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International audience; Delays appear in the dynamics of many systems due to non-vanishing reaction times of control systems. In biochemical systems, long sequences of repeated steps, especially in biopolymerization processes, can be modeled by delays. However, modelling systems with delays is often complicated by physical constraints, such as the requirement that solutions representing concentrations of chemical species remain positive. In this work, we consider a model for a detoxifying enzyme whose synthesis is controlled by its substrate. The model includes bindingsite clearance delays, caused by the time required for an RNA polymerase or ribosome to clear its binding site before another such machine can bind. The existence of a positive equilibrium and the positivity and boundedness of solutions of the corresponding delay-differential equations are proven. In addition, the stability of the model is studied using the "small-gain" theorem.

The conditions for the validity of the standard quasi-steady-state approximation in the Michaelis--Menten mechanism in a closed reaction vessel have been well studied, but much less so the conditions for the validity of this approximation for the system with substrate inflow. We analyze quasi-steady-state scenarios for the open system attributable to singular perturbations, as well as less restrictive conditions. For both settings we obtain distinguished invariant slow manifolds and time scale estimates, and we highlight the special role of singular perturbation parameters in higher order approximations of slow manifolds. We close the paper with a discussion of distinguished invariant manifolds in the global phase portrait., 32 pages, 6 figures

Delay-differential equations belong to the class of infinite-dimensional dynamical systems. However, it is often observed that the solutions are rapidly attracted to smooth manifolds embedded in the finite-dimensional state space, called inertial manifolds. The computation of an inertial manifold yields an ordinary differential equation (ODE) model representing the long-term dynamics of the system. Note in particular that any attractors must be embedded in the inertial manifold when one exists, therefore reducing the study of these attractors to the ODE context, for which methods of analysis are well developed. This contribution presents a study of a previously developed method for constructing inertial manifolds based on an expansion of the delayed term in small powers of the delay, and subsequent solution of the invariance equation by the Fraser functional iteration method. The combined perturbative-iterative method is applied to several variations of a model for the expression of an inducible enzyme, where the delay represents the time required to transcribe messenger RNA and to translate that RNA into the protein. It is shown that inertial manifolds of different dimensions can be computed. Qualitatively correct inertial manifolds are obtained. Among other things, the dynamics confined to computed inertial manifolds display Andronov&ndash, Hopf bifurcations at similar parameter values as the original DDE model.

A parametric sensitivity analysis for periodic solutions of delay-differential equations is developed. Because phase shifts cause the sensitivity coefficients of a periodic orbit to diverge, we focus on sensitivities of the extrema, from which amplitude sensitivities are computed, and of the period. Delay-differential equations are often used to model gene expression networks. In these models, the parametric sensitivities of a particular genotype define the local geometry of the evolutionary landscape. Thus, sensitivities can be used to investigate directions of gradual evolutionary change. An oscillatory protein synthesis model whose properties are modulated by RNA interference is used as an example. This model consists of a set of coupled delay-differential equations involving three delays. Sensitivity analyses are carried out at several operating points. Comments on the evolutionary implications of the results are offered.

Differential equation models of chemical or biochemical systems usually display multiple, widely varying time scales, i.e. they are stiff. After the decay of transients, trajectories of these systems approach low-dimensional invariant manifolds on which the eventual attractor (an equilibrium point in a closed system) is approached, and in which this attractor is embedded. Computing one of these slow invariant manifolds (SIMs) results in a reduced model of dimension equal to the dimension of the SIM. Another approach to model reduction involves lumping, the formulation of a reduced set of variables that combine the original model variables and in terms of which the reduced model is framed. In this study, we combine lumping with a constructive method for SIMs based on the iterative solution of the invariance equation. We illustrate these methods using a simple model of a linear metabolic pathway, and a model for hydrogen oxidation. The former is treated with a linear lumping function, while a nonlinear lumping function based on a Lyapunov function is used in the latter.

