Current and eigenvector analyses of chemical reaction networks at Hopf bifurcations

Constrained stoichiometric network analysis

Stoichiometric network analysis (SNA) is a method for studying the stability of steady states of stoichiometric systems by decomposing the corresponding network into elementary subnetworks (also known as extreme currents) and identifying those that may cause loss of a network's stability via interplay of positive and negative feedback. Experimentally studied complex (bio)chemical reactions often display dynamical instabilities leading to oscillations or bistable switches. When modelling such systems, a frequently met case is that an assumed detailed mechanism in terms of power law kinetics is available, but some of the rate coefficients are unknown and obtaining them by traditional kinetic methods based on a least-square fit is cumbersome or unfeasible. We propose a method combining the SNA and experimental data at the point of instability, which provides an estimate of the unknown rate coefficients along with unknown steady state concentrations. The core of the method rests in using constrained linear optimization to find a combination of the elementary subnetworks such that the dominant instability-causing subnetwork is just counter-balanced by stabilizing effects of all other subnetworks to obtain the instability threshold, and at the same time, the experimentally available data (inflow constraints, measured steady state concentrations of some species, frequency of emerging oscillations, etc.) are exactly matched. We illustrate this approach by examining two classical chemical oscillators: the Brusselator chosen as the simplest model for illustration of our methods and the Belousov-Zhabotinsky reaction and its mechanism represented by the Oregonator model as a more advanced example.

Reduction of a biochemical model with preservation of its basic dynamic properties

The complexity of full-scale metabolic models is a major obstacle for their effective use in computational systems biology. The aim of model reduction is to circumvent this problem by eliminating parts of a model that are unimportant for the properties of interest. The choice of reduction method is influenced both by the type of model complexity and by the objective of the reduction; therefore, no single method is superior in all cases. In this study we present a comparative study of two different methods applied to a 20D model of yeast glycolytic oscillations. Our objective is to obtain biochemically meaningful reduced models, which reproduce the dynamic properties of the 20D model. The first method uses lumping and subsequent constrained parameter optimization. The second method is a novel approach that eliminates variables not essential for the dynamics. The applications of the two methods result in models of eight (lumping), six (elimination) and three (lumping followed by elimination) dimensions. All models have similar dynamic properties and pin-point the same interactions as being crucial for generation of the oscillations. The advantage of the novel method is that it is algorithmic, and does not require input in the form of biochemical knowledge. The lumping approach, however, is better at preserving biochemical properties, as we show through extensive analyses of the models.

Nonlinear kinetics and new approaches to complex reaction mechanisms

This paper reviews recent developments in the field of nonlinear chemical kinetics. Five topics are dealt with: (a) new approaches to complex reaction mechanisms, stoichiometric network analysis, classification of chemical oscillators and formulation of their mechanisms by deduction from experiments, and correlation metric construction of reaction pathways from measurements; (b) thermodynamic and stochastic theory of nonequilibrium processes, the eikonal approximation, the evaluation of stochastic potentials, experimental tests of the thermodynamic and stochastic theory of relative stability, and fluctuation-dissipation relations in nonequilibrium chemical systems; (c) chemical kinetics and cellular automata and lattice gas automata; (d) theoretical approaches and experimental studies of stochastic resonance in chemical kinetics; and (e) rate processes in disordered systems, stochastic Liouville equations, stretched exponential relaxation in disordered systems, and universality classes for rate processes in systems with static or dynamic disorder.

Full-scale model of glycolysis in Saccharomyces cerevisiae

We present a powerful, general method of fitting a model of a biochemical pathway to experimental substrate concentrations and dynamical properties measured at a stationary state, when the mechanism is largely known but kinetic parameters are lacking. Rate constants and maximum velocities are calculated from the experimental data by simple algebra without integration of kinetic equations. Using this direct approach, we fit a comprehensive model of glycolysis and glycolytic oscillations in intact yeast cells to data measured on a suspension of living cells of Saccharomyces cerevisiae near a Hopf bifurcation, and to a large set of stationary concentrations and other data estimated from comparable batch experiments. The resulting model agrees with almost all experimentally known stationary concentrations and metabolic fluxes, with the frequency of oscillation and with the majority of other experimentally known kinetic and dynamical variables. The functional forms of the rate equations have not been optimized.

On the deduction of chemical reaction pathways from measurements of time series of concentrations

We discuss the deduction of reaction pathways in complex chemical systems from measurements of time series of chemical concentrations of reacting species. First we review a technique called correlation metric construction (CMC) and show the construction of a reaction pathway from measurements on a part of glycolysis. Then we present two new improved methods for the analysis of time series of concentrations, entropy metric construction (EMC), and entropy reduction method (ERM), and illustrate (EMC) with calculations on a model reaction system. (c) 2001 American Institute of Physics.

Sustained oscillations in glycolysis: an experimental and theoretical study of chaotic and complex periodic behavior and of quenching of simple oscillations

We report sustained oscillations in glycolysis conducted in an open system (a continuous-flow, stirred tank reactor; CSTR) with inflow of yeast extract as well as glucose. Depending on the operating conditions, we observe simple or complex periodic oscillations or chaos. We report the response of the system to instantaneous additions of small amounts of several substrates as functions of the amount added and the phase of the addition. We simulate oscillations and perturbations by a kinetic model based on the mechanism of glycolysis in a CSTR. We find that the response to particular perturbations forms an efficient tool for elucidating the mechanism of biochemical oscillations.