where Si is the concentration of the metabolite i. Notice that the expression contains the normalization of reaction rate and metabolite concentration with respect to the corresponding steady-state value of Si. (See Eqn [5.9] below for a relaxation of this condition.)

ConcControlMatrix^Ç, N, Returns a matrix where the element v, p, SteadyState 0 steadystate] mj is the concentration control coefficient of metabolite i with respect to reaction j.

Calculating concentration control coefficients.

Q: Consider the reaction scheme shown in the previous question/answer. Calculate the concentration control coefficient for M in this scheme.

A: We begin the solution by specifying S, N, and v in the same form as they were used above, and then we implement the function ConcControlMatrix from the MetabolicControlAnalysis package (which we have already loaded).

ConcControlMatrix[S, N, v,

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

Using the same definitions for e as above, we obtain

In a manner like flux control coefficients, there is also a summation theorem for concentration control coefficients. It is

Hence entries along each row of the matrix of concentration control coefficients add up

-'vj to 0. This is illustrated in the previous question/answer where Cf + CM = 0.

Q: Consider the reaction scheme below in Eqn [5.8] that is characterized by the system of differential equations: v[l] = ki si [t], v[2] = k2 s2 [t], and v[3] = k3 s3 [t].

Calculate the concentration control matrix for this system. A: A suitable program to solve this problem is

v[1] : = k1s1[t]; v[2] : = k2 s2 [t] ; v[3] : = k3 s3[t];

(*Parameter vector.*)

Before we can calculate the matrix of concentration control coefficients, we must calculate the metabolite concentrations at the steady state. This is achieved with the SteadyState function from the MetabolicControlAnalysis add-on package.

There is only one steady state for this reaction scheme, and the matrix of concentration control coefficients for this steady state is given by using the ConcControlMatrix function, that is also from the MetabolicControlAnalysis add-on package.

ConcControlMatrix[S, N, v, p,

SteadyStateConc -> solutionpj ] // MatrixForm

From this matrix it can be seen that the concentration of S2 depends only on reactions 1 and 2 (elements 1 an -1 in the first two columns of the first row, respectively), while the concentration of S3 depends only on reactions 1 and 3 (elements 1 and -1 in the first and third columns of the second row, respectively).

Q: Consider again the reaction scheme in Eqn [5.8]. What happens to the concentration control coefficient for S2when reaction vi becomes subject to feedback inhibition by

A: The feedback inhibition can be modelled by modifying the equation for v1 by including an inhibition term that is akin to the competitive inhibition factor in a Michaelis-Menten equation (Section 2.2.2). Hence we suppose that the inhibition of v1 is described by the equation v[1] = k1 s1[t]/(1+s3[t]/Ki,1). Then we redefine the rate vector and parameter vector from the previous question/answer as follows:

The steady state for this new metabolic system is obtained by using the function SteadyState from the MetabolicControlAnalysis add-on package.

{{s2[t] ^-0.358945, s3 [t] 0.717891}, {s2[t] ^ 0.208945, s3[t] ^ 0.417891}}

Of the two pairs of lists in solution only one is physically meaningful, namely, the one having non-negative values of the steady state concentrations of S2 and S3. These realistic concentrations are next used via the replacement rule, SteadyStateConc-> solution^], in the function ConcControlMatrix that calculates the matrix of concentration control coefficients.

ConcControlMatrix[S, N, v, p,

SteadyStateConc -> solution^] ] // MatrixForm

The main conclusion that can be drawn from this matrix is that the reaction v3 that consumes the metabolite S3 that inhibits reaction 1 now shares some of the control over the concentration of S2. Specifically, the element in the third column of the first row is now non-zero, being 0.367932.

The functions FluxControlMatrix and ConcControlMatrix also have the option of returning non-normalized control coefficients. Non-normalized flux and concentration control coefficients are defined by v : = Table[v[j] , {j, 3}] ;

I^Vi ajj /dPk and ^ = d S, / a Pi d vk / d pk v d vk / d pk

respectively, and can be calculated from the functions by including the option Normalized ^ False.

Q: Calculate non-normalized control coefficients for the reaction scheme analyzed in the previous question/answer.

A: For this operation we apply the ConcControlMatrix function as previously but this time we add the option Normalized ^ False, as follows:

One notable feature of this normalized matrix is that the elements in each row sum to 0; this is expected according to the corresponding summation theorem (Eqn [5.7]).

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