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

## TL;DR

This paper introduces mathematical programming methods to approximate stationary distributions of the chemical master equation with computable error bounds, enabling verification of accuracy and handling non-uniqueness in biochemical network models.

## Contribution

It develops semidefinite and linear programming techniques to bound stationary distributions and moments, providing a rigorous, verifiable approximation framework for complex biochemical systems.

## Key findings

- Provides tighter bounds on moments of stationary distributions.
- Yields converging approximations of stationary distributions and ergodic distributions.
- Demonstrates methodology on various biochemical models.

## Abstract

The stochastic dynamics of biochemical networks are usually modelled with the chemical master equation (CME). The stationary distributions of CMEs are seldom solvable analytically, and numerical methods typically produce estimates with uncontrolled errors. Here, we introduce mathematical programming approaches that yield approximations of these distributions with computable error bounds which enable the verification of their accuracy. First, we use semidefinite programming to compute increasingly tighter upper and lower bounds on the moments of the stationary distributions for networks with rational propensities. Second, we use these moment bounds to formulate linear programs that yield convergent upper and lower bounds on the stationary distributions themselves, their marginals and stationary averages. The bounds obtained also provide a computational test for the uniqueness of the distribution. In the unique case, the bounds form an approximation of the stationary distribution with a computable bound on its error. In the non-unique case, our approach yields converging approximations of the ergodic distributions. We illustrate our 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 multiple stationary distributions.