  Stochastic fluctuations in the copy number of gene products have perceivable
effects on the functioning of gene regulatory networks (GRN). 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 GRN. 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 the separation of the mRNAs from proteins. The GF for the
mRNAs is obtained using a formalism involving operators and vector states.
Using the same formalism, the GF for the proteins is solved for a particular
path taken by all mRNAs in the time-copy number plane; then, 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.
