# Metabolic control analysis

The theory of metabolic control determines sensitivity of steady metabolic fluxes (reaction rates) *v ^{ss}* and steady state

*C*of an ODE model under perturbation of its parameters

^{ss}*K*[1]. Unlike the Sensitivity analysis, this method does not require the computation of values

*C*(

^{ss}*p*+ Δ

*p*),

*p*∈

*K*, that can accelerate the model research. Instead, to analyze matrixes

*Ɗ*and

_{K}C^{ss}*Ɗ*of partial derivatives of

_{K}v^{ss}*C*and

^{ss}*v*with respect to parameters

^{ss}*K*, the method considers a matrix

*Ɗ*|

_{K}v_{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

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*, and

_{R}*Ɗ*is a matrix of elasticity coefficients

_{Css}vConsider 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 _{Css}v*:

Since *N _{R}* =

*N*and

*L*is the identity matrix of 3 by 3, we can calculate the matrix product

*N*·

*Ɗ*:

_{Css}vApplying 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 _{K}v*|

_{C(t) = Css}gives the same matrix

*Ɗ*, as in the example for Sensitivity analysis:

_{K}C^{ss}## References

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