4.2 Maximizing regime of oscillations in repressilator system
Motivated by biological applications such as increasing the robustness of rhythmic pacemakers, we consider maximizing the region of oscillations with respect to a given operating point. We utilize a case of the generalized repressilator [7] as our test system.
After non-dimensionalizing, an ODE system for the protein and RNA concentrations xi and yi is obtained, with the following dimensionless parameters: αi, βi are the ratios of degradation rates, δi represents the leakiness of the gene transcription, hi the Hill-coefficient reflecting the degree of cooperativity of gene transcription,
In this example, we take n = 3 and examine the symmetric case, i.e., αi = α, β = β, etc. Analysis shows that for ranges of parameters α and β, the system can exhibit oscillations or be in a stable, steady-state [7]. Thus, we take pi = (α, β) as the input parameters and seek values of ps = (δ, h) within the constraint set,
(10-4, 0) ≤ (δ, h) ≤ (10-1, 2)
so as to maximize the minimal distance to a Hopf bifurcation with respect to a reference point (α = 102.5, β = 100). Figure 9 shows the initial bifurcation diagram and the computed minimal distance from the reference point. The result of the inverse bifurcation analysis shows that the optimum solution is found at (δ, h) = (10-4, 2); thus, the inequality constraints are active equality constraints. Moreover, the signs of the Lagrange multipliers for the inequality constraints show that the (local) optimum distance to the Hopf bifurcation manifold increases if the upper bound hupp is increased and/or the lower bound δlow is decreased. Figure 10 shows the bifurcation diagram with the optimized parameters.
Figure 9  Initial bifurcation diagram for symmetric repressilator: δ = 10-3, h = 1.5.
Figure 10  Optimized bifurcation diagram for symmetric repressilator: δ = 10-4, h = 2. Table 2 shows the iteration counts required to obtain the result. The tolerances for optimization are the same as those of Section 4.1.
Table 2  Iteration summary
Optim. iter.  Func. eval.  One-param contin.
Symmetric repressilator  2  5  48