On a stochastic gene expression with pre-mRNA, mRNA and protein contribution

• We model stochastic gene expression with pre-mRNA, mRNA and protein contribution.
• This mechanism is mathematically based on Piece-wise Deterministic Markov Process.
• We show that if the initial distribution of the process has a density, then it stabilizes in time with some invariant density.
• We find a set on which the invariant density is supported.
• We investigate bistability state and the existence of limit cycle in the adiabatic limit.

In this paper we develop a model of stochastic gene expression, which is an extension of the model investigated in the paper [T. Lipniacki, P. Paszek, A. Marciniak-Czochra, A.R. Brasier, M. Kimmel, Transcriptional stochasticity in gene expression, J. Theor. Biol. 238 (2006) 348–367]. In our model, stochastic effects still originate from random fluctuations in gene activity status, but we precede mRNA production by the formation of pre-mRNA, which enriches classical transcription phase. We obtain a stochastically regulated system of ordinary differential equations (ODEs) describing evolution of pre-mRNA, mRNA and protein levels. We perform mathematical analysis of a long-time behavior of this stochastic process, identified as a piece-wise deterministic Markov process (PDMP). We check exact results using numerical simulations for the distributions of all three types of particles. Moreover, we investigate the deterministic (adiabatic) limit state of the process, when depending on parameters it can exhibit two specific types of behavior: bistability and the existence of the limit cycle. The latter one is not present when only two kinds of gene expression products are considered.