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Register a new userCritical parameters for singular perturbation reductions of chemical reaction networks
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We are concerned with polynomial ordinary differential systems that arise from modelling chemical reaction networks. For such systems, which may be of high dimension and may depend on many parameters, it is frequently of interest to obtain a reduction of dimension in certain parameter ranges. Singular perturbation theory, as initiated by Tikhonov and Fenichel, provides a path toward such reductions. In the present paper we discuss parameter values that lead to singular perturbation reductions (so-called Tikhonov-Fenichel parameter values, or TFPVs). An algorithmic approach is known, but it is feasible for small dimensions only. Here we characterize conditions for classes of reaction networks for which TFPVs arise by turning off reactions (by setting rate parameters to zero), or by removing certain species (which relates to the classical quasi-steady state approach to model reduction). In particular, we obtain definitive results for the class of complex balanced reaction networks (of deficiency zero) and first order reaction networks.
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Many biochemical and industrial applications involve complicated networks of simultaneously occurring chemical reactions. Under the assumption of mass action kinetics, the dynamics of these chemical reaction networks are governed by systems of polynomial ordinary differential equations. The steady states of these mass action systems have been analysed via a variety of techniques, including elementary flux mode analysis, algebraic techniques (e.g. Groebner bases), and deficiency theory. In this paper, we present a novel method for characterizing the steady states of mass action systems. Our method explicitly links a networks capacity to permit a particular class of steady states, called toric steady states, to topological properties of a related network called a translated chemical reaction network. These networks share their reaction stoichiometries with their source network but are permitted to have different complex stoichiometries and different network topologies. We apply the results to examples drawn from the biochemical literature.
We derive a reduction formula for singularly perturbed ordinary differential equations (in the sense of Tikhonov and Fenichel) with a known parameterization of the critical manifold. No a priori assumptions concerning separation of slow and fast variables are made, or necessary.We apply the theoretical results to chemical reaction networks with mass action kinetics admitting slow and fast reactions. For some relevant classes of such systems there exist canonical parameterizations of the variety of stationary points, hence the theory is applicable in a natural manner. In particular we obtain a closed form expression for the reduced system when the fast subsystem admits complex balanced steady states.
We study two specific measures of quality of chemical reaction networks, Precision and Sensitivity. The two measures arise in the study of sensory adaptation, in which the reaction network is viewed as an input-output system. Given a step change in input, Sensitivity is a measure of the magnitude of the response, while Precision is a measure of the degree to which the system returns to its original output for large time. High values of both are necessary for high-quality adaptation. We focus on reaction networks without dissipation, which we interpret as detailed-balance, mass-action networks. We give various upper and lower bounds on the optimal values of Sensitivity and Precision, characterized in terms of the stoichiometry, by using a combination of ideas from matroid theory and differential-equation theory. Among other results, we show that this class of non-dissipative systems contains networks with arbitrarily high values of both Sensitivity and Precision. This good performance does come at a cost, however, since certain ratios of concentrations need to be large, the network has to be extensive, or the network should show strongly different time scales.
It is becoming widely accepted that very early in the origin of life, even before the emergence of genetic encoding, reaction networks of diverse small chemicals might have manifested key properties of life, namely self-propagation and adaptive evolution. To explore this possibility, we formalize the dynamics of chemical reaction networks within the framework of chemical ecosystem ecology. To capture the idea that life-like chemical systems are maintained out of equilibrium by fluxes of energy-rich food chemicals, we model chemical ecosystems in well-mixed containers that are subject to constant dilution by a solution with a fixed concentration of food chemicals. Modelling all chemical reactions as fully reversible, we show that seeding an autocatalytic cycle (AC) with tiny amounts of one or more of its member chemicals results in logistic growth of all member chemicals in the cycle. This finding justifies drawing an instructive analogy between an AC and the population of a biological species. We extend this finding to show that pairs of ACs can show competitive, predator-prey, or mutualistic associations just like biological species. Furthermore, when there is stochasticity in the environment, particularly in the seeding of ACs, chemical ecosystems can show complex dynamics that can resemble evolution. The evolutionary character is especially clear when the network architecture results in ecological precedence (survival of the first), which makes the path of succession historically contingent on the order in which cycles are seeded. For all its simplicity, the framework developed here is helpful for visualizing how autocatalysis in prebiotic chemical reaction networks can yield life-like properties. Furthermore, chemical ecosystem ecology could provide a useful foundation for exploring the emergence of adaptive dynamics and the origins of polymer-based genetic systems.
Delay differential equations are used as a model when the effect of past states has to be taken into account. In this work we consider delay models of chemical reaction networks with mass action kinetics. We obtain a sufficient condition for absolute delay stability of equilibrium concentrations, i.e., local asymptotic stability independent of the delay parameters. Several interesting examples on sequestration networks with delays are presented.
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