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This paper considers adiabatic reduction in a model of stochastic geneexpression with bursting transcription considered as a jump Markov process. Inthis model, the process of gene expression with auto-regulation is described byfast/slow dynamics. The production of mRNA is assumed to follow a compoundPoisson process occurring at a rate depending on protein levels (the phenomenacalled bursting in molecular biology) and the production of protein is a linearfunction of mRNA numbers. When the dynamics of mRNA is assumed to be a fastprocess (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 stochasticdifferential equation with a jump Poisson process or a deterministic ordinarydifferential equation. These results are significant because adiabaticreduction techniques seem to have not been rigorously justified for astochastic differential system containing a jump Markov process. We expect thatthe results can be generalized to adiabatic methods in more general stochastichybrid systems.

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