Circadian Rhythmicity by Autocatalysis
Figure 2
The rate constant values are: k1 = 1 × 10−4 μM−1 h−1, k2 = 0.4 h−1, k3 = 0.45 μM−2 h−1, k4 = 3.65 μM−1 h−1, k5 = 4.0 h−1, k6 = 0.9 h−1, and k7 = 0.18 h−1.
(A) Unphosphorylated KaiC and phosphorylated KaiC (KaiC*) oscillate with the largest amplitudes. Oscillations of free unbound KaiA and KaiB concentrations are not shown, but they have amplitudes similar to the KaiBC* oscillations and peak at approximately the same phase as KaiC.
(B) Six different approaches to the limit cycle, each obeying mass balances in total KaiA, total KaiB, and total KaiC, i.e., [KaiA]tot = [KaiA] + [KaiAC] + [KaiAC*] + [KaiABC*] = 3 μM, [KaiB]tot = [KaiB] + [KaiABC*] + [KaiBC*] = 1 μM, and [KaiC]tot = [KaiC] + [KaiAC] + [KaiAC*] + [KaiABC*] + [KaiBC*] + [KaiC*] = 3.5 μM. (1) [KaiA] = 3 μM, [KaiB] = 1 μM, and [KaiC*] = 3.5 μM; (2) [KaiA] = 2.49 μM, [KaiABC*] = 0.137 μM, [KaiAC] = 0.0028 μM, [KaiAC*] = 0.369 μM, [KaiB] = 0.419 μM, [KaiBC*] = 0.444 μM, [KaiC] = 0.25 μM, and [KaiC*] = 2.3 μM; (3) [KaiA] = 2.54 μM, [KaiABC*] = 0.137 μM, [KaiAC] = 0.050 μM, [KaiAC*] = 0.269 μM, [KaiB] = 0.819 μM, [KaiBC*] = 0.444 μM, [KaiC] = 2.0 μM, and [KaiC*] = 1.0 μM; (4) [KaiA] = 2.74 μM, [KaiABC*] = 0.137 μM, [KaiAC] = 0.050 μM, [KaiAC*] = 0.069 μM, [KaiB] = 0.819 μM, [KaiBC*] = 0.444 μM, [KaiC] = 2.1 μM, and [KaiC*] = 1.1 μM; (5) [KaiA] = 2.7641 μM, [KaiABC*] = 0.0833 μM, [KaiAC] = 0.0036 μM, [KaiAC*] = 0.1490 μM, [KaiB] = 0.7280 μM, [KaiBC*] = 0.1887 μM, [KaiC] = 2.9296 μM, and [KaiC*] = 0.14557 μM; and (6) [KaiA] = 3 μM, [KaiB] = 1 μM, and [KaiC] = 3.5 μM.