Metabolic regulation and mathematical models

The first three-dimensional structure of phosphofructokinase from Saccharomyces cerevisiae determined by electron microscopy of single particles

Phosphofructokinaseis a key regulatory enzyme of the glycolytic pathway. We have determined the structure of this enzyme from Saccharomyces cerevisiae to a resolution of 2.0 nm. This is the first structure available for this family of enzymes in eukaryotic organisms. Phosphofructokinase is an octamer composed of 4alpha and 4beta subunits arranged in a dihedral point group symmetry D(2). The enzyme has a very open and elongated structure, with dimensions of 24 nm in length and 17 nm in width. The final structure, calculated from 0 degrees tilt projections of the molecule at random orientations using as reference the volume obtained by the random conical reconstruction technique in ice, has allowed us to discern the shapes of the subunits and their mutual arrangement in the octamer.

Twofold reduction of phosphofructokinase activity in Lactococcus lactis results in strong decreases in growth rate and in glycolytic flux

Two mutant strains of Lactococcus lactis in which the promoter of the las operon, harboring pfk, pyk, and ldh, were replaced by synthetic promoters were constructed. These las mutants had an approximately twofold decrease in the activity of phosphofructokinase, whereas the activities of pyruvate kinase and lactate dehydrogenase remained closer to the wild-type level. In defined medium supplemented with glucose, the growth rate of the mutants was reduced to 57 to 70% of wild-type levels and the glycolytic flux was reduced to 62 to 76% of wild-type levels. In complex medium growth was even further reduced. Surprisingly, the mutants still showed homolactic fermentation, which indicated that the limitation was different from standard glucose-limited conditions. One explanation could be that the reduced activity of phosphofructokinase resulted in the accumulation of sugar-phosphates. Indeed, when one of the mutants was starved for glucose in glucose-limited chemostat, the growth rate could gradually be increased to 195% of the growth rate observed in glucose-saturated batch culture, suggesting that phosphofructokinase does affect the concentration of upstream metabolites. The pools of glucose-6-phosphate and fructose-6-phosphate were subsequently found to be increased two- to fourfold in the las mutants, which indicates that phosphofructokinase exerts strong control over the concentration of these metabolites.

Computer modelling and experimental evidence for two steady states in the photosynthetic Calvin cycle

We present observations of photosynthetic carbon dioxide assimilation, and leaf starch content from genetically modified tobacco (Nicotiana tabacum) plants in which the activity of the Calvin cycle enzyme, sedoheptulose-1,7-bisphosphatase, is reduced by an antisense construct. The measurements were made on leaves of varying ages and used to calculate the flux control coefficients of sedoheptulose-1,7-bisphosphatase over photosynthetic assimilation and starch synthesis. These calculations suggest that control coefficients for both are negative in young leaves, and positive in mature leaves. This behaviour is compared to control coefficients obtained from a detailed computer model of the Calvin cycle. The comparison demonstrates that the experimental observations are consistent with bistable behaviour exhibited by the model, and provides the first experimental evidence that such behaviour in the Calvin cycle occurs in vivo as well as in silico.

A modelling study of feedforward activation in human erythrocyte glycolysis

Though feedforward activation (FA) is a little known principle of control in metabolic networks, there is one well-known example; namely, the activation of pyruvate kinase (PK) by fructose-1,6-biphosphate (FBP) in glycolysis. The effects of this activation on the enzyme's kinetics are well characterised, but its possible role in glycolytic control has not been determined, and, experimentally, there is as yet no direct way of modifying the enzyme to remove just the FBP activation without affecting other aspects of the enzyme's kinetics. Given this limitation, we used a detailed numerical simulation of human erythrocyte glycolysis to simulate the effects of selective removal of the activation of PK by FBP on steady-state metabolite concentrations and on the dynamic response of glycolytic flux to a sudden increase of the cell's demand for ATP. Our modelling results predict that in the absence of FA steady-state levels of metabolites within the activation loop, i.e. from FBP to phosphoenolpyruvate, would be four- to thirteen-fold higher than normal, whereas levels of ATP and metabolites outside the loop, i.e. glucose-6-phosphate, fructose-6-phosphate and pyruvate, would be lower than normal. Existing clinical evidence in a patient with haemolytic anaemia, correlated with a lack of activation of PK by FBP (Paglia D.E., Valentine W.N., Holbrook C.T., Brockway R., Blood (1983) 62 972-979), is consistent with this prediction. In response to changing demand for ATP, the model predicts that the corresponding change of glycolytic flux would entail changes of metabolite concentrations in the absence of FA, but that in its presence the levels of metabolites within the activation loop remain essentially unperturbed. Thus, our results suggest that by stabilising metabolite pools in the face of variable glycolytic flux, FA may serve to avoid perturbations of the oxygen affinity of haemoglobin (sensitive to the levels of 2,3-phosphoglycerate) and of cell osmolality that would otherwise occur during variations in the cell's demand for ATP. In addition, by significantly raising the steady-state setpoint of intermediate metabolite pools, the productivity index (ratio of glycolytic flux to total metabolites in the pathway) of glycolysis would fall almost four-fold in the absence of forward activation.

