## Response Coefficients

The final type of coefficients of major importance in MCA are the response coefficients. These characterize the effect of an infinitesimally small change in a parameter value in the system on concentrations or fluxes in the system of reactions. Thus, the concentration response coefficient is defined by

S pk d Si and the flux response coefficient is defined by

 ConcResponseMatrixf Returns a matrix where the S, N, v, {paramterlist}, p, element mik is the concentration SteadyState 0 steadystate] response coefficient of the concentration of metabolite i with respect to the kth parameter of parameter list. FluxResponseMatrix[ Returns a matrix where the element mjk is the flux S, N, v, {paramterlist}, p, response coefficient of SteadyState 0 steadystate] the flux through reaction j with respect to the kth parameter of parameter list.

Calculating response coefficients.

### Calculating response coefficients.

Q: (1) For the modified reaction scheme of Eqn [5.8] (S1 ^ S2 ^ S3 ^ ; v = k1 s1[t] /(1+s3 [t] /Ky), v = k2 s2[t], and v = k3 s3[t]) calculate the concentration response matrix for all parameters. (2) What would be the expected response in [S3] if the parameter k2 were changed?

A: (1) This problem is solved by defining the metabolite vector, S, the stoichiometry matrix, N , the three rate expressions that are placed in the vector, v , the parameter vector, p, and the vector of corresponding parameter values, pv. Then a function-call to ConcResponseMatrix, from the MetabolicControlAnalysis add-on package, yields the solution.

kiSi[t]

Ki,1

p = {k1 ,k2 ,k3 , Ki,1, S1 [t]} , pv = {1, 2, 1, 0.3, 1}; p = TranSpoSe[{p, pv}] ;

SteadyStateConc -> solution^] ] // MatrixForm j0.632068 -1. 0.367932 0.367932 0.632068) J0.632068 0. -0.632068 0.367932 0.632068)

The matrix represents the concentration response coefficients with those pertaining to S2 (in the column order specified in the vector p) in the first row and to S3 in the second row. While not intending to discuss all 10 coefficients, it is useful to note that the concentration response coefficient of S3 with respect to v = k2 s2 [t] is 0 (second row and second column of the matrix). This outcome is as expected simply on the grounds that the expression for v has no term in S3 and hence the derivative of v with respect to S3 is 0. Or, physically, the reaction characterized by v2 does not have any mechanistic involvement of S3 .

(2) Hence the answer to the second part of the question is that the entry in row 2 column 2 gives the response coefficient for S3 with respect to the parameter k2. Since this entry is 0, a perturbation of k2 would be expected to have no effect on [S3] under all conditions of reactant concentrations.

From MCA and the definition of the control coefficients it follows that response coefficients can be written in terms of control coefficients and p-elasticities. This outcome is as follows:

Thus by definition the key MCA functions in the MetabolicControlAnalysis add-on package have the following interrelationship:

ConcResponseMatrix = ConcControlMatrix.PiElasticityMatrix and

FluxResponseMatrix = FluxControlMatrix.PiElasticityMatrix.

Q: The matrix of concentration response coefficients given in the previous question/answer can also be calculated by using the following relationship between the key matrices:

ConcResponseMatrix = ConcControlMatrix • PiElasticityMat-rix.

A: The Mathematica implementation of this analysis for the reaction scheme used in the previous question/answer is

ConcControlMatrix[S, N, v, p, SteadyStateConc-> solution^] ] • PiElasticityMatrix[S, N, v, p, p, SteadyStateConc -> solutionpj ] // MatrixForm

Verify this.

ConcControlMatrix[S, N, v, p, SteadyStateConc-> solution^] ] • PiElasticityMatrix[S, N, v, p, p, SteadyStateConc -> solutionpj ] // MatrixForm

### This gives the same result as previously obtained.

From Eqns [5.19] and [5.20] it is seen that the total response from the perturbation of a parameter is the sum of the individual responses from each reaction. These individual responses cSi. ppk or cVi pvL are defined as the partial response coefficients.

 [email protected] Returns a matrix of partial S, N, v, n, {parameter list}, concentration response coefficients for a p, SteadyState 0 steadystate] metabolite at position n in S. Each entry mjk in the matrix gives the product of the concentration control coefficient with respect to reaction J and the p-elasticity coefficient with respect to reaction J and parameter k. PartialFluxResponsef Returns a matrix of S, N, v, n, {parameter list}, partial flux response coefficients for a p, SteadyState 0 steadystate] flux at position n in v. Each entry mjk in the matrix gives the product of the flux control coefficient with respect to reaction J and the p-elasticity coefficient with respect to reactionJ and parameter k.

Calculating partial response coefficients.

Calculating partial response coefficients.

Q: For the reaction scheme given in Eqn [5.8] calculate the matrix of partial concentration response coefficients for S2.

A: This is achieved with the function PartialConcResponse from the MetabolicControlAnalysis add-on package.

PartialConcResponse[S, N, v, 1, p, p, SteadyStateConc-> solution^] ] //MatrixForm

0.632068 0. 0. 0.367932 0.632068

The interpretation of this matrix is as follows. Each column corresponds to the consecutive members of the parameter set {k1, k2, k3, Kj,1, s1 [t]} and each row to the reactions v1, v2, and v3. Each column in the matrix describes how the total response to a parameter perturbation is partitioned amongst the reactions of the system. Hence the sum of entries in each column will add up to the total response coefficient.

For example, from the concentration response matrix calculated in the previous question/answer, the response coefficient of [S2] to k1 is 0.632. From column 1 in the partial concentration response matrix for [S2] calculated above, it is seen that this response is entirely due to reaction 1. Although this result is hardly surprising, this type of analysis can be very useful in more complicated metabolic networks (see Chapter 8).