State space truncation with quantified errors for accurate solutions to discrete Chemical Master Equation

TL;DR

This paper introduces a novel state space truncation method for the discrete Chemical Master Equation, providing theoretical error bounds and an efficient way to ensure accurate solutions for complex stochastic reaction networks.

Contribution

It presents a new truncation approach based on molecular equivalence groups and develops a theoretical framework for quantifying and bounding truncation errors in steady state solutions.

Findings

Truncation error can be asymptotically bounded by boundary state probabilities.

The method accurately estimates errors without solving the full dCME.

Validated on four stochastic networks, ensuring reliable solutions.

Abstract

The discrete chemical master equation (dCME) provides a general framework for studying stochasticity in mesoscopic reaction networks. Since its direct solution rapidly becomes intractable due to the increasing size of the state space, truncation of the state space is necessary for solving most dCMEs. It is therefore important to assess the consequences of state space truncations so errors can be quantified and minimized. Here we describe a novel method for state space truncation. By partitioning a reaction network into multiple molecular equivalence groups (MEG), we truncate the state space by limiting the total molecular copy numbers in each MEG. We further describe a theoretical framework for analysis of the truncation error in the steady state probability landscape using reflecting boundaries. By aggregating the state space based on the usage of a MEG and constructing an aggregated Markov process, we show that the truncation error of a MEG can be asymptotically bounded by the probability of states on the reflecting boundary of the MEG. Furthermore, truncating states of an arbitrary MEG will not undermine the estimated error of truncating any other MEGs. We then provide an error estimate for networks with multiple MEGs. To rapidly determine the appropriate size of an arbitrary MEG, we introduce an a priori method to estimate the upper bound of its truncation error, which can be rapidly computed from reaction rates, without costly trial solutions of the dCME. We show results of applying our methods to four stochastic networks. We demonstrate how truncation errors and steady state probability landscapes can be computed using different sizes of the MEG(s) and how the results validate out theories. Overall, the novel state space truncation and error analysis methods developed here can be used to ensure accurate direct solutions to the dCME for a large class of stochastic networks.