Metabolic control analysis for large changes: extension to variable elasticity coefficients

Modular approaches are powerful systems biology strategies to deal with complexity. They consist in lumping conceptually all that is irrelevant to the problem under study, leaving explicit the portions of interest. Modular (or top-down) metabolic control analysis is a theoretical and experimental approach to study the sensitivity properties of complex metabolic systems. Initially, it was conceived for infinitesimal changes but, recently, it started to be developed for large metabolic changes. A central result of this approach is that the systemic properties, represented by control coefficients, can be expressed as a function of the properties of isolated modules, the elasticity coefficients. Here we extend the theory for large changes to the case that the elasticity coefficients depend on the extent of the change. The novel theory is used to analyse experimental data related to the control of glycolytic flux in Escherichia coli. Our analysis shows that the pattern of control for large changes is quantitatively and qualitatively different from the one obtained applying the infinitesimal treatment.     

Modular metabolic control analysis of large responses in branched systems--application to aspartate metabolism

Organisms subject to changing environmental conditions or experimental protocols show complex patterns of responses. The design principles behind these patterns are still poorly understood. Here, modular metabolic control analysis is developed to deal with large changes in branched pathways. Modular aggregation of the system dramatically reduces the number of explicit variables and modulation sites. Thus, the resulting number of control coefficients, which describe system responses, is small. Three properties determine the pattern for large changes in the variables: the values of infinitesimal control coefficients, the effect of large rate changes on the control coefficients and the range of rate changes preserving feasible intermediate concentrations. Importantly, this pattern gives information about the possibility of obtaining large variable changes by changing parameters inside the module, without the need to perform any parameter modulations. The framework is applied to a detailed model of Asp metabolism. The system is aggregated in one supply module, producing Thr from Asp (SM1), and two demand modules, incorporating Thr (DM2) and Ile (DM3) into protein. Their fluxes are: J(1), J(2), and J(3), respectively. The analysis shows similar high infinitesimal control coefficients of J(2) by the rates of SM1 and DM2 (C(v1)(J2) = 0.6 and C(v2)(J2) = 0.7, respectively). In addition, these coefficients present only moderate decreases when the rates of the corresponding modules are increased. However, the range of feasible rate changes in SM1 is narrow. Therefore, for large increases in J(2) to be obtained, DM2 must be modulated. Of the rich network of allosteric interactions present, only two groups of inhibitions generate the control pattern for large responses.     

Steady-state analysis of metabolic pathways: comparing the double modulation method and the lin-log approach

The increasing interest in studying enzyme kinetics under in vivo conditions requires practical methods to estimate control parameters from experimental data. In contrast to currently established approaches of dynamic modelling, this paper addresses the steady-state analysis of metabolic pathways. Within the framework of metabolic control analysis (MCA), elasticity coefficients are used to describe the control properties of a local enzyme reaction. The double modulation method is one of the first experimental approaches to estimate elasticity coefficients from measurements of steady-state flux rates and metabolite concentrations. We propose a generalized form of the double modulation method and compare it to the recently developed linear-logarithmic approach.     

In vivo modular control analysis of energy metabolism in contracting skeletal muscle

We used (31)P MRS (magnetic resonance spectroscopy) measurements of energetic intermediates [ATP, P(i) and PCr (phosphocreatine)] in combination with the analytical tools of metabolic control analysis to study in vivo energy metabolism in the contracting skeletal muscle of anaesthetized rats over a broad range of workload. According to our recent MoCA (modular control analysis) used to describe regulatory mechanisms in beating heart, we defined the energetic system of muscle contraction as two modules (PCr-Producer and PCr-Consumer) connected by the energetic intermediates. Hypoxia and electrical stimulation were used in this in vivo study as reasonably selective modulations of Producer and Consumer respectively. As quantified by elasticity coefficients, the sensitivities of each module to PCr determine the control of steady-state contractile activity and metabolite concentrations. The magnitude of the elasticity of the producer was high (4.3+/-0.6) at low workloads and decreased 5-fold (to 0.9+/-0.2) at high workloads. By contrast, the elasticity of the consumer remained low (0.5-1.2) over the range of metabolic rates studied. The control exerted by each module over contraction was calculated from these elasticities. The control of contraction was found on the consumer at low workloads and then swung to the producer, due to the workload-dependent decrease in the elasticity of producer. The workload-dependent elasticity and control pattern of energy production in muscle is a major difference from heart. Since module rate and elasticity depend on the concentrations of substrates and products, the absence of homoeostasis of the energetic intermediates in muscle, by contrast with heart, is probably the origin of the workload-dependent elasticity of the producer module.     

