  The dynamics of cellular chemical reactions are variable due to stochastic
noise from intrinsic and extrinsic sources. The intrinsic noise is the
intracellular fluctuations of molecular copy numbers caused by the
probabilistic encounter of molecules and is modeled by the chemical master
equation. The extrinsic noise, on the other hand, represents the intercellular
variation of the kinetic parameters due to the variation of global factors
affecting gene expression. The objective of this paper is to propose a
computational framework to analyze the combined effect of the intrinsic and the
extrinsic noise modeled by the chemical master equation with uncertain
parameters. More specifically, we formulate a convex optimization problem to
compute the intervals of the stationary solution of uncertain moment equations
whose parameters are given only partially in the form of the statistics of
their distributions. The optimization program is derived without approximating
the governing equation in contrast with many existing approaches. Thus, we can
obtain guaranteed intervals of the worst possible values of the moments for all
parameter distributions satisfying the given statistics, which is useful for
model-based rational engineering of biomolecular circuits in synthetic biology,
where the robustness of synthetic reactions is important. We demonstrate the
proposed optimization approach using two examples of stochastic chemical
reactions and show that the solution of the optimization problem gives
practically useful upper and lower bounds of the statistics of the stationary
copy number distributions.
