An alternative interpretation of the slow KaiB-KaiC binding of the cyanobacterial clock proteins

The biological clock of cyanobacteria is composed of three proteins, KaiA, KaiB, and KaiC. The KaiB-KaiC binding brings the slowness into the system, which is essential for the long period of the circadian rhythm. However, there is no consensus as to the origin of the slowness due to the pre-binding conformational transition of either KaiB or KaiC. In this study, we propose a simple KaiB-KaiC binding scheme in a hexameric form with an attractive interaction between adjacent bound KaiB monomers, which is independent of KaiB’s conformational change. We then show that the present scheme can explain several important experimental results on the binding, including that used as evidence for the slow conformational transition of KaiB. The present result thus indicates that the slowness arises from KaiC rather than KaiB.

www.nature.com/scientificreports www.nature.com/scientificreports/ is dilute. Therefore, an explanation that satisfies both of the two experiments, in particular, the slow binding observed in Chang's experiment 21 , is needed.
In this article, we propose an alternative binding scheme that can explain both the initial rapid and the subsequent slow KaiB-KaiC bindings observed in Chang's experiment 21 . In contrast to the original conformational-selection scheme proposed in Chang's experiment 21 , the present scheme works even when the conformational transition of KaiB is arbitrarily fast, which is thus also consistent with the conclusion of Mukaiyama's experiment 22 . The key points of the present scheme are three-fold: the low concentration of KaiB 21 , the hexameric form of KaiC, and the attractive adjacent KaiB-KaiB interaction in the KaiB-KaiC complex 17,24 .
In the next section, we show that the slowness of the binding in Chang's experiment 21 may arise from the low concentration of KaiB. Specifically, in the present scheme, the initial rapid and the subsequent slow bindings correspond to the bindings from C 6 B 0 to C 6 B 1 and from C 6 B 1 to the other larger KaiB-KaiC complexes C 6 B n (n > 1), respectively. Here, C 6 B n represents the complex of a KaiC hexamer and n KaiB monomers. Note that, without any stabilization mechanism, the formation of the larger KaiB-KaiC complexes C 6 B n (n > 1) seldom occurs at a low concentration of KaiB. In the present scheme, however, the larger KaiB-KaiC complexes are substantially formed due to the stabilization by the attractive adjacent KaiB-KaiB interaction.

