Stochastic kinetic models: Dynamic independence, modularity and graphs

TL;DR

This paper analyzes the independence structure of stochastic kinetic models (SKMs) using directed graphs, enabling modular analysis and efficient computation of complex biochemical reaction networks.

Contribution

It introduces the kinetic independence graph (KIG) for SKMs, linking graphical separation to local and global independence, and develops methods for modular decomposition and analysis.

Findings

KIG encodes local independence structure of SKMs.

Graphical separation implies conditional independence of subprocesses.

Application to red blood cell SKM enhances biochemical understanding.

Abstract

The dynamic properties and independence structure of stochastic kinetic models (SKMs) are analyzed. An SKM is a highly multivariate jump process used to model chemical reaction networks, particularly those in biochemical and cellular systems. We identify SKM subprocesses with the corresponding counting processes and propose a directed, cyclic graph (the kinetic independence graph or KIG) that encodes the local independence structure of their conditional intensities. Given a partition $[A,D,B]$ of the vertices, the graphical separation $A\perp B|D$ in the undirected KIG has an intuitive chemical interpretation and implies that $A$ is locally independent of $B$ given $A\cup D$. It is proved that this separation also results in global independence of the internal histories of $A$ and $B$ conditional on a history of the jumps in $D$ which, under conditions we derive, corresponds to the internal history of $D$. The results enable mathematical definition of a modularization of an SKM using its implied dynamics. Graphical decomposition methods are developed for the identification and efficient computation of nested modularizations. Application to an SKM of the red blood cell advances understanding of this biochemical system.