Invariant manifold methods for metabolic model reduction

After the decay of transients, the behavior of a set of differential equations modeling a chemical or biochemical system generally rests on a low-dimensional surface which is an invariant manifold of the flow. If an equation for such a manifold can be obtained, the model has effectively been reduced to a smaller system of differential equations. Using perturbation methods, we show that the distinction between rapidly decaying and long-lived (slow) modes has a rigorous basis. We show how equations for attracting invariant (slow) manifolds can be constructed by a geometric approach based on functional equations derived directly from the differential equations. We apply these methods to two simple metabolic models. (c) 2001 American Institute of Physics.

Purification, molecular and kinetic characterization of phosphofructokinase-1 from the yeast Schizosaccharomyces pombe: evidence for an unusual subunit composition

Phosphofructokinase-1 (Pfk-1) from Schizosaccharomyces pombe was purified by 54-fold enrichment to homogeneity elaborating the following steps: (a) Disruption of the cells with glass beads; (b) fractionated precipitation with polyethylene glycol 6000; (c) affinity chromatography on Cibacron-Blue F3G-A-Sephadex G 100; (d) ion exchange chromatography on Resource Q. The native enzyme exhibits a mass of 790+/-30 kDa, as detected by sedimentation equilibrium measurements. The apparent sedimentation coefficient was found to be s(20,c)=20.2+/-0.3 S. No significant dependence of the s-value on the protein concentration was observed in the range 0. 07-0.7 mg/ml. Polyacrylamide gel electrophoresis in presence of sodium dodecyl sulphate and MALDI-TOF spectra showed that the enzyme is composed of subunits of identical size of 100+/-5 kDa, forming an octameric structure. The N-terminus of the enzyme was found to be blocked. Sequences of tryptic and chymotryptic peptides of the subunit coincide with the proposed amino acid sequence as deduced from the gene from the EMBL library. The Pfk-1 coding sequence of S. pombe was transformed into a Pfk-1 double deletion mutants of Saccharomyces cerevisiae resulting in glucose-positive cells with enzyme activity in the crude cell extract. The kinetic analysis revealed less cooperativity to fructose 6-phosphate (n(H)=1.6) and less inhibition by ATP as compared to the enzyme from baker's yeast. Fructose 2,6-bisphosphate (in micromolar range) and AMP (in millimolar range) were found to overcome ATP inhibition and to increase the affinity to fructose 6-phosphate.

Pathway analysis and metabolic engineering in Corynebacterium glutamicum

The gram-positive bacterium Corynebacterium glutamicum is used for the industrial production of amino acids, e.g. of L-glutamate and L-lysine. During the last 15 years, genetic engineering and amplification of genes have become fascinating methods for studying metabolic pathways in greater detail and for the construction of strains with the desired genotypes. In order to obtain a better understanding of the central metabolism and to quantify the in vivo fluxes in C. glutamicum, the [13C]-labelling technique was combined with metabolite balancing to achieve a unifying comprehensive pathway analysis. These methods can determine the flux distribution at the branch point between glycolysis and the pentose phosphate pathway. The in vivo fluxes in the oxidative part of the pentose phosphate pathway calculated on the basis of intracellular metabolite concentrations and the kinetic constants of the purified glucose-6-phosphate and 6-phosphogluconate dehydrogenases determined in vitro were in full accordance with the fluxes measured by the [13C]-labelling technique. These data indicate that the oxidative pentose phosphate pathway in C. glutamicum is mainly regulated by the ratio of NADPH/NADP concentrations and the specific activity of glucose-6-phosphate dehydrogenase. The carbon flux via the oxidative pentose phosphate pathway correlated with the NADPH demand for L-lysine synthesis. Although it has generally been accepted that phosphoenolpyruvate carboxylase fulfills a main anaplerotic function in C. glutamicum, we recently detected that a biotin-dependent pyruvate carboxylase exists as a further anaplerotic enzyme in this bacterium. In addition to the activities of these two carboxylases three enzymes catalysing the decarboxylation of the C4 metabolites oxaloacetate or malate are also present in this bacterium. The individual flux rates at this complex anaplerotic node were investigated by using [13C]-labelled substrates. The results indicate that both carboxylation and decarboxylation occur simultaneously in C. glutamicum so that a high cyclic flux of oxaloacetate via phosphoenolpyruvate to pyruvate was found. Furthermore, we detected that in C. glutamicum two biosynthetic pathways exist for the synthesis of DL-diaminopimelate and L-lysine. As shown by NMR spectroscopy the relative use of both pathways in vivo is dependent on the ammonium concentration in the culture medium. Mutants defective in one pathway are still able to synthesise enough L-lysine for growth, but the L-lysine yields with overproducers were reduced. The luxury of having these two pathways gives C. glutamicum an increased flexibility in response to changing environmental conditions and is also related to the essential need for DL-diaminopimelate as a building block for the synthesis of the murein sacculus.

