The requirement for net balance of synthesis, degradation and transport for all intermediates in the pathways from glucose to fat imposes constraints on the balance of fluxes between different pathways. Linear programming has been used to examine the interactions between these constraints on metabolism in adipocytes and the requirement for efficiency in the conversion of glucose into fat. The circumstances under which excessive ATP synthesis would accompany this conversion have been investigated.
This first paper in a series investigates the problem of predicting and analysing the effects of large changes in enzyme activities or external nutrients/effectors on metabolic fluxes. We introduce the concept of a deviation index, D, which gives a measure of the relative change in a metabolic variable (e.g. flux) due to a large (non-infinitesimal) relative change in a parameter (e.g. enzyme). Using simplifying kinetic assumptions we have found, for an unbranched metabolic chain, a direct relationship between deviation indices and flux control coefficients. This relationship provides a method to estimate flux control coefficients using a single large change in enzyme activity. We also provide a method of predicting the effects of, for example, DNA manipulation or other techniques for enzyme activity/concentration changes on metabolic fluxes. Up-modulations of single enzymes rarely produce significant changes in fluxes. We show that combined changes of activity of a group of enzymes will produce a more than 'additive' response. We provide a method of predicting the effects of these combined changes, given either the flux control coefficients of the group of enzymes or the effects on the flux of changing the enzymes individually. A similar analysis is carried out for large changes in external nutrients or effectors. These amplification factors, f, give experimentally accessible estimates of the expected changes in metabolic variables. We provide three 'case studies' to illustrate our results.The investigation of metabolism has had two principal aims : to understand how organisms work and to change them in desired directions. An understanding will be achieved if the physiological responses of the organism can be described in terms of the many thousand of 'unit' parts; the enzymecatalysed reactions. These responses happen as a result of naturally occurring changes in external conditions or due to the experimenter's manipulations. The desire to change organisms, including man, is testified by the long history of medicine and agriculture: to change sickness into health, to change low yields into high yields.The well-known complexity of metabolic structures along with many experimental explorations has made it evident that simple predictions of the outcome of external, or internal, manipulations are rarely possible. This is so in spite of the considerable progress in the enzymology and molecular biology of individual steps in the system. It is now accepted that a solution must come by taking a systemic view, rather than the study of isolated parts.The development of genetic manipulation technology has high-lighted the possibility of altering the metabolic properties of organisms in specific ways. In particular, the possible use of DNA manipulation in order to increase the concenCorrespondence to H. Kacser,
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.Branched pathways are a universal feature of metabolism. In metabolic control analysis, which deals with infinitesimal changes, all control coefficients are identically defined irrespective of the structure of the system. There is additionally the branch point theorem [l -31 which relates the coefficients in branched systems to one another. To analyse the effects of large changes on a branched metabolic system we will use the concepts developed in the preceding paper in which we introduced the definition of the 'deviation index' to aid in the analysis and prediction of the effects of large changes. The index was defined as: ..=(LK).", J'where AJ = J'-J", and AEj = Ei-E;, with EL = r . E .Using the assumption of linear kinetics (see Eqn 3 of preceding paper), it was possible to make a direct comparison between the deviation index of a particular enzyme change and the flux control coefficient of that enzyme. In an unbranched chain it was found that the flux control coefficient was in fact numerically equal to the deviation index for any change in enzyme activity. As a consequence of this result we were able to introduce a new method of estimating flux control coefficients or, alternatively, a method to predict the effects of large changes in single enzymes, and in groups of enzymes, on the pathway flux as well as the effects of large changes in external effectors. This has a number of important applications in the area of DNA manipulation. We must now enquire how far the simple conclusions established for an unbranched chain apply to the m...
