Covalent modification and metabolic control analysis. Modification to the theorems and their application to metabolic systems containing covalently modifiable enzymes

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.     

Calculation errors of time-varying flux control coefficients obtained from elasticity coefficients by means of summation and connectivity theorems in metabolic control analysis

This paper investigates the accuracy of a matrix method proposed by other researchers to calculate time-varying flux control coefficients (dynamic FCCs) from elasticity coefficients by means of summation and connectivity theorems in the framework of metabolic control analysis. A mathematical model for the fed-batch penicillin V fermentation process is used as a case example for discussion. Calculated results reveal that this method produces significant calculation errors because the theorems are essentially valid only in steady state, although it may provide rough time-transient behaviors of FCCs. Strictly, therefore, dynamic FCCs should be directly calculated from the differential equations for metabolite concentrations and sensitivities.     

A generalization of metabolic control analysis to conditions of no proportionality between activity and concentration of enzymes

The assumption currently considered in the Metabolic Control Theory that velocity of every isolated step is proportional to enzyme concentration, is analysed by considering that in some metabolic systems that condition could not be accomplished. Analysis of the main core of this theory is carried out removing this hypothesis, expressed as "epsilon ei vi is not necessarily equal to one". The results obtained supply a more general formulation of the main theorems of the Control Theory, extending its possible application to more complex metabolic systems.     

Metabolic control and its analysis. Additional relationships between elasticities and control coefficients

Existing theorems from the analysis of metabolic control have been taken and embedded in a simple matrix algebra procedure for calculating the flux control coefficients of enzymes (formerly known as sensitivities) in a metabolic pathway from their kinetic properties (their elasticities). New theorems governing the flux control coefficients of branched pathways and substrate cycles have been derived to allow the procedure to be applied to complex pathway configurations. Modifications to the elasticity terms used in the equations have been theoretically justified so that the method remains valid for pathways with conserved metabolites (for example, the adenine nucleotide pool or the intermediates of a catalytic cycle such as the tricarboxylic acid cycle) or with pools of metabolites kept very near to equilibrium by very rapid reactions. The matrix equations generated using these theorems and relationships may be solved algebraically or numerically. Algebraic solutions have been used to determine the factors responsible for the degree of amplification of flux control coefficients by substrate cycles and to show that it is possible to derive expressions for the elasticities of a group of enzymes.     

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.     

Dynamic simulation and metabolic re-design of a branched pathway using linlog kinetics

This paper presents a new mathematical framework for modeling of in vivo dynamics and for metabolic re-design: the linlog approach. This approach is an extension of metabolic control analysis (MCA), valid for large changes of enzyme and metabolite levels. Furthermore, the presented framework combines MCA with kinetic modeling, thereby also combining the merits of both approaches. The linlog framework includes general expressions giving the steady-state fluxes and metabolite concentrations as a function of enzyme levels and extracellular concentrations, and a metabolic design equation that allows direct calculation of required enzyme levels for a desired steady state when control and response coefficients are available. Expressions giving control coefficients as a function of the enzyme levels are also derived. The validity of the linlog approximation in metabolic modeling is demonstrated by application of linlog kinetics to a branched pathway with moiety conservation, reversible reactions and allosteric interactions. Results show that the linlog approximation is able to describe the non-linear dynamics of this pathway very well for concentration changes up to a factor 20. Also the metabolic design equation was tested successfully.     

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.     

Metabolic control analysis. The effects of high enzyme concentrations

Differing views have been given in the literature as to whether the presence in a pathway of an enzyme at a concentration comparable to that of its substrate affects the values of control coefficients and the theorems of metabolic control analysis. Here we argue in favour of one of those views: that there is no effect unless the enzyme sequesters a substrate that contains a conserved moiety. In this particular case, we derive both a general criterion for estimating whether such an effect will be of a significant magnitude, and equations for determining the changes in the flux control coefficients. The nature of the phenomenom and the application of the equations are illustrated with a numerical simulation.     

Summation theorems for flux and concentration control coefficients of dynamic systems

Metabolic control analysis (MCA) was developed to quantify how system variables are affected by parameter variations in a system. In addition, MCA can express the global properties of a system in terms of the individual catalytic steps, using connectivity and summation theorems to link the control coefficients to the elasticity coefficients. MCA was originally developed for steady-state analysis and not all summation theorems have been derived for dynamic systems. A method to determine time-dependent flux and concentration control coefficients for dynamic systems by expressing the time domain as a function of percentage progression through any arbitrary fixed interval of time is reported. Time-dependent flux and concentration control coefficients of dynamic systems, provided that they are evaluated in this novel way, obey the same summation theorems as steady-state flux and concentration control coefficients, respectively.     

Control, regulation and thermodynamics of free-energy transduction

The quantitative formalism called Metabolic Control Theory makes it possible to be precise in discussions of metabolic control. To illustrate this, I will mention 2 experimental systems where free energy is converted from one form to another, i.e., bacteriorhodopsin liposomes and mitochondrial oxidative phosphorylation. More specifically I shall discuss how the distribution of the control of fluxes, concentrations and potentials, among the various enzymes (catalysts) in these systems has been measured and how this distribution can be understood in terms of the enzyme properties. From the outset, Metabolic Control Theory was valid for branched metabolic pathways with non-linear kinetics. Yet, it seemed to be limited to metabolic pathways without enzyme-enzyme interactions and to steady states. It is now clear that these limitations were apparent only and recent extensions to Metabolic Control Theory deal explicitly with enzyme-enzyme interaction and with transient-time analysis. Other limitations are inherent. For instance, Metabolic Control Theory pays for its clarity and exactness by being limited to small modulations. Mosaic Non Equilibrium Thermodynamics and Biochemical System Analysis are formalisms that deal with larger changes, at the cost of accuracy and exactness.     

