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.

Conditions for effective allosteric feedforward and feedback in metabolic pathways.

Whether an allosteric feedback or feedforward modifier actually has an effect on the steady-state properties of a metabolic pathway depends not only on the allosteric modifier effect itself, but also on the control properties of the affected allosteric enzyme in the pathway of which it is part. Different modification mechanisms are analysed: mixed inhibition, allosteric inhibition and activation of the reversible Monod-Wyman-Changeux and reversible Hill models. In conclusion, it is shown that, whereas a modifier effect on substrate and product binding (specific effects) can be an effective negative feedback mechanism, it is much less effective as a positive feedforward mechanism. The prediction is that catalytic effects that change the apparent limiting velocity would be more effective in feedforward activation.

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.

Comparing the regulatory behaviour of two cooperative, reversible enzyme mechanisms.

It is shown that both the reversible Hill equation and a generalised, reversible Monod-Wyman-Changeux equation can give analogous regulatory behaviour when embedded in a model metabolic pathway.

Experimental evidence for allosteric modifier saturation as predicted by the bi-substrate Hill equation.

The cooperative enzyme reaction rates predicted by the bi-substrate Hill equation and the bi-substrate Monod-Wyman-Changeux (MWC) equation when allosterically inhibited are compared in silico. Theoretically, the Hill equation predicts that when the maximum inhibitory effect at a certain substrate condition has been reached, an increase in allosteric inhibitor concentration will have no effect on reaction rate, that is the Hill equation shows allosteric inhibitor saturation. This saturating inhibitory effect is not present in the MWC equation. Experimental in vitro data for pyruvate kinase, a bi-substrate cooperative enzyme that is allosterically inhibited, are presented. This enzyme also shows inhibitor saturation, and therefore serves as experimental evidence that the bi-substrate Hill equation predicts more realistic allosteric inhibitor behaviour than the bi-substrate MWC equation.

Evaluation of a simplified generic bi-substrate rate equation for computational systems biology.

The evaluation of a generic simplified bi-substrate enzyme kinetic equation, whose derivation is based on the assumption of equilibrium binding of substrates and products in random order, is described. This equation is much simpler than the mechanistic (ordered and ping-pong) models, in that it contains fewer parameters (that is, no K(i) values for the substrates and products). The generic equation fits data from both the ordered and the ping-pong models well over a wide range of substrate and product concentrations. In the cases where the fit is not perfect, an improved fit can be obtained by considering the rate equation for only a single set of product concentrations. Due to its relative simplicity in comparison to the mechanistic models, this equation will be useful for modelling bi-substrate reactions in computational systems biology.

Software tools that facilitate kinetic modelling with large data sets: an example using growth modelling in sugarcane.

A solution to manage cumbersome data sets associated with large modelling projects is described. A kinetic model of sucrose accumulation in sugarcane is used to predict changes in sucrose metabolism with sugarcane internode maturity. This results in large amounts of output data to be analysed. Growth is simulated by reassigning maximal activity values, specific to each internode of the sugarcane plant, to parameter attributes of a model object. From a programming perspective, only one model definition file is required for the simulation software used; however, the amount of input data increases with each extra interrnode that is modelled, and likewise the amount of output data that is generated also increases. To store, manipulate and analyse these data, the modelling was performed from within a spreadsheet. This was made possible by the scripting language Python and the modelling software PySCeS through an embedded Python interpreter available in the Gnumeric spreadsheet program.

Is there an optimal ribosome concentration for maximal protein production?

A core model is presented for protein production in Escherichia coli to address the question whether there is an optimal ribosomal concentration for non-ribosome protein production. Analysing the steady-state solution of the model over a range of mRNA concentrations, indicates that such an optimum ribosomal content exists, and that the optimum shifts to higher ribosomal contents at higher specific growth rates.

Metabolic regulation: a control analytic perspective.

A possible basis for a quantitative theory of metabolic regulation is outlined. Regulation is defined here as the alteration of reaction properties to augment or counteract the mass-action trend in a network reactions. In living systems the enzymes that catalyze these reactions are the "handles" through which such alteration is effected. It is shown how the elasticity coefficients of an enzyme-catalyzed reaction with respect to substrates and products are the sum of a mass-action term and a regulatory kinetic term; these coefficients therefore distinguish between mass-action effects and regulatory effects and are recognized as the key to quantifying regulation. As elasticity coefficients are also basic ingredients of metabolic control analysis, it is possible to relate regulation to such concepts as control, signalling, stability, and homeostasis. The need for care in the choice of relative or absolute changes when considering questions of metabolic regulation is stressed. Although the concepts are illustrated in terms of a simple coupled reaction system, they apply equally to more complex systems. When such systems are divided into reaction blocks, co-response coefficients can be used to measure the elasticities of these blocks.

Steady-state modelling of metabolic pathways: a guide for the prospective simulator.

Steady-state modelling and control analysis by means of computer simulation provides valuable insight into the behavior of metabolic pathways. This review, which is aimed at the newcomer to this field, discusses the objectives of steady-state modelling and the steady-state properties of the four basic metabolic structures, namely linear and branched chains, loops and cycles. It is shown how the model definition in terms of stoichiometric reactions and rate equations leads to a set of balance equations from which the conservation constraints and flux relationships can be deduced, either informally or through a rigorous analysis of the stoichiometric matrix. The initial analysis of a steady-state metabolic model is summarized in an algorithm. Key references to the literature on metabolic modelling are given.

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.

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.

MetaModel: a program for modelling and control analysis of metabolic pathways on the IBM PC and compatibles.