Can yeast glycolysis be understood in terms of in vitro kinetics of the constituent enzymes? Testing biochemistry

This paper examines whether the in vivo behavior of yeast glycolysis can be understood in terms of the in vitro kinetic properties of the constituent enzymes. In nongrowing, anaerobic, compressed Saccharomyces cerevisiae the values of the kinetic parameters of most glycolytic enzymes were determined. For the other enzymes appropriate literature values were collected. By inserting these values into a kinetic model for glycolysis, fluxes and metabolites were calculated. Under the same conditions fluxes and metabolite levels were measured. In our first model, branch reactions were ignored. This model failed to reach the stable steady state that was observed in the experimental flux measurements. Introduction of branches towards trehalose, glycogen, glycerol and succinate did allow such a steady state. The predictions of this branched model were compared with the empirical behavior. Half of the enzymes matched their predicted flux in vivo within a factor of 2. For the other enzymes it was calculated what deviation between in vivo and in vitro kinetic characteristics could explain the discrepancy between in vitro rate and in vivo flux.

Tendency modeling: a new approach to obtain simplified kinetic models of metabolism applied to Saccharomyces cerevisiae

A novel approach to construct kinetic models of metabolic pathways, to be used in metabolic engineering, is presented: the tendency modeling approach. This approach greatly facilitates the construction of these models and can easily be applied to complex metabolic networks. The resulting models contain a minimal number of parameters; identification of their values is straightforward. Use of in vitro obtained information in the identification of the kinetic equations is minimized. The tendency modeling approach has been used to derive a dynamic model of primary metabolism for aerobic growth of Saccharomyces cerevisiae on glucose, in which compartmentation is included. Simulation results obtained with the derived model are satisfying for most of the carbon metabolites that have been measured. Compared to a more detailed model, the simulations of our model are less accurate, but taking into account the much smaller number of kinetic parameters (35 instead of 84), the tendency the modeling approach is considered promising.

ADP modulates the dynamic behavior of the glycolytic pathway of Escherichia coli

A mathematical model that includes biochemical interactions among the PTS system, phosphofructokinase (PFK), and pyruvate kinase (PK) is used to evaluate the dynamic behavior of the glycolytic pathway of Escherichia coli under steady-state conditions. The influence of ADP, phosphoenolpyruvate (PEP), and fructose-6-phosphate (F6P) on the dynamic regulation of this pathway is also analyzed. The model shows that the dynamic behavior of the system is affected significantly depending on whether ADP, PEP, or F6P is considered constant a steady state. Sustained oscillations are observed only when dADP/dt not equal 0 and completely suppressed if dADP/dt = 0 at any steady-state value. However, when PEP or F6P is constant, the system evolves toward the formation of stable limit cycles with periods ranging from 0.2 min to hours.

How yeast cells synchronize their glycolytic oscillations: a perturbation analytic treatment

Of all the lifeforms that obtain their energy from glycolysis, yeast cells are among the most basic. Under certain conditions the concentrations of the glycolytic intermediates in yeast cells can oscillate. Individual yeast cells in a suspension can synchronize their oscillations to get in phase with each other. Although the glycolytic oscillations originate in the upper part of the glycolytic chain, the signaling agent in this synchronization appears to be acetaldehyde, a membrane-permeating metabolite at the bottom of the anaerobic part of the glycolytic chain. Here we address the issue of how a metabolite remote from the pacemaking origin of the oscillation may nevertheless control the synchronization. We present a quantitative model for glycolytic oscillations and their synchronization in terms of chemical kinetics. We show that, in essence, the common acetaldehyde concentration can be modeled as a small perturbation on the "pacemaker" whose effect on the period of the oscillations of cells in the same suspension is indeed such that a synchronization develops.

Compartmentation protects trypanosomes from the dangerous design of glycolysis

Unlike in other organisms, in trypanosomes and other Kinetoplastida the larger part of glycolysis takes place in a specialized organelle, called the glycosome. At present it is impossible to remove the glycosome without changing much of the rest of the cell. It would seem impossible, therefore, to assess the metabolic consequences of this compartmentation. Therefore, we here develop a computer experimentation approach, which we call computational cell biology. A validated molecular kinetic computer replica was built of glycolysis in the parasite Trypanosoma brucei. Removing the glycosome membrane in that replica had little effect on the steady-state flux, which argues against the prevalent speculation that glycosomes serve to increase flux by concentrating the enzymes. Removal of the membrane did cause (i) the sugar phosphates to rise to unphysiologically high levels, which must have pathological effects, and (ii) a failure to recover from glucose deprivation. We explain these effects on the basis of the biochemical organization of the glycosome. We conclude (i) that the glycosome protects trypanosomes from the negative side effects of the "turbo" structure of glycolysis and (ii) that computer experimentation based on solid molecular data is a powerful tool to address questions that are not, or not yet, accessible to experimentation.

