G Protein-Coupled Receptor Signaling Networks from a Systems Perspective

The signal-transduction network of a mammalian cell integrates internal and external cues to initiate adaptive responses. Among the cell-surface receptors are the G protein-coupled receptors (GPCRs), which have remarkable signal-integrating capabilities. Binding of extracellular signals stabilizes intracellular-domain conformations that selectively activate intracellular proteins. Hereby, multiple signaling routes are activated simultaneously to degrees that are signal-combination dependent. Systems-biology studies indicate that signaling networks have emergent processing capabilities that go far beyond those of single proteins. Such networks are spatiotemporally organized and capable of gradual, oscillatory, all-or-none, and subpopulation-generating responses. Protein-protein interactions, generating feedback and feedforward circuitry, are generally required for these spatiotemporal phenomena. Understanding of information processing by signaling networks therefore requires network theories in addition to biochemical and biophysical concepts. Here we review some of the key signaling systems behaviors that have been discovered recurrently across signaling networks. We emphasize the role of GPCRs, so far underappreciated receptors in systems-biology research.

Domain-oriented reduction of rule-based network models

The coupling of membrane-bound receptors to transcriptional regulators and other effector functions is mediated by multi-domain proteins that form complex assemblies. The modularity of protein interactions lends itself to a rule-based description, in which species and reactions are generated by rules that encode the necessary context for an interaction to occur, but also can produce a combinatorial explosion in the number of chemical species that make up the signalling network. The authors have shown previously that exact network reduction can be achieved using hierarchical control relationships between sites/domains on proteins to dissect multi-domain proteins into sets of non-interacting sites, allowing the replacement of each 'full' (progenitor) protein with a set of derived auxiliary (offspring) proteins. The description of a network in terms of auxiliary proteins that have fewer sites than progenitor proteins often greatly reduces network size. The authors describe here a method for automating domain-oriented model reduction and its implementation as a module in the BioNetGen modelling package. It takes as input a standard BioNetGen model and automatically performs the following steps: 1) detecting the hierarchical control relationships between sites; 2) building up the auxiliary proteins; 3) generating a raw reduced model and 4) cleaning up the raw model to provide the correct mass balance for each chemical species in the reduced network. The authors tested the performance of this module on models representing portions of growth factor receptor and immunoreceptor-mediated signalling networks and confirmed its ability to reduce the model size and simulation cost by at least one or two orders of magnitude. Limitations of the current algorithm include the inability to reduce models based on implicit site dependencies or heterodimerisation and loss of accuracy when dynamics are computed stochastically.

Signal processing at the Ras circuit: what shapes Ras activation patterns?

A systems biology approach is applied to gain a quantitative understanding of the integration of signalling by the small GTPase Ras. The Ras protein acts as a critical switch in response to signals that determine the cell's fate. In unstimulated cells, Ras switching between an inactive GDP-binding and active GTP-binding state is controlled by the intrinsic catalytic activities of Ras. The calculated high sensitivity of the basal Ras-GTP fraction to changes in the rate constant of GTP-hydrolysis by Ras can account for the carcinogenic potential of Ras mutants with decreased GTPase activities. Extracelluar stimuli initiate Ras interactions with GDP/GTP exchange factors such as SOS, and GTP-hydrolysis activating proteins such as RasGAP. Our data on freshly isolated hepatocytes stimulated with epidermal growth factor (EGF) show transient SOS activation and sustained Ras-GTP patterns. We demonstrate that these dose-response data can only be explained by transient RasGAP activitation, and not by merely switching off the SOS signal, e.g. by inhibitory phosphorylation of SOS. A transient RasGAP activity can be brought about by a number of mechanisms. A comprehensive kinetic model of the EGF receptor (EGFR) network was developed to explore feasible molecular scenarios, including the receptor-mediated recruitment of SOS and RasGAP to the plasma membrane, phosphorylation of RasGAP and p190 RhoGAP by soluble tyrosine kinases, and RasGAP interactions with phosphoinositides and p190 RhoGAP. We show that a transient RasGAP association with EGFR followed by the capture of RasGAP through the formation of complexes with p190 RhoGAP can account for data on hepatocytes. In summary, our results demonstrate that a combination of experimental monitoring and integrated dynamic analysis is capable of dissecting regulatory mechanisms that govern cellular signal transduction.

