  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
messenger RNA and micro RNA content in a cell. We provide theoretical and
numerical evidences that the robustness of the gene expression is increased in
the presence of micro RNA. At the mathematical level, increased robustness
shows in a smaller coefficient of variation of the marginal density of the
messenger RNA in the presence of micro RNA. 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.
