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

Anderson, David F.; Crăciun, Gheorghe; Gopalkrishnan, Manoj; Wiuf, Carsten

doi:10.1007/s11538-015-0102-8

Cited by 54 publications

(104 citation statements)

References 39 publications

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“…We introduce what we term the stoichiometric decomposition of the Liouville operator in Eq. (40), which separates two dynamically different classes of mean-regressing and nonmean-regressing flows.…”

Section: Organization Of the Presentationmentioning

confidence: 99%

Flows, scaling, and the control of moment hierarchies for stochastic chemical reaction networks

Smith

Krishnamurthy

2017

Phys. Rev. E

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Stochastic chemical reaction networks (CRNs) are complex systems which combine the features of concurrent transformation of multiple variables in each elementary reaction event, and nonlinear relations between states and their rates of change. Most general results concerning CRNs are limited to restricted cases where a topological characteristic known as deficiency takes value 0 or 1, implying uniqueness and positivity of steady states and surprising, low-information forms for their associated probability distributions. Here we derive equations of motion for fluctuation moments at all orders for stochastic CRNs at general deficiency. We show, for the standard base-case of proportional sampling without replacement (which underlies the mass-action rate law), that the generator of the stochastic process acts on the hierarchy of factorial moments with a finite representation. Whereas simulation of high-order moments for many-particle systems is costly, this representation reduces solution of moment hierarchies to a complexity comparable to solving a heat equation. At steady states, moment hierarchies for finite CRNs interpolate between low-order and high-order scaling regimes, which may be approximated separately by distributions similar to those for deficiency-0 networks, and connected through matched asymptotic expansions. In CRNs with multiple stable or metastable steady states, boundedness of high-order moments provides the starting condition for recursive solution downward to low-order moments, reversing the order usually used to solve moment hierarchies. A basis for a subset of network flows defined by having the same mean-regressing property as the flows in deficiency-0 networks gives the leading contribution to low-order moments in CRNs at general deficiency, in a 1/n-expansion in large particle numbers. Our results give a physical picture of the different informational roles of mean-regressing and non-mean-regressing flows, and clarify the dynamical meaning of deficiency not only for first-moment conditions but for all orders in fluctuations.

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Section: Organization Of the Presentationmentioning

confidence: 99%

Flows, scaling, and the control of moment hierarchies for stochastic chemical reaction networks

Smith

Krishnamurthy

2017

Phys. Rev. E

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“…For this class of models, and under the assumption of mass action kinetics, the fixed points of the deterministic models (Anderson 2011, 2008; Craciun 2015; Feinberg 1979, 1987; Gunawardena 2003) and the stationary distributions for the stochastic models, have been fully characterized (Anderson et al. 2010, 2015; Cappelletti and Wiuf 2016; Van Kampen 1976). In fact, it is the study of this class of networks that is largely responsible for the development of the field of chemical reaction network theory (Feinberg 1972, 1979; Gunawardena 2003; Horn 1972), a branch of applied mathematics in which the dynamical properties of the mathematical model are related to the structural properties of the interaction network.…”

Section: Introductionmentioning

confidence: 99%

Product-Form Stationary Distributions for Deficiency Zero Networks with Non-mass Action Kinetics

Anderson

Cotter

2016

Bull Math Biol

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In many applications, for example when computing statistics of fast subsystems in a multiscale setting, we wish to find the stationary distributions of systems of continuous-time Markov chains. Here we present a class of models that appears naturally in certain averaging approaches whose stationary distributions can be computed explicitly. In particular, we study continuous-time Markov chain models for biochemical interaction systems with non-mass action kinetics whose network satisfies a certain constraint. Analogous with previous related results, the distributions can be written in product form.

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“…On the other hand, there is a growing body of mathematical literature linking a CN's topology to its dynamics, and still bearing no thermodynamic interpretation. In particular, it has been understood that a topological number called deficiency subtends the onset of complex behavior, such as bistability a) Electronic mail: matteo.polettini@uni.lu b) Electronic mail: artur.wachtel@uni.lu c) Electronic mail: massimilano.esposito@uni.lu and oscillations [18][19][20] , which are the mechanisms of chemical switches and clocks 21 . When intrinsic noise is important, a crucial result by Anderson, Craciun and Kurtz (ACK) 22 relates the deficiency of the CN to steady statistical properties of the chemical mixture.…”

Section: Introductionmentioning

confidence: 99%

Dissipation in noisy chemical networks: The role of deficiency

Polettini

Wachtel

Esposito

2015

The Journal of Chemical Physics

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We study the effect of intrinsic noise on the thermodynamic balance of complex chemical networks subtending cellular metabolism and gene regulation. A topological network property called deficiency, known to determine the possibility of complex behavior such as multistability and oscillations, is shown to also characterize the entropic balance. In particular, when deficiency is zero the average stochastic dissipation rate equals that of the corresponding deterministic model, where correlations are disregarded. In fact, dissipation can be reduced by the effect of noise, as occurs in a toy model of metabolism that we employ to illustrate our findings. This phenomenon highlights that there is a close interplay between deficiency and the activation of new dissipative pathways at low molecule numbers.

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

Cited by 54 publications

References 39 publications

Flows, scaling, and the control of moment hierarchies for stochastic chemical reaction networks

Flows, scaling, and the control of moment hierarchies for stochastic chemical reaction networks

Product-Form Stationary Distributions for Deficiency Zero Networks with Non-mass Action Kinetics

Dissipation in noisy chemical networks: The role of deficiency

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