Metabolic flux control analysis of branch points: an improved approach to obtain flux control coefficients from large perturbation data

An overview of published approaches for the metabolic flux control analysis of branch points revealed that often not all fundamental constraints on the flux control coefficients have been taken into account. This has led to contradictory statements in literature on the minimum number of large perturbation experiments required to estimate the complete set of flux control coefficients C(J) for a metabolic branch point. An improved calculation procedure, based on approximate Lin-log reaction kinetics, is proposed, providing explicit analytical solutions of steady state fluxes and metabolite concentrations as a function of large changes in enzyme levels. The obtained solutions allow direct calculation of elasticity ratios from experimental data and subsequently all C(J)-values from the unique relation between elasticity ratio's and flux control coefficients. This procedure ensures that the obtained C(J)-values satisfy all fundamental constraints. From these it follows that for a three enzyme branch point only one characterised or two uncharacterised large flux perturbations are sufficient to obtain all C(J)- values. The improved calculation procedure is illustrated with four experimental cases.     

Matrix method for determining steps most rate-limiting to metabolic fluxes in biotechnological processes

The metabolic control theory developed by Kacser, Burns, Heinrich, and Rapoport is briefly outlined, extended, and transformed so as optimally to address some biotechnological questions. The extensions include (i) a new theorem that relates the control of metabolite concentrations by enzyme activities to flux ratios at branches in metabolic pathways; (ii) a new theorem that does the same for the control of the distribution of the flux over two branches; (iii) a method that expresses these controls into properties (the so-called elasticity coefficients) of the enzymes in the pathway; and (iv) a theorem that relates the effects of changes in metabolite concentrations on reaction rates to the effects of changes in enzyme properties on the same rates. Matrix equations relating the flux control and concentration control coefficients to the elasticity coefficients of enzymes in simple linear and branched pathways incorporating feedback are given, together with their general solutions and a numerical example. These equations allow one to develop rigorous criteria by which to decide the optimal strategy for the improvement of a microbial process. We show how this could be used in deciding which property of which enzyme should be changed in order to obtain the maximal concentration of a metabolite or the maximal metabolic flux.     

Kinetic properties required for sustained or paradoxical control of metabolic fluxes under large changes in enzyme activities

The effect that an increase in the activity of an enzyme has on its flux normally decreases with activity increase. To achieve a large increase in flux by manipulating a single step would therefore require a high initial effect that maintains or increases when the activity is increased, what has been called sustained or paradoxical control. Using metabolic control analysis for large responses, we derive conditions for sustained or paradoxical control in terms of elasticity coefficients. These are used to characterise types of rate laws contributing to this behaviour. The result that simple pathways, with normal kinetics, subject to large activity changes can lead to paradoxical control behaviour suggests that this type of pattern may be much more ubiquitous than could have, in principle, been suspected.     

CONTROL: software for the analysis of the control of metabolic networks

The program CONTROL is based on metabolic control theory anduses the method developed by Reder (1988). In this theory, twosets of parameters are defined in the vicinity of a steady-state:the elasticity coefficients which describe the local behaviourof the isolated enzymes, and the control coefficients whichexpress the response of the whole metabolic network to perturbationsat a given step. The theory shows that relationships exist betweenthe control coefficients (summation relationships or structuralrelationships) and also between the two types of coefficients(control and elasticity coefficients: connectivity relationships).The program CONTROL is divided into two parts (sub-menus). Thefirst one calculates all the control coefficients (flux andconcentrations) of a metabolic network from the elasticity coefficients.Using the second menu, the symbolic relationships are obtainedbetween the control coefficients (summation relationships) andbetween the control coefficients and the elasticity coefficients(connectivity relationships). These two sub-menus can be appliedindependently to any metabolic network (to date limited to 19steps and 19 metabolites).     

