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.

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.

A model combining cell physiology and population genetics to explain Escherichia coli laboratory evolution.

BACKGROUND: Laboratory experiments under controlled conditions during thousands of generations are useful tools to assess the processes underlying bacterial evolution. As a result of these experiments, the way in which the traits change in time is obtained. Under these conditions, the bacteria E. coli shows a parallel increase in cell volume and fitness. RESULTS: To explain this pattern it is required to consider organismic and population contributions. For this purpose we incorporate relevant information concerning bacterial structure, composition and transformations in a minimal modular model. In the short time scale, the model reproduces the physiological responses of the traits to changes in nutrient concentration. The decay of unused catabolic functions, found experimentally, is introduced in the model using simple population genetics. The resulting curves representing the evolution of volume and fitness in time are in good agreement with those obtained experimentally. CONCLUSIONS: This study draws attention on physiology when studying evolution. Moreover, minimal modular models appear to be an adequate strategy to unite these barely related disciplines of biology.

Design of large metabolic responses. Constraints and sensitivity analysis.

Metabolic control analysis (Kacser & Burns (1973). Symp. Soc. Exp. Biol.27, 65-104; Heinrich & Rapoport (1974). Eur. J. Biochem.42, 89-95) has been extensively used to describe the response of metabolic concentrations and fluxes to small (infinitesimal) changes in enzyme concentrations and effectors. Similarly, metabolic control design (Acerenza (1993). J. theor. Biol.165, 63-85) has been proposed to design small metabolic responses. These approaches have the limitation that they were not devised to deal with large (non-infinitesimal) responses. Here we develop a strategy to design large changes in the metabolic variables. The only assumption made is that, for all the parameter values under consideration, the system has a unique stable steady state. The procedure renders the kinetic parameters of the rate equations that when embedded in the metabolic network produce the pattern of large changes in the steady-state variables that we aim to design. Structural and kinetic constraints impose restrictions on the type of responses that could be designed. We show that these conditions can be transformed into the language of mean-sensitivity coefficients and, as a consequence, a sensitivity analysis of large metabolic responses can be performed after the system has been designed. The mean-sensitivity coefficients fulfil conservation and summation relationships that in the limit reduce to the well-known theorems for infinitesimal changes. Finally, it is shown that the same procedure that was used to design metabolic responses and analyse their sensitivity properties can also be used to determine the values of kinetic parameters of the rate laws operating "in situ".

Optimal metabolic control design.

In a previous work [Acerenza, L. (1993). Metabolic Control Design. J. theor. Biol. 165, 63-85] we devised a procedure to design metabolic systems that respond according to pre-established patterns. This procedure includes the mandatory structural and kinetic constraints that narrow the spectrum of responses. In an evolutionary context, the structural and functional features shown during the history of the system would also be conditioned by other factors. Here we incorporate to the Metabolic Control Design procedure two additional requirements that could have influenced metabolic evolution. These are constraints that result from the adaptation to the environment (represented by independent control coefficients that take fixed values) and optimization of metabolic variables at constant total enzyme concentration. To illustrate the general strategy we consider metabolic systems consisting of r reaction steps where the variables are the fluxes, internal metabolite concentrations, enzyme concentrations and control coefficients. In our conditions the number of degrees of freedom, calculated as number of variables minus number of number of relationships, is r - 1. A detailed analysis of three particular schemes, unbranched chain of two and three steps and branch point, with simple linear rate laws is given. Novel results are obtained for the optimization of the input flux of the simple branch point. In the well studied case where there are no evolutionary constraints one of the limbs of the branch point disappears. However, for particular independent control coefficients, when we impose to the control coefficient a fixed value, the branched structure may or may not persist depending on the range to which the fixed value belongs.

Generalization of the double-modulation method for in situ determination of elasticities.

The double-modulation method [Kacser and Burns (1979) Biochem. Soc. Trans. 7, 1149-1160] was the first method proposed for determining elasticities in situ. It is based on measuring changes in steady-state metabolite concentrations and fluxes induced by parameter modulations. It has the important advantage that it is not necessary to know the values of the changes in the parameters. Here we develop a matrix formulation of the double-modulation method that allows it to be applied to metabolic systems of any structure and size. It also shows which parameters need to be modulated and which variables need to be measured in order to calculate the elasticities that correspond to particular rates. Some suggestions for the practical implementation of the method are given, including various ways of testing the reliability of the results.

