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The stochastic dynamics of biochemical networks are usually modelled with thechemical master equation (CME). The stationary distributions of CMEs are seldomsolvable analytically, and numerical methods typically produce estimates withuncontrolled errors. Here, we introduce mathematical programming approachesthat yield approximations of these distributions with computable error boundswhich enable the verification of their accuracy. First, we use semidefiniteprogramming to compute increasingly tighter upper and lower bounds on themoments of the stationary distributions for networks with rationalpropensities. Second, we use these moment bounds to formulate linear programsthat yield convergent upper and lower bounds on the stationary distributionsthemselves, their marginals and stationary averages. The bounds obtained alsoprovide a computational test for the uniqueness of the distribution. In theunique case, the bounds form an approximation of the stationary distributionwith a computable bound on its error. In the non-unique case, our approachyields converging approximations of the ergodic distributions. We illustrateour methodology through several biochemical examples taken from the literature:Schl\"ogl's model for a chemical bifurcation, a two-dimensional toggle switch,a model for bursty gene expression, and a dimerisation model with multiplestationary distributions.

Rigorous bounds on the stationary distributions of the chemical master equation via mathematical programming