Metabolic control analysis of glycolysis in trypanosomes as an approach to improve selectivity and effectiveness of drugs

Glycolysis is the only ATP-generating process in bloodstream form trypanosomes and is therefore a promising drug target. Inhibitors which decrease significantly the glycolytic flux will kill the parasites. Both computer simulation and experimental studies of glycolysis in bloodstream form Trypanosoma brucei indicated that the control of the glycolytic flux is shared by several steps in the pathway. The results of these analyses provide quantitative information about the prospects of decreasing the flux by inhibition of any individual enzyme. The plasma membrane glucose transporter appears the most promising target from this perspective, followed by aldolase, glyceraldehyde-3-phosphate dehydrogenase, phosphoglycerate kinase and glycerol-3-phosphate dehydrogenase. Non-competitive or irreversible inhibitors would be most effective, but it is argued that potent competitive inhibitors can be suitable, provided that the concentration of the competing substrate cannot increase unrestrictedly. Such is the case for inhibitors that compete with coenzymes or with blood glucose.

Simulation of prokaryotic genetic circuits

Biochemical and genetic approaches have identified the molecular mechanisms of many genetic reactions, particularly in bacteria. Now a comparably detailed understanding is needed of how groupings of genes and related protein reactions interact to orchestrate cellular functions over the cell cycle, to implement preprogrammed cellular development, or to dynamically change a cell's processes and structures in response to environmental signals. Simulations using realistic, molecular-level models of genetic mechanisms and of signal transduction networks are needed to analyze dynamic behavior of multigene systems, to predict behavior of mutant circuits, and to identify the design principles applicable to design of genetic regulatory circuits. When the underlying design rules for regulatory circuits are understood, it will be far easier to recognize common circuit motifs, to identify functions of individual proteins in regulation, and to redesign circuits for altered functions.

Equations for progress curves of some kinetic models of enzyme-single substrate-single slow binding modifier system

A procedure is described by means of which the equations for progress curves for the kinetic models that include fast and slow reaction steps can be derived. It is based on combined assumptions of equilibrium and steady-state and uses Laplace transformation for solving the systems of differential equations. The progress curve equations and the significance of the corresponding parameters are given for some most frequently occurring models describing the influence of a slow binding modifier on a single substrate enzyme reaction.

The modelling of metabolic systems. Structure, control and optimality

This article gives an overview of recent developments in the modelling of the structure, control and optimality of metabolic networks. In particular, methods of algebraically analysing the topology of such networks are presented. By these methods, conservation relations and elementary modes of functioning (biochemical routes) can be detected. The principles of metabolic control analysis are outlined. Various recent extensions of this theory are presented, such as an analysis in terms of time dependent variables and modular analysis. Evolutionary optimisation principles are applied to explain the catalytic efficiency of single enzymes as well as the structural design of metabolic pathways. Special results concern the optimal distribution of ATP consuming and ATP producing reactions in glycolysis.

Application of mathematical tools for metabolic design of microbial ethanol production

Many attempts to engineer cellular metabolism have failed due to the complexity of cellular functions. Mathematical and computational methods are needed that can organize the available experimental information, and provide insight and guidance for successful metabolic engineering. Two such methods are reviewed here. Both methods employ a (log)linear kinetic model of metabolism that is constructed based on enzyme kinetics characteristics. The first method allows the description of the dynamic responses of metabolic systems subject to spatiotemporal variations in their parameters. The second method considers the product-oriented, constrained optimization of metabolic reaction networks using mixed-integer linear programming methods. The optimization framework is used in order to identify the combinations of the metabolic characteristics of the glycolytic enzymes from yeast and bacteria that will maximize ethanol production. The methods are also applied to the design of microbial ethanol production metabolism. The results of the calculations are in qualitative agreement with experimental data presented here. Experiments and calculations suggest that, in resting Escherichia coli cells, ethanol production and glucose uptake rates can be increased by 30% and 20%, respectively, by overexpression of a deregulated pyruvate kinase, while increase in phosphofructokinase expression levels has no effect on ethanol production and glucose uptake rates.

