Abstract
We present elements of a stability theory for small, stochastic, nonlinear chemical reaction networks. Steady state probability distributions are computed with zero-information (ZI) closure, a closure algorithm that solves chemical master equations of small arbitrary nonlinear reactions. Stochastic models can be linearized around the steady state with ZI-closure, and the eigenvalues of the Jacobian matrix can be readily computed. Eigenvalues govern the relaxation of fluctuation autocorrelation functions at steady state. Autocorrelation functions reveal the time scales of phenomena underlying the dynamics of nonlinear reaction networks. In accord with the fluctuation-dissipation theorem, these functions are found to be congruent to response functions to small perturbations. Significant differences are observed in the stability of nonlinear reacting systems between deterministic and stochastic modeling formalisms.
Highlights
Lower order moments depend on higher order ones for nonlinear reaction systems, we have found that higher order moments contribute little information needed in reconstructing the master probability
We have shown that ZI-closure, based on the maximization of the system’s information entropy, offers a closure scheme for the chemical master equation (CME) of small, nonlinear, stochastic chemical reaction networks
Proof of concept that ZI-closure and eigenvalue analysis capture nonlinear stochastic chemical dynamics can be demonstrated with the reversible dimerization model
Summary
Improved understanding of compelling dynamic phenomena, from molecular chaos to relativistic cosmological models, has been possible to large extent, thanks to such mathematical theories as Lyapunov’s theory of stability of nonlinear dynamic systems. First developed over 120 yr ago, Lyapunov’s elegant theories propelled the development of the disciplines of systems and control engineering.2,3In the area of chemical dynamics, Lyapunov’s stability theory laid the foundation for understanding nonlinear chemical reaction systems, e.g., the Belousov-Zhabotinsky reaction pattern formation, or the non-isothermal, continuously stirred chemical reactor.5Stochasticity is often a significant, impactful feature in dynamic systems. Stochastic processes in physics and chemistry have been studied since the days of Brown, Einstein, and Langevin. In the recent past, stochasticity has become a significant focus of studies of biochemical reaction systems. Compellingly, understanding the principles that underlie cellular and biomolecular behaviors often requires accounting for the influence of fluctuations inherently present in small biological systems. Improved understanding of compelling dynamic phenomena, from molecular chaos to relativistic cosmological models, has been possible to large extent, thanks to such mathematical theories as Lyapunov’s theory of stability of nonlinear dynamic systems.. First developed over 120 yr ago, Lyapunov’s elegant theories propelled the development of the disciplines of systems and control engineering.. In the area of chemical dynamics, Lyapunov’s stability theory laid the foundation for understanding nonlinear chemical reaction systems, e.g., the Belousov-Zhabotinsky reaction pattern formation, or the non-isothermal, continuously stirred chemical reactor.. Stochasticity is often a significant, impactful feature in dynamic systems.. Stochasticity has become a significant focus of studies of biochemical reaction systems.. Understanding the principles that underlie cellular and biomolecular behaviors often requires accounting for the influence of fluctuations inherently present in small biological systems
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