Theory of metabolism regulation: a complete system of equations for regulation coefficients

Basic quantitative parameters of control in a metabolic system are considered: control coefficients of enzymes with respect to metabolic fluxes and concentrations, and in the case when there are conservation laws, the response coefficients of metabolic fluxes and concentrations to changes in the conserved sums of metabolite concentrations (e. g. conserved moieties). Relationships are obtained which generalize the well known connectivity relations for the case of metabolites binding by conservation laws. Additional relationships are obtained which complement the set of connectivity relations up to the complete system of equations for determining all the control coefficients. The control coefficients are expressed through the enzyme elasticity coefficients, steady state metabolic fluxes and concentrations. Formulas are derived which express response coefficients of flux and concentrations through the enzyme control and elasticity coefficients and metabolite concentrations.     

A new theorem in the theory of metabolism regulation: how external parameters control the flow and concentration of metabolites

A simple theorem is derived concerning "global" and "local" coefficients of metabolic network. The coefficients of the flux and concentration response to changes in the external parameters (temperature, ionic strength, pH, external effectors) are considered. These coefficients are expressed through the sum of the control coefficients multiplied by the elasticity coefficients for all the enzymes of the metabolic pathway.     

Use of the tac promoter and lacIq for the controlled expression of Zymomonas mobilis fermentative genes in Escherichia coli and Zymomonas mobilis.

The Zymomonas mobilis genes encoding alcohol dehydrogenase I (adhA), alcohol dehydrogenase II (adhB), and pyruvate decarboxylase (pdc) were overexpressed in Escherichia coli and Z. mobilis by using a broad-host-range vector containing the tac promoter and the lacIq repressor gene. Maximal IPTG (isopropyl-beta-D-thiogalactopyranoside) induction of these plasmid-borne genes in Z. mobilis resulted in a 35-fold increase in alcohol dehydrogenase I activity, a 16.7-fold increase in alcohol dehydrogenase II activity, and a 6.3-fold increase in pyruvate decarboxylase activity. Small changes in the activities of these enzymes did not affect glycolytic flux in cells which are at maximal metabolic activity, indicating that flux under these conditions is controlled at some other point in metabolism. Expression of adhA, adhB, or pdc at high specific activities (above 8 IU/mg of cell protein) resulted in a decrease in glycolytic flux (negative flux control coefficients), which was most pronounced for pyruvate decarboxylase. Growth rate and flux are imperfectly coupled in this organism. Neither a twofold increase in flux nor a 50% decline from maximal flux caused any immediate change in growth rate. Thus, the rates of biosynthesis and growth in this organism are not limited by energy generation in rich medium.     

Determining and understanding the control of flux. An illustration in submitochondrial particles of how to validate schemes of metabolic control.

Two complementary methods were used to determine how the rate of respiration and that of ATP hydrolysis were controlled in rat liver submitochondrial particles. In the first, 'direct control analysis' method, respiration was titrated with malonate, antimycin or cyanide at 20, 30 and 37 degrees C, to determine the flux control exerted by succinate dehydrogenase, cytochrome bc1 complex and cytochrome c oxidase, respectively. Together, the three respiratory complexes only controlled the flux by about 50%, leaving the other 50% of flux control to the H+ leak. In the second, 'elasticity based' method, the elasticity coefficients of the respiratory chain or the H+-ATPase and the H+ leak towards the H+ gradient were determined. Then, the flux control coefficients were calculated using the connectivity and summation laws of metabolic control theory. The correspondence between the flux control coefficients determined in the two ways validated the two methods. This allowed us to use the second method to analyse what was the kinetic origin of the observed distribution of control. Control of ATP hydrolysis by the ATPase decreased with increasing ATPase activity; hence, the control exerted by the H+ leak increased with increasing ATPase activity, due to a diminishing elasticity towards the H+ gradient. Reverse electron transport was mainly controlled by the ATPase; the sum of flux control coefficients of succinate dehydrogenase, NADH-CoQ oxidoreductase, and H+-ATPase yielded a value greater than one, indicating that the H+ leak exerted a significant negative control on this pathway.     

Control theory of regulatory cascades.

