Simplification of Markov chains with infinite state space and the mathematical theory of random gene expression bursts

Abstract

Here we develop an effective approach to simplify two-time-scale Markov chains with infinite state spaces by removal of states with fast leaving rates, which improves the simplification method of finite Markov chains. We introduce the concept of fast transition paths and show that the effective transitions of the reduced chain can be represented as the superposition of the direct transitions and the indirect transitions via all the fast transition paths. Furthermore, we apply our simplification approach to the standard Markov model of single-cell stochastic gene expression and provide a mathematical theory of random gene expression bursts. We give the precise mathematical conditions for the bursting kinetics of both mRNAs and proteins. It turns out that random bursts exactly correspond to the fast transition paths of the Markov model. This helps us gain a better understanding of the physics behind the bursting kinetics as an emergent behavior from the fundamental multi-scale biochemical reaction kinetics of stochastic gene expression.

20 pages, 4 figures

References in corpus (5)

• Analytical distributions for stochastic gene expression
• Exact distributions for stochastic gene expression models with bursting and feedback
• Multiple-scale stochastic processes: decimation, averaging and beyond
• Entropy production and coarse-graining in Markov processes
• Coarse graining of master equations with fast and slow states

Cited by in corpus (2)

• Emergent Lévy behavior in single-cell stochastic gene expression
• Efficient analysis of stochastic gene dynamics in the non-adiabatic regime using piecewise deterministic Markov processes