MetaModel is a user-friendly program for calculating steady-state fluxes and metabolite concentrations of metabolic systems on the IBM PC and compatible computers. For any steady state that is obtained, one can then calculate a matrix of elasticity coefficients at that steady state, or a matrix of control and response coefficients. It thus offers a simple way to calculate the control structure of a pathway: it provides not only an educational tool that allows the student to verify empirically the classic summation relationships of metabolic control analysis but also a research tool for addressing 'what if?' questions about the behaviour of metabolic systems. Results can not only be printed or stored in a file, but can also be written to a special file that can be read by popular spreadsheet programs, thereby giving access to rapid, flexible and powerful methods for subsequent analysis and plotting of these results.

Determination of control coefficients in intact metabolic systems.

The control structure of a metabolic system can in principle be determined without the need for purification of the component enzymes and study of their kinetic properties, provided that their activities can be perturbed by amounts sufficient to produce measurable changes in the steady-state variables, i.e. the fluxes through the system and the concentrations of the intermediates. Each perturbation is characterized in terms of the co-response coefficients of all pairs of variables, i.e. the slopes of the lines produced when the logarithm of one variable is plotted against the logarithm of another, both varying in response to the same perturbation. If all the co-response coefficients are assembled into a matrix, the inverse of this matrix can be transformed into a matrix containing all the component elasticities, which can be inverted to provide the complete matrix of control coefficients. In a simple three-enzyme pathway studied, the analysis proves not to require unrealistically high accuracy in the original co-response measurements: even with errors with standard deviation +/- 5.77 degrees in the angles to the horizontal of the lines in the co-response plots (equivalent at best to errors of +/- 20% in the corresponding co-response coefficients), the final control coefficient matrix may be adequate for assessing the control structure of the system. Examination of literature data from studies of mitochondrial respiration and of gluconeogenesis indicates that considerably higher precision than this is achievable.

Co-response analysis: a new experimental strategy for metabolic control analysis.

The formulation of the standard summation and connectivity relationships as a statement that the matrix of all the elasticities in a system is the inverse of the matrix of all the control coefficients is completely general, provided that only control coefficients for independent fluxes and concentrations are considered, and that the elasticity matrix is written to take account of the stoichiometry of the pathway and the implied dependences between concentrations. This generally implies that co-response analysis is also general, i.e. that all of the elasticities and all of the control coefficients in any system, regardless of branching, feedback effects, moiety conservation or other complications, can be determined by comparing the effects of perturbations of the enzyme activities on the steady-state fluxes and concentrations of the pathway. The approach requires no quantitative information about the magnitudes of the effects on the individual enzyme activities, and consequently no enzymes need to be studied in isolation from the pathway.

The reversible Hill equation: how to incorporate cooperative enzymes into metabolic models.

MOTIVATION: Realistic simulation of the kinetic properties of metabolic pathways requires rate equations to be expressed in reversible form, because substrate and product elasticities are drastically different in reversible and irreversible reactions. This presents no special problem for reactions that follow reversible Michaelis-Menten kinetics, but for enzymes showing cooperative kinetics the full reversible rate equations are extremely complicated, and anyway in virtually all cases the full equations are unknown because sufficiently complete kinetic studies have not been carried out. There is a need, therefore, for approximate reversible equations that allow convenient simulation without violating thermodynamic constraints. RESULTS: We show how the irreversible Hill equation can be generalized to a reversible form, including effects of modifiers. The proposed equation leads to behaviour virtually indistinguishable from that predicted by a kinetic form of the Adair equation, despite the fact that the latter is a far more complicated equation. By contrast, a reversible form of the Monod-Wyman-Changeux equation that has sometimes been used leads to predictions for the effects of modifiers at high substrate concentration that differ qualitatively from those given by the Adair equation.

Building the cellular puzzle: control in multi-level reaction networks.

Quantitative conceptual tools dealing with control and regulation of cellular processes have been mostly developed for and applied to the pathways of intermediary metabolism. Yet, cellular processes are organized in different levels, metabolism forming the lowest level in a cascade of processes. Well-known examples are the DNA-mRNA-enzyme-metabolism cascade and the signal transduction cascades consisting of covalent modification cycles. The reaction network that constitutes each level can be viewed as a "module" in which reactions are linked by mass transfer. Although in principle all of these cellular modules are ultimately linked by mass transfer, in practice they can often be regarded as "isolated" from each other in terms of mass transfer. Here modules can interact with each other only by means of regulatory or catalytic effects-a chemical species in one module may affect the rate of a reaction in another module by binding to an enzyme or transport system or by acting as a catalyst. This paper seeks to answer two questions about the control and regulation of such multi-level reaction networks: (i) How can the control properties of the system as a whole be expressed in terms of the control properties of individual modules and the effects between modules? (ii) How do the control properties of a module in its isolated state change when it is embedded in the whole system through its connections with the other modules? In order to answer these questions a quantitative theoretical framework is developed and applied to systems containing two, three or four fully interacting modules; it is shown how it can be extended in principle to n modules. This newly developed theory therefore makes it possible to quantitatively dissect intermodular, internal and external regulation in multi-level systems.

Controlled-release pheromone dispenser for use in traps to monitor flight activity of false codling moth.

Field trials were conducted in the western Cape Province, South Africa, to develop a sex pheromone dispenser suitable for monitoring the flight activity of false codling moth. A controlled-release dispenser capable of releasing sex pheromone at a predetermined and constant release rate without replacement for more than seven months was produced.

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.