The matrix method of metabolic control analysis: its validity for complex pathway structures

Control analysis in the identification of key enzymes driving metabolic adaptations: Towards drug target discovery

Metabolic Control Analysis (MCA) marked a turning point in understanding the design principles of metabolic network control by establishing control coefficients as a means to quantify the degree of control that an enzyme exerts on flux or metabolite concentrations. MCA has demonstrated that control of metabolic pathways is distributed among many enzymes rather than depending on a single rate-limiting step. MCA also proved that this distribution depends not only on the stoichiometric structure of the network but also on other kinetic determinants, such as the degree of saturation of the enzyme active site, the distance to thermodynamic equilibrium, and metabolite feedback regulatory loops. Consequently, predicting the alterations that occur during metabolic adaptation in response to strong changes involving a redistribution in such control distribution can be challenging. Here, using the framework provided by MCA, we illustrate how control distribution in a metabolic pathway/network depends on enzyme kinetic determinants and to what extent the redistribution of control affects our predictions on candidate enzymes suitable as targets for small molecule inhibition in the drug discovery process. Our results uncover that kinetic determinants can lead to unexpected control distribution and outcomes that cannot be predicted solely from stoichiometric determinants. We also unveil that the inference of key enzyme-drivers of an observed metabolic adaptation can be dramatically improved using mean control coefficients and ruling out those enzyme activities that are associated with low control coefficients. As the use of constraint-based stoichiometric genome-scale metabolic models (GSMMs) becomes increasingly prevalent for identifying genes/enzymes that could be potential drug targets, we anticipate that incorporating kinetic determinants and ruling out enzymes with low control coefficients into GSMM workflows will facilitate more accurate predictions and reveal novel therapeutic targets.

Integrating systemic and molecular levels to infer key drivers sustaining metabolic adaptations

Metabolic adaptations to complex perturbations, like the response to pharmacological treatments in multifactorial diseases such as cancer, can be described through measurements of part of the fluxes and concentrations at the systemic level and individual transporter and enzyme activities at the molecular level. In the framework of Metabolic Control Analysis (MCA), ensembles of linear constraints can be built integrating these measurements at both systemic and molecular levels, which are expressed as relative differences or changes produced in the metabolic adaptation. Here, combining MCA with Linear Programming, an efficient computational strategy is developed to infer additional non-measured changes at the molecular level that are required to satisfy these constraints. An application of this strategy is illustrated by using a set of fluxes, concentrations, and differentially expressed genes that characterize the response to cyclin-dependent kinases 4 and 6 inhibition in colon cancer cells. Decreases and increases in transporter and enzyme individual activities required to reprogram the measured changes in fluxes and concentrations are compared with down-regulated and up-regulated metabolic genes to unveil those that are key molecular drivers of the metabolic response.

Deciphering the essential events in the reprogramming of metabolic networks subjected to complex perturbations, including the response to pharmacological treatments in multifactorial diseases like cancer, is crucial for the design of efficient therapies. Yet, tools to infer the molecular drivers sustaining such metabolic responses remain elusive for large metabolic networks. Here we develop an efficient computational strategy that integrates measured changes at systemic and molecular levels and combines metabolic control analysis with linear programming tools to infer key molecular drivers sustaining the metabolic adaptations to complex perturbations, such as an antitumoral drug therapy. The collective behavior is approximated using linear expressions where the adaptation of systemic concentrations and fluxes to a perturbation is described as a function of the molecular reprogramming of transport and enzyme activities. Starting from measured changes in fluxes and concentrations, we identify changes in the reprogramming of transporter and enzyme activities that are required to orchestrate the metabolic adaptation of colon cancer cells to a cell cycle inhibitor.

What mRNA Abundances Can Tell us about Metabolism

Inferring decreased or increased metabolic functions from transcript profiles is at first sight a bold and speculative attempt because of the functional layers in between: proteins, enzymatic activities, and reaction fluxes. However, the growing interest in this field can easily be explained by two facts: the high quality of genome-scale metabolic network reconstructions and the highly developed technology to obtain genome-covering RNA profiles. Here, an overview of important algorithmic approaches is given by means of criteria by which published procedures can be classified. The frontiers of the methods are sketched and critical voices are being heard. Finally, an outlook for the prospects of the field is given.