Cooperativity: a unified view.

Cooperativity, the departure from hyperbolic behaviour of the fractional saturation of a receptor at equilibrium (Y) for different values of ligand concentration (L), is an essential property of many physiological mechanisms and a first clue to the existence of conformational transitions and allosteric interactions. Here we investigate the properties of a simple and sensitive procedure to test and quantify cooperative behaviour. The measure of cooperativity involved is kappa = dK(L)/dL where K(L) = (1- Y) L/Y= [free sites]L/[occupied sites] is called the 'global dissociation quotient' Cooperative behaviour appears when kappa is not equal to 0, i.e., K(L) is a function of L. We have shown, for several equilibrium models of cooperative behaviour (e.g., Monod-Wyman-Changeux and Koshland-Némethy-Filmer), that K(L) can be expressed as the weighted average of the microscopic dissociation constants (K(i)) where the weights are the corresponding fractions of occupied sites (X(i)), K(L)= sigmaK(i)X(i). As a consequence, the change in the global dissociation quotient with ligand concentration for a dimer is kappa = (K1 - K2)dX1/dL. This result shows that the quantitative importance of a cooperative behaviour in a dimer depends on two factors: (i) the difference of the microscopic dissociation constants of the sites and (ii) the change in the fraction of occupied sites with ligand concentration. We analyze the generality of this unified view concluding that it would be fulfilled by every equilibrium model where there is a one-to-one relationship between free and occupied sites.

How constrained is metabolic control?

The response of metabolic variables to small rate changes is quantitatively described by the control coefficients (Kacser & Burns, 1973; Heinrich & Rapoport, 1974). Owing to the existence of structural and kinetic constraints in metabolism, the control coefficients are not all independent. Two quantities are defined to evaluate how constrained metabolic control is: the fraction of independent control coefficients (fw) and the number of independent control coeffcients per independent variable [N(C/V)]. It is shown that fw can be expressed in terms of the fraction of independent metabolite concentrations, the fraction of independent fluxes and the average fraction of independent metabolites affecting each rate. N(C/V) is equal to the average number of metabolite concentrations affecting each rate. The estimation of these quantities using experimental information available leads to the following conclusions concerning cellular metabolism: (i) only a small fraction of the control coefficients are independent; (ii) the number of rates (in average) that independently controls each independent variable is much smaller than its theoretical maximum; and (iii) the kinetic constraints are the main cause of the low value showed by fw and N(C/V). Finally, some arguments are given that could explain why living organisms do not evolve to less constrained metabolic responses.

Sensitivity constraints in a chemical/biochemical highly responsive system.

The sensitivity properties of a reaction scheme that can show highly sensitive responses is studied (see e.g. Okamoto, M., Sakai, T. and Hayashi, K., 1987, Biosystems 21, 1-11). This model was previously proposed to represent a 'chemical diode', a 'chemical McCulloch Pitts neuron', the cycling of a cofactor or the interconversion of a covalently modifiable enzyme. The sensitivity of the steady-state flux and concentrations with respect to changes in a rate is quantified by the control coefficient (CC). Two types of constraints reduce the sensitivity patterns that the model can exhibit: the structural and kinetic constraints. The existence of these constraints substantially reduces the number of CC that can show arbitrary values. For instance, under extreme kinetic constraints, the value of one CC suffices to determine the values of the other thirty nine. The dependent CC are obtained as a function of the independent CC in two particular cases: the chemical case, governed by simple mass action rate laws, and the biochemical case, catalyzed by saturable enzymes. It is shown that the biochemical case exhibits a qualitatively richer repertoire of sensitivity patterns than the chemical case. Although the strategy developed in this work is restricted to a particular model its application is general. The usefulness of this type of analysis in the solution of problems ranging from design of chemical/biochemical devices to evolution of metabolism is discussed.

