F. Hynne

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.

Sustained oscillations in living cells

Glycolytic oscillations in yeast have been studied for many years simply by adding a glucose pulse to a suspension of cells and measuring the resulting transient oscillations of NADH. Here we show, using a suspension of yeast cells, that living cells can be kept in a well defined oscillating state indefinitely when starved cells, glucose and cyanide are pumped into a cuvette with outflow of surplus liquid. Our results show that the transitions between stationary and oscillatory behaviour are uniquely described mathematically by the Hopf bifurcation. This result characterizes the dynamical properties close to the transition point. Our perturbation experiments show that the cells remain strongly coupled very close to the transition. Therefore, the transition takes place in each of the cells and is not a desynchronization phenomenon. With these two observations, a study of the kinetic details of glycolysis, as it actually takes place in a living cell, is possible using experiments designed in the framework of nonlinear dynamics. Acetaldehyde is known to synchronize the oscillations. Our results show that glucose is another messenger substance, as long as the glucose transporter is not saturated.

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.

Coexistence Curve for 3‐Methylpentane–Nitroethane near the Critical Point

The coexistence curve for the liquid mixture 3‐methylpentane–nitroethane has been obtained using a visual and a float technique. A detailed analysis of the data shows that the composition differences of the coexisting liquid phases are proportional to (tc — rpar;β, where β = 0.340 with a standard deviation of 0.010. As a result of this analysis and an analysis of the earlier work of Rice and Zimm it is shown that the value of β for binary liquids is not significantly different from the value for gas–liquid systems.

Synchronization of glycolytic oscillations in a yeast cell population

The mechanism of active phase synchronization in a suspension of oscillatory yeast cells has remained a puzzle for almost half a century. The difficulty of the problem stems from the fact that the synchronization phenomenon involves the entire metabolic network of glycolysis and fermentation, and consequently it cannot be addressed at the level of a single enzyme or a single chemical species. In this paper it is shown how this system in a CSTR (continuous flow stirred tank reactor) can be modelled quantitatively as a population of Stuart-Landau oscillators interacting by exchange of metabolites through the extracellular medium, thus reducing the complexity of the problem without sacrificing the biochemical realism. The parameters of the model can be derived by a systematic expansion from any full-scale model of the yeast cell kinetics with a supercritical Hopf bifurcation. Some parameter values can also be obtained directly from analysis of perturbation experiments. In the mean-field limit, equations for the study of populations having a distribution of frequencies are used to simulate the effect of the inherent variations between cells.

Amplitude equations for reaction-diffusion systems with a Hopf bifurcation and slow real modes

Abstract Using a normal form approach described in a previous paper we derive an amplitude equation for a reaction–diffusion system with a Hopf bifurcation coupled to one or more slow real eigenmodes. The new equation is useful even for systems where the actual bifurcation underlying the description cannot be realized, which is typical of chemical systems. For a fold-Hopf bifurcation, the equation successfully handles actual chemical reactions where the complex Ginzburg–Landau equation fails. For a realistic chemical model of the Belousov–Zhabotinsky reaction, we compare solutions to the reaction–diffusion equation with the approximations by the complex Ginzburg–Landau equation and the new distributed fold-Hopf equation.

Complete optimization of models of the Belousov–Zhabotinsky reaction at a Hopf bifurcation

We analyze two reaction networks for the Belousov–Zhabotinsky reaction at a Hopf bifurcation by methods developed in a previous paper. One network is equivalent to the oregonator, the other is an extended oregonator with seven species and eleven reactions. For each model the current polytope has a simple geometry and the paper serves to illustrate the analytic methods and explain their relation to the chemistry of the reactions. All possible models based on each network are compared with experimental quenching results and other experimental data at a supercritical Hopf bifurcation. The best possible fit is obtained through a systematic search of the current cone and concentration space. For the oregonator, the optimization is essentially complete and results in a determination from scratch of all six rate constants of the model. The agreement of the optimum with the experiments is very good, and the agreement of the deduced rate constants with the most recent set determined from kinetic measurements on su...

Systematic derivation of amplitude equations and normal forms for dynamical systems

We present a systematic approach to deriving normal forms and related amplitude equations for flows and discrete dynamics on the center manifold of a dynamical system at local bifurcations and unfoldings of these. We derive a general, explicit recurrence relation that completely determines the amplitude equation and the associated transformation from amplitudes to physical space. At any order, the relation provides explicit expressions for all the nonvanishing coefficients of the amplitude equation together with straightforward linear equations for the coefficients of the transformation. The recurrence relation therefore provides all the machinery needed to solve a given physical problem in physical terms through an amplitude equation. The new result applies to any local bifurcation of a flow or map for which all the critical eigenvalues are semisimple (i.e., have Riesz index unity). The method is an efficient and rigorous alternative to more intuitive approaches in terms of multiple time scales. We illustrate the use of the method by deriving amplitude equations and associated transformations for the most common simple bifurcations in flows and iterated maps. The results are expressed in tables in a form that can be immediately applied to specific problems. (c) 1998 American Institute of Physics.

Amplitude Equations and Chemical Reaction–Diffusion Systems

The paper discusses the use of amplitude equations to describe the spatio-temporal dynamics of a chemical reaction–diffusion system based on an Oregonator model of the Belousov–Zhabotinsky reaction. Sufficiently close to a supercritical Hopf bifurcation the reaction–diffusion equation can be approximated by a complex Ginzburg–Landau equation with parameters determined by the original equation at the point of operation considered. We illustrate the validity of this reduction by comparing numerical spiral wave solutions to the Oregonator reaction–diffusion equation with the corresponding solutions to the complex Ginzburg–Landau equation at finite distances from the bifurcation point. We also compare the solutions at a bifurcation point where the systems develop spatio-temporal chaos. We show that the complex Ginzburg–Landau equation represents the dynamical behavior of the reaction–diffusion equation remarkably well, sufficiently far from the bifurcation point for experimental applications to be feasible.

Hopf bifurcation in chemical kinetics

We study Hopf bifurcations in chemical reaction systems for their potential use in quantitative experimental analysis of the kinetics of oscillatory reactions. Three models of the Belousov–Zhabotinsky reaction are investigated as examples. For these we have determined and characterized the sub‐ and supercritical Hopf bifurcations in their dependence on parameters of the models. For supercritical bifurcations we calculate a number of parameters that can be used for quantitative comparison of models and experiment. In particular, we calculate expansion coefficients of the flow rate, the frequency of oscillation, and a Floquet exponent; the small parameter of the expansions is the square of the amplitude of the fundamental Fourier component of the oscillations. We also calculate quenching amplitudes from an adjoint eigenvector of the Jacobi matrix of the kinetics. They determine the conditions under which the small amplitude oscillations can be quenched by addition of the species participating in the reactio...