Rational engineering of the TOL meta-cleavage pathway

The meta-cleavage pathway of Pseudomonas putida mt-2 was simulated using a biochemical systems simulation developed by Regan (1996). A non-competitive inhibition term for catechol-2,3-dioxygenase (C23O) by 2-OH-pent-2,4-dienoate (Ki = 150 μM) was incorporated into the model. The simulation predicted steady state accumulation levels in the μM range for metabolites pre-meta-cleavage, and in the mM range for metabolites post-meta-cleavage. The logarithmic gains L[V-i, Xj] and L[X-i, Xj] clearly indicated that the pathway was most sensitive to the concentration of the starting substrate, benzoate, and the first enzyme of the pathway, toluate-1, 2-dioxygenase (TO). The simulation was validated experimentally; it was found that the amplification of TO increased the steady state flux from 0.024 to 0.091 (mmol/g cell dwt)/h. This resulted in an increased accumulation of a number of the pathway metabolites (intra- and extracellularly), especially cis-diol, 4-OH-2-oxovalerate, and 4-oxalocrotonate. Metabolic control analysis indicated that C23O was, in fact, the major controling enzymic step of the pathway with a scaled control coefficient of 0.83. The amplification of TO resulted in a shift of some of the control away from C23O. Catechol-2,3-dioxygenase, however, remained as the major controling element of the pathway. Copyright 1998 John Wiley & Sons, Inc.

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 system's 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.

Internal regulation of a modular system: the different faces of internal control

The living cell houses a multitude of molecular processes that operate simultaneously in a mutually consistent fashion. A certain degree of organization stands out, e.g. in terms of the various metabolic pathways, transcription versus translation, signal transduction versus metabolism. This paper shows that by taking one of the aforementioned organizational principles into account, the complexity of understanding cell function quantitatively may be reduced significantly. To this aim the definition of the corresponding type of organization is refined and the conceptual tools used in the analysis of the control of cell function are adjusted. The approach is elaborated for a theoretical model of cell function, in which the latter depends on a constellation of interdependent but unconnected modules. The organization of a system is reduced to global control within a limited set of partaking modules and the links between them. Information about the systems total internal control and regulability is then drastically reduced to the information specifying global control and the regulability of the pathways that constitute the system. It is shown quantitatively how control at a lower level of organization bears on the control of the cell as a whole. The approach centers on writing the product of control (matrix) and elasticity (matrix) at a number of different levels of aggregation; these products equalling the identity (matrix) under different conditions. We demonstrate that there are at least three ways in which control and regulability of a system can be matched. In one, the true control within and between the modules of the systems is the inverse of the primary regulability (i.e. elasticity plus stoichiometry). In a second, the control internal to a module (but partly determined through the other modules) is matched by the inverse of newly defined 'global' regulabilities for each module separately, which comprise the regulatory impact of the remainder of the system. In the third, the regulabilities are the ones intrinsic to the module and the control is taken equal to the control that would reign in the absence of the regulatory interactions between the units. In making these distinctions, it becomes transparent how much control stems from control within the organizational modules, and how much derives from the regulatory interactions between them. Control through other modules turns out to be equivalent, at steady state, to control within a module. The implications of this type of cellular organization for the location of the steady-state operating point is discussed.

Effects of urea on M4-lactate dehydrogenase from elasmobranchs and urea-accumulating Australian desert frogs

We measured the effect of urea on M4-lactate dehydrogenase (M4-LDH) from elasmobranchs and Australian desert frogs (urea accumulators) and from two animals that do not accumulate urea, the axolotl and the rabbit. An analysis of the effect of urea on the Kd(NADH), V, V/K(m(prr)) and V/K(m(NADH)) shows that in all cases the major effect of urea was on the binding of pyruvate, which fits with data in the literature that show that urea acts as a competitive inhibitor of LDH. The characteristics of the elasmobranch enzymes are consistent with a proposed adaptation model, but the situation for the enzymes from the aestivating frogs is equivocal. Urea (400 mM) had less effect on the K(m(prr)) of M4-LDH from the urea accumulators than it did on the non-accumulators, suggesting a general adaptation and that the enzyme produced by the aestivating frogs (urea accumulators) is kinetically different from that of non-aestivating frogs (non-accumulators). A new approach is used to characterize the overall pattern of adaptation to urea. The pattern is similar in an enzyme from an elasmobranch and an aestivating frog despite the temporary presence of urea in the latter and the phylogenetic difference between these animals.

A new class of biochemical oscillator models based on competitive binding

It has been noted that single-enzyme systems can undergo strongly damped transient oscillations. In this paper, we present a nonlinear dynamics analysis of oscillations in undriven chemical systems. This analysis allows us to classify transient oscillations into two groups. In the first group, oscillations arise from rapid oscillatory relaxation to a slower transient relaxation mode. These oscillations are always strongly damped. In the second group, it is the slowest relaxation mode which is implicated in the oscillations so these can be very lightly damped. This second class of oscillations has not previously been studied in enzymology. We show that a remarkably simple single-enzyme system, namely competitive inhibition with substrate flow, generates transient oscillations which belong to the second class. In an attempt to design an experimentally realizable version of this model, we then discovered a system which is capable of sustained oscillations. In this experimentally realizable model, two substrates compete to bind to a macromolecule. The flow of one substrate is controlled by a simple feedback device. Sustained oscillations are observed over a very wide range of parameters. In both models, oscillations are favored by a wide disparity in rates of binding and dissociation of the two substrates to the macromolecule.

