Simon J. Fraser

The steady state and equilibrium approximations: A geometrical picture

We present a geometrical picture of the steady state (SSA) and equilibrium (EA) approximations for simple chemical reactions. Geometrically, SSA and EA correspond to surfaces in the phase space of species concentrations. In this space, chemical reaction is represented by trajectory motion governed by first order differential equations. Initially, during transient decay, trajectories move quickly into the (narrow) region R between the equilibrium (E) and steady state (S) surfaces, puncturing E or S orthogonal to a variable axis. This is rigorously what the differential equations say. Once in R, trajectories approach another surface, the slow manifold M, sandwiched between E and S. SSA or EA is the restriction of the system motion to S or E as a ‘‘shadow’’ of its true, nearby motion along M. We illustrate these ideas in relation to the Lindemann mechanism, showing how M is related to S and E, and how it may be obtained by iteration, yielding higher order SSA and EA formulas. This procedure is asymptotic wit...

Geometry of the steady‐state approximation: Perturbation and accelerated convergence methods

The time evolution of two model enzyme reactions is represented in phase space Γ. The phase flow is attracted to a unique trajectory, the slow manifold M, before it reaches the point equilibrium of the system. Locating M describes the slow time evolution precisely, and allows all rate constants to be obtained from steady‐state data. The line set M is found by solution of a functional equation derived from the flow differential equations. For planar systems, the steady‐state (SSA) and equilibrium (EA) approximations bound a trapping region containing M, and direct iteration and perturbation theory are formally equivalent solutions of the functional equation. The iteration’s convergence is examined by eigenvalue methods. In many dimensions, the nullcline surfaces of the flow in Γ form a prism‐shaped region containing M, but this prism is not a simple trap for the flow. Two of its edges are EA and SSA. Perturbation expansion and direct iteration are now no longer equivalent procedures; they are compared in a...

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.

ON THE GEOMETRY OF TRANSIENT RELAXATION

Coupled chemical reactions are often described by (stiff) systems of ordinary differential equations (ODEs) with widely separated relaxation times. In the phase space Γ of species concentration variables, relaxation can be represented as a cascade through a nested hierarchy of smooth hypersurfaces (inertial manifolds) {Σ}: If d is the number of independent concentration variables, then Γ≡Γd⊇Σd−1⊇Σd−2⋅⋅⋅. The last three sets in this hierarchy have special chemical importance: Σ0 is the stagnation point of the ODEs, i.e., chemical equilibrium; M(≡Σ1) is the linelike slow manifold describing the dynamical steady state in closed systems; Σ(≡Σ2) is the two‐dimensional surface containing the slowest transient flow that reaches M. Thus M and Σ are the structures underlying most steady‐state and transient kinetics experiments. The ODEs describe the velocity field in Γ, which may be used to define functional equations for M, Σ, and other members in the hierarchy {Σ}. These functional equations can be solved to giv...

Geometrical picture of reaction in enzyme kinetics

A generalization of the steady‐state and equilibrium approximations for solving the coupled differential equations of a chemical reaction is presented. This method determines the (steady‐state) reaction velocity in closed form. Decay from rapidly equilibrating networks is considered, since many enzyme mechanisms belong to this category. In such systems, after transients have died away, the phase‐space flow lies close to a unique trajectory, the slow manifold M. Locating M reduces the description of the system’s progress to a one‐dimensional integration. This manifold is a solution of a functional equation, derived from the differential equations for the reaction. As an example, M for Michaelis–Menten–Henri mechanism is found by direct iteration. The solution is very accurate and the appropriate boundary conditions are obeyed automatically. In an arbitrary mechanism, at vanishing decay rate, the slow manifold becomes a line of equilibrium states, which solves the functional equation exactly; it is thus a g...

Double perturbation series in the differential equations of enzyme kinetics

The connection between combined singular and ordinary perturbation methods and slow-manifold theory is discussed using the Michaelis-Menten model of enzyme catalysis as an example. This two-step mechanism is described by a planar system of ordinary differential equations (ODEs) with a fast transient and a slow “steady-state” decay mode. The systems of scaled nonlinear ODEs for this mechanism contain a singular (η) and an ordinary (e) perturbation parameter: η multiplies the velocity component of the fast variable and dominates the fast-mode perturbation series; e controls the decay toward equilibrium and dominates the slow-mode perturbation series. However, higher order terms in both series contain η and e. Finite series expansions partially decouple the system of ODEs into fast-mode and slow-mode ODEs; infinite series expansions completely decouple these ODEs. Correspondingly, any slow-mode ODE approximately describes motion on M, the linelike slow manifold of the system, and in the infinite series limit...

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.

Variance reduction for lattice walks grown with Markov chain sampling

A Markov chain procedure for growing self‐ and neighbor‐avoiding walks is described. By making the probabilities of all walks nearly equal, the variance in estimates of walk statistics is greatly reduced. A detailed analysis is given of the important example of next‐neighbor‐avoiding walks on the tetrahedral lattice: Such walks serve as a model for polymethylene chains. This case demonstrates that, in general, a similar variance reduction can be achieved at finite temperature for walks with conformationally dependent internal energy. The algorithm for constructing the sampling distributions for the steps of these finite temperature walks is summarized.

Dynamics of oscillators with periodic dichotomous noise

The dynamics of bistable oscillators driven by periodic dichotomous noise is described. The stochastic differential equation governing the flow implies smooth trajectories between noise switching events. The dynamics of the two-branched map induced by this flow is a Markov process. Harmonic and quartic models of the bistable potential are studied in the overdamped limit. In the linear (harmonic) case the dynamics can be reduced to a stochastic one-dimensional map with two branches. The moments decay exponentially in this case, although the invariant measure may be multifractal. For strong damping, relaxation induces a cascade leading to a Cantor set and anomalous decay of the density in this case is modeled by a Markov chain. For the physically more realistic case of a quartic potential many additional features arise since the contraction factor is distance dependent. By tuning the barrier-height parameter in the quartic potential, noise-induced transition rates with the characteristics of intermittency are found.

Monte Carlo simulation of a lattice model of intramolecular exciplex formation

Monte Carlo simulation on a diamond lattice is used to estimate the intramolecular cyclization probability of a bulky chromophore (benzophenone) joined to an amine quencher by a polymethylene chain. This probability determines the chain length dependence of relative rate constants in exciplex formation between photoexcited amine and chromophore when the intrinsic rate for this reaction is very low and therefore this cyclization is configurationally controlled. The chromophore is represented by suitable, fixed lattice sites in our model, the hydrocarbon chain by a (random) next‐neighbor‐avoiding walk, and the amine by an orientable group attached to the end of the polymethylene chain. Excluded volume and steric effects are incorporated by singly occupying lattice sites in the chain‐growing algorithm (equivalent to hard sphere interactions between the atoms comprising the system). The temperature dependence of cyclization is investigated by giving the chain segments conformational energy. The cyclization pr...