Figure 1.
A Dynamical Model of In Vitro Rhythmic KaiC Phosphorylation
See text for description.
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.
Figure 3.
Comparison of Model Oscillations with Experimental Results
Same rate constant values as in Figure 2. Initial concentrations as in curve 5, Figure 2B.
(A) Calculated oscillations of total phosphorylated KaiC (P-KaiC = [KaiAC*] + [KaiBC*] + [KaiC*]), total unphosphorylated KaiC (NP-KaiC = [KaiC] + [KaiAC]), and their antiphase behavior. Total protein is P-KaiC + NP-KaiC.
(B) Calculated oscillatory ratio of P-KaiC = P-KaiC/(P-KaiC + NP-KaiC).
(C) Experimental oscillations of P-KaiC and NP-KaiC with estimated KaiC concentrations. Data replotted from [12].
(D) Experimental oscillatory ratio of P-KaiC = P-KaiC/(P-KaiC + NP-KaiC). Data replotted from [12].
Figure 4.
Oscillatory and Bistable Regions
(A) Stable oscillations exist only for certain rate constant values as shown here for variations in k3 (0.373 μM−2 h−1 ≤ k3 ≤ 0.660 μM−2 h−1; all other rate constants as in Figure 2). The dashed line within the oscillatory region indicates the unstable steady state, which becomes stable outside the oscillatory region (solid lines).
(B) The oscillations are crucially dependent on the total concentrations of KaiA, KaiB, and KaiC. The figure shows the oscillatory regions as closed loops in the concentration space of total KaiC ([KaiC]tot) and total KaiB ([KaiB]tot) when total concentrations of KaiA ([KaiA]tot are varied from 4.0 μM to 1.5 μM (see numbers in graph). The model predicts that the region of oscillations shrinks as the total KaiA concentration is lowered.
(C) The oscillatory and bistable regions (Figure 3A and 3B) are found within a so-called “cross-shaped diagram” shown here in the k6–k3 parameter space. Such diagrams have been observed in chemical oscillatory systems [39]. In more technical terms, HB denotes a Hopf bifurcation, SNB a Saddle Node bifurcation, and TB the Takens-Bogdanov bifurcation. The solid heavy line inside the oscillatory region indicates oscillations with a period length of 24 h.
(D) The model can show bistability/hysteresis when for example k6 is reduced from 0.9 h−1 to 0.1 h−1. Steady state levels of KaiABC* are shown as a function of k3. The dashed line (0.0446 μM−2 h−1 ≤ k3 ≤ 0.297 μM−2 h−1) indicates unstable steady states and the bistable region. Solid lines show stable steady states.
Figure 5.
Parameter Sensitivities and Temperature Compensation
(A–D) Period sensitivity analyses by varying the following selected rate constants (A) k1, (B) k3, (C) k4, and (D) k7. As ki is varied, the other rate constants are kept constant at their initial values as described in Figure 2.
(E) Temperature dependence of the period. Open circles show calculations, and solid circles show experimental results replotted from [1]. Each rate constant obeys the Arrhenius equation, ki = Ai exp (−Ei/RT), where Ei is the activation energy, R is the gas constant, and T is the temperature in Kelvin. Ai is the pre-exponential factor, which is also treated as a constant. Initial values of rate constants (Figure 2) are defined at 25 °C, and the period lengths are calculated for temperatures between 25 °C and 35 °C. (1) Temperature compensation (Q10 = 1.06) using the following activation energies: E1 = 40 kJ/mol, E2 = 35 kJ/mol, E3 = 28 kJ/mol, E4 = 43 kJ/mol, E5 = 40 kJ/mol, E6 = 25 kJ/mol, and E7 = 25 kJ/mol. (2) All activation energies have the average value (34 kJ/mol) from the calculation in (1) showing a Q10 of 1.58. (3) All activation energies are 80 kJ/mol, and the oscillator shows a Q10 of 2.85.
(F) Dependence of period and P-KaiC ratio as a function of KaiABC* stability mimicking KaiC mutant behaviors. (1) Rate constants are as in Figure 2 (“wild-type behavior”); (2) Decreased P-KaiC ratios amplitudes and shorter periods (“short period KaiC mutant behavior”) are observed for a more stable KaiABC* complex relative to (1): k4 = 2.3 μM−1 h−1, k5 = 4.5 h−1, k6 = 1.0 h−1; and (3) Long-period KaiC mutant behavior is observed when the KaiABC* complex is more stable compared to (1): k4 = 4.0 μM−1 h−1, k5 = 1.5 h−1, and k6 = 0.8 h−1.
Figure 6.
An Expanded Model of the KaiC Oscillator Containing Explicit Stoichiometries
(A) KaiXkYlZm denotes a complex between k KaiX, l KaiY, and m KaiZ molecules.
(B) Calculated oscillations of total phosphorylated KaiC (P-KaiC) and total unphosphorylated KaiC (NP-KaiC).
(C) Calculated oscillatory ratio of P-KaiC = P-KaiC/(P-KaiC + NP-KaiC).
(D) Rate equations of the model. The following rate constants and initial concentrations were used in (B) and (C): k1 = 0.1 μM−5 h−1, k2 = 0.1 μM−1 h−1, k3 = 0.1 μM−3 h−1, k4 = 0.001 μM−1 h−1, k5 = 1.8 μM−1 h−1, k6 = 2.5 μM−1 h−1, k7 = 1.0 h−1, k8 = 1.0 h−1, k9 = 0.1 h−1, and k10 = 40 μM−1 h−1. [KaiA] = 0.0161 μM, [KaiA2 B4C*6] = 0.0611 μM, [KaiA2C*6] = 0.0205 μM, [KaiA2C*6C6] = 7.4 × 10−4 μM, [KaiA2] = 0.9094 μM, [KaiB] = 0.1402 μM, [KaiB4C*6] = 0.0894 μM, [KaiB4] = 0.7395 μM, [KaiC] = 1.0288 μM, [KaiC*6] = 1.1805 μM, and [KaiC6] = 0.7256 μM.