Model of the pH-dependence of the concentrations of complexes involving metabolites, haemoglobin and magnesium ions in the human erythrocyte

The rate of glucose consumption and the concentrations of glycolytic intermediates in human erythrocytes have long been known to be pH sensitive. Despite the extensive literature on modelling erythrocyte metabolism, no model developed so far can adequately describe all of these pH-dependent changes. None of these models have included all the significant association reactions between metabolites, Hb and Mg2+ that will influence metabolism. As part of a larger enterprise to develop a detailed model of erythrocyte glycolysis, we present a sub-model which predicts, as a function of pH and oxygenation state, the concentrations of free and Mg(2+)-bound metabolites that are substrates, co-factors and effectors of glycolysis. This model shows that pH changes around physiological values can cause large changes in the distribution of metabolites between free, bound and Mg(2+)-complexed forms, based on binding interactions alone; in oxygenated cells, at pH 7.2-7.6, many glycolytic intermediates undergo changes in concentration of 50-100%. The model also predicts intracellular concentrations of free Mg2+ in erythrocytes to be 0.4 mM and 0.64 mM in oxygenated and deoxygenated cells, respectively, assuming a total magnesium concentration of 3 mM (approximately 88% of the total magnesium usually found in erythrocytes). This is in close agreement with the values found by Flatman [Flatman, P. W. (1980) J. Physiol. 300, 19-30] and the finding by Flatman and Lew [Flatman, P. & Lew, V. L. (1977) Nature 267, 360-362] that the main Mg2+ buffer systems bind approximately 90% of Mg2+ in the cell. Hexokinase has a high 'flux control coefficient' in human erythrocyte glycolysis, so the dependence of its rate on the pH and oxygenation state of haemoglobin is important. With a low oxygen tension and an intracellular pH of 7.34, the major inhibitor of its activity (2,3-bisphosphoglycerate) is 85% bound to either haemoglobin or Mg2+, and the maximum possible flux of substrate via it would be 2.05 mmol L erythrocytes-1 h-1. However, if the haemoglobin were saturated with oxygen, and the pH were 7.2, it was calculated that the maximum rate would be 1.48 mmol L erythrocytes-1 h-1; this is primarily due to a doubling of the free 2,3-bisphosphoglycerate concentration. However, the full extent of the inhibition is counteracted because the concentration of the Mg(2+)-2,3-bisphosphoglycerate would be approximately doubled. Many other similar comparisons are possible with this new model, which highlights the complex network of interactions between haemoglobin, Mg2+, H+ and the metabolites as substrates and effectors of the glycolytic reactions.

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.

Control analysis of glycolytic oscillations

The principles involved in the control of the frequency of sustained metabolic oscillations are developed in the context of glycolytic oscillations in Saccharomyces cerevisiae. To this purpose, an existing mathematical model that describes the experimentally obtained oscillations was simplified to a core model. Frequency, relative phase, average concentrations and amplitudes of the oscillations were well approximated by writing the two remaining metabolic variables of the core model (representing [ATP] and [hexose]) as harmonic functions of time and by requiring them to fulfill the differential equations. The extent to which an enzyme (-conglomerate) controls the frequency in a sustained oscillation is defined as the log-log derivative of that frequency with respect to enzyme activity. In both the full model and the core model this control of frequency and the control over the average concentrations proved to be distributed over the enzymes. We identified a summation theorem, stating that the sum of such control coefficients over all processes equals unity for frequency and zero for the average concentrations.

Regulation of oxidative and glycogenolytic ATP synthesis in exercising rat skeletal muscle studied by 31P magnetic resonance spectroscopy

31P magnetic resonance spectroscopy measurements of pH and the concentrations of orthophosphate and phosphocreatine were used to estimate rates of glycogenolytic and oxidative ATP synthesis in rat leg muscle during 6 min sciatic nerve stimulation at different rates (1-4 Hz). To study the regulation of glycogenolysis during exercise, the apparent 'glycogenolytic capacity' (L(MAX)) was calculated from glycogenolytic ATP synthesis rate and orthophosphate concentration as a measure of the Ca2+-dependent activation of glycogen phosphorylase. This was found to be proportional to the total ATP synthesis rate (F), and to decline with time; expressed relative to total ATP turnover rate as L(MAX)/F, its initial value was 2.9+/-0.6, declining with half-time 1.4+/-0.4 min. The apparent 'mitochondrial capacity' (Q(MAX)), calculated from oxidative ATP synthesis rate and [ADP], was independent of ATP turnover rate, but increased with half-time 0.8+/-0.1 min to 29+/-2 mmol kg(-1) min(-1): thus [ADP] was the predominant but not the only influence on oxidative ATP synthesis. Numerical simulation shows that time-dependent changes in L(MAX)/F exert a strong influence on pH and on the concentrations of phosphocreatine and ADP.

