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

TL;DR

This paper introduces a method to simplify infinite state space Markov chains by removing fast states, and applies it to model gene expression bursts, revealing their connection to fast transition paths.

Contribution

It develops a new approach for simplifying two-time-scale Markov chains with infinite states and provides a mathematical theory linking gene expression bursts to fast transition paths.

Findings

Effective reduction of infinite Markov chains by removing fast states

Mathematical conditions for gene expression bursting kinetics

Bursts correspond to fast transition paths in the Markov model

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.