[Kinetic modeling of energy metabolism and generation of active forms of oxygen in hepatocyte mitochondria]

Direct nonenzymatic oxidation of semiquinone by oxygen is one of the main sources of superoxide radicals (O2.-) in mitochondria. By using all the known data on hepatocyte mitochondria, we have revealed the correlation between the rate of superoxide generation by the bc1 complex and the transmembrane potential (delta psi). If the main electrogenic stage of the Q cycle is suggested to be the electron transfer between the cytochrome b hemes, then the rate of superoxide generation sharply increases when delta psi grows from 150 mV to 180 mV. However, this interrelation is ambiguous. Indeed, the increase of the generation rate with the growth of the potential can occur faster when succinate dehydrogenase is inhibited by malonate than when external ADP is exhausted. When the potential is changed by adding phosphate or potassium (K+), the rate of O2.- production remains constant, although the comparison of the rate values at the same delta psi reveals the effect of phosphate or potassium. It turned out that the rate of O2.- generation is a function of delta mu H rather than any of its components. Phosphate and K+ have practically no influence on delta mu H, since the change in delta psi is compensated by delta pH. The rate of superoxide generation by the bc1 complex is a multiple function of the electron-transfer activity of enzymes, the processes determining the membrane potential (e.g., loading), and of the oxygen concentration. The kinetic model proposed in this work may serve a tool to understand how the superoxide production is regulated.

Occurrence of paradoxical or sustained control by an enzyme when overexpressed: necessary conditions and experimental evidence with regard to hepatic glucokinase

It is widely assumed that the control coefficient of an enzyme on pathway flux decreases as the concentration of enzyme increases. However, it has been shown [Kholodenko and Brown (1996) Biochem. J. 314, 753-760] that enzymes with sigmoidal kinetics can maintain or even gain control with an increase in enzyme activity or concentration. This has been described as 'paradoxical control'. Here we formulate the general requirements for allosteric enzyme kinetics to display this behaviour. We show that a necessary condition is that the Hill coefficient of the enzyme should increase with an increase in substrate concentration or decrease with an increase in product concentration. We also describe the necessary and sufficient requirements for the occurrence of paradoxical control in terms of the flux control coefficients and the derivatives of the elasticities. The derived expression shows that the higher the control coefficient of an allosteric enzyme, the more likely it is that the pathway will display this behaviour. Control of pathway flux is generally shared between a large number of enzymes and therefore the likelihood of observing sustained or increased control is low, even if the kinetic parameters are in the most favourable range to generate the phenomenon. We show that hepatic glucokinase, which has a very high flux control coefficient and displays sigmoidal behaviour within the hepatocyte in situ as a result of interaction with a regulatory protein, displays sustained or increased control over an extended range of enzyme concentrations when the regulatory protein is overexpressed.

Diffusion control of protein phosphorylation in signal transduction pathways

Multiple signalling proteins are phosphorylated and dephosphorylated at separate cellular locations, which potentially causes spatial gradients of phospho-proteins within the cell. We have derived relationships that enable us to estimate the extent to which a protein kinase, a phosphatase and the diffusion of signalling proteins control the protein phosphorylation flux and the phospho-protein gradient. Two different cellular geometries were analysed: (1) the kinase is located on one planar membrane and the phosphatase on a second parallel planar membrane, and (2) the kinase is located on the plasma membrane of a spherical cell and the phosphatase is distributed homogeneously in the cytoplasm. We demonstrate that the control contribution of protein diffusion is potentially significant, given the measured rates for protein kinases, phosphatases and diffusion. If the distance between the membranes is 1 microm or greater, the control by diffusion can reach 33% or more, with the rest of the control (67%) shared by the kinase and the phosphatase. At distances of less than 0.1 microm, diffusion does not limit protein phosphorylation. For a spherical cell of radius 10 microm, a protein diffusion coefficient of 10(-8) cm(2). s(-1) and rate constants for the kinase and the phosphatase of approx. 1 s(-1), control over the phosphorylation flux resides mainly with the phosphatase and protein diffusion, with approximately equal contributions of each of these. The ratio of phospho-protein concentrations at the cell membrane and the cell centre (the dynamic compartmentation of the phospho-protein) is shown to be controlled by the rates of the protein phosphatase and of diffusion. The kinase can contribute significantly to the control of the absolute value of the phospho-protein gradient.

Why cytoplasmic signalling proteins should be recruited to cell membranes

It has been suggested that localization of signal-transduction proteins close to the cell membrane causes an increase in their rate of encounter after activation. We maintain that such an increase in the first-encounter rate is too small to be responsible for truly enhanced signal transduction. Instead, the function of membrane localization is to increase the number (or average lifetime) of complexes between cognate signal transduction proteins and hence increase the extent of activation of downstream processes. This is achieved by concentrating the proteins in the small volume of the area just below the plasma membrane. The signal-transduction chain is viewed simply as operating at low default intensity because one of its components is present at a low concentration. The steady signalling level of the chain is enhanced 1000-fold by increasing the concentration of that component. This occurs upon 'piggyback' binding to a membrane protein, such as the activated receptor, initiating the signal-transduction chain. For the effect to occur, the protein translocated to the membrane cannot be free but has to remain organized by being piggyback bound to a receptor, membrane lipid(s) or scaffold. We discuss an important structural constraint imposed by this mechanism on signal transduction proteins that might also account for the presence of adaptor proteins.

