Metabolic control analysis example

The method description could be found in the section Metabolic Control Analysis. Here we give an example of the method application and using in BioUML.

The theory of metabolic control determines sensitivity of steady metabolic fluxes (reaction rates) v^ss and steady state C^ss of an ODE model under perturbation of its parameters K [1]. Unlike the Sensitivity analysis, this method does not require the computation of values C^ss(p + Δp), p ∈ K, that can accelerate the model research. Instead, to analyze matrixes Ɗ_KC^ss and Ɗ_Kv^ss of partial derivatives of C^ss and v^ss with respect to parameters K, the method considers a matrix Ɗ_Kv|_{C(t) = C^ss} of partial derivatives of a vector v(t) with respect to K in a steady state of the model, and searches for control matrices Γ and Λ, such that

In accordance with [1], these matrices can be calculated by formulas

where N_R is a matrix consisting of linearly independent rows of the model stoichiometric matrix N of n by m, L is a transition matrix such that: N = L · N_R, and Ɗ_C^ssv is a matrix of elasticity coefficients

.

Consider the model of p53 and Mdm2 proteins regulation described in the example for Sensitivity analysis. For this model, we get:

Next, we find the matrixes N and Ɗ_C^ssv:

Since N_R = N and L is the identity matrix of 3 by 3, we can calculate the matrix product N · Ɗ_C^ssv:

Applying the cofactor method to the resulting matrix, we deduce:

Thus, we have:

Take into account the formulas for calculation of the model steady state derived in the Sensitivity analysis example:

Using these formulas, we can convert matrix Γ to the form:

Multiplication of the calculated matrices Γ and Ɗ_Kv|_{C(t) = C^ss} gives the same matrix Ɗ_KC^ss, as in the example for Sensitivity analysis:

References

1. Reder C. Metabolic control theory: a structural approach. Journal of Theoretical Biology. 1988. V. 135. № 2. p. 175-201.