Systems Biology (Methods in Molecular Biology)

3.5 Inverse
Bifurcation

Bifurcation analysis has proven to be a powerful method for under-

standing the qualitative behavior of gene regulatory networks

[24]. In addition to the more traditional forward problem of

determining the mapping from parameter space to the space of

model behavior, the inverse problem of determining model para-

meters to result in certain desired properties of the bifurcation

diagram provides an attractive methodology for addressing impor-

tant biological problems. For a certain range of a bifurcation

parameter, three steady states coexist, whereas outside this interval

only a single steady state exists. This implies that slowly varying

parameters can induce a sudden jump. Manipulating the Tyson’s

model, assuming the total cdc2CT¼C 2 þCP+pM+M¼const

andCP¼(1w)CTC2 andC 2 1 one rewrites Eq.17 asu¼

[M]/[CT],ν¼([Y]þ[pM]þ[M])/[CT],w¼([pM]þ[M])/

[CT], andy¼[YT]/[CT] and that the first three equations can be

solved independently of the fourth because y does not appear in the

first three equations; and becausek3[CT]max {kiaa,k2,k6},

wchanges very rapidly compared to changes inν,sow¼νas long as

0 <ν<1. Thus, the cdc2-cyclin model reduces to a pair of

nonlinear ordinary differential equations

u_½tŠ¼k 4 ðνuÞðαþu^2 Þðαþu^2 Þk 6 u

ν_½tŠ¼ðk 1 aak 6 uÞ

ð 18 Þ

whereα¼k 4 prime/k4. Applying the mathematics rules shown in

Appendix under Bifurcation, results of such model are shown in

Fig.3.

Table 3
Parameters retrieval, standard error,tstatistic andPvalue

Value Estimate Standard error t-Statistic P-value

k1aa 0.015001 0.000938491 15.9841 4.01224 10 ^48

k2 2.37146 10 ^7 0.00522666 0.0000453724 0.999964

k3 200.001 0.960852 208.149 1.049133302926 10 ^558

k4 180. 1.4801 121.613 6.100160101569 10 ^423

k4prime 0.0179989 0.000843644 21.3347 1.81877 10 ^75

k5notP 3.1582 10 ^7 0.00397082 0.0000795352 0.999937

k6 1. 0.00343817 290.853 1.911259397235 10 ^644

k7 0.6 0.00281254 213.33 5.507756176045 10 ^565

k8notP 1.00001 106 5864.59 170.517 6.067726632465 10 ^508

k9 999.999 2.45854 406.745 8.26166736095 10 ^73

Inverse Problems in Systems Biology 81