Negative feedback and ultrasensitivity can bring about oscillations in the mitogen-activated protein kinase cascades

Functional organization of signal transduction into protein phosphorylation cascades, such as the mitogen-activated protein kinase (MAPK) cascades, greatly enhances the sensitivity of cellular targets to external stimuli. The sensitivity increases multiplicatively with the number of cascade levels, so that a tiny change in a stimulus results in a large change in the response, the phenomenon referred to as ultrasensitivity. In a variety of cell types, the MAPK cascades are imbedded in long feedback loops, positive or negative, depending on whether the terminal kinase stimulates or inhibits the activation of the initial level. Here we demonstrate that a negative feedback loop combined with intrinsic ultrasensitivity of the MAPK cascade can bring about sustained oscillations in MAPK phosphorylation. Based on recent kinetic data on the MAPK cascades, we predict that the period of oscillations can range from minutes to hours. The phosphorylation level can vary between the base level and almost 100% of the total protein. The oscillations of the phosphorylation cascades and slow protein diffusion in the cytoplasm can lead to intracellular waves of phospho-proteins.

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.

Engineering a living cell to desired metabolite concentrations and fluxes: pathways with multifunctional enzymes

With molecular genetics enabling modulation of the concentrations of cellular enzymes, metabolic engineering becomes limited by the question of which modulations of the enzyme concentrations are required to bring about a desired pattern of cellular metabolism. In an earlier paper (Kholodenko et al. (1998). Biotechnol. Bioeng. 59, 239-247) we derived a method to determine the required modulations. This method, however, cannot be immediately applied to cellular pathways with enzymes catalyzing more than one step in metabolism (multifunctional enzymes). In the present paper we show to which extent the presence of multifunctional enzymes limits biotechological ambitions, which one might otherwise pursue in vain. In particular, it is impossible to change the concentration of a single intermediate and leave the rest of metabolism unperturbed if that intermediate interacts directly with a multifunctional enzyme. The analytical machinery of Metabolic Control Analysis is used to relate the desired and ensuing changes in the metabolic pattern. An explicit solution to this problem of engineering metabolism is then given in the form of a single matrix equation.

Quantification of short term signaling by the epidermal growth factor receptor

During the past decade, our knowledge of molecular mechanisms involved in growth factor signaling has proliferated almost explosively. However, the kinetics and control of information transfer through signaling networks remain poorly understood. This paper combines experimental kinetic analysis and computational modeling of the short term pattern of cellular responses to epidermal growth factor (EGF) in isolated hepatocytes. The experimental data show transient tyrosine phosphorylation of the EGF receptor (EGFR) and transient or sustained response patterns in multiple signaling proteins targeted by EGFR. Transient responses exhibit pronounced maxima, reached within 15-30 s of EGF stimulation and followed by a decline to relatively low (quasi-steady-state) levels. In contrast to earlier suggestions, we demonstrate that the experimentally observed transients can be accounted for without requiring receptor-mediated activation of specific tyrosine phosphatases, following EGF stimulation. The kinetic model predicts how the cellular response is controlled by the relative levels and activity states of signaling proteins and under what conditions activation patterns are transient or sustained. EGFR signaling patterns appear to be robust with respect to variations in many elemental rate constants within the range of experimentally measured values. On the other hand, we specify which changes in the kinetic scheme, rate constants, and total amounts of molecular factors involved are incompatible with the experimentally observed kinetics of signal transfer. Quantitation of signaling network responses to growth factors allows us to assess how cells process information controlling their growth and differentiation.

Spatial gradients of cellular phospho-proteins

If a protein is rapidly phosphorylated and dephosphorylated at separate cellular locations and protein diffusion is slow, then a spatial gradient of the phosphorylated form of the protein may develop within the cell. We have estimated the potential size of such gradients using measured values of protein diffusion coefficients and protein kinase and phosphatase activities. We analysed two different cellular geometries: (1) where the kinases is located on the plasma membrane of a spherical cell and the phospatase is distributed homogenously in the cytoplasm and (2) where the kinase is located on one planar membrane and the phosphatase on a second parallel planar membrane. The estimated gradients of phospho-proteins were potentially very large, which has important implications for cellular signalling.