High efficiency of energy flux controls within mitochondrial interactosome in cardiac intracellular energetic units

The aim of our study was to analyze a distribution of metabolic flux controls of all mitochondrial complexes of ATP-Synthasome and mitochondrial creatine kinase (MtCK) in situ in permeabilized cardiac cells. For this we used their specific inhibitors to measure flux control coefficients (C(vi)(JATP)) in two different systems: A) direct stimulation of respiration by ADP and B) activation of respiration by coupled MtCK reaction in the presence of MgATP and creatine. In isolated mitochondria the C(vi)(JATP) were for system A: Complex I - 0.19, Complex III - 0.06, Complex IV 0.18, adenine nucleotide translocase (ANT) - 0.11, ATP synthase - 0.01, Pi carrier - 0.20, and the sum of C(vi)(JATP) was 0.75. In the presence of 10mM creatine (system B) the C(vi)(JATP) were 0.38 for ANT and 0.80 for MtCK. In the permeabilized cardiomyocytes inhibitors had to be added in much higher final concentration, and the following values of C(vi)(JATP) were determined for condition A and B, respectively: Complex I - 0.20 and 0.64, Complex III - 0.41 and 0.40, Complex IV - 0.40 and 0.49, ANT - 0.20 and 0.92, ATP synthase - 0.065 and 0.38, Pi carrier - 0.06 and 0.06, MtCK 0.95. The sum of C(vi)(JATP) was 1.33 and 3.84, respectively. Thus, C(vi)(JATP) were specifically increased under conditions B only for steps involved in ADP turnover and for Complex I in permeabilized cardiomyocytes within Mitochondrial Interactosome, a supercomplex consisting of MtCK, ATP-Synthasome, voltage dependent anion channel associated with tubulin βII which restricts permeability of the mitochondrial outer membrane.

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.

Time-dependent sensitivity analysis of biological networks: coupled MAPK and PI3K signal transduction pathways

Sensitivity analysis has been widely used in the studies of complicated chemical reaction and biological networks, for example, in combustion studies and metabolic control analysis of pathways. In the latter cases, the responses of system properties at steady states with respect to changes of parameters, such as initial concentrations and rate constants, are often expressed as sensitivities. Besides steady-state sensitivities, time-dependent sensitivities should be useful; however, the explicit use of them in analyzing complicated biological systems has so far been limited. Using the coupled mitogen activated protein kinase (MAPK)-phophatidylinoisitol 3'-kinase (PI3K) system of the Ras pathways, known to be involved in about 30% of human cancers, as an example, we show that time-dependent sensitivities are useful for the studies of complex biological systems. They provide, for example, the following information: (a) multiple time scales existing in a complex system involving cross-talks and feedback loops; (b) the signs and strengths of responses to perturbations (as system complication increases, the signs of global responses are not always easily determined; for example, response may change sign more than once as time evolves); (c) beyond concentration dynamics, sensitivities revealing further details about the intricate dynamics and the effects of the cross-talks; (d) ranking of vulnerability of nodes of a biological network using integrated sensitivity-a first step toward the identification of drug targets; (e) reduced sensitivity serving as a measure of the stability or robustness of pathways. Our results indicate that the role of the PI3K branch in the coupled pathways is to enhance the robustness of the MAPK pathway. More importantly, they demonstrate that time-dependent sensitivity analysis can be a valuable tool in system biology.

Sensitivity and control analysis of periodically forced reaction networks using the Green's function method

A general sensitivity and control analysis of periodically forced reaction networks with respect to small perturbations in arbitrary network parameters is presented. A well-known property of sensitivity coefficients for periodic processes in dynamical systems is that the coefficients generally become unbounded as time tends to infinity. To circumvent this conceptual obstacle, a relative time or phase variable is introduced so that the periodic sensitivity coefficients can be calculated. By employing the Green's function method, the sensitivity coefficients can be defined using integral control operators that relate small perturbations in the network's parameters and forcing frequency to variations in the metabolite concentrations and reaction fluxes. The properties of such operators do not depend on a particular parameter perturbation and are described by the summation and connectivity relationships within a control-matrix operator equation. The aim of this paper is to derive such a general control-matrix operator equation for periodically forced reaction networks, including metabolic pathways. To illustrate the general method, the two limiting cases of high and low forcing frequency are considered. We also discuss a practically important case where enzyme activities and forcing frequency are modulated simultaneously. We demonstrate the developed framework by calculating the sensitivity and control coefficients for a simple two reaction pathway where enzyme activities enter reaction rates linearly and specifically.

