4.3.2 Example 2: engineer irreversibility, fixed G1/S transition
In this example, we consider an extension of the previous problem. Here, we would like to engineer an irreversible switch with the additional constraint that the x-abscissa of the G1/S transition point SN1 should be fixed at the value of the nominal system. In addition to allowing 0.1 ≤ k25 ≤ 1.5 as before, we also modify the stability of E2F1 and AP-1. That is, we relax the values of φE2F1 and φAP-1 in (7) and (8) from from their fixed nominal values 0.1 and 0.01 respectively, to allow searching within the ranges,
0.08 ≤ φE2F1 ≤ 0.12
0.008 ≤ φAP-1 ≤ 0.012.
Mathematically, the inverse problem is formulated as an nonlinear optimization problem with equality constraint on the x-abscissa of the G1/S point, SN1. The bifurcation diagram corresponding to the solution (k25 = 1.182, φE2F1 = 0.0992, φAP-1 = 0.0109) is given in Figure 13. Row 2 of Table 3 summarizes the number of iterations required.
Figure 13  Example 2: initial and optimized gene switches.