Preben Graae Sørensen

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.

Dynamical quorum sensing: Population density encoded in cellular dynamics

Mutual synchronization by exchange of chemicals is a mechanism for the emergence of collective dynamics in cellular populations. General theories exist on the transition to coherence, but no quantitative, experimental demonstration has been given. Here, we present a modeling and experimental analysis of cell-density-dependent glycolytic oscillations in yeast. We study the disappearance of oscillations at low cell density and show that this phenomenon occurs synchronously in all cells and not by desynchronization, as previously expected. This study identifies a general scenario for the emergence of collective cellular oscillations and suggests a quorum-sensing mechanism by which the cell density information is encoded in the intracellular dynamical state.

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.

Size adaptation of turing prepatterns

Spontaneous pattern formation may arise in biological systems as primary and secondary bifurcations to nonlinear parabolic partial differential equations describing chemical reaction-diffusion systems. Such Turing prepatterns have a specified geometry as long as D/R2 (the diffusion coefficient of the morphogen D divided by the square of a characteristic length) is confined to a (usually) limited interval. As real biochemical systems like cleaving eggs or early embryos vary considerably in size, Turing prepatterns are unable to maintain a specified prepattern-geometry, unless D/R2 is varied as well. We show, that actual biochemical control systems may vary Dapp/R2, where Dapp(k) is an apparent diffusion constant, dependent on enzyme regulated rate constants, and that such simple control systems allow Turing structures to adapt to size variations of at least a factor 103 (linearly), not only in large connected cell systems, but in single cells as well.

On the mechanisms of glycolytic oscillations in yeast

This work concerns the cause of glycolytic oscillations in yeast. We analyse experimental data as well as models in two distinct cases: the relaxation‐like oscillations seen in yeast extracts, and the sinusoidal Hopf oscillations seen in intact yeast cells. In the case of yeast extracts, we use flux‐change plots and model analyses to establish that the oscillations are driven by on/off switching of phosphofructokinase. In the case of intact yeast cells, we find that the instability leading to the appearance of oscillations is caused by the stoichiometry of the ATP‐ADP‐AMP system and the allosteric regulation of phosphofructokinase, whereas frequency control is distributed over the reaction network. Notably, the NAD+/NADH ratio modulates the frequency of the oscillations without affecting the instability. This is important for understanding the mutual synchronization of oscillations in the individual yeast cells, as synchronization is believed to occur via acetaldehyde, which in turn affects the frequency of oscillations by changing this ratio.

Quantitative characterization of cell synchronization in yeast

Metabolic oscillations in bakers yeast serve as a model system for synchronization of biochemical oscillations. Despite widespread interest, the complexity of the phenomenon has been an obstacle for a quantitative understanding of the cell synchronization process. In particular, when two yeast cell populations oscillating 180° out of phase are mixed, it appears as if the synchronization dynamics is too fast to be explained. We have probed the synchronization dynamics by forcing experiments in an open-flow reactor, and we find that acetaldehyde has a very strong synchronization effect that can account quantitatively for the classical mixing experiment. The fast synchronization dynamics is explained by a general synchronization mechanism, which is dominated by a fast amplitude response as opposed to the expected slow phase change. We also show that glucose can mediate this kind of synchronization, provided that the glucose transporter is not saturated. This makes the phenomenon potentially relevant for a broad range of cell types.

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 description of chemical reaction–diffusion systems

Abstract We consider a new method for modeling waves in complex chemical systems close to bifurcation points. The method overcomes numerical problems connected with the high dimensional configuration phase space of realistic chemical systems without sacrificing the quantitative accuracy of the calculations. The efficiency is obtained by replacing the conventional use of kinetic equations considering just a few species by the use of amplitude equations for determining the evolution of the state. Coupled with calculation of an explicit function connecting the amplitude space and the concentration space this method permits the quantitative determination of the concentrations of all species. We also introduce a new method for calculating the boundaries of convective and absolute stability of waves for a chemical model at an operating point close to a supercritical Hopf bifurcation and with a slow stable mode.

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.