Implications of macromolecular crowding for signal transduction and metabolite channeling

The effect of different total enzyme concentrations on the flux through the bacterial phosphoenolpyruvate:carbohydrate phosphotransferase system (PTS) in vitro was determined by measuring PTS-mediated carbohydrate phosphorylation at different dilutions of cell-free extract of Escherichia coli. The dependence of the flux on the protein concentration was more than linear but less than quadratic. The combined flux-response coefficient of the four enzymes constituting the glucose PTS decreased slightly from values of approximately 1.8 with increasing protein concentrations in the assay. Addition of the macromolecular crowding agents polyethylene glycol (PEG) 6000 and PEG 35000 led to a sharper decrease in the combined flux-response coefficient, in one case to values of approximately 1. PEG 6000 stimulated the PTS flux at lower protein concentrations and inhibited the flux at higher protein concentrations, with the transition depending on the PEG 6000 concentration. This suggests that macromolecular crowding decreases the dissociation rate constants of enzyme complexes. High concentrations of the microsolute glycerol did not affect the combined flux-response coefficient. The data could be explained with a kinetic model of macromolecular crowding in a two-enzyme group-transfer pathway. Our results suggest that, because of the crowded environment in the cell, the different PTS enzymes form complexes that live long on the time-scale of their turnover. The implications for the metabolic behavior and control properties of the PTS, and for the effect of macromolecular crowding on nonequilibrium processes, are discussed.

Metabolic design: how to engineer a living cell to desired metabolite concentrations and fluxes

A biotechnological aim of genetic engineering is to increase the intracellular concentration or secretion of valuable compounds, while making the other concentrations and fluxes optimal for viability and productivity. Efforts to accomplish this based on over-expression of the enzyme, catalyzing the so-called "rate-limiting step," have not been successful. Here we develop a method to determine the enzyme concentrations that are required to achieve such an aim. This method is called Metabolic Design Analysis and is based on the perturbation method and the modular ("top-down") approach-formalisms that were first developed for the analysis of biochemical regulation such as, Metabolic Control Analysis. Contrary to earlier methods, the desired alterations of cellular metabolism need not be small or confined to a single metabolite or flux. The limits to the alterations of fluxes and metabolite concentrations are identified. To employ Metabolic Design Analysis, only limited kinetic information concerning the pathway enzymes is needed.

A model of O2.-generation in the complex III of the electron transport chain

Oxidation of semiquinone by O2 in the Q cycle is known to be one of the sources of superoxide anion (O2.-) in aerobic cells. In this paper, such a phenomenon was analyzed using the chemical kinetics model of electron transfer from succinate to cytochrome c, including coenzyme Q, the complex III non-heme iron protein FeSIII and cytochromes bl, bh and cl. Electron transfers from QH2 to FeSIII and cytochrome bl were assumed to occur according to direct transfer mechanism (dynamic channelling) involving the formation of FeS(red)III-Q.- and Q.--cytochrome bl complexes. For oxidation/reduction reactions involving cytochromes bh and bl, the dependence of the equilibrium and elementary rate constants on the membrane potential (deltapsi) was taken into consideration. The rate of O2.- generation was found to increase dramatically with increase in deltapsi above the values found in State 3. On the other hand, the rate of cytochrome c reduction decreased sharply at the same values of the membrane potential. This explains experimental data that the O2.- generation at State 4 appears to be very much faster than at State 3. A mild uncoupling in State 4 can markedly decrease the superoxide generation due to a decrease in deltapsi below the above mentioned critical level. DeltapH appears to be equally effective as deltapsi in stimulation of superoxide production which depends, in fact, upon the deltamuH+ level.

Subtleties in control by metabolic channelling and enzyme organization

Because of its importance to cell function, the free-energy metabolism of the living cell is subtly and homeostatically controlled. Metabolic control analysis enables a quantitative determination of what controls the relevant fluxes. However, the original metabolic control analysis was developed for idealized metabolic systems, which were assumed to lack enzyme-enzyme association and direct metabolite transfer between enzymes (channelling). We here review the recently developed molecular control analysis, which makes it possible to study non-ideal (channelled, organized) systems quantitatively in terms of what controls the fluxes, concentrations, and transit times. We show that in real, non-ideal pathways, the central control laws, such as the summation theorem for flux control, are richer than in ideal systems: the sum of the control of the enzymes participating in a non-ideal pathway may well exceed one (the number expected in the ideal pathways), but may also drop to values below one. Precise expressions indicate how total control is determined by non-ideal phenomena such as ternary complex formation (two enzymes, one metabolite), and enzyme sequestration. The bacterial phosphotransferase system (PTS), which catalyses the uptake and concomitant phosphorylation of glucose (and also regulates catabolite repression) is analyzed as an experimental example of a non-ideal pathway. Here, the phosphoryl group is channelled between enzymes, which could increase the sum of the enzyme control coefficients to two, whereas the formation of ternary complexes could decrease the sum of the enzyme control coefficients to below one. Experimental studies have recently confirmed this identification, as well as theoretically predicted values for the total control. Macromolecular crowding was shown to be a major candidate for the factor that modulates the non-ideal behaviour of the PTS pathway and the sum of the enzyme control coefficients.

