We study several Fokker-Planck equations arising from a stochastic chemical kinetic system modeling a gene regulatory network in biology. The densities solving the Fokker-Planck equations describe the joint distribution of the mRNA and µRNA content in a cell. We provide theoretical and numerical evidence that the robustness of the gene expression is increased in the presence of µRNA. At the mathematical level, increased robustness shows in a smaller coefficient of variation of the marginal density of the mRNA in the presence of µRNA. These results follow from explicit formulas for solutions. Moreover, thanks to dimensional analyses and numerical simulations we provide qualitative insight into the role of each parameter in the model. As the increase of gene expression level comes from the underlying stochasticity in the models, we eventually discuss the choice of noise in our models and its influence on our results.

中文翻译：

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Fokker-Planck方法研究基因表达的鲁棒性

我们研究了由Fokker-Planck方程产生的随机化学动力学系统，该系统模拟了生物学中的基因调控网络。求解Fokker-Planck方程的密度描述了细胞中mRNA和µ RNA含量的联合分布。我们提供了理论和数值证据，表明在存在µ RNA的情况下基因表达的鲁棒性增强。在数学水平上，增强的鲁棒性表示存在µ时，mRNA的边际密度的变化系数较小。RNA。这些结果来自解决方案的明确公式。此外，借助尺寸分析和数值模拟，我们可以定性地了解模型中每个参数的作用。由于基因表达水平的提高来自模型中潜在的随机性，因此我们最终讨论了模型中噪声的选择及其对结果的影响。