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In this paper we develop a model of stochastic gene expression, which is anextension of the model investigated in the paper [T. Lipniacki, P. Paszek, A.Marciniak-Czochra, A.R. Brasier, M. Kimmel, Transcriptional stochasticity ingene expression, J. Theor. Biol. 238 (2006) 348-367]. In our model, stochasticeffects still originate from random uctuations in gene activity status, but weprecede mRNA production by the formation of pre-mRNA, which enriches classicaltranscription phase. We obtain a stochastically regulated system of ordinarydifferential equations (ODEs) describing evolution of pre-mRNA, mRNA andprotein levels. We perform mathematical analysis of a long-time behaviour ofthis stochastic process, identified as a piece-wise deterministic Markovprocess (PDMP). We check exact results using numerical simulations for thedistributions of all three types of particles. Moreover, we investigate thedeterministic (adiabatic) limit state of the process, when depending onparameters it can exhibit two specific types of behavior: bistability and theexistence of the limit cycle. The latter one is not present when only two kindsof gene expression products are considered.

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