Metabolic control analysis

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The theory of metabolic control determines sensitivity of steady metabolic fluxes (reaction rates) vss and steady state Css of an ODE model under perturbation of its parameters K [1]. Unlike the Sensitivity analysis, this method does not require the computation of values Css(p + Δp), pK, that can accelerate the model research. Instead, to analyze matrixes ƊKCss and ƊKvss of partial derivatives of Css and vss with respect to parameters K, the method considers a matrix ƊKv|C(t) = Css 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

Mca formula 1.png

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

Mca formula 2.png

where NR 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 · NR, and ƊCssv is a matrix of elasticity coefficients

Mca formula 3.png.

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

Mca formula 4.png

Next, we find the matrixes N and ƊCssv:

Mca formula 5.png

Since NR = N and L is the identity matrix of 3 by 3, we can calculate the matrix product N · ƊCssv:

Mca formula 6.png

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

Mca formula 7.png

Thus, we have:

Mca formula 8.png

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

Sa formula 2.png

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

Mca formula 9.png

Multiplication of the calculated matrices Γ and ƊKv|C(t) = Css gives the same matrix ƊKCss, as in the example for Sensitivity analysis:

Mca formula 10.png


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