Metabolic control design.

The response of metabolic variables to small parameter changes is quantitatively described by the control coefficients. Their values (the control profile) depend on the rate equations of the enzymes which compose the metabolic system. A procedure, called Metabolic Control Design (MCD), is developed which achieves the "inverse", namely to calculate the kinetic properties of the enzymes which would produce a preconceived control profile when they are embedded in the network. It is shown that the lack of interaction between some variable metabolites and enzymes (i.e. some epsilon-elasticity coefficients are zero), together with the well-known conservation and summation relationships, reduce the number of control coefficients to which arbitrary values can be assigned. A choice of the values of these coefficients constitutes the pre-established control profile. The same procedure can also be used as an in situ method to detect unknown interactions between enzymes and metabolites. Finally, we discuss some implications of the results to the evolution of living organisms.

A universal method for achieving increases in metabolite production.

To increase the in vivo output of a chosen metabolite requires the increase of a number of enzyme concentrations. The enzymes are identified as those leading directly from output back to the input(s). This identification will reveal a number of branch points leading to other parts of metabolism which should not be perturbed. A simple calculation gives the enzyme multipliers, factors by which the identified enzyme concentrations should be increased to generate a given increase in the output flux. The net result of this procedure will be to extract any desired increase in flux while leaving the rest of metabolism to growth and other important functions of the cell unchanged. The method is entirely general.

Enzyme kinetics and metabolic control. A method to test and quantify the effect of enzymic properties on metabolic variables.

It is usual to study the sensitivity of metabolic variables to small (infinitesimal) changes in the magnitudes of individual parameters such as an enzyme concentration. Here, the effect that a simultaneous change in all the enzyme concentrations by the same factor alpha (Co-ordinate-Control Operation, CCO) has on the variables of time-dependent metabolic systems is investigated. This factor alpha can have any arbitrary large value. First, we assume, for each enzyme measured in isolation, the validity of the steady-state approximation and the proportionality between reaction rate and enzyme concentration. Under these assumptions, any time-invariant variable may behave like a metabolite concentration, i.e. S alpha = Sr (S-type), or like a flux, i.e. J alpha = alpha Jr (J-type). The subscripts r and alpha correspond to the values of the variable before and after the CCO respectively. Similarly, time-dependent variables may behave according to S alpha (t/alpha) = Sr (t) (S-type) or to J alpha (t/alpha) = alpha J r (t) (J-type). A method is given to test these relationships in experimental systems, and to quantify deviations from the predicted behaviour. A positive test for deviations proves the violation of some of the assumptions made. However, the breakdown of the assumptions in an enzyme-catalysed reaction, studied in isolation, may or may not affect significantly the behaviour of the system when the component reaction is embedded in the metabolic network.

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 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.

Time delays in metabolic control systems.

In this work we use mathematical models with discrete and distributed time delays to analyse the stability of metabolic pathways controlled by end product. We assume the kinetics of the intermediates of the path to be unknown, and we cover the lack of information by using a time delay. We find that above a definite substrate value, there is a critical delay Tc in which a transition from stability to instability occurs. For discrete delays, we find that even if the interaction of the end product with the first (allosteric) enzyme is not cooperative, the pathway can potentially become unstable and oscillate. We then show that the existence of cooperative inhibition extends the parametric domain of instability. The introduction of distributed delays shows, when the kernels are not monotonically decreasing, that the dispersion increases the critical delay Tc. Finally, we comment on the possibility that metabolic oscillations are physiological signals useful for triggering adaptive strategies in cell behavior.

Viscoelastic models for enzymes with multiple conformational states.

In this article we represent an enzyme capable of exhibiting more than one conformational state as a viscoelastic unit embedded in a fluid medium. We show how this viscoelastic unit is thermally activated to make transitions between equilibrium states, and propose this model as a mesoscopic representation for transitions between conformational states of an enzyme. In this representation, kinetic constants for transitions between conformations are given explicitly in terms of interactions between the enzyme and the medium. Two applications of this model are discussed: mnemonic enzymes, and co-operative enzymes exhibiting half-of-the-sites reactivity.