Application of computer algebra-techniques to metabolic control analysis

For practical purposes the calculation of rate constants is not particularly valuable, since their physical significance is not clear. Of greater practical use are metabolic control coefficients and elasticities. Given the definition of the flux control coefficients C(E)(J), concentration control coefficient C(E)(X) and elasticity epsilon (X)(v(1)). We can calculate symbolic formulae for these using computer algebra-techniques. These are then functions of V(max), K(m), K(i) enzyme and concentrations. Having derived estimates of V(max), K(m), K(i) using the fitting method we can then calculate values of the control coefficients and elasticities. Furthermore we can calculate the metabolic control parameters using symbolic values for the conventional kinetic parameters. Using these we have verified the summation and connectivity theorems. This is a useful cross check on the reliability of the calculations.

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.

Automatic analysis of the control of metabolic networks

In this paper we apply computer algebra techniques to analyze the control of metabolic networks. For this purpose, a computer program based on metabolic control theory was developed. When a stoichiometry matrix of the metabolic networks is given, the program calculates all the control coefficients (flux and metabolic control coefficients, summation and connectivity relationships) using elasticity coefficients. The program can be applied to any metabolic network which includes unlimited steps and intermediate metabolites.

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 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.

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.

Modular analysis of the control of complex metabolic pathways

The understanding of the functioning of the intact cell would be simplified appreciably if it were possible first to analyze particular modules of cell physiology separately, and then to integrate the information so as to yield understanding of the control structure in terms of the mutual regulation of the modules. Here we develop a quantitative method based on Metabolic Control Analysis that makes this possible: The relevant properties of the modules are contained in "overall" elasticity coefficients, which reflect the changes in fluxes in the module upon a small variation of the environment of the module, allowing the latter to attain steady state. We show how overall control coefficients, which reflect the control exerted by the processes catalyzed by each module, can be expressed into the overall elasticity coefficients. We derive corresponding summation and connectivity theorems. A number of possible divisions of physiological systems into modules is discussed. This work is a generalization of previous analyses of overall control properties in that it allows for multiple fluxes to connect the modules, and reaction stoichiometries of any complexity.

A computer program for the algebraic determination of control coefficients in Metabolic Control Analysis

A computer program (MetaCon) is described for the evaluation of flux control, concentration control and branch-point distribution control coefficients of a metabolic pathway. Requiring only the reaction scheme as input, the program produces algebraic expressions for the control coefficients in terms of elasticity coefficients, metabolite concentrations and pathway fluxes. Any of these variables can be substituted by numeric or simple algebraic expressions; the expressions will then be automatically rearranged in terms of the remaining unknown variables. When all variables have been substituted, numeric values will be obtained for the control coefficients. The program is a computerized implementation of the matrix method for the determination of control coefficients. The features of MetaCon are compared with those of other programs available to workers in Metabolic Control Analysis. Potential benefits of, and methods of using, MetaCon are discussed. The mathematical background and validity of the matrix method rules are discussed, and the algorithm used by MetaCon is described. The matrix method is shown to be a specific case of a previously described general formalism for calculating control coefficients.

Control involving metabolism and gene expression: the square-matrix method for modular decomposition

Control of DNA supercoiling by the free-energy of hydrolysis of ATP that involves gene expression is analyzed in terms of three levels of unconnected metabolic pathways. These are synthesis and breakdown of topoisomerase mRNAs, synthesis and breakdown of topoisomerase proteins and supercoiling and relaxation of DNA. The so-called square-matrix method previously developed for the control of metabolic pathways, is extended to deal with this hierarchical control system. It turns out that also in this case, the matrix of control coefficients is equal to the inverse of the so-called elasticity matrix, which contains all relevant elasticity coefficients as well as information about the structure and connectedness of the pathways involved. For a simpler case of a hierarchy of two systems, we demonstrate that the explicit matrix inversion method may be replaced by an implicit in which the regulatory effects that run through the other level are described by an additional elasticity coefficient which may then be treated as if local.