Modular metabolic control analysis of large responses

Deciphering the laws that govern metabolic responses of complex systems is essential to understand physiological functioning, pathological conditions and the outcome of experimental manipulations of intact cells. To this aim, a theoretical and experimental sensitivity analysis, called modular metabolic control analysis (MMCA), was proposed. This field was previously developed under the assumptions of infinitesimal changes and/or proportionality between parameters and rates, which are usually not fulfilled in vivo. Here we develop a general MMCA for two modules, not relying on those assumptions. Control coefficients and elasticity coefficients for large changes are defined. These are subject to constraints: summation and response theorems, and relationships that allow calculating control from elasticity coefficients. We show how to determine the coefficients from top-down experiments, measuring the rates of the isolated modules as a function of the linking intermediate (there is no need to change parameters inside the modules). The novel formalism is applied to data of two experimental studies from the literature. In one of these, 40% increase in the activity of the supply module results in less than 4% increase in flux, while infinitesimal MMCA predicts more than 30% increase in flux. In addition, it is not possible to increase the flux by manipulating the activity of demand. The impossibility of increasing the flux by changing the activity of a single module is due to an abrupt decrease of the control of the modules when their corresponding activities are increased. In these cases, the infinitesimal approach can give highly erroneous predictions.     

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.     

Temperature effects on a whole metabolic reaction cannot be inferred from its components

Changes in temperature affect the kinetic energy of the constituents of a system at the molecular level and have pervasive effects on the physiology of the whole organism. A mechanistic link between these levels of organization has been assumed and made explicit through the use of values of organismal Q10 to infer control of metabolic rate. To be valid this postulate requires linearity and independence of the isolated reaction steps, assumptions not accepted by all. We address this controversy by applying dynamic systems theory and metabolic control analysis to a metabolic pathway model. It is shown that temperature effects on isolated steps cannot rigorously be extrapolated to higher levels of organization.     

Instantaneous flux control analysis for biochemical systems.

Linear sensitivity analysis of steady-state control of enzymic systems has been extended to non-steady states yielding sensitivity coefficients which provide non-intuitive insights into the behavior of the system and the sites of metabolic control, and which are quantitative counterparts to traditional qualitative concepts. Because this information is provided in a readily understood format, these coefficients serve as convenient indices of metabolic control. This treatment was applied to a simple test system, consisting of two enzymes and one non-enzymatic reaction, which exhibits oscillatory behavior. The results indicate that oscillations in the concentrations of the intermediate metabolites are regulated almost exclusively by the second enzyme. Control of the flux through the pathway is apportioned equally among the three reactions during periods of low net flux, but it is due almost exclusively to the second enzyme during periods of high net flux.     

Effects of epistasis on phenotypic robustness in metabolic pathways

It is an open question whether phenomena such as phenotypic robustness to mutation evolve as adaptations or are simply an inherent property of genetic systems. As a case study, we examine this question with regard to dominance in metabolic physiology. Traditionally the conclusion that has been derived from Metabolic Control Analysis has been that dominance is an inevitable property of multi-enzyme systems and hence does not require an evolutionary explanation. This view is based on a mathematical result commonly referred to as the flux summation theorem. However it is shown here that for mutations involving finite changes (of any magnitude) in enzyme concentration, the flux summation theorem can only hold in a very restricted set of conditions. Using both analytical and simulation results we show that for finite changes, the summation theorem is only valid in cases where the relationship between genotype and phenotype is linear and devoid of non-linearities in the form of epistasis. Such an absence of epistasis is unlikely in metabolic systems. As an example, we show that epistasis can arise in scenarios where we assume generic non-linearities such as those caused by enzyme saturation. In such cases dominance levels can be modified by mutations that affect saturation levels. The implication is that dominance is not a necessary property of metabolic systems and that it can be subject to evolutionary modification.     

Use of implicit methods from general sensitivity theory to develop a systematic approach to metabolic control. I. Unbranched pathways

It is shown that metabolic control theory (MCT), is its present form, is a particular case of general sensitivity theory, which studies the effects of parameter variations on the behavior of dynamic systems. It has been shown that metabolic control theory is obtained from this more general theory for the particular case of steady-state and linear relationships between velocities and enzyme concentrations. In such conditions the relationships between elasticities and flux control coefficients are easily obtained. These relationships are in the form of a matrix product constructed in a priori form. Relationships between combined response coefficients and concentration control coefficients are presented. The use of implicit methodology from general sensitivity theory provides a generalization of MCT, which is applied to unbranched pathways. For this particular case, provided the matrices have been properly constructed, the matrix of global properties (flux and concentration control coefficients) can be obtained by inversion of the matrix of local properties (elasticities). The theorems of MCT (concentration summation, flux summation, flux connectivity, and concentration connectivity) applicable for unbranched pathways are directly obtained by inspection of the matrix product. With these results, the present theoretical basis of MCT is extended with a more structured framework that allows a wider range of application. The results make clearer the relatedness of MCT to the more general approach provided by biochemical systems theory (BST).