We have extended Metabolic Control Theory to include cascades consisting of several modules controlling each other solely via regulatory effects. We derive several theorems that determine how the control properties of a cascade derive from (1) the control properties of each module, taken in isolation and (2) the regulatory interactions between the modules. Two cases are treated explicitly. The first concerns cascades in the absence of feed-back: in this case the internal control behaviour of each module is unaffected by external regulatory interactions. The second includes one feed-back loop and gives a quantitative expression of how feed-back modifies control properties: the internal control matrix within one module can be calculated as if the elasticity matrix of this module was the sum of its intrinsic elasticity matrix and a cyclic regulation matrix. More complex cascades can be analysed recursively by subdividing them into simpler modules, which can be treated individually. The theoretical framework developed here should facilitate quantitative experimental analysis of the control of cell physiology where the latter involves regulatory cascades.     

Control analysis of time-dependent metabolic systems

Metabolic Control Analysis is extended to time dependent systems. It is assumed that the time derivative of the metabolite concentrations can be written as a linear combination of rate laws, each one of first order with respect to the corresponding enzyme concentration. The definitions of the control and elasticity coefficients are extended, and a new type of coefficient ("time coefficient", "T") is defined. First, we prove that simultaneous changes in all enzyme concentrations by the same arbitrary factor, is equivalent to a change in the time scale. When infinitesimal changes are considered, these arguments lead to the derivation of general summation theorems that link control and time coefficients. The comparison of two systems with identical rates, that only differ in one metabolite concentration, leads to a method for the construction of general connectivity theorems, that relate control and elasticity coefficients. A mathematical proof in matrix form, of the summation and connectivity relationships, for time dependent systems is given. Those relationships allow one to express the control coefficients in terms of the elasticity and time coefficients for the case of unbranched pathway.     

A 'top-down' approach to the determination of control coefficients in metabolic control theory

A new approach to the determination of flux and concentration control coefficients in metabolic pathways is outlined. Linear pathways are conceptually divided in two around an intermediate metabolite (or group or metabolites) and the control coefficients of the two parts are derived from the elasticity coefficients of the two parts to the intermediate. Branched pathways are treated similarly, the control coefficients of the branches being derived either from the elasticities of the branches to their common intermediate or from the relative flux changes of the branches. Repeating this analysis around other intermediates in the pathway allows the control coefficients of smaller and smaller groups of enzymes to be determined. In complex systems this approach to describing control may have several advantages over determining the control coefficients of individual enzymes and is a potentially useful complementary approach.     

Systems analysis of the tricarboxylic acid cycle in Dictyostelium discoideum. II. Control analysis.

A steady-state computer model of the tricarboxylic cycle in Dictyostelium discoideum was analyzed using metabolic control theory. The steady state had variations of less than 0.04% over the last half of the simulation for both metabolite concentrations and fluxes. Metabolite and flux control coefficients were determined by varying enzymatic activities within 2% of their initial values and simulating the responses of metabolite concentrations and fluxes to these changes. Under these conditions, summation properties were met for most metabolite and all flux control coefficients. Maximum flux control coefficients were found for succinate dehydrogenase (0.35), malic enzyme (0.24), and malate dehydrogenase (-0.18). Comparable control was found for the reaction supplying pyruvate (0.14) and for the sum of the input amino acids (0.43), which serve as an energy source for D. discoideum. The time-dependent processes by which a new steady state was established were examined after increasing malic enzyme or malate dehydrogenase activities. This provided a method for an analysis of the mechanisms by which the observed control coefficients were generated. In addition, the effects of increasing the stimuli within 5-20% of the original enzyme activity were examined. Under these conditions, more typical of experimental stimuli and measurable responses, the metabolic model failed to return to steady state, and thus summation properties were not met. Whether "true" steady states ever occur or whether metabolic control theory can be applied in vivo is discussed.     

Total cell protein concentration as an evolutionary constraint on the metabolic control distribution in cells.

Cell protein occupies 15-35% of cell volume. This level is argued to be the maximum compatible with cell function. Because of this constraint, selection pressure during evolution is likely to have maximized pathway fluxes for minimum total protein level. Pathways optimized in this way are shown to have the following characteristics: (1) the "simple" flux control coefficients of all enzymes are equal, (2) the normal flux control coefficients depend on the relative kinetic constants of the enzymes, such that enzymes with low specific activity are present at relatively high levels and have high flux control, (3) the normal flux control coefficients are proportional to enzyme levels. A single rate limiting step located at the first step in a pathway is likely to be inefficient in terms of protein levels, and the major metabolic pathways are therefore expected to have control distributed throughout the pathway. This has important implications for metabolic control.     

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.