Elasticities, metabolic control analysis

To reconcile the nomenclature with Metabolic Control Analysis, we note that the kinetic orders corresponds to the (scaled) elasticities of the reaction... 

The interpretation of the elements of the matrix 0 is slightly more subtle, as they represent the derivatives of unknown functions fi(x) with respect to the variables x at the point x° = 1. Nevertheless, an interpretation of these parameters is possible and does not rely on the explicit knowledge of the detailed functional form of the rate equations. Note that the definition corresponds to the scaled elasticity coefficients of Metabolic Control Analysis, and the interpretation is reminiscent to the interpretation of the power-law coefficients of Section VII.C Each element 6% of the matrix measures the normalized degree of saturation, or likewise, the effective kinetic order, of a reaction v, with respect to a substrate Si at the metabolic state S°. Importantly, the interpretation of the elements of does again not hinge upon any specific mathematical representation of specific... 

To highlight the relationship of the matrices A and to the quantities discussed in Section VILA (Dynamics of Metabolic Systems) and Section VII.B (Metabolic Control Analysis), we briefly outline an alternative approach to the parameterization of the Jacobian matrix. Note the correspondence between the saturation parameter and the scaled elasticity ... 

Considering that f, g, x, and u are vectors, the differentiation leads to formation of matrices. The matrix A is well known in stability analysis as the jacobian matrix it quantifies the effects of all state variables on their rates of change. A matrix similar to B turns up in metabolic control analysis, as N3v/3p [48, 108], where it denotes the immediate effects of parameter perturbations on the rates of change of all variables. If the function y is scalar and denotes a rate, then C becomes a row vector c harboring unsealed elasticity coefficients and D becomes a row vector d containing so-called n-elasticities - sensitivities of the rates with respect to the parameters [109]. The linearized system is ... 

Metabolic control analysis (MCA) is a specialized theory that is concerned with particular sensitivity coefficients, elasticity coefficients and control coefficients. These coefficients tell us how a steady state of a biochemical system shifts in response to perturbation in enzyme activities or external (clamped) substrate concentration [53, 209],... 

As briefly outlined in Section 6.3, one of the theoretical frameworks in quantitative analysis of metabolic networks is metabolic control analysis. In metabolic control analysis, the enzyme elasticity coefficients provide empirical constraints between the metabolites concentrations and the reaction fluxes. These constraints can be considered in concert with the interdependencies in the J and c spaces that are imposed by the network stoichiometry. If the coefficients elk = (c / Ji)dJi/dck are known, then these values bind the fluxes and concentrations to a hyperplane in the (J, c) space. 

The metabolic control analysis determines quantitatively the effects of various metabolic pathway reactions on flows and on metabolic concentrations. The analysis defines two coefficients (i) the control coefficients, which characterize the response of the system flows, concentrations, and other variables after parameter perturbations and (ii) the elasticity coefficients, which quantify the changes of reaction rates after perturbations of substrate concentrations or kinetic parameters under specified conditions. 

Kholodenko, B.N., Sauro, H.M., Westerhoff, H.V. and Cascante, M. (1995b) Coenzyme cycles and metabolic control analysis the determination of the elasticity coefficients from the generalised connectivity theorem. Biochem. Mol. Biol. Int. 35, 615-625. 

Fell, D. A., and H. M. Sauro. 1985. Metabolic control analysis Additional relationships between elasticities and control coefficients. Eur J Biochem 148 555-61. 

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