Volume 1, Issue 1 of Mathematical Biosciences was the venue for a now-classic paper on the application of singular perturbation theory in enzyme kinetics, "On the mathematical status of the pseudo-steady state hypothesis of biochemical kinetics" by F. G. Heineken, H. M. Tsuchiya and R. Aris. More than 50 years have passed, and yet this paper continues to be studied and mined for insights. This perspective discusses both the strengths and weaknesses of the work presented in this paper. For many, the justification of the pseudo-steady-state approximation using singular perturbation theory is the main achievement of this paper. However, there is so much more material here, which laid the foundation for a great deal of research in mathematical biochemistry in the intervening decades. The parameterization of the equations, construction of the first-order uniform singular-perturbation solution, and an attempt to apply similar principles to the pseudo-equilibrium approximation are discussed in particular detail.

Network conditions for Turing instability in biochemical systems with two biochemical species are well known and involve autocatalysis or self-activation. On the other hand general network conditions for potential Turing instabilities in large biochemical reaction networks are not well developed. A biochemical reaction network with any number of species where only one species moves is represented by a simple digraph and is modeled by a reaction–diffusion system with non-mass action kinetics. A graph-theoretic condition for potential Turing-Hopf instability that arises when a spatially homogeneous equilibrium loses its stability via a single pair of complex eigenvalues is obtained. This novel graph-theoretic condition is closely related to the negative cycle condition for oscillations in ordinary differential equation models and its generalizations, and requires the existence of a pair of subnetworks, each containing an even number of positive cycles. The technique is illustrated with a double-cycle Goodwin type model.

A chemical mechanism is a model of a chemical reaction network consisting of a set of elementary reactions that express how molecules react with each other. In classical mass-action kinetics, a mechanism implies a set of ordinary differential equations (ODEs) which govern the time evolution of the concentrations. In this article, ODE models of chemical kinetics that have the potential for multiple positive equilibria or oscillations are studied. We begin by considering some methods of stability analysis based on the digraph of the Jacobian matrix. We then prove two theorems originally given by A. N. Ivanova which correlate the bifurcation structure of a mass-action model to the properties of a bipartite graph with nodes representing chemical species and reactions. We provide several examples of the application of these theorems.

We study methods for reducing chemical master equations using the Michaelis-Menten mechanism as an example. The master equation consists of a set of linear ordinary differential equations whose variables are probabilities that the realizable states exist. For a master equation with s(0) initial substrate molecules and e(0) initial enzyme molecules, the manifold can be parametrized by s(0) of the probability variables. Fraser's functional iteration method is found to be difficult to use for master equations of high dimension. Building on the insights gained from Fraser's method, techniques are developed to produce s(0)-dimensional manifolds of larger systems directly from the eigenvectors. We also develop a simple, but surprisingly effective way to generate initial conditions for the reduced models.

Although the theory of delay-differential equations (DDEs) is generally best set in a function space, some systems of DDEs have solutions which, after the decay of transients, lie on a low-dimensional manifold in their state space. When the delay is small, highly accurate approximations to the state-space manifold which attracts the solutions can be constructed by a simple functional equation treatment. This allows the reduction of the original system of DDEs to a smaller system of ordinary differential equations. The simplified model obtained may be used to facilitate bifurcation analysis. The method is applied to two biochemical models, namely to a delay-differential version of Michaelis–Menten kinetics (the Brown model) and to a simple inducible operon model.

In many dynamical systems, an invariant manifold attracts the phase‐space flow. These manifolds can be approximated by an iterative method based on a functional equation treatment. However, a convergent mapping is not automatically generated from the functional equation. Nevertheless, it is possible to construct a convergent mapping by a simple modification of the original functional equation. As an example, a convergent sequence of approximations to the slow manifold of the Michaelis–Menten system is constructed.

A simple epidemiological model whose dynamics are described by a pair of nonlinearly coupled Lotka--Volterra oscillators is shown to have a two-dimensional center manifold. This center manifold turns out to be identical to the center eigenspace and is thus analytically determinable. On the center manifold, the system is reduced to a single Lotka--Volterra oscillator.