On the analysis of the inverse problem of metabolic pathways using artificial neural networks

Here we develop the use of artificial neural networks for solving the inverse metabolic problem, in other words, given a set of steady-state metabolite levels and fluxes in a pathway of known structure to obtain the parameters of the system, in this case the enzymatic limiting rate and Michaelis constants. This requires two main procedures: first the development of a computer program with which one can model metabolism in the forward direction (i.e. given the internal and parameters to determine the steady-state fluxes and metabolite concentrations), and second, given arrays of associated parameters and variables thereby obtained, to exploit artificial neural networks to form a model capable of obtaining the parameters from the variables. We studied 2-step pathways exhibiting first-order kinetics, 2-step pathways exhibiting reversible Michaelis-Menten kinetics and then 3-step pathways (again exhibiting reversible Michaelis-Menten kinetics), modelled using the program Gepasi. Whilst it was fairly easy for the networks to learn most of the parameters in the 2-step pathway, it was found helpful for the Michaelis-Menten case to vary the concentration of the starting pathway substrate for each set of internal parameters, and to train separate networks for each parameter. Some parameters were much easier to learn than others, reverse K(m) and V(max) values normally being the most difficult. For the 3-step pathway learning sometimes required as much as 3 days, and occasionally convergence was not obtained. Overall, neural networks of the present type, with fully interconnected feedforward architectures and trained according to the backpropagation algorithm, scaled poorly as the problem size was increased.

Experimental determination of control by the H(+)-ATPase in Escherichia coli

Strains carrying deletions in the atp genes, encoding the H(+)-ATPase, were unable to grow on nonfermentable substrates such as succinate, whereas with glucose as the substrate the growth rate of an atp deletion mutant was surprisingly high (some 75-80% of wild-type growth rate). The rate of glucose and oxygen consumption of these mutants was increased compared to the wild-type rates. In order to analyze the importance of the H(+)-ATPase at its physiological level, the cellular concentration of H(+)-ATPase was modulated around the wild-type level, using genetically manipulated strains. The control coefficient by the H(+)-ATPase with respect to growth rate and catabolic fluxes was measured. Control on growth rate was absent at the wild-type concentration of H(+)-ATPase, independent of whether the substrate for growth was glucose or succinate. Control by the H(+)-ATPase on the catabolic fluxes, including respiration, was negative at the wild-type H(+)-ATPase level. Moreover, the turnover number of the individual H(+)-ATPase enzymes increased as the H(+)-ATPase concentration was lowered. The negative control by the H(+)-ATPase on catabolism may thus be involved in a homeostatic control of ATP synthesis and, to some extent, explain the zero control by the H(+)-ATPase on E. coli growth rate.

Defining control coefficients in non-ideal metabolic pathways

The extent to which an enzyme controls a flux has been defined as the effect on that flux of a small modulation of the activity of that enzyme divided by the magnitude of the modulation. We here show that in pathways with metabolic channelling or high enzyme concentrations and conserved moieties involving both enzymic and non-enzymic species, this definition is ambiguous; the magnitude of the corresponding flux control coefficient depends on how the enzyme activity is modulated. This is illustrated with two models of biochemically relevant pathways, one in which dynamic metabolite channelling plays a role, and one with a moiety-conserved cycle. To avoid such ambiguity, we view biochemical pathways in a more detailed manner, i.e., as a network of elemental steps. We define 'elemental control coefficients' in terms of the effect on a flux of an equal modulation of the forward and reverse rate constant of any such elemental step (which may correspond to transitions between enzyme states). This elemental control coefficient is independent of the method of modulation. We show how metabolic control analysis can proceed when formulated in terms of the elemental control coefficients and how the traditional control coefficients are related to these elemental control coefficients. An 'impact' control coefficient is defined which quantifies the effect of an activation of all elemental processes in which an enzyme is involved. It equals the sum of the corresponding elemental control coefficients. In ideal metabolic pathways this impact control coefficient reduces to the traditional flux control coefficient. Differences between the traditional control coefficients are indicative of non-ideality of a metabolic pathway, i.e. of channelling or high enzyme concentrations.

Elusive control

The concept of a single rate-limiting step was proven to be too simplistic for understanding control and regulation of metabolism. Consequently, searches have identified relatively few steps with high control. Here we review a number of such searches and indicate what mechanisms may be responsible for this elusiveness of control. It turns out that this elusiveness of control has itself led to increased understanding of the roles played in metabolic control and regulation of such diverse factors as distributiveness of control, condition dependence, enzyme elasticity, homeostasis, control hierarchies, the input into a pathway, coenzyme sequestration, and redundancy and diversity of control function.