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.

Quantification of information transfer via cellular signal transduction pathways

A conceptual framework is developed for the quantitative analysis of signal transfer through cellular signal transduction pathways and networks. This approach is referred to as signal transfer analysis and is based on formalisms that were first developed for the analysis of metabolic networks. Signal transduction is quantified as the sensitivity, known as the response coefficient of a target (e.g. an ion channel or transcription factor) to a signal (e.g. a hormone, growth factor or neurotransmitter). This response coefficient is defined in terms of the fractional change in the activated target brought about by a small fractional change in the signal. Quantifying the signal transduction in this way makes it possible to prove that for an idealized signaling cascade without feedback loops, the total response equals the product of all the local response coefficients, one for each level of the cascade. We show under which conditions merely having more levels in a cascade can boost the sensitivity of a target to a signal. If a signal propagates to a target through two different routes, these routes contribute independently to the total response, provided there is no feedback from the target. This independence makes the behavior of signaling cascades different from that of metabolic pathways, where different branches are connected through Kirchhoffs law. The relations between the total response and the local kinetics at each level are given for a number of network structures, such as branched signaling pathways and pathways with feedback. The formalism introduced here may provide a general approach to quantify cellular information transfer.

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.

Molecular control analysis: control within proteins and molecular processes

Molecular control analysis is a method for analysing the extent to which the different elementary steps or rate constants within a molecular process limit the steady-state rate (or other variables) of that process. Any process which may be described by a kinetic diagram of transitions between states of the system may be analysed by molecular control analysis, and this approach has previously been used to analyse control within enzymes, transporters, enzyme complexes, channelled pathways, and group-transfer pathways. We outline the theory of molecular control analysis here, and illustrate its use by analysing control within enzymes (three beta-lactamases). Further potential applications include signal-transduction processes, protein folding, and chemical reactions.

Strong control on the transit time in metabolic channelling

A suite of different characteristic times is used to describe the temporal behavior of a metabolic pathway. Here we focus on the 'transit' time, that is the average time it takes for a molecule, entering the steady-state pathway as a substrate, to exit the pathway as a product. We show that metabolic channelling results in dramatic changes in control exerted by pathway enzymes on the transit time. In an 'ideal' pathway a doubling of the enzyme concentrations halves the transit time. In a dynamic channel such an increase can reduce the transit time by a factor of four or more.

[Control of the rate of the glucokinase reaction by its elementary stages]

Limitation of an enzymatic reaction by its elemental steps is characterized by so-called control coefficients for (quasi)stationary reaction rate. Established paradigm of "limiting step" cannot describe the control of glucokinase reaction. Flux control is distributed between several elemental steps (ATP binding, ADP release, etc.); this distribution significantly changes when reaction conditions are modified, i.e., changes in glucose or ATP concentrations occur. At about 1 mM ATP and 2-5 mM glucose, impact of ATP binding stage on total flux control is positive and it exceeds 50% if ATP binds after glucose. However, if ATP binds before glucose then control coefficient for ATP binding is negative. At high substrate concentrations, impact of product dissociation on flux control becomes significant. Distribution of control between elemental steps cannot be predicted by changes of Gibbs energy or by the rate constants of elemental steps.

Paradoxical control properties of enzymes within pathways: can activation cause an enzyme to have increased control?

It is widely assumed that within a metabolic pathway inhibition of an enzyme causes the control exerted by that enzyme over the flux through its own reaction to increase, whereas activation causes its control to decrease. This assumption forms the basis of a number of experimental methods. For a pathway conceptually divided into two enzyme groups connected via a single metabolite we have derived a general condition under which this assumption is false, and thus the pathway shows paradoxical control behaviour, i.e. increased control with activation and decreased control with inhibition of an enzyme or group of enzymes. Paradoxical control behaviour occurs widely when enzyme activity is altered by changing Km (if an enzyme is already close to saturation by its substrate), but may also occur with changes in Vmax. when the elasticity to the linking metabolite increases with its concentration (as in some cases of sigmoidal and exponential kinetics or for reactions catalysed by isoenzymes). These findings suggest that enzymes with sigmoidal kinetics may have low control in the absence of activation, but may gain control with activation, and thus have beneficial regulatory properties.

