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UKPMC Funders Author Manuscripts logoLink to UKPMC Funders Author Manuscripts
. Author manuscript; available in PMC: 2011 Jan 26.
Published in final edited form as: Ann Stat. 2010 Aug 1;38(4):2242–2281. doi: 10.1214/09-AOS779

STOCHASTIC KINETIC MODELS: DYNAMIC INDEPENDENCE, MODULARITY AND GRAPHS1

Clive G Bowsher 1
PMCID: PMC3027064  EMSID: UKMS32199  PMID: 21278808

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 AB|D in the undirected KIG has an intuitive chemical interpretation and implies that A is locally independent of B given AD. 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.

Keywords: Stochastic kinetic model, kinetic independence graph, counting and point processes, dynamic and local independence, graphical decomposition, reaction networks, systems biology

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