Noise and delay can shape distribution functions in stochastic reaction dynamics

Noise can drive the dynamics of stochastic systems to different important states. Delay is another significant parameter that may impart non-Markovian behavior in the system dynamics. The interplay of noise and delay can exhibit interesting, complex behaviors in stochastic systems. In this work, we considered the stochastic gene expression model and studied this interplay of noise and delay in describing the functioning of a gene via transcription and translation processes. The calculated probability distributions of mRNA and protein, both in non-delay and delay, are found to obey certain universal classes, namely Poisson distribution at $$u,N\rightarrow large$$ limit, and Normal distribution at $$u,\langle u\rangle ,N\rightarrow large$$ limit. Analytical result of noise, measured by the Fano factor, indicates that, both in delay and non-delay cases, the gene expression system follows sub-Poissonian processes when the values of parameters are far from asymptotic values and that it becomes Poissonian at asymptotic values of the system parameters. We provided a detailed study of the noise using the Fano Factor with respect to different parameters such as mean, initial population, and time delay for the gene expression process. Again, the stochastic simulation results of the model indicate the transition of mRNA states (low and high transcription and translation) driven by the translation rate.