Effect of channelling on the concentration of bulk-phase intermediates as cytosolic proteins become more concentrated

This paper shows that metabolic channelling can provide a mechanism for decreasing the concentration of metabolites in the cytoplasm when cytosolic proteins become more concentrated. A dynamic complex catalysing the direct transfer of an intermediate is compared with the analogous pathway lacking a channel (an "ideal" pathway). In an ideal pathway a proportional increase in protein content does not result in a change in the steady-state concentration of the bulk-phase intermediate, whereas in a channelling pathway the bulk-phase intermediate either decreases or increases depending on the elemental rate constants within the enzyme mechanisms. When the concentration of the enzymes are equal, the pool size decreases with increasing protein concentration if the elemental step depleting the bulk-phase intermediate exerts more control on its concentration than the step supplying the intermediate. Results are illustrated numerically, and a simplified dynamic channel is analysed in which the concentration of the enzyme-enzyme forms. For such a "hit-and-run" channel it is shown that, when the product-releasing step of the enzyme located upstream is close to equilibrium, the pool size decreases as the concentrations of the enzymes increase in proportion, regardless of the rate, equilibrium constants and concentration ratios of the two sequential enzymes.

Calcium indirectly increases the control exerted by the adenine nucleotide translocator over 2-oxoglutarate oxidation in rat heart mitochondria

The effect of calcium on the control exerted by the adenine nucleotide translocator over respiration in isolated heart mitochondria was investigated in order to determine whether calcium directly stimulates the translocator. At respiration rates intermediate between states 3 and 4, Ca2+ is shown to increase the control over 2-oxoglutarate oxidation exerted by the adenine nucleotide translocator in rat heart mitochondria. This did not occur when succinate was the respiratory substrate, even though the control exerted by the translocator was substantial, indicating that Ca2+ does not have a direct effect on the adenine nucleotide translocator. Ca2+ increased the uncoupled oxidation rate of 2-oxoglutarate, but not succinate. Using the summation theorem for flux control, the effect of Ca2+ is explained by a shift of the control over respiration rate toward the adenine nucleotide translocator, from the respiratory chain, presumably as the result of the activation of the 2-oxoglutarate dehydrogenase complex.

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.

Composite control of cell function: metabolic pathways behaving as single control units

This paper shows that under some conditions the control exerted by a part of a metabolic network (a pathway) on a flux or concentration in any other part can be described through a single (overall) control coefficient. This has the following implications: (i) the relative contributions of a pathway enzyme to the regulation of the pathway (output) flux and of any flux or concentration outside are identical; therefore, the control analysis of the pathway 'in isolation' allows one to determine the control exerted by any pathway enzyme on the rest of the cell by estimation of the control efficient of just one, arbitrarily chosen enzyme; (ii) the relative control of any two metabolic variables outside the pathway (measured as the ratio of the control coefficients over these two variables outside) is the same for all pathway enzymes. These properties allow one to substitute effectively a pathway by a single (super)reaction and make it possible to consider such a pathway as a metabolic unit within the cellular enzyme network.

Coenzyme cycles and metabolic control analysis: the determination of the elasticity coefficients from the generalised connectivity theorem

Metabolic control analysis allows one to express the elasticity coefficients (which describe the "local" kinetic features of enzymes) in terms of the control coefficients (quantitative indicators of the "global" control properties). However, when coenzymes (or metabolites linked by conservation constraints) are present in the pathway this procedure yields the "apparent" values of elasticity coefficients that correspond to the kinetic responses of the enzymes to such a simultaneous change of the coenzyme forms which leaves the total concentration of these forms unchanged (e.g., NAD+ + NADH in the glycolysis). We show that a generalised connectivity theorem (Kholodenko et al, Eur. J. Biochem. (1994) 225, 179-186) makes it possible to express the elasticity coefficients with respect to every coenzyme form separately. Such expressions include (i) the control coefficients and (ii) the responses to changes in the total concentrations of the coenzymes.

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 macroworld versus the microworld of biochemical regulation and control

Our understanding of cell physiology has been helped greatly by viewing metabolism as a set of reactions catalysed by independent catalysts (enzymes) in ideal solutions. Yet the differences between this idealized cell and reality have strong implications for biochemical regulation and control. We show here that in the real cell an enzyme controls cell physiology in more than a single way. These different controlling modes in the real 'macroworld' can be related to one another by implementing a new type of control analysis, which is formulated in terms of the 'microworld' of the elemental processes.

