Path integral approach to generating functions for multistep post-transcription and post-translation processes and arbitrary initial conditions.

Stochastic fluctuations in the copy number of gene products have perceivable effects on the functioning of gene regulatory networks (GRNs). The master equation (ME) provides a theoretical basis for studying such effects. However, solving the ME can be a task that ranges from simple to difficult to impossible using conventional methods. Therefore, discovering new techniques for solving the ME is an important part of research on stochastic GRNs. In this paper, we present a novel approach to obtaining the generating function (GF), which contains the same information as the ME, for a one gene system that includes multi-step post-transcription and post-translation processes. The novelty of the approach lies in solving the GF for proteins in terms of a particular path taken by the partially and fully processed mRNAs in the time-copy number plane, after which the GF is summed over all possible paths. We prove a theorem that shows the summation of all paths to be equivalent to an equation similar to the ME for the mRNAs. On a system with six gene products in total and randomly selected initial conditions, we confirm the validity of our results by comparing them with Gillespie simulations.