Markov chain aggregation and its applications to combinatorial reaction networks

TL;DR

This paper introduces a new framework for aggregating continuous-time Markov chains using a variant of weak lumpability, with applications to biochemical reaction networks, enabling efficient analysis of complex stochastic systems.

Contribution

It develops a sufficient condition for Markov chain aggregation and demonstrates its application to rule-based biochemical reaction network models.

Findings

Aggregation preserves the Markov property under certain conditions.

The approach allows for efficient computation of aggregate distributions.

Case studies validate the effectiveness of the methods.

Abstract

We consider a continuous-time Markov chain (CTMC) whose state space is partitioned into aggregates, and each aggregate is assigned a probability measure. A sufficient condition for defining a CTMC over the aggregates is presented as a variant of weak lumpability, which also characterizes that the measure over the original process can be recovered from that of the aggregated one. We show how the applicability of de-aggregation depends on the initial distribution. The application section is a major aspect of the article, where we illustrate that the stochastic rule-based models for biochemical reaction networks form an important area for usage of the tools developed in the paper. For the rule-based models, the construction of the aggregates and computation of the distribution over the aggregates are algorithmic. The techniques are exemplified in three case studies.