Control in channelled pathways. A matrix method calculating the enzyme control coefficients

The usual equations expressing the enzyme control coefficients (quantitative indicators of 'global' control properties of a pathway) via the elasticity coefficients (reflecting local kinetic properties of an enzyme reaction), cannot be applied to a variety of 'non-ideal' pathways, in particular to pathways with metabolic channelling. Here we show that the relationship between the control and elasticity coefficients can be obtained by considering such a metabolic pathway as a network of elemental chemical conversions (steps). To calculate the control coefficients of enzymes one should first determine the elasticity coefficients of such elemental steps and then take their appropriate combinations. Although the method is illustrated for a channelled pathway it can be used for any non-ideal pathway including those with high enzyme concentrations where the sequestration of metabolites by enzymes cannot be neglected.

Control theory of one enzyme

The analogue of metabolic control theory is developed for the control of reactions catalyzed by single enzymes. The control exerted by any of the elemental transitions of enzyme catalytic cycles on reaction rate and on concentrations (probabilities) of enzyme states is quantified in line with the principle of detailed balance. For enzyme reactions with arbitrary kinetic schemes, e.g., with several enzyme cycles, reflecting coupling and slipping of reactions, it is derived what the various sums of the control coefficients are equal to (cycle summation theorems). Total control on flux, state probability and ratios of branch fluxes are 1, 0 and 0, respectively. The general connectivity theorems are derived which indicate how control is determined by the kinetics of the elemental steps. In addition, for enzymes catalyzing single (or completely coupled) processes the control coefficients are expressed in terms of actual and standard free energy differences across the steps. The prevalent qualitative contention that the step with the smallest forward rate constant, or with the largest free energy drop is the step limiting the performance of the enzyme is shown to fail. The new theory should allow subtle analysis of the control of an enzyme catalyzed reaction.

Control by enzymes, coenzymes and conserved moieties. A generalisation of the connectivity theorem of metabolic control analysis

The control and regulation of metabolic systems are determined by their responses to changes in the internal metabolites (the internal state) and parameters of the system. In many cases, the concentrations of the intermediates are constrained by moiety conservations, for example those requiring that all intermediate forms of any enzyme sum to the conserved total concentration of that enzyme. In this study, we show how responses to changes in the internal state are related to responses to changes in the total amounts of conserved moieties. The relationship between these two different measures of control leads to a generalisation of the connectivity theorems. The results have important implications for the study of a variety of phenomena such as metabolite (coenzyme) sequestration, group-transfer and channelling. The relationships we derive make it possible to determine the control features of these pathways. As an illustration, two examples are chosen. The first shows the effect of sequestration of substrate moiety while the second deals with the sequestration of the enzyme moieties and enzyme/enzyme interactions.

How to determine control of growth rate in a chemostat. Using metabolic control analysis to resolve the paradox

The chemostat makes it possible to study microbial physiology at steady state. However, because growth rate in a chemostat is set by the experimenter, it seems impossible to employ the chemostat to study the control of microbial growth by processes within the microorganism. In this paper we show how, paradoxically, one can determine control of growth rate, of growth yield and of other fluxes in a chemostat. We develop metabolic control analysis for the chemostat. This analysis does not depend on the particular way in which specific growth rate varies with the concentration of the growth limiting substrate.

Rate limitation within a single enzyme is directly related to enzyme intermediate levels

The extents to which different rate constants limit the steady-state rate of an isolated enzyme can be quantified as the control coefficients of those constants and elemental steps. We have found that the sum of the control coefficients of rate constants characterising unidirectional rates depleting a particular enzyme intermediate is equal to the concentration of that enzyme intermediate as a fraction of the total enzyme concentration. Together with simple measurements this powerful relation may be used (i) to estimate certain enzyme intermediate levels, in particular the free enzyme concentration, and (ii) to estimate the control coefficients of rate constants and steps.

Getting to the inside of cells using metabolic control analysis

Metabolic control analysis can relate control properties of an intact system to kinetic properties (elasticity coefficients) of the enzymes within that system. The method formulating the former as matrix inverse of the latter is elaborated here for the general case and founded in standard metabolic control theory. Then a method is developed that accomplishes the reverse: it is shown that a matrix containing all elasticity coefficients and information concerning the pathway structure equals the inverse of a matrix containing flux and concentration control coefficients. As a consequence, by measuring the control properties of an intact system, one is able to deduce its in situ pathway structure and enzyme kinetic properties: This solves the ever-present question of whether the kinetic properties of enzymes in their isolated state differ from those under the conditions prevailing in the cell.

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 chapter 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 form 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 methods functionally to diagnose the extent to which channelling and metabolite sequestration occur.

