  In biophysics, the search for analytical solutions of stochastic models of
cellular processes is often a challenging task. In recent work on models of
gene expression, it was shown that a mapping based on partitioning of Poisson
arrivals (PPA-mapping) can lead to exact solutions for previously unsolved
problems. While the approach can be used in general when the model involves
Poisson processes corresponding to creation or degradation, current
applications of the method and new results derived using it have been limited
to date. In this paper, we present the exact solution of a variation of the
two-stage model of gene expression (with time dependent transition rates)
describing the arbitrary partitioning of proteins. The methodology proposed
makes full use of the the PPA-mapping by transforming the original problem into
a new process describing the evolution of three biological switches. Based on a
succession of transformations, the method leads to a hierarchy of reduced
models. We give an integral expression of the time dependent generating
function as well as explicit results for the mean, variance, and correlation
function. Finally, we discuss how results for time dependent parameters can be
extended to the three-stage model and used to make inferences about models with
parameter fluctuations induced by hidden stochastic variables.