Metabolic regulation: a control analytic perspective

A possible basis for a quantitative theory of metabolic regulation is outlined. Regulation is defined here as the alteration of reaction properties to augment or counteract the mass-action trend in a network reactions. In living systems the enzymes that catalyze these reactions are the "handles" through which such alteration is effected. It is shown how the elasticity coefficients of an enzyme-catalyzed reaction with respect to substrates and products are the sum of a mass-action term and a regulatory kinetic term; these coefficients therefore distinguish between mass-action effects and regulatory effects and are recognized as the key to quantifying regulation. As elasticity coefficients are also basic ingredients of metabolic control analysis, it is possible to relate regulation to such concepts as control, signalling, stability, and homeostasis. The need for care in the choice of relative or absolute changes when considering questions of metabolic regulation is stressed. Although the concepts are illustrated in terms of a simple coupled reaction system, they apply equally to more complex systems. When such systems are divided into reaction blocks, co-response coefficients can be used to measure the elasticities of these blocks.

Energy, control and DNA structure in the living cell

Maintenance (let alone growth) of the highly ordered living cell is only possible through the continuous input of free energy. Coupling of energetically downhill processes (such as catabolic reactions) to uphill processes is essential to provide this free energy and is catalyzed by enzymes either directly or via "storage" in an intermediate high energy form, i.e., high ATP/ADP ratio or H+ ion gradient. Although maintenance of a sufficiently high ATP/ADP ratio is essential to overcome the thermodynamic burden of uphill processes, it is not clear to what degree enzymes that control this ratio also control cell physiology. Indeed, in the living cell homeostatic control mechanisms might exist for the free-energy transduction pathways so as to prevent perturbation of cellular function when the Gibbs energy supply is compromised. This presentation addresses the extent to which the intracellular ATP level is involved in the control of cell physiology, how the elaborate control of cell function may be analyzed theoretically and quantitatively, and if this can be utilized selectively to affect certain cell types.

Application of metabolic control analysis to the pathways of carbohydrate breakdown in Hymenolepis diminuta

The application of metabolic control theory to carbohydrate breakdown in the tapeworm Hymenolepis diminuta shows that it is not necessary for both phosphoenolpyruvate carboxykinase and pyruvate kinase to be modulated in order to control the relative fluxes through the two arms of the phosphoenolpyruvate branchpoint. Changes in activity of enzymes outside of the two branches also influence the flux ratio. Control coefficients of individual enzymes for the fluxes through phosphoenolpyruvate carboxykinase and pyruvate kinase are not fixed, but vary as the flux ratio between the two arms of the branchpoint changes. The metabolic model can also be used to evaluate the role of the fructose-1,6-bisphosphate loop and to calculate metabolite transition time control coefficients.

Control theory of metabolic channelling

Various factors appear to control muscle energetics, often in conjunction. This calls for a quantitative approach of the type provided by Metabolic Control Analysis for intermediary metabolism and mitochondrial oxidative phosphorylation. To the extent that direct transfer of high energy phosphates and spatial organization plays a role in muscle energetics however, the standard Metabolic Control Theory does not apply, neither do its theorems regarding control. This paper develops the Control Theory that does apply to the muscle system. It shows that direct transfer of high energy phosphates bestows a system with enhanced control: the sum of the control exerted by the participating enzymes on the flux of free energy from the mitochondrial matrix to the actinomyosin may well exceed the 100% mandatory for ideal metabolic pathways. It is also shown how sequestration of high energy phosphates may allow for negative control on pathway flux. The new control theory gives method functionally to diagnose the extent to which channelling and metabolite sequestration occur.

The sum of flux control coefficients in the electron-transport chain of mitochondria

The sum of the flux control coefficients for group-transfer reactions such as electron transport has been proposed to be two when the coefficients are calculated from experiments in which the concentrations of the electron carriers are changed (CE) but one when they are calculated from changes in the rates of the electron-transfer processes (Cv). We tested this proposal using electron transport in uncoupled beef heart, potato tuber and rat liver mitochondria. First, with ascorbate plus N,N,N',N"-tetramethyl-p-phenylenediamine as substrate, the CE flux control coefficients of ascorbate, N,N,N',N"-tetramethyl-p-phenylenediamine, mitochondria and oxygen over electron-transport rate were measured by direct titration of the concentrations of these electron carriers. CE values were close to zero, one, one and zero, respectively, giving a sum of CE flux control coefficients of approximately two. At higher concentrations of N,N,N',N'-tetramethyl-p-phenylenediamine, its CE control decreased and the sum decreased towards one as predicted. Secondly, the Cv control coefficients of groups of electron-transfer processes with succinate or ascorbate plus N,N,N',N'-tetramethyl-p-phenylenediamine as substrate were measured. This was achieved by measuring the effects of KCN (or malonate or N,N,N',N'-tetramethyl-p-phenylenediamine) on system flux when intermediates were allowed to relax and on local flux when intermediates were held constant. The Cv flux control coefficients were calculated as the ratio of the effects on system flux and on local flux. The sum of the Cv flux control coefficients was approximately one. Whether a sum of one or a sum of two was obtained depended entirely on the definition of control coefficients that was used, since either sum was obtained from the same set of data depending on the method of calculation. Both definitions are valid, but they give different information. It is important to be aware of which definition is being used when analysing control coefficients in electron-transport chains and other group-transfer systems.