Metabolic Control Analysis

Preliminaries

Metabolic control analysis is the study of how sensitive the system is to pertubrations in parameters and how those perturbations progagate through the network. Two kinds of sensitivity are defined, system and local. The local sensitivities are described by the elasticities. These are defined as follows:

\varepsilon^v_S = \frac{\partial v}{\partial S} \frac{S}{v} = \frac{\partial \ln v}{\partial \ln S}

Given a reaction rate v_i, the elasticity describes how a given effector of the reaction step affects the reaction rate. Because the definition is in terms of partial deriviatives, any effector that is perturbed assumes that all other potential effectors are unchanged.

The system sensitivities are described by the control and response coefficients. These come in two forms, flux and concentration. The flux control coefficients measures how senstive a given flux is to a perturbation in the local rate of a reaction step. Often the local rate is perturbed by changing the enzyme concentration at the step. In this siutation the flux control coefficicent with respect to enzyme E_i is defined as follow:

C^J_{E_i} = \frac{dJ}{dE_i} \frac{E_1}{J} = \frac{d\ln J}{d\ln E_i}

Likewise the concentration control coefficient is defined by:

C^S_{E_i} = \frac{dS}{d E_i} \frac{E_1}{S} = \frac{d\ln S}{d\ln E_i}

where S is a given species. The response coefficients measure the sensitivity of a flux or species concentration to a perturbation in some external effector. These are defined by:

R^J_{X} = \frac{dJ}{dX} \frac{X}{J} = \frac{d\ln J}{d\ln X}

R^S_{X} = \frac{dJ}{dX} \frac{X}{S} = \frac{d\ln S}{d\ln X}

where X is the external effector.

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