[Paper Review] Fixed Points for Stochastic Open Chemical Systems

V. A. Malyshev|arXiv (Cornell University)|Dec 16, 2011

Gene Regulatory Network Analysis26 references3 citations

TL;DR

This paper establishes the existence of attracting fixed points in a class of stochastic open chemical reaction systems, including Michaelis-Menten kinetics, by modeling compartments as thermodynamic networks with matter exchange as the sole source of nonreversibility. It introduces a network-of-networks framework with delayed transport between compartments, proving that such systems can exhibit stable, homeostatic fixed points under specific transport dynamics.

ABSTRACT

In the first part of this paper we give a short review of the hierarchy of stochastic models, related to physical chemistry. In the basement of this hierarchy there are two models --- stochastic chemical kinetics and the Kac model for Boltzman equation. Classical chemical kinetics and chemical thermodynamics are obtained as some scaling limits in the models, introduced below. In the second part of this paper we specify some simple class of open chemical reaction systems, where one can still prove the existence of attracting fixed points. For example, Michaelis ire Menten kinetics belongs to this class. At the end we present a simplest possible model of the biological network. It is a network of networks (of closed chemical reaction systems, called compartments), so that the only source of nonreversibility is the matter exchange (transport) with the environment and between the compartments. Keywords: chemical kinetics, chemical thermodynamics, Kac model, mathematical biology

Motivation & Objective

• To bridge the gap between physical chemistry and mathematical biology by analyzing stochastic open chemical systems with nonreversible dynamics.
• To identify conditions under which open chemical reaction systems exhibit stable, attracting fixed points despite the absence of reversibility.
• To model biological networks as networks of thermodynamic compartments connected by matter transport, capturing nonreversibility through exchange with environment and between compartments.
• To analyze the role of time delays and transport rates in inducing phase transitions and homeostatic regulation in multi-compartment systems.
• To lay a rigorous mathematical foundation for studying logical and dynamical behavior in biological networks using stochastic thermodynamic models.

Proposed method

• Models molecular dynamics using a stochastic process on countable configurations of particles with position, velocity, and type, governed by a measure space of initial conditions.
• Defines a Markov process for individual particles with time-dependent transition kernels depending on concentration densities, using Kolmogorov equations for the evolution of joint distributions.
• Introduces a network-of-networks structure where each compartment is a closed, thermodynamically stable system, and inter-compartment transport is modeled via stochastic arrival times with random delays.
• Derives mean-field limit equations for concentration dynamics in each compartment, incorporating input/output rates, transport from other compartments with time delays, and internal reaction terms.
• Uses the Kac model and stochastic chemical kinetics as foundational models, showing that classical chemical kinetics and thermodynamics emerge as scaling limits.
• Applies methods from stochastic processes and ergodic theory to prove null recurrence and existence of fixed points in single-compartment systems, extending to multi-compartment networks.

Experimental results

Research questions

• RQ1 Under what conditions does an open chemical reaction system with nonreversible dynamics admit an attracting fixed point?
• RQ2 How do time delays in inter-compartmental transport affect the stability and convergence of concentration dynamics in a network of thermodynamic compartments?
• RQ3 Can homeostatic regulation emerge in a network of compartments due to controlled transport rates, and how do fixed points depend on these rates?
• RQ4 What phase transitions occur in a network of compartments when transport parameters or time delays are varied?
• RQ5 How does the system behave under periodic or stochastic input rates, particularly in cyclic topologies, and what convergence properties emerge?

Key findings

• A class of open chemical reaction systems, including Michaelis-Menten kinetics, admits attracting fixed points under appropriate conditions, even without reversibility.
• In single compartments (rank-1 networks), the system converges to a fixed point governed by thermodynamic equilibrium, with null recurrence proven via methods from [7].
• For networks of compartments (rank-2 networks), the limiting equations for concentration dynamics include transport terms with random time delays, which preserve randomness at macroscopic scales.
• The system exhibits homeostatic behavior when transport rates are varied smoothly, with fixed points depending continuously on transport parameters.
• Phase transitions can occur when transport rates cross critical thresholds, leading to abrupt changes in the basin of attraction of fixed points.
• Time desynchronization due to random delays can significantly alter long-term behavior, especially in cyclic networks, suggesting a mechanism for dynamical regulation in biological systems.