A general formalism for Metabolic Control Analysis

-A general formalism for Metabolic Control Analysis is derived using general sensitivity analysis and structural information of the metabolic pathway inherent in the stoichiometry matrix. The equations derived provide a general procedure for calculating the control coefficients from the elasticity coefficients using matrix algebra, and is valid for any pathway stoichiometry. The procedure diminishes the risk of deriving erroneous relations and is, due to its generality, well suited for computer handling. The formalism is mathematically stringent and is a complement to the original theorems of Metabolic Control Analysis, which were derived using ad hoc reasoning. © 1997 Elsevier Science Ltd I N T R O D U C T I O N Metabolic Control Analysis (MCA) was developed independently in the early seventies by Kacser and Burns (1973) and by Heinrich and Rapoport (1974) [for a review see Fell (1992)] and is now a well established and important framework for quantifying how the control of steady-state fluxes and concentrations is distributed between the different reactions in a metabolic pathway. In MCA, three types of coefficients are defined; flux control coefficients, concentration control coefficients, and elasticity coefficients. The flux control coefficients are defined as C] j d J j e i dlnJ j (1) de/ Jj dlnel where Jj is the steady-state flux through enzyme E~ and ei is the concentration of enzyme E/. The value of a flux control coefficient is a measure of how a change in the concentration of enzyme E/affects the steady-state flux through enzyme E j, i.e. the degree of control exerted by enzyme E/on this steady-state flux. The concentration control coefficients are similarly defined as CX j _ d x j e i _ d In xj (2) dei x j dln e/ where xj is the steady-state concentration of metabolite Xj. The value of the concentration control coefficient is a measure of the degree of control *Corresponding author. exerted by enzyme Ei on the steady-state concentration xj. The third type of coefficients, elasticity coefficients, are defined as i (~/)i xj _ ~ In vi ~J = t?x i vi t3 In xj (3) where v/is the reaction rate of enzyme Ei and xj is the concentration of metabolite Xj. An elasticity coefficient is a measure of how the reaction rate vi will respond to variations in the concentration xj when all other concentrations are kept constant. Both types of control coefficients are global properties in the sense that they reflect the state of the whole pathway. Making a change in a parameter of the pathway (e.g. a K,, value, an enzyme concentration, etc.) will affect all control coefficients of the pathway. An elasticity coefficient, on the other hand, is a local property (hence the partial derivative). It only reflects the state of a single enzyme and its effectors. An enzyme has as many non-zero elasticity coefficients as there are metabolites affecting it (substrate, product, inhibitor, etc.). In the early papers of MCA (Kacser and Burns, 1973; Heinrich and Rapoport, 1974) two types of relationships were derived, called summation theorems and connectivity theorems. The flux summation theorem states that the flux control coefficients of a metabolic pathway always add up to unity and the concentration summation theorem states that the concentration control coefficients add up to zero. The flux summation theorem indicates that the term 'ratelimiting step' implies that the flux control coefficient for this step would have a value of 1, while all other flux control coefficients would be 0. This is a very