Sune Danø

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.

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.

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.

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.

Complexity reduction of biochemical rate expressions

MOTIVATION The current trend in dynamical modelling of biochemical systems is to construct more and more mechanistically detailed and thus complex models. The complexity is reflected in the number of dynamic state variables and parameters, as well as in the complexity of the kinetic rate expressions. However, a greater level of complexity, or level of detail, does not necessarily imply better models, or a better understanding of the underlying processes. Data often does not contain enough information to discriminate between different model hypotheses, and such overparameterization makes it hard to establish the validity of the various parts of the model. Consequently, there is an increasing demand for model reduction methods. RESULTS We present a new reduction method that reduces complex rational rate expressions, such as those often used to describe enzymatic reactions. The method is a novel term-based identifiability analysis, which is easy to use and allows for user-specified reductions of individual rate expressions in complete models. The method is one of the first methods to meet the classical engineering objective of improved parameter identifiability without losing the systems biology demand of preserved biochemical interpretation. AVAILABILITY The method has been implemented in the Systems Biology Toolbox 2 for MATLAB, which is freely available from http://www.sbtoolbox2.org. The Supplementary Material contains scripts that show how to use it by applying the method to the example models, discussed in this article.

Quantitative evaluation of respiration induced metabolic oscillations in erythrocytes

The changes in the partial pressures of oxygen and carbon dioxide (P(O(2)) and P(CO(2))) during blood circulation alter erythrocyte metabolism, hereby causing flux changes between oxygenated and deoxygenated blood. In the study we have modeled this effect by extending the comprehensive kinetic model by Mulquiney and Kuchel [P.J. Mulquiney, and P.W. Kuchel. Model of 2,3-bisphosphoglycerate metabolism in the human erythrocyte based on detailed enzyme kinetic equations: equations and parameter refinement, Biochem. J. 1999, 342, 581-596.] with a kinetic model of hemoglobin oxy-/deoxygenation transition based on an oxygen dissociation model developed by Dash and Bassingthwaighte [R. Dash, and J. Bassingthwaighte. Blood HbO(2) and HbCO(2) dissociation curves at varied O(2), CO(2), pH, 2,3-DPG and temperature levels, Ann. Biomed. Eng., 2004, 32(12), 1676-1693.]. The system has been studied during transitions from the arterial to the venous phases by simply forcing P(O(2)) and P(CO(2)) to follow the physiological values of venous and arterial blood. The investigations show that the system passively follows a limit cycle driven by the forced oscillations of P(O(2)) and is thus inadequately described solely by steady state consideration. The metabolic system exhibits a broad distribution of time scales. Relaxations of modes with hemoglobin and Mg(2+) binding reactions are very fast, while modes involving glycolytic, membrane transport and 2,3-BPG shunt reactions are much slower. Incomplete slow mode relaxations during the 60 s period of the forced transitions cause significant overshoots of important fluxes and metabolite concentrations - notably ATP, 2,3-BPG, and Mg(2+). The overshoot phenomenon arises in consequence of a periodical forcing and is likely to be widespread in nature - warranting a special consideration for relevant systems.

Chemical interpretation of oscillatory modes at a Hopf point.

We present two complementary methods for studying the oscillatory mechanisms in a chemical reaction network in the neighbourhood of a supercritical Hopf bifurcation. The first method is a modification of metabolic control analysis (a form of sensitivity analysis), and focuses on the reactions rather than the chemical species. By rephrasing metabolic control analysis in terms of the amplitude equation of the Hopf bifurcation, we show that control of amplitude and frequency of the oscillations should be considered separately, and that the amplitude control is directly related to the control of the stability of the stationary state. Generally, the frequency of the oscillations is controlled by more reactions than the amplitude is, and those reactions controlling amplitude will generally also exert control of the frequency. The second method focuses on the role of the chemical species. By considering their relative phases and amplitudes, the method reveals to what extent a simple activator-inhibitor interpretation of the amplitude equation associated with the Hopf bifurcation corresponds to an equally simple chemical interpretation. If applicable, the method identifies the activating and inhibiting modes chemically. Prior knowledge of the underlying reaction network is not needed, only phase and amplitude measurements are used in the analysis. Hence, this method is a top-down approach well suited for systems biology. Both methods are exemplified by calculations on the Oregonator model for the Belousov-Zhabotinsky reaction.