A Second-order Approach to Metabolic Control Analysis

In the mathematical formalism of metabolic control analysis the changes in metabolite concentrations and fluxes have been related to the underlying parameter perturbations by a linear approximation, thus restricting the analysis to very small parameter perturbations. Obviously, the response of the system variables to larger parameter perturbations can be described more accurately if, in addition to the linear terms, the second-order terms are considered. The basic equations of this approach are derived. The second-order effects can be expressed (i) by the second-order elasticity coefficients of the reaction rates with respect to the kinetic parameters and/or metabolite concentrations, and (ii) by the control coefficients and π-elasticities of the linear theory. Parameter-independent second-order control coefficients can be defined if the reaction rates depend linearly on the perturbation parameters. These coefficients satisfy summation theorems similar to those of the linear theory. The formalism is applied to an unbranched chain of monomolecular reactions and to a skeleton model of glycolysis. The second-order approximation turns out to be more accurate for a relatively wide range of rate perturbations. However, the use of the second- and higher-order expansions for tackling practical problems seems limited since the required higher-order elasticity coefficients may be hard, if not impossible, to obtain experimentally. These difficulties, as well as the general limitations of the local approach of metabolic control analysis, may be compensated for by a combination with kinetic modelling.