Reinhart Heinrich

A Linear Steady‐State Treatment of Enzymatic Chains

A theoretical analysis of linear enzymatic chains is presented. By linear approximation simple analytical solutions can be obtained for the metabolite concentrations and the flux through the chain for steady-state conditions. The equations are greatly simplified if the common kinetic constants are expressed as functions of two parameters, i.e. the thermodynamic equilibrium constant and the “characteristic time”. Three cardinal terms are proposed for the quantitative description of enzyme systems. The first two are the control strength and the control matrix; these indicate the dependence of the flux and the metabolite concentrations, respectively, on the kinetic properties of a given enzyme. The third is the effector strength, which defines the dependence of the velocity of an enzyme on the concentration of an effector; it expresses the importance of an effector. By linear approximation simple analytical expressions were derived for the control strength, the control matrix and the mass-action ratios. The effector strength was calculated for two cases: for a competitive inhibitor and for allosteric effectors according to the Monod (1965) model. The influence of an effector on the concentrations of the metabolites was considered.

Mathematical Models of Protein Kinase Signal Transduction

We have developed a mathematical theory that describes the regulation of signaling pathways as a function of a limited number of key parameters. Our analysis includes linear kinase-phosphatase cascades, as well as systems containing feedback interactions, crosstalk with other signaling pathways, and/or scaffolding and G proteins. We find that phosphatases have a more pronounced effect than kinases on the rate and duration of signaling, whereas signal amplitude is controlled primarily by kinases. The simplest model pathways allow amplified signaling only at the expense of slow signal propagation. More complex and realistic pathways can combine high amplification and signaling rates with maintenance of a stable off-state. Our models also explain how different agonists can evoke transient or sustained signaling of the same pathway and provide a rationale for signaling pathway design.

Biological Control through Regulated Transcriptional Coactivators

Gene activation in higher eukaryotes requires the concerted action of transcription factors and coactivator proteins. Coactivators exist in multiprotein complexes that dock on transcription factors and modify chromatin, allowing effective transcription to take place. While biological control focused at the level of the transcription factor is very common, it is now quite clear that a substantial component of gene control is directed at the expression of coactivators, involving pathways as diverse as B-cell development, smooth muscle differentiation, and hepatic gluconeogenesis. Quantitative control of coactivators allows the functional integration of multiple transcription factors and facilitates the formation of distinct biological programs. This coordination and acceleration of different steps in linked pathways has important kinetic considerations, enabling outputs of particular pathways to be increased far more than would otherwise be possible. These kinetic aspects suggest opportunities and concerns as coactivators become targets of therapeutic intervention.

Transduction of Intracellular and Intercellular Dynamics in Yeast Glycolytic Oscillations

Under certain well-defined conditions, a population of yeast cells exhibits glycolytic oscillations that synchronize through intercellular acetaldehyde. This implies that the dynamic phenomenon of the oscillation propagates within and between cells. We here develop a method to establish by which route dynamics propagate through a biological reaction network. Application of the method to yeast demonstrates how the oscillations and the synchronization signal can be transduced. That transduction is not so much through the backbone of glycolysis, as via the Gibbs energy and redox coenzyme couples (ATP/ADP, and NADH/NAD), and via both intra- and intercellular acetaldehyde.

DYNAMICS OF TWO-COMPONENT BIOCHEMICAL SYSTEMS IN INTERACTING CELLS ; SYNCHRONIZATION AND DESYNCHRONIZATION OF OSCILLATIONS AND MULTIPLE STEADY STATES

Systems of interacting cells containing a metabolic pathway with an autocatalytic reaction are investigated. The individual cells are considered to be identical and are described by differential equations proposed for the description of glycolytic oscillations. The coupling is realized by exchange of metabolites across the cell membranes. No constraints are introduced concerning the number of interacting systems, that is, the analysis applies also to populations with a high number of cells. Two versions of the model are considered where either the product or the substrate of the autocatalytic reaction represents the coupling metabolite (Model I and II, respectively). Model I exhibits a unique steady state while model II shows multistationary behaviour where the number of steady states increases strongly with the number of cells. The characteristic polynomials used for a local stability analysis are factorized into polynomials of lower degrees. From the various factors different Hopf bifurcations may result in leading for model I, either to asynchronous oscillations with regular phase shifts or to synchronous oscillations of the cells depending on the strength of the coupling and on the cell density. The multitude of steady states obtained for model II may be grouped into one class of states which are always unstable and another class of states which may undergo bifurcations leading to synchronous oscillations within subgroups of cells. From these bifurcations numerous different oscillatory regimes may emerge. Leaving the near neighbourhood of the boundary of stability, secondary bifurcations of the limit cycles occur in both models. By symmetry breaking the resulting oscillations for the individual cells lose their regular phase shifts. These complex dynamic phenomena are studied in more detail for a low number of interacting cells. The theoretical results are discussed in the light of recent experimental data on the synchronization of oscillations in populations of yeast cells.

