The subject of the paper is reciprocal influence of pure mathematics and applied sciences. We illustrate the idea by giving a review of mathematical results obtained recently, related to the model of stochastic gene expression due to Lipniacki et al. . In this model, featuring mRNA and protein levels, and gene activity, the stochastic part of processes involved in gene expression is distinguished from the part that seems to be mostly deterministic, and the dynamics is expressed by means of a piece-wise deterministic Markov process. Mathematical results pertain to asymptotic behavior of the process in time as well as limit behavior when certain parameters may be assumed to be large. These results are but an inspiration to considering the ways applied sciences influence pure mathematics by supplying fresh ideas and providing new challenges. On the other hand, they may also be seen as an exemplification of the fact that statements that seem to be almost obvious and are often taken for granted in applied sciences may require mathematical scrutiny and non-standard proofs.