Results
Design of binding scheme. In this study, we discuss the KaiB-KaiC binding with a binding scheme in which the hexameric form of KaiC is explicitly considered (Fig. 1). We distinguish all six binding sites on a KaiC hexamer and consider 2 6 types of the KaiB-KaiC complex, i.e. whether KaiB is bound at each binding site (not explicitly shown in Fig. 1). The conformational transition of KaiB is assumed to be fast and is treated by the so-called rapid equilibrium approximation. Thus, the mutagenesis of KaiB to stabilize the binding-competent conformation 21 is effectively represented by the increase in the association rate constant of the KaiB-KaiC binding. To focus on the binding in Chang's experiment 21 , we ignore the conformational transition of KaiC. Moreover, to focus on post-phosphorylation processes, we assume that all KaiC hexamers are phosphorylated and take their binding-competent conformational state.
We set the association and dissociation rate constants as follows. For simplicity, we assume that the association rate constant of a KaiB monomer to a KaiB-KaiC complex is always k on (Fig. 2). On the other hand, we assume that the dissociation of a KaiB monomer from a KaiB-KaiC complex becomes slow when another KaiB monomer is bound at an adjacent binding site. This is because the attractive KaiB-KaiB interaction 17,24 may stabilize the complex. Specifically, the dissociation rate constant k off is modeled as  Fig. 3A is consistent with the binding observed in Chang's experiment 21 in terms of the following three features. First, the present scheme can reproduce both the initial rapid and the subsequent slow bindings. Second, an increase in k on , which corresponds to the mutagenesis of KaiB to stabilize the binding-competent conformation, leads to the increases in the amount of bound KaiB both in the initial and the subsequent bindings. Lastly, the KaiB-KaiC binding is accelerated as k on increases. Next, we explain why the present scheme can reproduce the KaiB-KaiC binding in Chang's experiment 21 . The initial rapid and the subsequent slow bindings can be explained by the low concentration of KaiB (B tot = 0.05 μM, C tot = 10.0 μM) and the hexameric form of KaiC. The rate of the KaiB-KaiC binding at an unoccupied binding site of C 6 B n ( =  n 0, 1, , 5), v n , is given by  On the other hand, [C 6 B n ] (n ≥ 1) never exceeds B tot , i.e.
t ot Therefore, we obtain n 0 which indicates that the initial binding from C 6 B 0 to C 6 B 1 is much faster than the subsequent binding from C 6 B n to C 6 B n+1 (n ≥ 1). This is why the subsequent binding is much slower than the initial binding in the present scheme. Figure 3B indeed shows that the initial rapid binding is attributed to the binding from C 6 B 0 to C 6 B 1 and the subsequent slow binding to the binding from C 6 B 1 to the larger KaiB-KaiC complexes.
In the present scheme, we consider the KaiB-KaiB interaction for the stabilization of larger KaiB-KaiC complexes 24 . Without the KaiB-KaiB interaction, the KaiB-KaiC binding at a binding site becomes independent of other binding sites. Thus the distribution of [C 6 B n ] (0 ≤ n ≤ 6) at equilibrium obeys a binominal distribution. In this case, the amount of larger KaiB-KaiC complexes becomes much smaller than [C 6 B 1 ] when KaiB is dilute, which means that the binding from C 6 B n to C 6 B n + 1 (n ≥ 1) seldom occurs. The numerical results without the stabilization by the KaiB-KaiB interaction, in which α is set to be 1 and the other parameter values are the same as those used in Figs  www.nature.com/scientificreports www.nature.com/scientificreports/ ( ≥ n 2) becomes comparable to [C B ] 6 1 at equilibrium, as we have already seen in Fig. 3A and B. This is why the amount of the subsequent binding is comparable to the initial binding in the present scheme.
Equations (2) and (6) can also explain k on dependence of the binding. Equation (2) indicates that the binding is accelerated as k on increase, and Eq. (6) means that the amount of the binding increase with k on both in the initial and subsequent bindings due to the equilibrium shift toward larger KaiB-KaiC complexes.

KaiB-KaiC binding at other protein concentrations.
In this subsection, we investigate the KaiB-KaiC binding at protein concentrations different from the low KaiB concentration. A recent experiment by mass spectrometry has shown that C B 6 6 is the most abundant in C B n 6 =  n ( 1, 2, , 6) six hours after mixing KaiB with a phospho-mimic mutant of KaiC, even when B tot is smaller than C tot ( = . B /C 025 tot tot ) 24 . The present scheme can describe this experimental result (Fig. 4A) because the ring-shaped alignment of KaiB monomers in C B 6 6 acquires the KaiB-KaiB interaction more efficiently than C B n 6 (n < 6) as follows. When n < 6, the number of KaiB-KaiB interfaces in C B n 6 is not more than − n 1. On the other hand, the number of interfaces is n when n = 6 because the bound KaiB monomers form a ring. Therefore, if the stabilization effect α is small enough to be comparable to k k [B]/ on off,0 at equilibrium, i.e.
on off,0 Equation (6) yields    6). Lastly, we investigate the KaiB-KaiC binding at the so-called standard condition of the KaiABC oscillator (B tot = C tot = 3.5 μM). As Eq. (2) indicates that the binding is accelerated with increasing protein concentrations, the result of the present scheme exhibits a rapid binding with the half-life time t 1/2 = 0.06 h at the standard condition (Fig. 4B). Although this half-life time is considerably shorter than that observed in Mukaiyama's experiment 22 , the present result does not contradict with the experimental result. This is because the present scheme ignores the conformational transition of KaiC. If we incorporate a slow pre-binding conformational transition of KaiC into the present scheme in a conformational-selection manner, the binding rate could be regulated by KaiC as suggested in Mukaiyama's experiment 22 . Rather, the present result (Fig. 4B) indicates that the slowness of the binding arising from protein concentration may be negligible at the standard condition compared to that from the conformational transition of KaiC, i.e. the KaiB-KaiC binding itself is rapid and may not be a rate-limiting process in the KaiABC oscillator around the standard condition. This result is also consistent with the experimental result that an equimolar increase in the concentrations of KaiA, KaiB, and KaiC unalters the phosphorylation oscillation 25 . If the binding process is so slow that it depends on the protein concentration, the change of the protein concentration would modulate the oscillation.

