The circadian KaiABC oscillator is driven by a sequestration feedback, which is biochemically realized by a strong affinity of KaiA to exclusively serine phosphorylated KaiBC complexes.A highly non-linear feedback model explains the time courses of the phosphorylation states and the robustness under concerted changes of all Kai proteins.Native mass spectrometry reveals the existence of two KaiA binding sites on KaiC, confirming the theoretical predictions.Temperature entrainment arises from a temperature-dependent change in the abundance of KaiAC and KaiBC complexes.
The circadian rhythm of the cyanobacterium Synechococcus elongatus is controlled by three proteins, KaiA, KaiB, and KaiC. In a test tube, these proteins form complexes of various stoichiometry and the average phosphorylation level of KaiC exhibits robust circadian oscillations in the presence of ATP (Nakajima et al, 2005). Although the three component oscillator is apparently simple, it is highly precise and shows in-phase oscillations over several days (Mihalcescu et al, 2004). If we assume that cyanobacteria gain an evolutionary advantage from predicting the time of maximal sunlight intensity in a very reliable way, a highly robust but entrainable oscillator is likely the optimal solution. To elucidate the mechanism of the opposing properties of the clockwork—robustness and tenability—a mathematical modeling approach is necessary. The model must account for both the measured invariance of the phosphorylation level under concerted concentration changes of all Kai proteins (Kageyama et al, 2006) and for the experimentally observed temperature entrainment (Yoshida et al, 2009).
Here, we show by mathematical modeling, in combination with the measurements of the KaiABC complex formation dynamics from native mass spectrometry, that oscillations in the Kai system are a consequence of KaiA sequestration by KaiC hexamers and KaiBC complexes. Our in vitro model includes the characteristic KaiC phospho-form cycle, originating from two KaiC residues, serine (431) and threonine (432) (Nishiwaki et al, 2000). We allow for three pools that represent different forms of the KaiC complex. Together with the four phosphorylation states of KaiC, this gives a 3 × 4-dimensional model. We used data from the time course experiments of the O'Shea laboratory (Rust et al, 2007) to determine the unknown parameters of the KaiABC system. A global optimization algorithm is used to scan a large range of parameter values, resulting in a mathematical quantitative model of the KaiABC clockwork, see Figure 5. The KaiB response experiment (Rust et al, 2007) turned out to be crucial for identifying the mechanism by which individual KaiC hexamers can be synchronized in their phosphorylation dynamics. Importantly, the dephosphorylation phase can only be explained by a highly non-linear dependency of the KaiBC complex formation on the actual phosphorylation state. This can be realized by allowing only KaiBC complexes with exclusively serine phosphorylated KaiC, [S–KaiBC]6, to inactivate KaiA with a high efficiency.
This theoretical prediction is confirmed by native mass spectrometry, generating semi-quantitative time courses of the KaiABC complex formation dynamics. Our experiments show the existence of two different KaiC binding sites to KaiA. The constant sequestration of free KaiA by the KaiA2C6 complexes is the molecular realization of the dynamic invariance condition because it requires most of the KaiA to be inactive at every instant of time, regardless of the phosphorylation state. The second binding site KaiA4C6 reflects the KaiA-binding domain at the catalytic active center of the KaiC hexamer. This hypothesis is confirmed by comparison of the mass spectrometry signal for KaiA4C6 with predictions from the mathematical model (Figure 7A and B). In the late phosphorylation phase, KaiBC complexes rapidly build up and sequestrate KaiA (Figure 7C), which represent an additional binding site. The relative amount of KaiA6B6C6 confirms the sequestration hypotheses and corresponds to the theoretically estimated amount of sequestrated KaiA. The time of maximum sequestration—as defined by the appearance of the largest observed sequestration complex KaiA10B6C6 (Figure 7E)—agrees with the theoretical expected sequestration maximum (Figure 7F) where [S-KaiBC]6 is maximal.
To test further the predictive power of the mathematical model, we reproduced the observed phase synchronization dynamics on entrainment by temperature cycles. The response to a sudden temperature change results in a phase shift of phospho-KaiC, whereas the circadian period does not show any temperature dependency within a physiologically relevant range (Yoshida et al, 2009). As phosphorylation and dephosphorylation dynamics of KaiC alone and incubated with KaiA do not show significant temperature dependence (Tomita et al, 2005), phase entrainment is likely a consequence of temperature-induced changes in binding constants associated with the various KaiABC complexes. From thermodynamic arguments, we expect that an increasing temperature will enhance dissociation of KaiA and KaiB from KaiC. Indeed, a reduction in the net complex formation rate for S-KaiBC, D-KaiBC, and for KaiAC on temperature increase results in the experimentally observed differences in phase response, which compensate during the circadian cycle to assure temperature invariance of the ciracadian period.
The circadian rhythm of the cyanobacterium Synechococcus elongatus is controlled by three proteins, KaiA, KaiB, and KaiC. In a test tube, these proteins form complexes of various stoichiometry and the average phosphorylation level of KaiC exhibits robust circadian oscillations in the presence of ATP. Using mathematical modeling, we were able to reproduce quantitatively the experimentally observed phosphorylation dynamics of the KaiABC clockwork in vitro. We thereby identified a highly non-linear feedback loop through KaiA inactivation as the key synchronization mechanism of KaiC phosphorylation. By using the novel method of native mass spectrometry, we confirm the theoretically predicted complex formation dynamics and show that inactivation of KaiA is a consequence of sequestration by KaiC hexamers and KaiBC complexes. To test further the predictive power of the mathematical model, we reproduced the observed phase synchronization dynamics on entrainment by temperature cycles. Our model gives strong evidence that the underlying entrainment mechanism arises from a temperature-dependent change in the abundance of KaiAC and KaiBC complexes.