Stefan Schuster

The Regulation of Cellular Systems

Introduction Fundamentals of biochemical modeling Balance equations Rate laws Generalized mass-action kinetics Various enzyme kinetic rate laws Thermodynamic flow-force relationships Power-law approximation Steady states of biochemical networks General considerations Stable and unstable steady states Multiple steady states Metabolic oscillations Background Mathematical conditions for oscillations Glycolytic oscillations Models of intracellular calcium oscillations A simple three-variable model with only monomolecular and bimolecular reactions Possible physiological significance of oscillations Stoichiometric analysis Conservation relations Linear dependencies between the rows of the stoichiometry matrix Non-negative flux vectors Elementary flux modes Thermodynamic aspects A generalized Wegscheider condition Strictly detailed balanced subnetworks Onsagers reciprocity reactions for coupled enyme reactions Time hierarchy in metabolism Time constants The quasi-steady-state approximation The Rapid equilibrium approximation Modal analysis Metabolic control analysis Basic definitions A systematic approach Theorems of metabolic control analysis Summation theorems Connectivity theorems Calculation of control coefficients using the theorems Geometrical interpretation Control analysis of various systems General remarks Elasticity coefficients for specific rate laws Control coefficients for simple hypothetical pathways Unbranched chains A branched system Control of erythrocyte energy metabolism The reaction system Basic model Interplay of ATP production and ATP consumption Glycolytic energy metabolism and osmotic states A simple model of oxidative phosphorylation A three-step model of serine biosynthesis Time-dependent control coefficients Are control coefficients always parameter independent? Posing the problem A system without conserved moieties A system with a conserved moiety A system including dynamic channeling Normalized versus non-normalized coefficients Analysis in terms of variables other than steady-state concentrations and fluxes General analysis Concentration ratios and free-energy-differences as state variables Entropy production as response variable Control of transient times Control of oscillations A second-order approach A quantitative approach to metabolic regulations Co-response coefficients Fluctuations of internal variables versus parameter perturbations Internal response coefficients Rephrasing the basic equations of metabolic control analysis in terms of co-response coefficients and internal response coefficients Control within and between subsystems Modular approach Overall elasticities Overall control coefficients Flux control insusceptibility Control exerted by elementary steps in enzyme catalysis Control analysis of metabolic channeling Comparison of metabolic control analysis and power-law formalism Computational aspects Application of optimization methods and the interrelation with evolution Optimization of the catalytic properties of single enzymes Basic assumptions Optimal values of elementary rate constants Optimal Michaelis constants Optimization of multienzyme systems Maximization of steady-state flux Influence of osmotic constraints and minimization of intermediate concentrations Minimization of transient times Optimal stoichiometries.

A general definition of metabolic pathways useful for systematic organization and analysis of complex metabolic networks

A set of linear pathways often does not capture the full range of behaviors of a metabolic network. The concept of ‘elementary flux modes’ provides a mathematical tool to define and comprehensively describe all metabolic routes that are both stoichiometrically and thermodynamically feasible for a group of enzymes. We have used this concept to analyze the interplay between the pentose phosphate pathway (PPP) and glycolysis. The set of elementary modes for this system involves conventional glycolysis, a futile cycle, all the modes of PPP function described in biochemistry textbooks, and additional modes that are a priori equally entitled to pathway status. Applications include maximizing product yield in amino acid and antibiotic synthesis, reconstruction and consistency checks of metabolism from genome data, analysis of enzyme deficiencies, and drug target identification in metabolic networks.

ON ELEMENTARY FLUX MODES IN BIOCHEMICAL REACTION SYSTEMS AT STEADY STATE

A mathematical definition of the concept of elementary mode is given so as to apply to biochemical reaction systems subsisting at steady state. This definition relates to existing concepts of null-space vectors and includes a condition of simplicity. It is shown that for systems in which all flux- have fixed signs, all elementary modes are given by the generating vectors of a convex cone and can, thus, be computed by an existing algorithm. The present analysis allows for the more general case that some reactions can proceed in either direction. Basic ideas on how to compute the complete set of elementary modes in this situation are outlined and verified by way of several examples, with one of them repraenting glycolysis and gluconeogenesis. These examples show that the elementary modes can be interpreted in terms of the particular biochemical functions of the network. The relationships to (futile) substrate cycles are elucidated.

Metabolic pathway analysis: basic concepts and scientific applications in the post-genomic era.

This article reviews the relatively short history of metabolic pathway analysis. Computer‐aided algorithms for the synthesis of metabolic pathways are discussed. Important algebraic concepts used in pathway analysis, such as null space and convex cone, are explained. It is demonstrated how these concepts can be translated into meaningful metabolic concepts. For example, it is shown that the simplest vectors spanning the region of all admissible fluxes in stationary states, for which the term elementary flux modes was coined, correspond to fundamental pathways in the system. The concepts are illustrated with the help of a reaction scheme representing the glyoxylate cycle and adjacent reactions of aspartate and glutamate synthesis. The interrelations between pathway analysis and metabolic control theory are outlined. Promising applications for genome annotation and for biotechnological purposes are discussed. Armed with a better understanding of the architecture of cellular metabolism and the enormous amount of genomic data available today, biochemists and biotechnologists will be able to draw the entire metabolic map of a cell and redesign it by rational and directed metabolic engineering.

METATOOL: for studying metabolic networks.

