Figure 4. Weak chaotic dynamics for elements with two negative feedback loops.

Left panels show the results for the element of the gene A with two delayed self-inhibitions (circuit diagram in inset of (a)). Panel (a) shows bifurcation of the maximum levels of protein A with respect to the values of time delay τ₂ while fixing the time delay τ₁ = 18 minutes. The dynamics is periodic when τ₂ < 4.2 minutes or τ₂ > 4.9 minutes, and quasi-periodic or weak chaotic dynamics otherwise. Panels (b) and (c) show the A(t) – dA(t)/dt maps and the full spectra of the protein level A for the case when τ₁ = 18 minutes, τ₂ = 4.65 minutes (navy arrow in (a)). Right panels show the results for the two-gene flip-flop element with time delays (circuit diagram in inset of (d)). Panels (d–f) are similar to panels (a–c). In the bifurcation diagram in (d), the time delay τ₁ is fixed to be 5.3 minutes. The dynamics is periodic when τ₂ < 8.0 minutes or τ₂ > 9.18 minutes, and quasi-periodic or weak chaotic dynamics otherwise. Panels (e) and (f) are for the case when τ₁ = 5.3 minutes, τ₂ = 8.2 minutes (navy arrow in (d)). Circuits in both cases exhibit weak chaotic dynamics as illustrated by the maps (b,e), the full spectra (c,f) and the maximum-minimum spectra (see Supplementary section SI2).