Formation of Kai protein complexes during the in vivo phosphorylation cycle. (A) Proteins (0.2 μg ml⁻¹ KaiA, 0.2 μg ml⁻¹ KaiB, and 0.8 μg ml⁻¹KaiC; corresponding to a molar ratio of about 1:3:3) were incubated under standard conditions for different periods of time. The protein samples were immediately applied to a 4–16% BN-PAGE gel. Molecular weight marker proteins are indicated on the left. (B) Several standard incubation mixtures were supplemented with [γ-³²P]ATP (10 μCi per reaction mixture), subjected to BN-PAGE, and the dried gel was autoradiographed. (C) Formation of protein complexes after 12 h incubation of a standard KaiC and KaiA mixture with varying concentrations of KaiB (1 × corresponds to the standard reaction mixture). (D) Lanes of interest (2, 8, and 18 h) were excised from a 1D BN gel, placed on top of a resolving gel for Tricine–SDS–PAGE and subsequently separated by electrophoresis. Arrows indicate newly arising KaiA and KaiB protein spots associated with larger KaiC-containing complexes.

Core reaction scheme of KaiC phosphorylation. The KaiC hexamers, C₆, undergo phosphorylation and dephosphorylation depending on the actual reaction rates. Higher (lower) level of KaiA increases (decreases) the phosphorylation rates. Hexamers in the highest phosphorylation state bind six KaiB dimers to form a stable complex. The [B₁₂C₆] complex is assumed to dephosphorylate and releases KaiB dimers in the lowest phosphorylation state. The superscript indicates the number of phosphate groups added, starting from the minimal physiological phosphorylation level.

Three examples out of eight reaction networks that show oscillatory behaviour with high amplitude. In the reaction schemes (left panels), bar ends indicate repressions and arrows enzymatic activations. Feedbacks are allowed to work only between the groups that are indicated by dashed boxes in (A). Each link that targets a group of low- or high-phosphorylated hexamers (C₆) or KaiBC complexes ([B₁₂C₆]) is assigned one additional reaction constant to be optimised (see Supplementary information). (A) Left panel: mechanism of KaiABC clock as proposed by Emberly and Wingreen (2006). The negative feedback arises from the assumption that hexamers can exchange monomers. The middle graph shows the time course of the KaiC phosphorylation level for protein concentrations taken from in vitro experiments with N=6 (green line) and N=20 (blue line) phosphorylation steps. Changes in phase and frequency owing to a collective five-fold increase or 10-fold decrease in protein concentrations are shown by the solid black (only partially drawn) and dashed red line, respectively, for N=20. The right panel shows the fraction of KaiC in the C₆ (blue line) and in the [B₁₂C₆] (green line) state (N=20). (B) Left panel: positive feedback by assigning the highest phosphorylated hexamer, C₆⁶, a kinase activity (Mehra et al, 2006). Middle and right panels: as in (A) but for N=6 phosphorylation steps. (C) Left panel: negative feedback induced by sequestration of KaiA proteins by the KaiBC complexes. Middle and right panels as in (B).

Time course of phosphorylated KaiC (upper panels) and free hexamers and KaiBC complexes (lower panels) for the model using KaiA sequestration as a negative feedback. Red lines arise from a concerted five-fold increase of the native protein concentrations (blue lines) after optimisation. Lower panels: solid and dashed lines indicate the amount of free hexamers and KaiBC complexes, respectively. (A) With reaction constants from optimisation for both highest amplitude and robustness against concerted five-fold elevation of protein concentrations in the absence of monomer exchange. (B) Optimisation as in (A) but accounting also for monomer exchange. (C) Optimisation for highest amplitude and accounting for monomer exchange but without any constraints on robustness.

Upper panels: abundances of the complexes KaiAC, KaiBC, and KaiABC over time, relative to the total amount of hexamers in the system. Lower panels: relative concentrations of free KaiA and KaiB. (A) Optimisation as in Figure 4A. (B) Optimisation as in Figure 4B.

Time course of KaiC phosphorylation and complex formation in comparison with experiments. Upper panels: results from the mathematical modelling using the optimisation approach from Figure 4A; lower panels: experimental results from Kageyama et al (2006). (A) Relative level of phosphorylated KaiC for different protein levels. (B) Complex formation of KaiAC, KaiBC, and KaiABC.

Cartoon of the KaiC phosphorylation cycle. Phosphorylated KaiC monomers are marked by filled circles. The relative amount of molecules involved in the transitions between the indicated states is indicated by the line thickness of the arrows. After incubation, autophosphorylation of KaiC hexamers is enhanced by KaiA. After 6 h, KaiBC complexes start to build. The KaiBC complexes dephosphorylate, presumably by inhibiting access of KaiA to its active sites for KaiC phosphorylation. Low phosphorylated KaiBC can bind KaiA with high affinity and this leads to dephosphorylation also of the KaiC hexamers. The synchronisation of the KaiBC phosphorylation level by monomer exchange plays only a minor role.