MOTIVATIONnTo reconstruct metabolic pathways from biochemical and/or genome sequence data, the stoichiometric and thermodynamic feasibility of the pathways has to be tested. This is achieved by characterizing the admissible region of flux distributions in steady state. This region is spanned by what can be called a convex basis. The concept of elementary flux modes provides a mathematical tool to define all metabolic routes that are feasible in a given metabolic network. In addition, we define enzyme subsets to be groups of enzymes that operate together in fixed flux proportions in all steady states of the system.nnnRESULTSnAlgorithms for computing the convex basis and elementary modes developed earlier are briefly reviewed. A newly developed algorithm for detecting all enzyme subsets in a given network is presented. All of these algorithms have been implemented in a novel computer program named METATOOL, whose features are outlined here. The algorithms are illustrated by an example taken from sugar metabolism.nnnAVAILABILITYnMETATOOL is available from ftp://bmsdarwin.brookes.ac. uk/pub/software/ibmpc/metatool.nnnSUPPLEMENTARY INFORMATIONnhttp://www. biologie.hu-berlin.de/biophysics/Theory/tpfeiffer/metatoo l.html

Modern trends in biothermokinetics

Proceedings of the Fifth International Biothermokinetics Meeting held in Bordeaux-Bombannes, France, September 1992. The volume is divided into seven sections: thermodynamics and kinetics of transport processes and biological energy transduction; modeling of cell processes with applications to biote

Control analysis of metabolic systems involving quasi-equilibrium reactions

Reactions for which the rates are extremely sensitive to changes in the concentrations of variable metabolite concentrations contribute little to the control of biochemical reaction networks. Yet they do interfere with the calculation of the systems behaviour, both in terms of numerical integration of the rate equations and in terms of the analysis of metabolic control. We here present a way to solve this problem systematically for systems with time hierarchies. We identify the fast reactions and fast metabolites, group them apart from the other (slow) reactions and metabolites, and then apply the appropriate quasi-equilibrium condition for the fast subsystem. This then makes it possible to eliminate the fast reactions and their elasticity coefficients from the calculations, allowing the calculation of the control coefficients of the slow reactions in terms of the elasticity coefficients of the slow reactions. As expected, the elasticity coefficients of the fast reactions drop out of the calculations, and they are irrelevant for control at the time resolution of the steady state of the slow reactions. The analysis, when applied iteratively, is expected to be particularly valuable for the control analysis of living cells, where a time hierarchy exists, the fastest being at the level of enzyme kinetics and the slowest at gene expression.

Cellular information transfer regarded from a stoichiometry and control analysis perspective.

Metabolic control analysis (MCA) allows one to formalize important aspects of information processing in living cells. For example, information processing via multi-level enzyme cascades can be quantified in terms of the response coefficient of a cellular target to a signal. In many situations, control and response coefficients cannot be determined exactly for all enzymes involved, owing to difficulties in observing all enzymes experimentally. Here, we review a number of qualitative approaches that were developed to cope with such situations. The usefulness of the concept of null-space of the stoichiometry matrix for analysing the structure of intracellular signaling networks is discussed. It is shown that signal transduction operates very efficiently when the network structure is such that the null-space matrix can be block-diagonalized (which may or may not imply that the network consists of several disconnected parts) and some enzymes have low elasticities to their substrates.

Modelling the interrelation between the transmembrane potential and pH difference across membranes with electrogenic proton transport upon build-up of the proton-motive force

Abstract The wide-spread belief that the electric potential difference (Δφm) across membranes with electrogenic proton transport varies nearly linearly with the pH difference (ΔpH) upon build-up of the proton-motive force is critically examined. First, we analyse experimental literature data concerning rat liver and yeast mitochondria, and E. coli. We then present a model describing the interrelation between the ΔpH and Δφm as the activities of the proton pumps or H+-ATPase or the influence of the proton leak vary. It is based on the quasi-electroneutrality condition, the dissociation equilibrium of impermeant weak acids, a simple description of the cation–proton antiporters and cation leak, and the Nernst equation applied to all those ions subsisting in equilibrium. The model yields a nonlinear equation giving Δφm as a function of ΔpH. In various situations this function is quasi-linear in physiologically relevant ranges of ΔpH. Thus, the linearity hypothesis can be substantiated theoretically, but is not necessarily justified under all circumstances. It is shown that the slope of the Δφm vs. ΔpH curve is, in the quasi-linear regions, about −2.303 RT/F (thus having the same value, but the opposite sign as in the Nernst equation) when the cation–proton antiporters are absent or completely inhibited, and can be much higher in absolute value when these antiporters are operative.

Treatment of multifunctional enzymes in metabolic pathway analysis

In metabolic pathway analysis, it should be considered that many enzymes operate with low specificity (e.g. nucleoside diphosphokinase, uridine kinase, transketolase, aldolase), so that various substrates and products can be converted. Here, we analyze the effect of enzymes with low substrate specificity on the elementary flux modes (pathways). We also study the benefits of two different approaches to describing multifunctional enzymes. The usual description is in terms of (overall) enzymatic reactions. At a more detailed level, the reaction steps (half-reactions, hemi-reactions) of the formation and conversion of enzyme-substrate complexes are considered. Multifunctional enzymes operate according to various mechanisms. This is illustrated here by the reaction schemes for the different enzyme mechanisms of bifunctional enzymes. For enzymes with two or more functions, it is important to consider only linearly independent functions, because otherwise cyclic elementary modes would occur which do not perform any net transformation. However, the choice of linearly independent functions is not a priori unique. We give a method for making this choice unique by considering the extreme pathways of the hemi-reactions system. A formal application of the algorithm for computing elementary flux modes (pathways) yields the result that the number of such modes sometimes depend on the level of description if some reactions are reversible and the products of the multifunctional enzymes are external metabolites or some multifunctional enzymes partly share the same metabolites. However, this problem can be solved by appropriate interpretation of the definition of elementary modes and the correct choice of independent functions of multifunctional enzymes. The analysis is illustrated by a biochemical example taken from nucleotide metabolism, comparing the two ways of description for nucleoside diphosphokinase and adenylate kinase, and by several smaller examples.