Double perturbation series in the differential equations of enzyme kinetics

Heineken, Tsuchiya and Aris on the mathematical status of the pseudo-steady state hypothesis: A classic from volume 1 of Mathematical Biosciences

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.

Spatial dynamics of steady flames 2. Low-dimensional manifolds and the role of transport processes

The study of the spatial dynamics of steady one-dimensional H 2/O 2 flames is continued. Algorithms for generating low-dimensional manifolds for these systems are presented and used to find low-dimensional manifolds for the flames and the corresponding adiabatic, isobaric chemical-kinetic systems. It is demonstrated that these algorithms generate manifolds that are more accurate than the ILDM algorithm for two-dimensional manifolds of the flames. The manifolds are then employed to study the relationship between the manifolds of the flame and the manifolds of the chemical-kinetic system. It is shown that the one-dimensional manifolds of the flame match well with the composite manifolds of the chemical kinetics, but that for two-dimensional manifolds there are discrepancies between the flame manifolds and the chemical-kinetic manifolds.

Spatial dynamics of steady flames 1. Phase space structure and the dynamics of individual trajectories

The spatial dynamics of steady, one-dimensional premixed H 2/O 2 flames are studied. The emphasis in this Article is the geometry of the phase space of the dynamical system describing the steady flames. It is shown that steady flames are described by trajectories on the stable manifolds of saddle fixed points. The saddle fixed points correspond to equilibrium points of time-dependent chemical-kinetic systems that are adiabatic and isobaric and whose constant enthalpy matches the asymptotic enthalpy of the flames. The dimensions of the stable manifolds match the dimensions of the chemical-kinetic systems under most conditions, although the dynamics on them are different. It is further shown that the stable manifolds have low-dimensional attractive submanifolds near the saddlepoint. An algorithm for generating trajectories over the spatial domain of these flames is proposed, and it is used to study individual trajectories and trajectory ensembles, whose collective behavior suggests that there are low-dimensional submanifolds away from the saddlepoint.

Low-dimensional manifolds in reaction-diffusion equations. 1. Fundamental aspects

The approach to equilibrium for systems of reaction-diffusion equations on bounded domains is studied geometrically. It is shown that equilibrium is approached via low-dimensional manifolds in the infinite-dimensional function space for these dissipative, parabolic systems. The fundamental aspects of this process are mapped out in some detail for single species cases and for two-species cases where there is an exact solution. It is shown how the manifolds reduce the dimensionality of the system from infinite dimensions to only a few dimensions.

Low-Dimensional Manifolds in Reaction−Diffusion Equations. 2. Numerical Analysis and Method Development

Calculations are undertaken to study the approach to equilibrium for systems of reaction-diffusion equations on bounded domains. It is demonstrated that a number of systems approach equilibrium along attractive low-dimensional manifolds over significant ranges of parameter space. Numerical methods for generating the manifolds are adapted from methods that were developed for systems of ordinary differential equations. The truncation of the infinite spectrum of the partial differential equations makes it necessary to devise a new version of one of these methods, the well-known algorithm of Maas and Pope.

Slow manifold for a bimolecular association mechanism

Finding the slow manifold for two-variable ordinary differential equation (ODE) models of chemical reactions with a single equilibrium is generally simple. In such planar ODEs the slow manifold is the unique trajectory corresponding to the slow relaxation of the system as it moves towards the equilibrium point. One method of finding the slow manifold is to use direct iteration of a functional equation; another method is to obtain a series solution of the trajectory differential equation of the system. In some cases these two methods agree order-by-order in the singular perturbation parameter controlling the fast relaxation of the intermediate (complex). However, de la Llave has found a model ODE where the series method always diverges. Bimolecular association is an example of a chemical reaction where the series method for finding the slow manifold diverges but the iterative method converges. In this mechanism a complex is formed which can then undergo unimolecular decay, i.e., [reaction: see text]. The kinetics of this reaction are investigated and its properties compared with two other two-step mechanisms where series expansion and iteration methods are equivalent: the Michaelis-Menten mechanism for enzyme kinetics, and the Lindemann-Christiansen mechanism of unimolecular decay in gas kinetics.

Invariant manifold methods for metabolic model reduction

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.