Figure 4.

Robust perfect adaptation in A with A_set = 1.0. (a) Model described in Fig. 3a with rate constants as given in the Appendix. At t = 5.0 time units, k_pert is increased from 0.0 to 1.0. (b) Initial conditions as given in the Appendix with k_pert = 1.0. At t = 5.0 time units, k_pert is increased from 1.0 to 10³ a.u. (c) Initial conditions as in panel b. At t = 5.0 time units, k_pert is decreased from 1.0 to 10⁻³ a.u. (d) Same initial conditions as in panel b, but E_tr is successively increased leading eventually to the breakdown in homeostasis indicated by the decreasing A_ss values. This breakdown can be opposed to a certain degree by increasing the values of k_pert or k_synth. In the figure, k_pert or k_synth were increased from their original values 1.0 and 3.0 to 10.0 and 12.0, respectively, thereby extending the homeostasis to larger E_tr values (dashed line). However, at higher E_tr concentrations the homeostasis fails again with decreasing A_ss values (data not shown). (e) Calculated A_ss values for varying $\log k_{\mathrm{f}}^{E_{\text{set}}}$ with k_synth = 3.0 a.u. and k_pert = 1.0 a.u. (solid circles), or with k_synth = 3.0 a.u. and k_pert = 5.0 a.u. (open circles). For $k_{\mathrm{f}}^{E_{\text{set}}}$ < 10⁹ a.u., perfect homeostasis in A is lost (indicated by the condition that A_ss < A_set), because for decreasing $k_{\mathrm{f}}^{E_{\text{set}}}$ the $K_{\mathrm{M}}^{E_{\text{set}}}=\left(k_{\text{cat}}^{E_{\text{set}}}+k_{\mathrm{r}}^{E_{\text{set}}}\right)/k_{\mathrm{f}}^{E_{\text{set}}}$ associated with the removal of E_adapt by E_set increases, which eventually leads to the loss of the zero-order kinetics in the E_adapt degradation. (f) Time profiles in A with two different $k_{\mathrm{f}}^{E_{\text{set}}}$ values. At t = 5.0 time units, k_pert is increased from 1.0 to 5.0 a.u. 1 = Perfect homeostasis in A for $k_{\mathrm{f}}^{E_{\text{set}}}$ = 10¹² a.u.; 2 = Loss of perfect homeostasis in A when $k_{\mathrm{f}}^{E_{\text{set}}}$ = 10⁶ a.u., which is due to the loss of zero-order kinetics in the degradation of E_adapt.