Mass-action Networks Research Articles

Bifurcations in planar, quadratic mass-action networks with few reactions and low molecularity

In this paper we study bifurcations in mass-action networks with two chemical species and reactant complexes of molecularity no more than two. We refer to these as planar, quadratic networks as they give rise to (at most) quadratic differential equations on the nonnegative quadrant of the plane. Our aim is to study bifurcations in networks in this class with the fewest possible reactions, and the lowest possible product molecularity. We fully characterise generic bifurcations of positive equilibria in such networks with up to four reactions, and product molecularity no higher than three. In these networks we find fold, Andronov–Hopf, Bogdanov–Takens and Bautin bifurcations, and prove the non-occurrence of any other generic bifurcations of positive equilibria. In addition, we present a number of results which go beyond planar, quadratic networks. For example, we show that mass-action networks without conservation laws admit no bifurcations of codimension greater than m-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m-2$$\\end{document}, where m is the number of reactions; we fully characterise quadratic, rank-one mass-action networks admitting fold bifurcations; and we write down some necessary conditions for Andronov–Hopf and cusp bifurcations in mass-action networks. Finally, we draw connections with a number of previous results in the literature on nontrivial dynamics, bifurcations, and inheritance in mass-action networks.

Application of Parameter Optimization to Search for Oscillatory Mass-Action Networks Using Python

Biological systems can be described mathematically to model the dynamics of metabolic, protein, or gene-regulatory networks, but locating parameter regimes that induce a particular dynamic behavior can be challenging due to the vast parameter landscape, particularly in large models. In the current work, a Pythonic implementation of existing bifurcation objective functions, which reward systems that achieve a desired bifurcation behavior, is implemented to search for parameter regimes that permit oscillations or bistability. A differential evolution algorithm progressively approximates the specified bifurcation type while performing a global search of parameter space for a candidate with the best fitness. The user-friendly format facilitates integration with systems biology tools, as Python is a ubiquitous programming language. The bifurcation–evolution software is validated on published models from the BioModels Database and used to search populations of randomly-generated mass-action networks for oscillatory dynamics. Results of this search demonstrate the importance of reaction enrichment to provide flexibility and enable complex dynamic behaviors, and illustrate the role of negative feedback and time delays in generating oscillatory dynamics.

Precision and Sensitivity in Detailed-Balance Reaction Networks

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 nondissipative 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 contain many species and reactions, or the network should show strongly different time scales.

The influence of empirical contact networks on modelling diseases in cattle

We present two stochastic models of the passage of an SEIR (susceptible–latent–infected–resistant) disease through herds of cattle. One model is based on a contact network constructed via continuously recorded interaction data from two herds of cattle, the other, a matching network constructed using the principles of mass-action mixing. The recorded contact data were produced by attaching proximity data loggers to two separate herds of cattle during two separate recording periods. The network constructed using the principles of mass-action mixing uses the same number of contacts as the recorded network but distributes them randomly amongst the animals. The recorded networks had a greater number of repeated contacts, lower closeness and clustering scores and greater average path length than the mass-action networks. A lower proportion of simulations of the recorded network produce any disease spread when compared to those simulations of the mass-action network and, of those that did, fewer infected animals were predicted. For all parameter values tested, within the sensitivity analysis, similar differences were found between the recorded and mass-action network models.

Switching in Mass Action Networks Based on Linear Inequalities

Many biochemical processes can successfully be described by dynamical systems allowing some form of switching when, depending on their initial conditions, solutions of the dynamical system end up in different regions of state space (associated with different biochemical functions). Switching is often realized by a bistable system (i.e., a dynamical system allowing two stable steady state solutions) and, in the majority of cases, bistability is established numerically. In our view, this approach is too restrictive. On the one hand, due to predominant parameter uncertainty, numerical methods are generally difficult to apply to realistic models originating in systems biology. On the other hand, switching already arises with the occurrence of a saddle-type steady state (characterized by a Jacobian where exactly one eigenvalue is positive and the remaining eigenvalues have negative real part). Consequently we derive conditions based on linear inequalities that allow the analytic computation of states and param...

Multistationarity in mass action networks with applications to ERK activation

Ordinary Differential Equations (ODEs) are an important tool in many areas of Quantitative Biology. For many ODE systems multistationarity (i.e. the existence of at least two positive steady states) is a desired feature. In general establishing multistationarity is a difficult task as realistic biological models are large in terms of states and (unknown) parameters and in most cases poorly parameterized (because of noisy measurement data of few components, a very small number of data points and only a limited number of repetitions). For mass action networks establishing multistationarity hence is equivalent to establishing the existence of at least two positive solutions of a large polynomial system with unknown coefficients. For mass action networks with certain structural properties, expressed in terms of the stoichiometric matrix and the reaction rate-exponent matrix, we present necessary and sufficient conditions for multistationarity that take the form of linear inequality systems. Solutions of these inequality systems define pairs of steady states and parameter values. We also present a sufficient condition to identify networks where the aforementioned conditions hold. To show the applicability of our results we analyse an ODE system that is defined by the mass action network describing the extracellular signal-regulated kinase (ERK) cascade (i.e. ERK-activation).

Constraints on Reciprocal Flux Sensitivities in Biochemical Reaction Networks

We identify a connection between the structural features of mass-action networks and the robustness of their steady-state fluxes against rate constant variations. We find that in all positive steady states of so-called injective networks—networks that arise, for example, in metabolic and gene regulation contexts—there are certain firm bounds on the flux control coefficients. In particular, the control coefficient of the flux of a reaction, with respect to variation in its own rate constant, is delimited in a precise way. Moreover, for each pair of reactions, the flux of at least one of them must have a precisely delimited control coefficient with respect to variation in the rate constant of the other. The derived bounds can, however, be violated in noninjective networks, so for them a more pronounced lack of robustness could be exhibited. These results, which indicate a mechanism by which some degree of robustness is induced in the injective setting, also shed light on how robustness might evolve.

Understanding bistability in complex enzyme-driven reaction networks.

Much attention has been paid recently to bistability and switch-like behavior that might be resident in important biochemical reaction networks. There is, in fact, a great deal of subtlety in the relationship between the structure of a reaction network and its capacity to engender bistability. In common physicochemical settings, large classes of extremely complex networks, taken with mass action kinetics, cannot give rise to bistability no matter what values the rate constants take. On the other hand, bistable behavior can be induced in those same settings by certain very simple and classical mass action mechanisms for enzyme catalysis of a single overall reaction. We present a theorem that distinguishes between those mass action networks that might support bistable behavior and those that cannot. Moreover, we indicate how switch-like behavior results from a well-studied mechanism for the action of human dihydrofolate reductase, an important anti-cancer target.

On Topological‐Dynamical Equivalent Representations of Reaction Networks: The Omega‐Equation and a Canonical Class of Mass Action Kinetics

AbstractAn unifying concept ‐ the Omega‐equation ‐ is introduced to describe analogies of dynamical systems such as for chemical reaction systems, transport and metabolism of coupled compartments, cell populations, population genetics and animal behavior in ecology. The phenomenology of rate‐laws of such systems are frequently described, perhaps after an equivalent transformation, in terms of ordinary polynomial differential equations, which can be incorporated in the strong formalism of mass action reaction networks. Furthermore, we show that the set of all mass action kinetics can be reduced to the smaller class of so‐called μ‐molar mass action networks, which generate homogeneous polynomial Ω‐equations of order μ.