Systems biology may work when we learn to understand the parts in terms of the whole.

The first point to note about whether systems biology will work is that the essential idea of systems biology is not new: there has been interest in it, as well as efforts to apply it, since the middle of the 20th century. The difference now is that it has become fashionable, with an explosion in the number of publications using the two words, albeit not always with the same meaning. The reductionist approach remains dominant, however, and systems biology is often seen as no more than integration of diverse data into models of systems. This way of thinking needs to be changed if systems biology is to lead to an understanding of life and to provide the benefits that are expected from it. The emphasis ought to be on the needs of the system as a whole for understanding the components, not the converse. General properties of metabolic systems, such as feed-back inhibition, can be properly understood by taking account of supply and demand, i.e. the requirements of the system as a whole, but this is often overlooked. Metabolism tends to be viewed as static, although enzymes (and proteins in general) are continuously synthesized and degraded. The fact that they are themselves therefore metabolites introduces great complexity to metabolism, including an implication of infinite regress; understanding how living organisms escape from this will be an essential step towards understanding life.

The systems biology markup language (SBML): a medium for representation and exchange of biochemical network models.

MOTIVATION: Molecular biotechnology now makes it possible to build elaborate systems models, but the systems biology community needs information standards if models are to be shared, evaluated and developed cooperatively. RESULTS: We summarize the Systems Biology Markup Language (SBML) Level 1, a free, open, XML-based format for representing biochemical reaction networks. SBML is a software-independent language for describing models common to research in many areas of computational biology, including cell signaling pathways, metabolic pathways, gene regulation, and others. AVAILABILITY: The specification of SBML Level 1 is freely available from http://www.sbml.org/

Information transfer in metabolic pathways. Effects of irreversible steps in computer models.

Various metabolic models have been studied by computer simulation in an effort to understand why allowing for the reversibility of the reaction catalysed by pyruvate kinase, normally considered as irreversible for all practical purposes, significantly altered the behaviour of the model of glycolysis in Trypanosoma brucei [Eisenthal, R. & Cornish-Bowden, A. (1998) J. Biol. Chem. 273, 5500-5505]. Studies of several much simpler models indicate that the enzymes catalysing early steps in a pathway must receive information about the concentrations of the metabolites at the end of the pathway if a model is to be able to reach a steady state; treating all internal steps as reversible is just one way of ensuring this. Feedback inhibition provides a much better way, and as long as feedback loops are present in a model it makes almost no difference to the behaviour whether the intermediate steps with large equilibrium constants are treated as irreversible. In the absence of feedback loops, ordinary product inhibition of all the enzymes in the chain can also transfer information; this is efficient for regulating fluxes but very inefficient for regulating intermediate concentrations. More complicated patterns of regulation, such as activation of a competing branch or forcing flux through a parallel route, can also serve to some degree as ways of passing information around an irreversible step. However, they normally do so less efficiently than inhibition, because the extent to which an enzyme or a pathway can be activated always has an upper limit (which may be below what is required), whereas most enzymes are inhibited completely at saturating concentrations of inhibitor.

Detection of errors of interpretation in experiments in enzyme kinetics.

Although modern statistical computing will often be the method of choice for analyzing kinetic data, graphic methods provide an important supplement that ought not to be neglected. Residual plots, or plots of differences between observed and calculated values against variables not expected to be correlated with these differences, permit a rapid judgment of whether data have been correctly interpreted and analyzed. The rapid increase in the frequency with which artificially modified or mutated enzymes are studied is making it less and less safe to assume that enzymes are stable under assay conditions, and there is thus an increased need for methods to check for enzyme stability, and a method for doing this is briefly described. Finally, the Scatchard plot (together with the Eadie-Hofstee plot) is used as an example to discuss the dangers of publishing derived information unaccompanied by any primary data.

Relationships between inhibition constants, inhibitor concentrations for 50% inhibition and types of inhibition: new ways of analysing data.

The concentration of an inhibitor that decreases the rate of an enzyme-catalysed reaction by 50%, symbolized i(0.5), is often used in pharmacological studies to characterize inhibitors. It can be estimated from the common inhibition plots used in biochemistry by means of the fact that the extrapolated inhibitor concentration at which the rate becomes infinite is equal to -i(0.5). This method is, in principle, more accurate than comparing the rates at various different inhibitor concentrations, and inferring the value of i(0.5) by interpolation. Its reciprocal, 1/i(0.5), is linearly dependent on v(0)/V, the uninhibited rate divided by the limiting rate, and the extrapolated value of v(0)/V at which 1/i(0.5) is zero allows the type of inhibition to be characterized: this value is 1 if the inhibition is strictly competitive; greater than 1 if the inhibition is mixed with a predominantly competitive component; infinite (i.e. 1/i(0.5) does not vary with v(0)/V) if the inhibition is pure non-competitive (i.e. mixed with competitive and uncompetitive components equal); negative if the inhibition is mixed with a predominantly uncompetitive component; and zero if it is strictly uncompetitive. The type of analysis proposed has been tested experimentally by examining inhibition of lactate dehydrogenase by oxalate (an uncompetitive inhibitor with respect to pyruvate) and oxamate (a competitive inhibitor with respect to pyruvate), and of cytosolic malate dehydrogenase by hydroxymalonate (a mixed inhibitor with respect to oxaloacetate). In all cases there is excellent agreement between theory and experiment.