Discussion
In this study, we proposed an alternative interpretation of the KaiB-KaiC binding based on a novel binding scheme, which is consistent with the three experimental results: the initial rapid and the subsequent slow bindings at a low KaiB concentration 21 , the protein concentration dependence of the binding rate 22 , and the predominant formation of C B 6 6 even when B tot < C tot 24 . To the best of our knowledge, these experiments have not been explained in a unified way before.
The binding scheme used in this study simply models the hexameric form of the KaiB-KaiC complex and the attractive adjacent KaiB-KaiB interaction in the complex. As the structural study has revealed a strong electrostatic interaction between two adjacent KaiB monomers in the complex 17 , it is reasonable to assume that the dissociation of a KaiB monomer from the complex becomes slow when the KaiB-KaiB interaction is formed. It should be noted that the parameter values used in the present study are determined by hand, i.e. without any numerical fitting to experimental data. Thus, the values themselves are of little importance. However, the framework of the present scheme is still of importance because it gives the possible comprehensive mechanism of the KaiB-KaiC binding summarized below.
We showed that the present scheme can qualitatively reproduce the initial rapid and the subsequent slow bindings 21 at a low KaiB concentration along the following scenario. When KaiB is much fewer than KaiC, the amount of the bare KaiC hexamer C B 6 0 substantially exceeds the KaiB-KaiC complexes C B n 6 ( ≥ n 1). Thus, the initial binding from C B 6 0 to C B 6 1 becomes much faster than the subsequent binding form C B 6 1 to the larger complexes C B n 6 ( > n 1). Moreover, due to the stabilization by the adjacent KaiB-KaiB interaction, a comparable amount of the larger complex to C B 6 1 is produced. Note that this binding mechanism does not rely on the structural transition of KaiB, which indicates that the rate of the transition can be arbitrarily fast. In the present study, by assuming that the transition is very fast and effectively changes the association rate constant k on , we showed that the present scheme can reproduce the mutagenesis dependence of the binding in Chang's experiment 21 . Therefore, the slow binding observed in Chang's experiment 21 does not necessarily mean that the structural transition of KaiB is slow. www.nature.com/scientificreports www.nature.com/scientificreports/ The present scheme can further explain why C B 6 6 is predominantly produced even when B tot < C tot 24 . This is because the ring-shaped alignment of bound KaiB monomers in C B 6 6 has an advantage in the formation of KaiB-KaiB interaction compared to other alignments consisting of one or more linear segments.
We also showed that, without the slow pre-binding conformational transition of KaiC suggested in Mukaiyama's experiment 22 , the KaiB-KaiC binding may become rapid at the standard condition of the KaiABC oscillator i.e. B tot = C = 3.5 μM (Fig. 4B). This means that the conformational transition of KaiC could be the slowest process in the whole KaiB-KaiC binding process. In this sense, the present scheme is consistent with the experimental study indicating that the slowness of the binding arises from KaiC 22 .
In summary, the present study indicates that the slowness arises from KaiC and that the conformational transition of KaiB is of less importance in the binding. This conclusion is also supported by a recent mathematical reaction model of the KaiABC oscillator, which shows that various experimental results can be reproduced without considering the conformational transition of KaiB 26 .
It must be noted, however, that the conformational transition of KaiB can still play an essential role in the KaiABC oscillator. Indeed, the mutagenesis of KaiB to stabilize the binding-competent state of KaiB results in arrhythmic phosphorylation of KaiC in the presence of both KaiA and KaiB 21 . Thus, further investigation of the functional role of the conformational transition of KaiB is still needed in the future.

Methods
Numerical calculation. The rate equations of the present binding scheme are integrated by the fourth-order Runge-Kutta method with the time step Δt = 0.001 s.

Data availability
All data generated in this study are available from the corresponding authors upon request.