## Control Coefficients

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A major focus of MCA is the quantification of the role that individual enzymic reactions play in determining metabolite concentrations and pathway fluxes at a particular steady state of metabolite concentrations. Fundamental to the theory is the definition and calculation of elasticity coefficients, response coefficients, and control coefficients. These three types of coefficient are dealt with, in turn, in this and the following few sections. It is also important to note that MCA was developed primarily for the analysis of systems under steady-state conditions. Hence all the coefficients are defined for a particular reference steady state.

The description of the MCA coefficients begins with the one that, historically, was the first to be defined: for a metabolic system of i metabolites, j metabolite fluxes, and k reactions, at a particular steady state, the flux control coefficients (FCC) are defined as

where pk is a parameter that only affects the velocity of reaction k (vk), and dvkIdpk is evaluated with the parameter values and concentrations associated with the particular steady state. Note that the change in flux is normalized by dividing it by the reference value at the steady state, and the change in reaction velocity is also normalized by dividing it by the value at the steady state. Thus the FCC expresses quantitatively the effect that varying the parameter pk has on the flux through the system, Jj , if the effect of pk on the local enzyme rate, vk, is known. In other words, the FCC is a measure of the extent to which the intrinsic rate of reaction k controls the steady-state flux J.

 << Reads in the addon package MetabolicControlAnalysi'-. MetabolicControlAnalysis. FluxControlMatrix^, N, Calculates a matrix for the metabolic v, p, SteadyState 0 steadystate] system defined by the substrate vector, S, the stoichiometric matrix, N, and the parameter matrix, p, at the steady state given by the replacement rule steadystate; where the element mp is the flux control coefficient of the flux through reaction i with respect to reaction j. Note that the last two arguments are optional.

Calculating flux control coefficients.

### Calculating flux control coefficients.

Q: Consider the example from Hofmeyer and Cornish-Bowden;(7) a metabolite, M, is produced at a rate, v, by a supply (source) pathway and consumed at a rate, vd, by a demand (sink) pathway.

Assuming that both vs and vd are influenced by M, what are the FCC values for pathway flux with respect to the supply and demand pathways?

A: First we load the add-on package MetabolicControlAnalysis and define the reactant vector S, the stoichiometry matrix N, and the vector of velocity or rate expressions v, for this system.

<< MetabolicControlAnalysis*;

By assuming that in the steady state M[t] = Mss, we apply the function FluxControlMatrix to return a matrix of flux control coefficients.

FluxControlMatrix[S, N, v,

SteadyStateConc 0 {M[t] -> Mss}] //MatrixForm i ^__v [ s ] ' [M s s ] _v[ d ] [ Ms s ] v[ s ]' [Mss ]_

1 -v [ d] ' [Mss ] + v [ s ] ' [Mss ] v [s ][Mss ] ( - v [ d] ' [Mss ] + v [ s ] ' [Mss ])

__v [ s ] [ Mss ] v [ d ]' [ Ms s ]__"1 + v [ d ] ' [ Ms s ]

v [d][Mss ] ( - v [d] ' [Mss ] + v [s] ' [Mss ]) -v [d] ' [Mss ]+v [s] ' [Mss ]

From the elements in row 2 column 1, and row 1 column 2, we can write the following expressions for the flux control coefficients:

Vd eM eM eM eM

where ¿M = v [d] " [M^] / v[d][Mss] and ¿M = v [s] " [Mss] / v[s][Mss]. Note that these expressions have been simplified by using the fact that in the steady-state vs = vd. Be careful to look closely at the meanings of these two expressions: the numerator specifies the dependence of a change in reaction rate with respect to [M] when [M] is equal to the steady-state concentration, and the denominator expresses the velocity of the reaction at the steady-state concentration of M. These terms, ¿M and ¿M, are called elasticity coefficients (see Section 5.3).

An important point regarding this question/answer is that CJ + CJ = 1. This is an example of the summation theorem for flux control coefficients.H1-3,5,6L Formally, the theorem states that the sum of all flux control coefficients in a metabolic system for a particular metabolic flux, Jj, is equal to 1, i.e.,

Hence, the entries along each row of the matrix of flux control coefficients should add up to 1. This relationship is only true for normalized flux control coefficients, namely, flux control coefficients that are defined according to Eqn [5.1].

In a manner similar to the flux control coefficient, a concentration control coefficient is defined as

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