## Calculation of Control Coefficients by Numerical Perturbation

The functions presented in the previous section, in the MetabolicControl-Analysis add-on package, allow control coefficients to be conveniently and rapidly calculated. The algorithms used in these functions are given in Appendix 2 and are based on the matrix methods described by Heinrich and Schuster.(5) However, for most realistic metabolic systems, application of these methods can lead to significant numerical errors. (For a discussion of this point see Section 8.4.) Hence it is useful to have another more generally applicable method for estimating control coefficients.

This alternative method of calculating control coefficients is to use numerical perturbation. This involves replacing the partial differentials in the above equations (Eqns [5.1], [5.5], [5.9]) by finite differences. For example, we can numerically approximate the differential, dJj /dpk, in Eqn [5.1] by making a small (say, 0.1%) change in the value or the parameter and then resimulating a time course to determine system fluxes. d Jj / d pk will then be approximated by the difference in the flux values before and after perturbation divided by 0.001. This technique is illustrated in the following example.

Q: For the reaction scheme given in Eqn [5.8], calculate the value of CV by numerical perturbation.

ConcControlMatrix[S, N, v, p, SteadyStateConc-> solution^] , Normalized-> False] //MatrixForm

0.316034 -0.5 0.183966

### 0.316034 -0.5 0.183966

A: A parameter that we can perturb in order to affect v1 is k1. Hence we must set up two parameter tables, one containing the original parameter values and the other exactly the same except that the value of k1 is increased by 0.1% of its original value. This is done as follows:

pv1 = {1, 2, 1, 0.3, 1}; (*Vector of parameter values.*) pv2 = {1 * 1.001, 2, 1, 0.3, 1}; (*k multiplied by 1.001.*)

p1 = Transpose[{p, pv1}]; (*Parameter table.*) p2 = Transpose [{p, pv2}] ;

(*Second parameter table with k increased by 0.1%*)

The effect of the perturbation of the parameter k1 on the steady-state concentration of S2 can now be determined by evaluating steady states using the original and then the perturbed parameter tables. This is done with the function SteadyState in the MetabolicControlAnalysis add-on package.

solution1 = SteadyState[S, N, v, p1] ; solution2 = SteadyState[S, N, v, p2] ;

s2/1 = s2 [t] /. solution1p2j ; s2,2 = s2 [t] /. solution2p2j ;

concControlCoeff = - x-

s2/1 0.001

0.631921

The value of 0.631921 of the concentration control coefficient compares well with the value of 0.632068 calculated previously by the matrix method.

An important point to note when applying the numerical perturbation method for calculating control coefficients is that for many enzymic reactions the following relationship holds:

ek O vk

Vk O ek and hence Eqns [5.1] and [5.5] reduce to j. ek dJj S ek dSi cVi = j net ^ Cv = S Ok- • [5 l1i respectively. Thus, for many reactions control coefficients can be estimated simply by perturbing the total enzyme concentration. For an example of this see Section 8.4.