Dramatic changes in control properties that accompany channelling and metabolite sequestration

A simple summation theorem describes the control of fluxes in 'ideal' metabolic pathways. This paper shows how this theorem and the control properties of a pathway change when direct transfer of intermediates and/or sequestration of metabolites involved in moiety conservations (by enzymes present at high concentrations) take place. The derived generalized summation theorem quantifies the extent to which metabolite sequestration decreases and direct metabolite transfer can increase the control exerted by enzymes on the flux. The implications of metabolite channelling for the control of fluxes are discussed quantitatively.

Kinetic models of coupling between H+ and Na(+)-translocation and ATP synthesis/hydrolysis by F0F1-ATPases: can a cell utilize both delta mu H+ and delta mu Na+ for ATP synthesis under in vivo conditions using the same enzyme?

Kinetic models of the F0F1-ATPase able to transport H+ or/and Na+ ions are proposed. It is assumed that (i) H+ and Na+ compete for the same binding sites, (ii) ion translocation through F0 is coupled to the rate-limiting step of the F1-catalyzed reaction. The main characteristics of the dependences of ATP synthesis and hydrolysis rates on delta psi, delta pH, and delta pNa are predicted for various versions of the coupling model. The mechanism of the switchover from delta mu H(+)-dependent synthesis to the delta mu Na(+)-dependent one is demonstrated. It is shown that even with a drastic drop in delta mu H+, ATP hydrolysis by the proton mode of catalysis can be effectively inhibited by delta psi and delta pNa. The results obtained strongly support the possibility that the same F0F1-ATPase in bacterial cells can utilize both delta muH+ and delta muNa+ for ATP synthesis under in vivo conditions.

Metabolic channelling and control of the flux

Metabolic control theory is extended to include channelled metabolism in general. A simple relationship between the flux control by the enzymes and the degree of metabolite channelling is derived. This relationship suggests experiments in which modulation of gene expression allows one to quantify channelling.

'Channelled' pathways can be more sensitive to specific regulatory signals

In 'simple' metabolic pathways the response to an external signal is readily described in terms of the effect of the signal on its receptor enzyme and the control exerted by that enzyme. We show here that in the response of 'channelled' pathways to such a signal, additional terms appear that reflect the direct enzyme-enzyme interactions. They tend to enhance the responsiveness of the pathway. The normalized value of the response is called the signal transduction coefficient. We show that in channelled pathways these coefficients are usually larger than in corresponding non-channelled (simple) pathways.

The sum of the control coefficients of all enzymes on the flux through a group-transfer pathway can be as high as two

In simple metabolic pathways the control exerted by enzyme concentrations on the pathway flux adds up to one when the control is quantified in terms of control coefficients. In this paper we demonstrate that this classical summation theorem has to be modified in pathways where the enzymes participate by transferring a group between each other. We derive the corresponding new control theorem and show how it is consistent with standard metabolic control analysis. In group-transfer pathways lacking enzyme complexes, the sum of the flux control by enzyme concentrations and by the donor and acceptor couples of the pathway, equals two. In group-transfer pathways with enzyme-enzyme interactions the flux control by the dissociation rate constants of the enzyme-enzyme complexes must be added to obtain this sum of two. In all cases, the sum of the controls by all reaction activities remains one. Both by using the new theorem and by numerical simulations, we then demonstrate that, in group-transfer pathways with or without enzyme interactions, the sum of the control of enzymes on the pathway flux is higher than one and can reach a value of two. The total control of all enzymes on the concentration of any intermediate either with or without the transferred group can be equal to one, rather than to the zero found in the classical case. Examples of group-transfer pathways are the bacterial phosphoenolpyruvate:sugar phosphotransferase system, the main pathway for uptake of sugars in Enterobacteriaceae, and the electron-transfer chain in free-energy transducing membranes.

Control of the metabolic flux in a system with high enzyme concentrations and moiety-conserved cycles. The sum of the flux control coefficients can drop significantly below unity

In a number of metabolic pathways enzyme concentrations are comparable to those of substrates. Recently it has been shown that many statements of the 'classical' metabolic control theory are violated if such a system contains a moiety-conserved cycle. For arbitrary pathways we have found: (a) the equation connecting coefficients CEiJ (obtained by varying the Ei concentration) and CviJ (obtained by varying the kicat), and (b) modified summation equations. The sum of the enzyme control coefficients (equal to unity under the 'classical' theory) appears always to be below unity in the systems considered. The relationships revealed were illustrated by a numerical example where the sum of coefficients CEiJ reached negative values. A method for experimental measurements of the above coefficients is proposed.