Figure 1.
Fundamental schemes proposed in the model.
(A) Reaction network for KaiC phosphorylation without KaiA and KaiB. The full multi-layer network can be found in Text S1 (section 1.1). (B) Thermodynamic box for analyzing the apparent free energy of activation in KaiC phosphorylation or dephosphorylation. On the left side, the six KaiC monomers are independent from each other. The subunit interaction that is driven by ATP hydrolysis results in the formation of stabilized and coordinated hexamers in ground state and transition state (on the right). ATP hydrolysis provides sufficient amount of extra negative free energy to make all the reactions and interactions thermodynamically possible.
Figure 2.
Circadian oscillations of Kai proteins in the full reaction network using deterministic method.
(A) Circadian oscillation of fractional saturation of phosphates for S431 (YS, dashed line), T432 (YT, dotted line), and overall (Y, solid line). We defined ,
and
, where [C]T is the total amount of KaiC hexamers. (B) The circadian profiles of m01 (S431 phosphorylated only), m10 (T432 phosphorylated only), m11 (dual phosphorylated) and their total amount (m01+m10+m11). (C) Oscillatory profiles of free KaiC hexamer and its complexes. The standard protein concentrations are used: KaiA (dimer): 0.58 µM; KaiB (dimer): 1.75 µM; KaiC (hexamer): 0.58 µM.
Figure 3.
Circadian KaiC phosphorylation at low protein numbers.
The ratio of the amounts of Kai proteins are fixed by NA ∶ NB ∶ NC≈1 ∶ 3 ∶ 1, where NA, NB and NC stand for the total numbers of KaiA dimers, KaiB dimers and KaiC hexamers, respectively. The ratio of total Kai protein numbers to the cell volume is set to be constant. KaiC phosphorylation cycle can be generated as NC = 1 (A), NC = 10 (B), NC = 30 (C) and NC = 100 (D).
Figure 4.
Snapshots of node-to-node mass evolution in the network.
Panels A–D show the snapshots for LC layer, and panels E–H for LBC layer. The time for minimum Y is set to be the zero circadian time. Three other time points are chosen for comparison: T = 7.1 h (at phosphorylation phase), T = 10.1 h (the peak time) and T = 16 h (at dephosphorylation phase). The color bar shown on right corresponds to the fractions of all nodes in the contour map.
Figure 5.
Stereotypes of diverse waveforms exhibited in KaiC phosphoforms from both deterministic and stochastic simulations.
Stereotypes of four groups of waveforms of KaiC are selected: for group 1 (A and E);
for group 2 (B and F);
for group 3 (C and G) and
for group 4 (D and H). (A–D) for deterministic model, while (E–H) for stochastic model. The total number of KaiC hexamer is 2000 in stochastic simulation (see section 2.3 in Text S1).
Figure 6.
Phase distributions of hexameric KaiC phosphoforms in one circadian period.
The time of minimum Y (the overall fractional saturation of phosphates) is taken as the zero time point of one circadian period. The sample are 49 KaiC phosphoforms defined by . These 49 point are identified by colors (as s, the numbers of phosphorylated S431) and radiuses (as t, the numbers of phosphorylated T432) in the diagram. The phase of each phosphoforms is defined in two ways: first, by the time of maximum
in one circadian cycle (A); the other by the time of minimum
(B).
Figure 7.
Robustness in Kai reaction network.
(A) Resilience of Kai oscillator examined by mixing two anti-phased (trough and peak) samples with equal amount. The thin black and grey lines are the applied samples, and the bold black line the mixture. (B) Contour map of the initial reaction rate of Y after equal-amount sample mixing. The contour map is established in the interior region of the limit cycle of YS and YT. (C–D) Bifurcation analysis for the period (C) and amplitude (D) of KaiC phosphorylation by varying total Kai protein concentration. is the concentration of the standard mixture.
Figure 8.
Further examination of the model by recent experimental data.
(A) The distribution of KaiC phosphoforms determines the phase of KaiC phosphorylation. The KaiC sample enriched in m10 (monomer with T432 phosphorylated only) is obtained as follows: simulating binary KaiA and unphosphorylated KaiC reaction for 3 hours and then removing KaiA; then adding KaiB for KaiB-KaiC binary reaction for another 3 hours. The m01 (monomer with S431 phosphorylated only) enriched KaiC is prepared by mixing KaiA and KaiC for 20 hours, removing KaiA, and allowing KaiC dephosphorylate for 2.4 hours. (B) Standard amount of KaiB is introduced at different time points during KaiA-KaiC reaction process. (C) Extra unphosphorylated KaiC with 10% standard concentration of KaiC is introduced at various circadian time points in KaiC phosphorylation. Colored profiles (not black) show the phosphorylation dynamics for the added KaiC samples after their introductions. (D) KaiC phosphorylation profiles with 5× standard KaiA added at different circadian time points.
Figure 9.
Phase response curves obtained by transient variations in KaiA concentration.
Nonstandard KaiA concentrations are used as the 4-hour stimulus pulses, i.e. 1/3×, 2/3×, 1.5× and 3× standard amount of KaiA. Parameter set used is the new one obtained in section 2.6 in Text S1.