Complex Balanced Networks Research Articles

On Non-ideal Chemical-Reaction Networks and Phase Separation

Much of the theory on chemical-reaction networks (CRNs) has been developed in the ideal-solution limit, where interactions between the solutes are negligible. However, there is a large variety of phenomena in biological cells and soft-matter physics which appear to deviate from the ideal-solution behaviour. Particularly striking is the case of liquid-liquid phase separation, which is typically caused by inter-particle interactions. Here, we revisit a number of known results in the domain of ideal CRNs, and we generalise and adapt them to arbitrary interactions between the solutes which stem from a given free energy. We start by reviewing the theory of chemical reaction networks, linking it to concepts in statistical physics. Then we obtain a number of new results for non-ideal complex-balanced networks, where the creation and annihilation rates are equal for all chemical complexes which appear as reactants or products in the CRN. Among these is the form of the steady-state probability distribution and Lyapunov functions for such networks. Finally, this allows us to draw a phase diagram for complex-balanced reaction-diffusion systems based on the minimisation of such Lyapunov function, with a rationale similar to that of equilibrium thermodynamics but for systems that may sustain non-equilibrium chemical currents at steady state. In addition, we show that for complex-balanced networks at steady-state, there are no diffusion currents.

Diffusion-dynamics laws in stochastic reaction networks.

Many biological activities are induced by cellular chemical reactions of diffusing reactants. The dynamics of such systems can be captured by stochastic reaction networks. A recent numerical study has shown that diffusion can significantly enhance the fluctuations in gene regulatory networks. However, the universal relation between diffusion and stochastic system dynamics remains veiled. Within the approximation of reaction-diffusion master equation (RDME), we find general relation that the steady-state distribution in complex balanced networks is diffusion-independent. Here, complex balance is the nonequilibrium generalization of detailed balance. We also find that for a diffusion-included network with a Poisson-like steady-state distribution, the diffusion can be ignored at steady state. We then derive a necessary and sufficient condition for networks holding such steady-state distributions. Moreover, we show that for linear reaction networks the RDME reduces to the chemical master equation, which implies that the stochastic dynamics of networks is unaffected by diffusion at any arbitrary time. Our findings shed light on the fundamental question of when diffusion can be neglected, or (if nonnegligible) its effects on the stochastic dynamics of the reaction network.

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

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.

Computational Identification of Irreducible State-Spaces for Stochastic Reaction Networks

Stochastic models of reaction networks are becoming increasingly important in systems biology. In these models, the dynamics is generally represented by a continuous-time Markov chain whose states denote the copy-numbers of the constituent species. The state-space on which this process resides is a subset of a nonnegative integer lattice, and for many examples of interest, this state-space is countably infinite. This causes numerous problems in analyzing the Markov chain and understanding its long-term behavior. These problems are further confounded by the presence of conservation relations among species which constrain the dynamics in complicated ways. In this paper we provide a linear-algebraic procedure to disentangle these conservation relations and represent the state-space in a special decomposed form, based on the copy-number ranges of various species and dependencies among them. This decomposed form is advantageous for analyzing the stochastic model, and for a large class of networks we demonstrate how this form can be used for finding all the closed communication classes for the Markov chain within the infinite state-space. Such communication classes are irreducible state-spaces for the dynamics and they support all the extremal stationary distributions for the Markov chain. Hence our results provide important insights into the long-term behavior and stability properties of stochastic models of reaction networks. We discuss how the knowledge of these irreducible state-spaces can be used in many ways such as speeding-up stochastic simulations of multiscale networks or in identifying the stationary distributions of complex-balanced networks. We illustrate our results with several examples of gene-expression networks from systems biology.

Reduction of multiscale stochastic biochemical reaction networks using exact moment derivation.

Biochemical reaction networks (BRNs) in a cell frequently consist of reactions with disparate timescales. The stochastic simulations of such multiscale BRNs are prohibitively slow due to high computational cost for the simulations of fast reactions. One way to resolve this problem uses the fact that fast species regulated by fast reactions quickly equilibrate to their stationary distribution while slow species are unlikely to be changed. Thus, on a slow timescale, fast species can be replaced by their quasi-steady state (QSS): their stationary conditional expectation values for given slow species. As the QSS are determined solely by the state of slow species, such replacement leads to a reduced model, where fast species are eliminated. However, it is challenging to derive the QSS in the presence of nonlinear reactions. While various approximation schemes for the QSS have been developed, they often lead to considerable errors. Here, we propose two classes of multiscale BRNs which can be reduced by deriving an exact QSS rather than approximations. Specifically, if fast species constitute either a feedforward network or a complex balanced network, the reduced model based on the exact QSS can be derived. Such BRNs are frequently observed in a cell as the feedforward network is one of fundamental motifs of gene or protein regulatory networks. Furthermore, complex balanced networks also include various types of fast reversible bindings such as bindings between transcriptional factors and gene regulatory sites. The reduced models based on exact QSS, which can be calculated by the computational packages provided in this work, accurately approximate the slow scale dynamics of the original full model with much lower computational cost.

Nonequilibrium Thermodynamics of Chemical Reaction Networks: Wisdom from Stochastic Thermodynamics

We build a rigorous nonequilibrium thermodynamic description for open chemical reaction networks of elementary reactions. Their dynamics is described by deterministic rate equations with mass action kinetics. Our most general framework considers open networks driven by time-dependent chemostats. The energy and entropy balances are established and a nonequilibrium Gibbs free energy is introduced. The difference between this latter and its equilibrium form represents the minimal work done by the chemostats to bring the network to its nonequilibrium state. It is minimized in nondriven detailed-balanced networks (i.e., networks that relax to equilibrium states) and has an interesting information-theoretic interpretation. We further show that the entropy production of complex-balanced networks (i.e., networks that relax to special kinds of nonequilibrium steady states) splits into two non-negative contributions: one characterizing the dissipation of the nonequilibrium steady state and the other the transients due to relaxation and driving. Our theory lays the path to study time-dependent energy and information transduction in biochemical networks.Received 23 February 2016DOI:https://doi.org/10.1103/PhysRevX.6.041064Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasChemical kinetics, dynamics & catalysisNonequilibrium & irreversible thermodynamicsNonequilibrium statistical mechanicsStatistical PhysicsNonlinear DynamicsNetworksInterdisciplinary Physics

A graph-theoretical approach for the analysis and model reduction of complex-balanced chemical reaction networks

In this paper we derive a compact mathematical formulation describing the dynamics of chemical reaction networks that are complex-balanced and are governed by mass action kinetics. The formulation is based on the graph of (substrate and product) complexes and the stoichiometric information of these complexes, and crucially uses a balanced weighted Laplacian matrix. It is shown that this formulation leads to elegant methods for characterizing the space of all equilibria for complex-balanced networks and for deriving stability properties of such networks. We propose a method for model reduction of complex-balanced networks, which is similar to the Kron reduction method for electrical networks and involves the computation of Schur complements of the balanced weighted Laplacian matrix.