Responses of metabolic systems to large changes in enzyme activities and effectors. 2. The linear treatment of branched pathways and metabolite concentrations. Assessment of the general non-linear case

We extend the analysis of unbranched chains (preceding paper) to large parameter changes in branched systems using linear kinetic assumptions. More complex relationships between flux control coefficients and deviation indices are established. In particular, the deviation index in such systems depends on more than one control coefficient as well as on the magnitude of the enzyme change. Non-additivity of the indices is the general rule. Combined changes of groups of enzymes, whether co-ordinate or not, have also been formulated. Control coefficients can be estimated from a small number of independent large-change experiments. Alternatively, the amplification factors can be calculated given the knowledge of the control coefficients. A 'case study' using published data is presented. The movement of intermediate metabolites as a consequence of large parameter changes can be dealt with in a similar manner. Experimental methods for showing the admissibility of assuming the simplifying assumptions used are summarised. Some simulation studies show possible limits of the application of the approach and some aspects of the general, non-linear, case are discussed. It is concluded that, although metabolic systems are in principle non-linear, many behave, in practice, as quasi-linear systems. The relationships established between deviation indices and control coefficients therefore provide a practical way of predicting the effects of large-scale changes in parameters for many metabolic systems.

Taking enzyme kinetics out of control; putting control into regulation

The matrix formulation of metabolic control analysis, which states that multiplying the elasticity matrix for any system by the corresponding control matrix yields an identity matrix, can be transformed into a statement that multiplying a matrix expressing internal regulatory properties by a matrix expressing external regulatory properties also yields an identity matrix. This transformation supplies the formal basis for metabolic regulation analysis, and provides the key to determining the control structure of a system without the need to know the exact changes in enzyme activities that are made to measure control coefficients.

Metabolic control analysis: a survey of its theoretical and experimental development

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The definitions of metabolic control analysis revisited

Various definitions of coefficients in metabolic control analysis are examined with respect to their theoretical consistency and practical applicability. We suggest agreement upon a definition for control coefficients which is clearly distinct from that for response coefficients, in such a way that the former describe inherent properties of the metabolic system while the latter refer to the influence of special parameters. Advantages and drawbacks of using normalized or non-normalized control coefficients are studied. It is shown that normalized control coefficients have the advantage of being invariant to a different rescaling of the particular fluxes. We demonstrate that some problems are easier to tackle if the consistency of time-independent control coefficients with their time-dependent counterparts is taken into account. It is shown that the matrix of flux control coefficients is an indempotent matrix. This allows an interpretation in terms of the transduction of the effect of parameter perturbations. Several aspects of the experimental measurement of control coefficients are discussed, with special reference to the different definitions.

Quantitative approaches to the analysis of the control and regulation of microbial metabolism

Recently, a number of novel ways of considering the control, regulation and thermodynamics of microbial physiology have been developed and applied. We here present an overview of the new concepts involved, of their limitations and of the most recent attempts to deal with those limitations. We conclude that there no longer exist reasons of principle for vagueness in discussions of the control of microbial physiology and energetics. Further, the novel conceptual methods serve to remove part of the discordance between holistic and reductionistic views of microbial physiology.

Quantitative assessment of regulation in metabolic systems

We show how metabolic regulation as commonly understood in biochemistry can be described in terms of metabolic control analysis. The steady-state values of the variables of metabolic systems (fluxes and concentrations) are determined by a set of parameters. Some of these parameters are concentrations that are set by the environment of the system; they can act as external regulators by communicating changes in the environment to the metabolic system. How effectively a system is regulated depends both on the degree to which the activity of the regulatory enzyme with which a regulator interacts directly can be altered by the regulator (its regulability) and on the ability of the regulatory enzyme to transmit the changes to the rest of the system (its regulatory capacity). The regulatory response of a system also depends on its internal organisation around key variable metabolites that act as internal regulators. The regulatory performance of the system can be judged in terms of how sensitivity the fluxes respond to the external stimulus and to what degree homeostasis in the concentrations of the internal regulators is maintained. We show how, on the level of both external and internal regulation, regulability can be quantified in terms of an elasticity coefficient and regulatory capacity in terms of a control coefficient. Metabolic regulation can therefore be described in terms of metabolic control analysis. The combined response relationship of control analysis relates regulability and regulatory capacity and allows quantification of the regulatory importance of the various interactions of regulators with enzymes in the system. On this basis we propose a quantitative terminology and analysis of metabolic regulation that shows what we should measure experimentally and how we should interpret the results. Analysis and numerical simulation of a simple model system serves to demonstrate our treatment.

