Rudnicki Ryszard, Tomski Andrzej
Institute of Mathematics, Polish Academy of Sciences, Bankowa 14, 40-007 Katowice, Poland.
Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland.
J Theor Biol. 2015 Dec 21;387:54-67. doi: 10.1016/j.jtbi.2015.09.012. Epub 2015 Oct 3.
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.
在本文中，我们开发了一种随机基因表达模型，它是对论文[T. Lipniacki, P. Paszek, A. Marciniak-Czochra, A.R. Brasier, M. Kimmel, Transcriptional stochasticity in gene expression, J. Theor. Biol. 238 (2006) 348 - 367]中所研究模型的扩展。在我们的模型中，随机效应仍然源于基因活性状态的随机波动，但我们在mRNA产生之前先形成前体mRNA，这丰富了经典转录阶段。我们得到了一个描述前体mRNA、mRNA和蛋白质水平演化的随机调控常微分方程组（ODEs）。我们对这个被识别为分段确定性马尔可夫过程（PDMP）的随机过程的长期行为进行了数学分析。我们使用数值模拟来检验所有三种类型粒子分布的精确结果。此外，我们研究了该过程的确定性（绝热）极限状态，根据参数不同，它可以表现出两种特定类型的行为：双稳态和极限环的存在。当只考虑两种基因表达产物时，后者不存在。
