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To what extent do the characteristic features of a chemical reaction network reflect its purpose and function? In general, one argues that correlations between specific features and specific functions are key to understanding a complex…

Statistical Mechanics · Physics 2012-03-22 Sang Hoon Lee , Sebastian Bernhardsson , Petter Holme , Beom Jun Kim , Petter Minnhagen

Chemical reaction networks in living cells maintain precise control over thousands of metabolites despite operating far from equilibrium under constant perturbations. While mass action kinetics accurately describe the underlying dynamics,…

Optimization and Control · Mathematics 2025-08-27 Steven Diamond

Analyzing qualitative behaviors of biochemical reactions using its associated network structure has proven useful in diverse branches of biology. As an extension of our previous work, we introduce a graph-based framework to calculate steady…

Classical Analysis and ODEs · Mathematics 2014-04-28 Inom Mirzaev , David Matthew Bortz

Chemical reaction networks (CRNs) exhibit complex dynamics governed by their underlying network structure. In this paper, we propose a novel approach to study the dynamics of CRNs by representing them on species graphs (S-graphs). By…

Molecular Networks · Quantitative Biology 2024-04-24 Keisuke Sugie , Dimitri Loutchko , Tetsuya J. Kobayashi

Many physical systems--from mechanical lattices and electrical circuits to biological tissues and architected metamaterials--can be understood as networks transmitting physical quantities. We present a unified mathematical framework for…

Soft Condensed Matter · Physics 2025-08-07 José M. Ortiz-Tavárez , William Stephenson , Xiaoming Mao

We provide a new decomposition of the Laplacian matrix (for labeled directed graphs with strongly connected components), involving an invertible $\textit{core matrix}$, the vector of tree constants, and the incidence matrix of an auxiliary…

Combinatorics · Mathematics 2023-11-21 Stefan Müller

In this paper, we provide a graphic formulation of non-isothermal reaction systems and show that a non-isothermal detailed balanced network system converges (locally) asymptotically to the unique equilibrium within the invariant manifold…

Dynamical Systems · Mathematics 2020-10-01 Zhou Fang , Arjan van der Schaft , Chuanhou Gao

Within a fully microscopic setting, we derive a variational principle for the non-equilibrium steady states of chemical reaction networks, valid for time-scales over which chemical potentials can be taken to be slowly varying: at…

Molecular Networks · Quantitative Biology 2015-06-05 Andrea De Martino , Daniele De Martino , Roberto Mulet , Guido Uguzzoni

Very often, models in biology, chemistry, physics, and engineering are systems of polynomial or power-law ordinary differential equations, arising from a reaction network. Such dynamical systems can be generated by many different reaction…

Dynamical Systems · Mathematics 2020-01-01 Gheorghe Craciun , Jiaxin Jin , Polly Y. Yu

We present a systematic mathematical analysis of the qualitative steady-state response to rate perturbations in large classes of reaction networks. This includes multimolecular reactions and allows for catalysis, enzymatic reactions,…

Dynamical Systems · Mathematics 2017-11-22 Bernhard Brehm , Bernold Fiedler

A reaction network is a chemical system involving multiple reactions and chemical species. Stochastic models of such networks treat the system as a continuous time Markov chain on the number of molecules of each species with reactions as…

Probability · Mathematics 2007-05-23 Karen Ball , Thomas G. Kurtz , Lea Popovic , Greg Rempala

We further clarify the relation between detailed-balanced and complex-balanced equilibria of reversible chemical reaction networks. Our results hold for arbitrary kinetics and also for boundary equilibria. Detailed balance, complex balance,…

Dynamical Systems · Mathematics 2020-09-29 Stefan Müller , Badal Joshi

It has recently been shown that structural conditions on the reaction network, rather than a 'fine-tuning' of system parameters, often suffice to impart 'absolute concentration robustness' on a wide class of biologically relevant,…

Probability · Mathematics 2014-01-20 David F. Anderson , German Enciso , Matthew Johnston

We present a simple model of network dynamics that can be solved analytically for uniform networks. We obtain the dynamics of response of the system to perturbations. The analytical solution is an excellent approximation for random…

Cellular Automata and Lattice Gases · Physics 2009-11-11 M. A. M. de Aguiar , Irving R. Epstein , Yaneer Bar-Yam

With an increasing amount of observations on the dynamics of many complex systems, it is required to reveal the underlying mechanisms behind these complex dynamics, which is fundamentally important in many scientific fields such as climate,…

Neurons and Cognition · Quantitative Biology 2025-03-19 Zhendong Yu , Haiping Huang

The graph-related symmetries of a reaction network give rise to certain special equilibria (such as complex balanced equilibria) in deterministic models of dynamics of the reaction network. Correspondingly, in the stochastic setting, when…

Probability · Mathematics 2018-05-21 Daniele Cappelletti , Badal Joshi

It has been recently observed that the dynamical properties of mass action systems arising from many models of biochemical reaction networks can be derived by considering the corresponding properties of a related generalized mass action…

Numerical Analysis · Mathematics 2014-07-22 Matthew D. Johnston

We propose a unified framework that allows for the full mechanistic reconstruction of chemical reaction networks (CRNs) from concentration data. The framework utilizes an integral formulation of the differential equations governing the…

Numerical Analysis · Mathematics 2026-02-13 Abraham Reyes-Velazquez , Stefan Güttel , Igor Larrosa , Jonas Latz

This paper develops the concept of decomposition for chemical reaction networks, based on which a network decomposition technique is proposed to capture the stability of large-scale networks characterized by a high number of species, high…

Dynamical Systems · Mathematics 2021-02-23 Yafei Lu , Chuanhou Gao , Denis Dochain

The analysis of non-equilibrium steady states of biochemical reaction networks relies on finding the configurations of fluxes and chemical potentials satisfying stoichiometric (mass balance) and thermodynamic (energy balance) constraints.…

Molecular Networks · Quantitative Biology 2011-07-13 Daniele De Martino , Matteo Figliuzzi , Andrea De Martino , Enzo Marinari