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Lyapunov Functions, Stationary Distributions, and Non-equilibrium Potential for Reaction Networks.

ABSTRACT: We consider the relationship between stationary distributions for stochastic models of reaction systems and Lyapunov functions for their deterministic counterparts. Specifically, we derive the well-known Lyapunov function of reaction network theory as a scaling limit of the non-equilibrium potential of the stationary distribution of stochastically modeled complex balanced systems. We extend this result to general birth-death models and demonstrate via example that similar scaling limits can yield Lyapunov functions even for models that are not complex or detailed balanced, and may even have multiple equilibria.

SUBMITTER: Anderson DF

PROVIDER: S-EPMC8408763 | biostudies-literature | 2015 Sep

REPOSITORIES: biostudies-literature

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Lyapunov Functions, Stationary Distributions, and Non-equilibrium Potential for Reaction Networks.

Anderson David F DF Craciun Gheorghe G Gopalkrishnan Manoj M Wiuf Carsten C

Bulletin of mathematical biology 20150916 9

We consider the relationship between stationary distributions for stochastic models of reaction systems and Lyapunov functions for their deterministic counterparts. Specifically, we derive the well-known Lyapunov function of reaction network theory as a scaling limit of the non-equilibrium potential of the stationary distribution of stochastically modeled complex balanced systems. We extend this result to general birth-death models and demonstrate via example that similar scaling limits can yiel ...[more]

PMID: 26376889

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Project description:Complex molecular biological processes such as transcription and translation, signal transduction, post-translational modification cascades, and metabolic pathways can be described in principle by biochemical reactions that explicitly take into account the sophisticated network of chemical interactions regulating cell life. The ability to deduce the possible qualitative behaviors of such networks from a set of reactions is a central objective and an ongoing challenge in the field of systems biology. Unfortunately, the construction of complete mathematical models is often hindered by a pervasive problem: despite the wealth of qualitative graphical knowledge about network interactions, the form of the governing nonlinearities and/or the values of kinetic constants are hard to uncover experimentally. The kinetics can also change with environmental variations. This work addresses the following question: given a set of reactions and without assuming a particular form for the kinetics, what can we say about the asymptotic behavior of the network? Specifically, it introduces a class of networks that are "structurally (mono) attractive" meaning that they are incapable of exhibiting multiple steady states, oscillation, or chaos by virtue of their reaction graphs. These networks are characterized by the existence of a universal energy-like function called a Robust Lyapunov function (RLF). To find such functions, a finite set of rank-one linear systems is introduced, which form the extremals of a linear convex cone. The problem is then reduced to that of finding a common Lyapunov function for this set of extremals. Based on this characterization, a computational package, Lyapunov-Enabled Analysis of Reaction Networks (LEARN), is provided that constructs such functions or rules out their existence. An extensive study of biochemical networks demonstrates that LEARN offers a new unified framework. Basic motifs, three-body binding, and genetic networks are studied first. The work then focuses on cellular signalling networks including various post-translational modification cascades, phosphotransfer and phosphorelay networks, T-cell kinetic proofreading, and ERK signalling. The Ribosome Flow Model is also studied.

Dataset Information

Lyapunov Functions, Stationary Distributions, and Non-equilibrium Potential for Reaction Networks.

Publications

Lyapunov Functions, Stationary Distributions, and Non-equilibrium Potential for Reaction Networks.

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