Adiabatic reduction of a piecewise deterministic markov model of stochastic gene expression with bursting transcription

Details Adiabatic reduction of a piecewise deterministic markov model of stochastic gene expression with bursting transcription Adiabatic reduction of a piecewise deterministic markov model of stochastic gene expression with bursting transcription

2012-02-24

arxiv.org/abs/1202.5411v1

Mathematics

Probability

39 pages

Scientific paper

This paper considers adiabatic reduction in a model of stochastic gene expression with bursting transcription. We prove that an adiabatic reduction can be performed in a stochastic slow/fast system with a jump Markov process. In the gene expression model, the production of mRNA (the fast variable) is assumed to follow a compound Poisson process (the phenomena called bursting in molecular biology) and the production of protein (the slow variable) is linear as a function of mRNA. When the dynamics of mRNA is assumed to be faster than the protein dynamics (due to a mRNA degradation rate larger than for the protein) we prove that, with the appropriate scaling, the bursting phenomena can be transmitted to the slow variable. We show that the reduced equation is either a stochastic differential equation with a jump Markov process or a deterministic ordinary differential equation depending on the scaling that is appropriate. These results are significant because adiabatic reduction techniques seem to have not been developed for a stochastic differential system containing a jump Markov process. We hope that they are generalizable to give insight into adiabatic methods in more general stochastic hybrid systems. In particular, a similar proof gives us a mathematical justification of the bursting of mRNA through an appropriate scaling of an "ON-OFF" switching gene model. Last but not least, for our particular system, the adiabatic reduction allows us to understand what are the necessary conditions for the bursting production-like of mRNA and protein to occur.

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