A slow manifold is a low-dimensional invariant manifold to which trajectories nearby are rapidly attracted on the way to the equilibrium point. The exact computation of the slow manifold simplifies the model without sacrificing accuracy on the slow time scales of the system. The Maas-Pope intrinsic low-dimensional manifold (ILDM) [Combust. Flame 88, 239 (1992)] is frequently used as an approximation to the slow manifold. This approximation is based on a linearized analysis of the differential equations and thus neglects curvature. We present here an efficient way to calculate an approximation equivalent to the ILDM. Our method, called functional equation truncation (FET), first develops a hierarchy of functional equations involving higher derivatives which can then be truncated at second-derivative terms to explicitly neglect the curvature. We prove that the ILDM and FET-approximated (FETA) manifolds are identical for the one-dimensional slow manifold of any planar system. In higher-dimensional spaces, the ILDM and FETA manifolds agree to numerical accuracy almost everywhere. Solution of the FET equations is, however, expected to generally be faster than the ILDM method.

The conditions for diffusion-driven (Turing) instabilities in systems with two reactive species are well known. General methods for detecting potential Turing bifurcations in larger reaction schemes are, on the other hand, not well developed. We prove a theorem for a graph-theoretic condition originally given by Volpert and Ivanova [Mathematical Modeling (Nauka, Moscow, 1987) (in Russian), p. 57] for Turing instabilities in a mass-action reaction-diffusion system involving n substances. The method is based on the representation of a reaction mechanism as a bipartite graph with two types of nodes representing chemical species and reactions, respectively. The condition for diffusion-driven instability is related to the existence of a structure in the graph known as a critical fragment. The technique is illustrated using a substrate-inhibited bifunctional enzyme mechanism which involves seven chemical species.

In the small-number limit, we must abandon the description of chemical systems in terms of continuous concentration variables which evolve according to deterministic rate equations in favor of a discrete stochastic formulation. The probability distribution for the molecular populations however does obey a deterministic equation called the chemical master equation. Any desired population statistic (mean, standard deviation, etc.) can be obtained from the probability distribution. Unfortunately, the master equation consists of a huge set of differential equations, and it is thus in general impractical to use it directly. In this paper, we review the ideas underlying discrete population modeling and the chemical master equation. We then develop methods for reducing the chemical master equation to a much smaller set of differential equations by exploiting the same time-scale separation which leads to the emergence of a hierarchy of attracting manifolds in the mass-action case. Finally, we develop a method for generating an initial condition for the reduced model based on a generalization of the stationary reactant approximation.

A theoretical analysis of the distinguishability problem of two rival models of the single enzyme-single substrate reaction, the Michaelis-Menten and Henri mechanisms, is presented. We also outline a general approach for analysing the structural indistinguishability between two mechanisms. The approach involves constructing, if possible, a smooth mapping between the two candidate models. Evans et al. [N.D. Evans, M.J. Chappell, M.J. Chapman, K.R. Godfrey, Structural indistinguishability between uncontrolled (autonomous) nonlinear analytic systems, Automatica 40 (2004) 1947-1953] have shown that if, in addition, either of the mechanisms satisfies a particular criterion then such a transformation always exists when the models are indistinguishable from their experimentally observable outputs. The approach is applied to the single enzyme-single substrate reaction mechanism. In principle, mechanisms can be distinguished using this analysis, but we show that our ability to distinguish mechanistic models depends both on the precise measurements made, and on our knowledge of the system prior to performing the kinetics experiments.

We study invariant manifold methods for reducing chemical master equations using the Michaelis‐Menten mechanism as an example. We try Fraser’s functional iteration method first, but find that it is difficult to use for master equations of high dimension. Using the insights gained from Fraser’s method, we develop a technique to produce reduced chemical master equations directly from the eigenvectors of the state‐to‐state transition rate matrix. The dimension of the original chemical master equation grows quadratically with number of molecules, while the dimension of the reduced one we obtain is linear in the number of molecules. Additionally, a simple, effective way is developed to generate initial conditions for the reduced models.

After the decay of transients, the behavior of a set of differential equations modeling a chemical or biochemical system generally rests on a low-dimensional surface which is an invariant manifold of the flow. If an equation for such a manifold can be obtained, the model has effectively been reduced to a smaller system of differential equations. Using perturbation methods, we show that the distinction between rapidly decaying and long-lived (slow) modes has a rigorous basis. We show how equations for attracting invariant (slow) manifolds can be constructed by a geometric approach based on functional equations derived directly from the differential equations. We apply these methods to two simple metabolic models. (c) 2001 American Institute of Physics.