Topological analysis of metabolic control

A topological approach is presented for the analysis of control and regulation in metabolic pathways. In this approach, the control structure of a metabolic pathway is represented by a weighted directed graph. From an inspection of the topology of the graph, the control coefficients of the enzymes are evaluated in a heuristic manner in terms of the enzyme elasticities. The major advantage of the topological approach is that it provides a visual framework for (1) calculating the control coefficients of the enzymes, (2) analyzing the cause-effect relationships of the individual enzymes, (3) assessing the relative importance of the enzymes in metabolic regulation, and (4) simplifying the structure of a given pathway, from a regulatory viewpoint. Results are obtained for (a) an unbranched pathway in the absence of feedback the feedforward regulation and (b) an unbranched pathway with feedback inhibition. Our formulation is based on the metabolic control theory of Kacser and Burns (1973) and Heinrich and Rapoport (1974).

Control analysis of systems having two steps catalyzed by the same protein molecule in unbranched chains

The analysis of the control of a metabolic pathway having an enzyme catalyzing two different reactions (or a protein displaying two different activities) has been performed. For such systems although the summation theorems are valid, the flux and concentration connectivity theorems of the metabolic control analysis are not valid. Another general relationship of control analysis is shown to be more widely obeyed and holds in these systems. An exemplary case, where the enzyme catalyzes two irreversible reactions, demonstrates that the level of one internal intermediate is constant, i.e. it does not depend upon the independent variables of the system.

Metabolic control analysis. Sensitivity of control coefficients to elasticities

This paper illustrates a method to calculate the sensitivities of control coefficients to the elasticities which determine their values and it is shown that these sensitivities are systemic properties. We show, both theoretically and with a practical example, how they can be used to investigate: (a) the relative importance of a particular elasticity in the determination of the value of a control coefficient; (b) the effect of experimental error on the values of the control coefficients and (c) the construction of confidence limits around the values of the control coefficients.

Enzyme-enzyme interactions and control analysis. 1. The case of non-additivity: monomer-oligomer associations

Two usual assumptions of the treatment of metabolism are: (a) the rates of isolated enzyme reactions are additive, i.e., that rate is proportional to enzyme concentration; (b) in a system, the rates of individual enzyme reactions are not influenced by interactions with other enzymes, i.e. that they are acting independently, except by being coupled through shared metabolites. On this basis, control analysis has established theorems and experimental methods for studying the distribution of control. These assumptions are not universally true and it is shown that the theorems can be modified to take account of such deviations. This is achieved by defining additional elasticity coefficients, designated by the symbol pi, which quantify the effects of homologous and heterologous enzyme interactions. Here we show that for the case of non-proportionality of rate with enzyme concentration, (pi ii not equal to 1), the summation theorems are given by (Formula: see text). The example of monomer-oligomer equilibria is used to illustrate non-additive behaviour and experimental methods for their study are suggested.

Control-pattern analysis of metabolic pathways. Flux and concentration control in linear pathways

Metabolic control analysis [Kacser and Burns (1973) Symp. Soc. Exp. Biol. 27, 65-104; Heinrich and Rapoport (1974) Eur. J. Biochem. 42, 89-95] leads to a description of the systemic properties of a metabolic system (expressed as control coefficients) in terms of the local kinetic properties of the individual enzyme-catalyzed reactions (expressed as elasticity coefficients). This paper describes a non-algebraic diagrammatic method which generates the mathematical expressions for flux or concentration-control coefficients in terms of elasticity coefficients. According to a set of simple rules, 'flux-control patterns' or 'concentration-control patterns' are drawn on a metabolic diagram. Each control pattern represents a product of elasticity coefficients that occurs as a term in the expression for a control coefficient. The rules also generate the correct sign that precedes each term. The control patterns are then used to build the expressions for control coefficients. The procedure was developed in such a way that each control pattern can be understood in terms of a 'chain of local effects' which shows how a perturbation in the activity of an enzyme is propagated through the metabolic pathway.

A simple matrix method for determining flux control coefficients in complex pathways

Flux control coefficients express in quantitative terms the extent to which the steady state flux through a metabolic pathway is controlled by a particular parameter. Enzyme flux control coefficients can be calculated using matrix algebra methods which express the control coefficients in terms of parameters which can be determined experimentally (enzyme elasticities, flux ratios, metabolite ratios). This paper describes an algorithm based on a 'constraint' matrix which enables expressions for enzyme control coefficients to be written for pathways of any complexity.