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

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