MODELLING THE INTERRELATIONS BETWEEN CALCIUM OSCILLATIONS AND ER MEMBRANE POTENTIAL OSCILLATIONS

A refined electrochemical model accounting for intracellular calcium oscillations and their interrelations with oscillations of the potential difference across the membrane of the endoplasmic reticulum (ER) or other intracellular calcium stores is established. The ATP dependent uptake of Ca2+ from the cytosol into the ER, the Ca2+ release from the ER through channels following a calcium-induced calcium release mechanism, and a potential-dependent Ca2+ leak flux out of the ER are included in the model and described by plausible rate laws. The binding of calcium to specific proteins such as calmodulin is taken into account. The quasi-electroneutrality condition allows us to express the transmembrane potential in terms of the concentrations of cytosolic calcium and free binding sites on proteins, which are the two independent variables of the model. We include monovalent ions in the model, because they make up a considerable portion in the balance of electroneutrality. As the permeability of the endoplasmic membrane for these ions is much higher than that for calcium ions, we assume the former to be in Nernst equilibrium. A stability analysis of the steady-state solutions (which are unique or multiple depending on parameter values) is carried out and the Hopf bifurcation leading from stable steady states to self-sustained oscillations is analysed with the help of appropriate mathematical techniques. The oscillations obtained by numerical integration exhibit the typical spike-like shape found in experiments and reasonable values of frequency and amplitude. The model describes the process of switching between stationary and pulsatile regimes as well as changes in oscillation frequency upon parameter changes. It turns out that calcium oscillations can arise without a permanent influx of calcium into the cell, when a calcium-buffering system such as calmodulin is included.

Mathematical analysis of a mechanism for autonomous metabolic oscillations in continuous culture of Saccharomyces cerevisiae

Autonomous metabolic oscillations were observed in aerobic continuous culture of Saccharomyces cerevisiae. Experimental investigation of the underlying mechanism revealed that several pathways and regulatory couplings are involved. Here a hypothetical mechanism including the sulfate assimilation pathway, ethanol degradation and respiration is transformed into a mathematical model. Simulations confirm the ability of the model to produce limit cycle oscillations which reproduce most of the characteristic features of the system.

Influence of calcium binding to proteins on calcium oscillations and ER membrane potential oscillations: a mathematical model

Oscillations of the cytosolic calcium concentration have turned out to be an important phenomenon in a variety of living cells and have recently been intensely investigated (Woods et al., 1986; De Young and Keizer, 1992; Dupont and Goldbeter, 1993; Li and Rinzel, 1994; Jafri and Gillo, 1994; Jouaville et al., 1995; Goldbeter, 1996). They play an important role in intracellular information processing (Cuthbertson, 1989; Goldbeter et al., 1990). The risk of overloading the cell with calcium in the process of signalling is believed to be avoided by oscillatory mechanisms instead of adjustable stationary messenger concentrations (Berridge, 1989). Another advantage is that a pulsatile signal can carry two types of information: a digital signal (oscillation or stationarity) acting as a switch, for example, between two modes of metabolism, and an analogue signal which may determine the flux of some metabolic pathway (Cuthbertson, 1989).

Dynamics of biochemical oscillators in a large number of interacting cells

6. Summary We investigated here minimal models for coupled metabolic oscillators in cell suspensions. They are based on a feedback-activation mechanism, proposed for the explanation of the glycolytic oscillations. Despite the simplicity of the models we found a wide variety of complex dynamical phenomena. Depending on the kinetic parameters interacting cells may oscillate synchronous or asynchronous. It was shown, that there are different possibilities for asynchronous oscillations. (a) Depending on the type of coupling regular asynchronous behaviour may occur near to the boundary of stability. This type of behaviour was only possible, if the coupling metabolite belongs to the pool of products of the autocatalytic reaction. (b) Leaving the near neighbourhood of the points of Hopf-bifurcations in both models nonregular asynchronous behaviour may arise by secondary symmetry breaking bifurcations, either from the branch of synchronous oscillations or from the branch of regular asynchronous oscillations. In model II the coupling opens the possibility of multiple steady states. The whole population may be within different stable steady states or different oscillatory modes. The results of our study may be helpful for the interpretation of experimental data in this field. The mechanism of interaction, which was investigated here, has been proposed for many biological systems. Especially for yeast cell suspensions the question of the coupling intermediates is studied intensively. In recent time acetaldehyde was reported to be the synchronizer of the oscillations [5], but this substance was shown to desynchronize the oscillations too [4]. Without discussing the problem, whether or not this substance mediates the coupling, we want to emphasize that any metabolite may play the role of a synchronizer as well as that of a desynchronizer. Moreover we have demonstrated that the regular asynchronous oscillations may lead to the phenomenon of hidden oscillations, where it may be difficult to observe cellular oscillations.

Fundamentals of Biochemical Modeling

In this book, we deal with deterministic kinetic modeling of biochemical reaction systems. The principal notions are the concentration (i.e., the number of moles of a given substance per unit volume) and the reaction rate (expressed as concentration change per unit time). This type of modeling is sometimes referred to as macroscopic or phenomenological approach, at variance with microscopic approaches, where molecules and their interactions are considered as fundamental concepts. In the latter approaches, rate constants are calculated in terms of molecular quantities, for example, in the Transition State and Kramers Rate Theories (cf. Haggi et al., 1990).