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Detailed balance, local detailed balance, and global potential for stochastic chemical reaction networks

Part of: Limit theorems Mathematical biology in general Chemistry Physiological, cellular and medical topics Markov processes

Published online by Cambridge University Press:  08 October 2021

Chen Jia ,
Da-Quan Jiang  and
Youming Li
Chen Jia*
Affiliation:
Beijing Computational Science Research Center
Da-Quan Jiang*
Affiliation:
Peking University
Youming Li*
Affiliation:
Peking University
*
*Postal address: Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing 100193, China.
**Postal address: LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China; Center for Statistical Science, Peking University, Beijing 100871, China. Email: jiangdq@math.pku.edu.cn
***Postal address: LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China.
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Abstract

Detailed balance of a chemical reaction network can be defined in several different ways. Here we investigate the relationship among four types of detailed balance conditions: deterministic, stochastic, local, and zero-order local detailed balance. We show that the four types of detailed balance are equivalent when different reactions lead to different species changes and are not equivalent when some different reactions lead to the same species change. Under the condition of local detailed balance, we further show that the system has a global potential defined over the whole space, which plays a central role in the large deviation theory and the Freidlin–Wentzell-type metastability theory of chemical reaction networks. Finally, we provide a new sufficient condition for stochastic detailed balance, which is applied to construct a class of high-dimensional chemical reaction networks that both satisfies stochastic detailed balance and displays multistability.

Keywords

Chemical reaction systemsmicroscopic reversibilityKolmogorov’s cycle conditionquasi-potentiallarge deviationmetastability

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces 60J28: Applications of continuous-time Markov processes on discrete state spaces 60F10: Large deviations 92E20: Classical flows, reactions, etc.
Secondary: 92C40: Biochemistry, molecular biology 92C45: Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.) 92B05: General biology and biomathematics
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 3 , September 2021 , pp. 886 - 922
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Copyright
© The Author(s) 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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