  A widely used approach to describe the dynamics of gene regulatory networks
is based on the chemical master equation, which considers probability
distributions over all possible combinations of molecular counts. The analysis
of such models is extremely challenging due to their large discrete state
space. We therefore propose a hybrid approximation approach based on a system
of partial differential equations, where we assume a continuous-deterministic
evolution for the protein counts. We discuss efficient analysis methods for
both modeling approaches and compare their performance. We show that the hybrid
approach yields accurate results for sufficiently large molecule counts, while
reducing the computational effort from one ordinary differential equation for
each state to one partial differential equation for each mode of the system.
Furthermore, we give an analytical steady-state solution of the hybrid model
for the case of a self-regulatory gene.
