Adiabatic reduction of a model of stochastic gene expression with jump Markov process

TL;DR

This paper rigorously justifies adiabatic reduction in a stochastic gene expression model with bursting, showing how fast mRNA dynamics influence slow protein dynamics, resulting in either stochastic or deterministic reduced equations.

Contribution

It provides the first rigorous justification of adiabatic reduction for stochastic hybrid systems with jump Markov processes in gene expression models.

Findings

Reduced equations can be stochastic or deterministic depending on scaling.

Fast mRNA dynamics transmit bursting phenomena to slow variables.

Results can be generalized to other stochastic hybrid systems.

Abstract

This paper considers adiabatic reduction in a model of stochastic gene expression with bursting transcription considered as a jump Markov process. In this model, the process of gene expression with auto-regulation is described by fast/slow dynamics. The production of mRNA is assumed to follow a compound Poisson process occurring at a rate depending on protein levels (the phenomena called bursting in molecular biology) and the production of protein is a linear function of mRNA numbers. When the dynamics of mRNA is assumed to be a fast process (due to faster mRNA degradation than that of protein) we prove that, with appropriate scalings in the burst rate, jump size or translational rate, the bursting phenomena can be transmitted to the slow variable. We show that, depending on the scaling, the reduced equation is either a stochastic differential equation with a jump Poisson process or a deterministic ordinary differential equation. These results are significant because adiabatic reduction techniques seem to have not been rigorously justified for a stochastic differential system containing a jump Markov process. We expect that the results can be generalized to adiabatic methods in more general stochastic hybrid systems.