Sensitivity analysis

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Consider the model of p53 and Mdm2 proteins regulation described by Geva-Zatorsky et al. [1]. The model includes the negative feedback loop in which p53 transcriptionally activates an Mdm2 precursor (pMdm2) representing, for example Mdm2 mRNA, and produces subsequent synthesis of Mdm2. Active Mdm2 increases the degradation rate of p53. The list of the model reactions is done in the table below, where x, y and y0 denote concentrations of p53, Mdm2 and pMdm2 respectively.

The model of p53 and Mdm2 interactions
    
ID Reactions Reaction rates
r1 → p53 βx
r2 -p53 → pMdm2 x
r3 pMdm2 → Mdm2 α0 · y0
r4 p53 -Mdm2 → αxy · x · y
r5 Mdm2 → αy · y

Assume βx = 0.3, βy = 0.4, α0 = αy = 0.1 and αxy = 3.2, and solve the algebraic system:

Sa formula 1.png

As a result we obtain the following steady state of the model:

Sa formula 2.png

Exploring the sensitivity of this state to perturbations of parameters βx, βy, α0, αxy and αy, we can find the matrix of partial derivatives:

Sa formula 3.png

Substituting the values of the investigated parameters, we get:

Sa formula 4.png

References

  1. Geva-Zatorsky N., Rosenfeld N., Itzkovitz S., Milo R., Sigal A., Dekel E., Yarnitzky T., Liron Y., Polak P., Lahav G., Alon U. Oscillations and variability in the p53 system. Molecular Systems Biology. 2006. V. 2, № 2006.0033.
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