Regulating the cellular economy of supply and demand.

Cellular metabolism is a molecular economy that is functionally organised into supply and demand blocks linked by metabolic products and cofactor cycles. Supply-demand analysis allows the behaviour, control and regulation of metabolism as a whole to be understood quantitatively in terms of the elasticities of supply and demand, which are experimentally measurable properties of the individual blocks. The kinetic and thermodynamic aspects of regulation are clearly distinguished. One important result is the demonstration that when flux is controlled by one block, the other block determines to which degree the concentration of the linking metabolite is homeostatically maintained.

Advantages and disadvantages of aggregating fluxes into synthetic and degradative fluxes when modelling metabolic pathways.

It is now widely accepted that mathematical models are needed to predict the behaviour of complex metabolic networks in the cell, in order to have a rational basis for planning metabolic engineering with biotechnological or therapeutical purposes. The great complexity of metabolic networks makes it crucial to simplify them for analysis, but without violating key principles of stoichiometry or thermodynamics. We show here, however, that models for branched complex systems are sometimes obtained that violate the stoichiometry of fluxes at branch points and as a result give unrealistic metabolite concentrations at the steady state. This problem is especially important when models are constructed with the S-system form of biochemical systems theory. However, the same violation of stoichiometry can occur in metabolic control analysis if control coefficients are assumed to be constant when trying to predict the effects of large changes. We derive the appropriate matrix equations to analyse this type of problem systematically and to assess its extent in any given model.

Prospects for antiparasitic drugs. The case of Trypanosoma brucei, the causative agent of African sleeping sickness.

Glycolysis in the bloodstream form of Trypanosoma brucei provides a convenient context for studying the prospects for using enzyme inhibitors as antiparasitic drugs. As the recently developed model of this system (Bakker, B. M., Michels, P. A. M., Opperdoes, F. R., and Westerhoff, H. V. (1997) J. Biol. Chem. 272, 3207-3215) contains 20 enzyme-catalyzed reactions or transport steps, there are apparently numerous potential targets for drugs. However, as most flux control resides in the glucose-transport step, this is the only step for which inhibition can be expected to produce large effects on flux, and in the computer model such effects prove to be surprisingly small (although larger than those obtained by inhibiting any other step). It follows that there is little prospect of killing trypanosomes by depressing their glycolysis to a level incapable of sustaining life. The alternative is to use inhibition to increase the concentration of a metabolite sufficiently to interfere with the viability of the organism. For this purpose, only uncompetitive inhibition of pyruvate export proves effective in the model; in all other cases studied, the effects on metabolite concentrations are little more than trivial. This observation can be explained by the fact that nearly all of the metabolite concentrations in the system are held within relatively narrow ranges by stoichiometric constraints.

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.

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.

Kinetics of membrane-bound nitrate reductase A from Escherichia coli with analogues of physiological electron donors--different reaction sites for menadiol and duroquinol.

We have compared the steady-state kinetics of wild-type nitrate reductase A and two mutant forms with altered beta subunits. To mimic conditions in vivo as closely as possible, we used analogues of the physiological quinols as electron donors and membranes with overexpressed nitrate reductase A in preference to a purified alpha beta gamma complex. With the wild-type enzyme both menadiol and duroquinol supply their electrons for the reduction of nitrate at rates that depend on the square of the quinol concentration, menadiol having the higher catalytic constant. The results as a whole are consistent with a substituted-enzyme mechanism for the reduction of nitrate by the quinols. Kinetic experiments suggest that duroquinol and menadiol deliver their electrons at different sites on nitrate reductase, with cross-inhibition. Menadiol inhibits the duroquinol reaction strongly, suggesting that menaquinol may be the preferred substrate in vivo. To examine whether electron transfer from menadiol and duroquinol for nitrate reduction requires the presence of all of the Fe-S centres, we have studied the steady-state kinetics of mutants with beta subunits that lack an Fe-S centre. The loss of the highest-potential Fe-S centre results in an enzyme without menadiol activity, but retaining duroquinol activity; the kinetic parameters are within a factor of two of those of the wild-type enzyme, indicating that this centre is not required for the duroquinol activity. The loss of a low-potential Fe-S centre affects the activity with both quinols: the enzyme is still active but the catalytic constants for both quinols are decreased by about 75%, indicating that this centre is important but not essential for the activity. The existence of a specific site of reaction on nitrate reductase for each quinol, together with the differences in the effects on the two quinols produced by the loss of the Fe-S centre of +80 mV, suggests that the pathways for transfer of electrons from duroquinol and menadiol are not identical.

Extending double modulation: combinatorial rules for identifying the modulations necessary for determining elasticities in metabolic pathways.

The double modulation method for determining the elasticities of pathway enzymes, originally devised by Kacser & Burns (Biochem. Soc. Trans. 7, 1149-1160, 1979), is extended to pathways of complex topological structure, including branching and feedback loops. An explicit system of linear equations for the unknown elasticities is derived. The constraints imposed on this linear system imply that modulations of more than one enzyme are not necessarily independent. Simple combinatorial rules are described for identifying without using any algebra the set of independent modulations that allow the determination of the elasticities of any enzyme. By repeated application, the minimum numbers of modulations required to determine the elasticities of all enzymes of a given pathway can be determined. The procedure is illustrated with numerous examples.

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.