Phase reduction of stochastic biochemical oscillators
File(s)
Published version
Author(s)
Bressloff, Paul C
MacLaurin, James N
Type
Journal Article
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 \epsilon = \Omega - 1
, where
\Omega characterizes the system size. For example, \Omega could be the mean number of proteins produced by
a gene regulatory network. In the deterministic limit \Omega \rightarrow \infty , 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 \Omega . This contrasts
with the phase equation obtained under the system-size expansion, which is only accurate up to
times O(\Omega ). In particular, we show that for a constant C, the probability that the system leaves an
O(\zeta ) neighborhood of the limit cycle before time T scales as T exp( - C\Omega b\zeta 2
), where b is the rate of
attraction to the limit cycle. We illustrate our analysis using the example of a chemical Brusselator.
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 \epsilon = \Omega - 1
, where
\Omega characterizes the system size. For example, \Omega could be the mean number of proteins produced by
a gene regulatory network. In the deterministic limit \Omega \rightarrow \infty , 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 \Omega . This contrasts
with the phase equation obtained under the system-size expansion, which is only accurate up to
times O(\Omega ). In particular, we show that for a constant C, the probability that the system leaves an
O(\zeta ) neighborhood of the limit cycle before time T scales as T exp( - C\Omega b\zeta 2
), where b is the rate of
attraction to the limit cycle. We illustrate our analysis using the example of a chemical Brusselator.
Date Issued
2020-01
Date Acceptance
2019-10-01
Citation
SIAM Journal on Applied Dynamical Systems, 2020, 19 (1), pp.151-180
ISSN
1536-0040
Publisher
Society for Industrial and Applied Mathematics
Start Page
151
End Page
180
Journal / Book Title
SIAM Journal on Applied Dynamical Systems
Volume
19
Issue
1
Copyright Statement
© 2020 Society for Industrial and Applied Mathematics
Identifier
http://dx.doi.org/10.1137/18m1221205
Publication Status
Published
Date Publish Online
2020-01-08