Model Reduction for the Chemical Master Equation: an Information-Theoretic Approach

TL;DR

This paper introduces an information-theoretic framework for model reduction of the Chemical Master Equation, unifying existing methods through Kullback-Leibler divergence minimization and enabling numerical optimization.

Contribution

It reformulates model reduction as a variational problem using KL divergence, providing a general approach and new expressions for reduced system propensities.

Findings

KL divergence effectively measures model discrepancy

The approach unifies existing reduction techniques

Validated on biological examples including feedback loops and enzyme systems

Abstract

The complexity of mathematical models in biology has rendered model reduction an essential tool in the quantitative biologist's toolkit. For stochastic reaction networks described using the Chemical Master Equation, commonly used methods include time-scale separation, the Linear Mapping Approximation and state-space lumping. Despite the success of these techniques, they appear to be rather disparate and at present no general-purpose approach to model reduction for stochastic reaction networks is known. In this paper we show that most common model reduction approaches for the Chemical Master Equation can be seen as minimising a well-known information-theoretic quantity between the full model and its reduction, the Kullback-Leibler divergence defined on the space of trajectories. This allows us to recast the task of model reduction as a variational problem that can be tackled using standard numerical optimisation approaches. In addition we derive general expressions for the propensities of a reduced system that generalise those found using classical methods. We show that the Kullback-Leibler divergence is a useful metric to assess model discrepancy and to compare different model reduction techniques using three examples from the literature: an autoregulatory feedback loop, the Michaelis-Menten enzyme system and a genetic oscillator.