Systematic Measures of Biological Networks I: Invariant Measures and Entropy

Li, Yao; Yi, Yingfei

doi:10.1002/cpa.21647

Cited by 14 publications

(14 citation statements)

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“…It is possible to find a complex balanced or detailed balanced realization for the network in question, then its asymptotical stability holds based on the dynamics equivalence between the network and its realization.Different from all of the above investigations stemming from macroscopic deterministic analysis, some literature contributes to explaining the system properties from a microscopic stochastic viewpoint. Li and Yi [20,21] connected the strength of attractions to the global attractor with the stationary distribution of a diffusion system, in which a white noise is added to the deterministic case. In the meanwhile, Anderson et.…”

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confidence: 99%

Lyapunov Function Partial Differential Equations for Chemical Reaction Networks: Some Special Cases

Fang¹,

Gao²

2019

SIAM J. Appl. Dyn. Syst.

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In this paper we develop a method to generate the Lyapunov function for stability analysis for chemical reaction networks. Based on the Chemical Master Equation, we derive the Lyapunov Function partial differential equations (PDEs), whose solution approximates the scaling non-equilibrium potential and serves as the candidate Lyapunov function for the given network. We further prove that for any chemical reaction network the solution (if exists) of the PDEs is dissipative. Moreover, the proposed method of Lyapunov Function PDEs is qualified for analyzing the asymptotic stability of complex balanced networks, all networks with 1-dimensional stoichiometric subspace and some special networks with more than 2-dimensional stoichiometric subspace if some moderate conditions are added. Several examples are presented to illustrate the efficiency of the method.1. Introduction. Chemical reactions networks (CRNs) arise abundantly in the fields including chemistry, systems biology, process industry, and even those seemingly irrelevant to chemistry such as mechanics and ecology. The dynamics of a CRN often appears to be extremely complex due to chemical interactions of cellular processes but sometimes still exhibiting certain regular behaviors like period solutions and stable-fixed-points. As a special subclass, mass-action CRNs (CRNs assigned mass action kinetics, often named mass action systems) have the dynamics of the concentrations of the various species captured by polynomial ordinary differential equations (ODEs), and have received much attention since the pioneering work [10-12,18] emerged. A major concern over this class of systems is to understand the relations between network structures and/or parameters and dynamical properties [6,7,28], especially in characterizing the stability (in the sense of Lyapunov) property [1,2,18,24,27,29]. Following this line of study, we also focus on capturing stability of equilibria in mass action systems (MASs) in the current work. Naturally, Lyapunov functions are desirable objects to prove stability of equilibria. In the field of CRNs, one important example is the pseudo-Helmholtz free energy function, proposed by Horn and Jackson [18]. This Lyapunov function can be derived from the microscopic level using potential theory [2]. Here, we build further on this, and provide a general theory that bridges between the microscopic and the macroscopic level, thermodynamics and potential theory. Based on it, it is possible to derive or find a Lyapunov function for any MAS.The early work on stability analysis mainly focused on exploring causal association from the network topology to the distribution of equilibria, and further to stability of equilibria. Thereinto, the weakly reversible structure, a requirement of complex balanced MAS, is the * most active one. Horn et al. [18] proved the well-known Deficiency Zero Theorem that states a weakly reversible deficiency zero MAS to be complex balanced and to have only one equilibrium in each positive stoichiometric compatibility class. Moreover, ...

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Lyapunov Function Partial Differential Equations for Chemical Reaction Networks: Some Special Cases

Fang¹,

Gao²

2019

SIAM J. Appl. Dyn. Syst.

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“…Part I [24] gives several examples of regular family µ with respect to A. We conjecture that the family µ is regular with respect to A for a much larger class of systems.…”

Section: 1mentioning

confidence: 93%

“…Existence and concentration of stationary measures. It was shown in Part I of the series [24] that the condition H 1 ) is implied by H 0 ) together with the following condition:…”

Section: 1mentioning

confidence: 99%

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Systematic Measures of Biological Networks II: Degeneracy, Complexity, and Robustness

2016

Comm Pure Appl Math

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This paper is part II of a two‐part series devoted to the study of systematic measures in a complex bionetwork modeled by a system of ordinary differential equations. In this part, we quantify several systematic measures of a biological network including degeneracy, complexity, and robustness. We will apply the theory of stochastic differential equations to define degeneracy and complexity for a bionetwork. Robustness of the network will be defined according to the strength of attractions to the global attractor. Based on the study of stationary probability measures and entropy made in part I of this series, we will investigate some fundamental properties of these systematic measures, in particular the connections between degeneracy, complexity, and robustness.© 2016 Wiley Periodicals, Inc.

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“…The resolution of the numerical solution imposes additional challenges. When the strength of a random perturbation is 0 < σ 1, it is known that the probability density function should concentrate on a O(σ)-neighborhood of the attractor [21]. Hence the grid size of the discretization cannot be larger than σ.…”

Section: Introductionmentioning

confidence: 99%

A data-driven method for the steady state of randomly perturbed dynamics

2019

Communications in Mathematical Sciences

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We demonstrate a data-driven method to solve for the invariant probability density function of a randomly perturbed dynamical system. The key idea is to replace the boundary condition of numerical schemes by a least squares problem corresponding to a reference solution, which is generated by Monte Carlo simulation. With this method we can solve for the invariant probability density function in any local area with high accuracy, regardless of whether the attractor is covered by the numerical domain.

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Systematic Measures of Biological Networks I: Invariant Measures and Entropy

Cited by 14 publications

References 51 publications

Lyapunov Function Partial Differential Equations for Chemical Reaction Networks: Some Special Cases

Lyapunov Function Partial Differential Equations for Chemical Reaction Networks: Some Special Cases

Systematic Measures of Biological Networks II: Degeneracy, Complexity, and Robustness

A data-driven method for the steady state of randomly perturbed dynamics

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