The matrix algebra procedure for determining the flux control coefficients of enzymes in metabolic pathways has been extended to allow determination of the concentration control coefficients. Although it is shown that the procedure is essentially unchanged in most cases, the presence of moiety-conserved cycles in a pathway places additional limitations on the form of the equations that can be used in the matrix formulation for concentration control coefficients. In the case of branched pathways, a new coefficient has been defined, the branch distribution control coefficient, which can be obtained via the matrix procedure. Thus a single matrix equation permits calculation or algebraic evaluation of the control coefficients for flux, concentration and distribution of flux at branches, so that the complete response of a pathway to alteration of enzyme content, or to modulation by an effector, can be determined.The relationships have been determined between flux control coefficients in isolated sections of metabolic pathways and the coefficients for the same enzymes when part of a larger metabolic system. It is shown that the control analysis of the isolated system provides useful information towards determining the control properties of the extended system. As a contribution to this theory, we recently proposed [5] a collection of rules that related flux control coefficients to elasticities and to flux distributions in a variety of pathway structures. These rules allowed calculation of flux control coefficients, or their algebraic evaluation, by matrix algebra. We did not consider the evaluation of substrate concentration control coefficients in that paper. However, concentration control coefficients are governed by summation and connectivity relationships [2, 4, 261 that are a counterpart of the equivalent relationships between flux control Coefficients [l] on which we based our approach. More recently, Hofmeyr et al. [6] showed that the connectivity relationship for concentration control coefficients must be modified when applied to moiety conserved cycles in the same way that we had shown to be necessary for flux control coefficients [5]. Accordingly, we have examined the applicability of other aspects of our method for obtaining flux control coefficients, in particular the treatment of branch points and substrate cycles, and have found that they also apply to the concentration control coefficients in pathways with these features. Thus in this paper we demonstrate an extension of our analysis that allows the determination of the flux control coefficients and the concentration control coefficients by the solution of one matrix equation. We also show that, in branched pathways, the matrix equation contains information about the distribution of flux at the branch point. Finally, we examine the relationships between the control coefficients for a subsection of a large pathway and the control coefficients with respect to the same steps in the complete system. The significance of this is that an experimenter always is...
A study of the sensitivity properties of metabolic systems containing covalently modifiable enzymes and cascades has been carried out with the aid of metabolic control analysis. We have considered how the theorems of metabolic control analysis must be modified to take into account covalently modifiable enzymes, and have used these results to investigate the effects of increasing the total amount of modifiable enzyme. The sensitivity of system variables to an effector acting through a covalent-modification cycle has also been investigated.Reversible covalent modification is often cited as a means by which enzyme activities, and hence metabolism, can be regulated. In this paper we shall be concerned with the case where an enzyme has two forms, with different catalytic activities, and these forms are interconverted by a pair of enzymes. This type of system is known as a covalent-modification cycle. Although many forms of covalent modification have been discovered most of the current literature is concerned with reversible phosphorylation [l], and there is much evidence to suggest that regulation of many enzymes may occur by thisExperimental results have demonstrated that covalent modification may change the kinetic properties of an enzyme, but have told us little about the specific regulatory properties which covalent modification can confer on a system possessing such a feature. Due to the complexity of such a system it is difficult to see any inherent regulatory features, beyond general possibilities. Mathematical modelling and computer simulation have demonstrated some of the potential properties of a covalent-modification cycle [5 -131. One such property, that of increased sensitivity to an effector acting through a covalent-modification cycle compared to an effector acting on a single enzyme (in the sense of a large change in the activity of the target enzyme relative to the change in the effector), has been proposed by these groups. Goldbeter and Koshland [lo-131 carried out an algebraic analysis of sensitivity of a covalent-modification cycle in terms of discrete changes in effector concentrations and defined a 'response coefficient' and an 'amplification factor' to give a measure of the sensitivity. , on the other hand, used infinitesimal changes in effector concentration to study the sensitivity of the system to the effector. Both of these groups have investigated the properties which the converting enzymes must show to enable an increased sensitivity to an effector, as has the group of Stadtman and Chock [5 -91. All these investigations involved the study of covalent-modification cycles with specified reaction mechanisms, and so one of the aims of this paper is to provide a Correspondence to D . A. Fell, School of Biological and Molecular Sciences, Oxford Polytechnic, Oxford, OX3 OBP, England general, non-mechanism-dependent, approach to the analysis of covalent-modification cycles and their relationships with the complete metabolic system using the framework of metabolic control analysis [15 -181. Applic...
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