system-size expansion, which is only accurate up to times O(Ω). In particular, we show that for a constant C, the probability that the system leaves an O(ζ) neighborhood of the limit cycle before time T scales as T exp(C Ω b ζ 2), where b is the rate of attraction to the limit cycle. We illustrate our analysis using the example of a chemical Brusselator.

Identifier

85080039007 (Scopus)

Publication Title

SIAM Journal on Applied Dynamical Systems

External Full Text Location

https://doi.org/10.1137/18M1221205

e-ISSN

15360040

First Page

151

Last Page

180

Issue

1

Volume

19

Grant

1755431

Fund Ref

National Science Foundation

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