Cross-scale analysis of temperature compensation in the cyanobacterial circadian clock system

Circadian clock proteins often reveal temperature-compensatory responses that counteract temperature influences to keep their enzymatic activities constant over a physiological range of temperature. This temperature-compensating ability at the reaction level is likely crucial for circadian clock systems, to which the clock proteins are incorporated, to achieve the system-level temperature compensation of the oscillation frequency. Nevertheless, temperature compensation is yet a puzzling phenomenon, since side chains that make up the clock proteins fluctuate more frequently due to greater thermal energy at higher temperature. Here, we investigated temperature influences on the dynamics of KaiC, a temperature-compensated enzyme (ATPase) that hydrolyzes ATP into ADP in the cyanobacterial circadian clock system, using quasielastic neutron scattering. The frequency of picosecond to sub-nanosecond incoherent local motions in KaiC was accelerated by a factor of only 1.2 by increasing the temperature by 10 °C. This temperature insensitivity of the local motions was not necessarily unique to KaiC, but confirmed also for a series of temperature-sensitive mutants of KaiC and proteins other than clock-related proteins. Rather, the dynamics associated with the temperature-compensatory nature of the reaction- and system-level was found in global diffusional motions, which was suggested to regulate the temperature dependence of ATPase activity and dephosphorylation process presumably through changes in the hexamer conformation of KaiC. The spatiotemporal scale at which cross-scale causality of the temperature sensitivity is established is finite, and extends down to picosecond to sub-nanosecond dynamics only in a very limited part of KaiC, not in its entire part. The mechanism by which circadian clock proteins compensate for changes in temperature to maintain regular operation remains unclear. Here, temperature compensation from the reaction to cell level is explained by large-scale motions inside the clock protein observed via neutron scattering spectroscopy.

C ircadian clocks are endogenous timing systems that rhythmically control various biological processes with añ 24-h period 1 . This rhythm persists stably even without any external cues, and the period length is kept constant even when ambient temperature changes (temperature compensation). The phase of the clock system can be shifted upon receiving external stimuli such as light and temperature and then synchronized to the phase of the outer world. These unique characteristics enable organisms to optimize their fitness during day/night environmental cycles [2][3][4][5] .
Temperature compensation is a remarkable characteristic of the circadian clock systems. Q10 values, the factor by which reaction speed or cycle frequency is accelerated by increasing the ambient temperature by 10°C, are mostly in the range of 0.9-1.1 for circadian clock systems, whereas those of most biochemical reactions 6 and the Belousov-Zhabotinsky oscillator 7-10 range from 2 to 3. As schematically shown in Fig. 1a, circadian clocks exhibiting system-level temperature compensation often comprise unique clock proteins with temperature-compensated biochemical activities such as adenosine triphosphate (ATP) hydrolysis (ATPase) [11][12][13] , and protein phosphorylation (kinase) [14][15][16] and dephosphorylation (phosphatase) 14,15 . A simple but attractive idea that emerged from studies 12,13,15,17,18 of these biochemical activities is that the reaction-level temperature compensability is somehow correlated to system-or even higher cell-level temperature compensability (Fig. 1a). A great deal of effort has been devoted to elucidating the mechanism of this connectivity via experimental 12,13,15,16,18 and modeling [19][20][21][22][23] approaches.
Nevertheless, temperature compensation remains a puzzling phenomenon in terms of protein dynamics, as atoms and side chains in proteins should fluctuate more frequently at higher temperature due to their greater thermal energy. Three extreme cases can be considered (inset in Fig. 1a). First, thermal fluctuation in these key clock proteins is accelerated in a temperaturedependent manner, as observed for ordinary proteins. Second, the atoms and side chains in temperature-compensated clock proteins fluctuate in a temperature-insensitive manner via unknown mechanisms. Third, some opposing but balancing contributions Fig. 1 Potential cross-scale causal relationship in the circadian clock system of cyanobacteria. a Spatiotemporal hierarchy spanning from atomic-scale dynamics, molecular-scale reaction, molecular system-scale frequency, and cellular-scale frequency. When the ATPase activity of KaiC doubles as a result of amino acid substitutions (orange arrow), both the in vitro system-and cell-level frequencies also double 12,17,25 . In cyanobacteria, the temperature (T) sensitivity of ATPase is correlated with those of the system and cell levels. (Inset) Three extreme cases of temperature influence on clock-protein dynamics. Thermal fluctuations are sensitively accelerated, insensitively compensated, or sensitively compensated through a balance of opposing contributions. b Temperature-dependent ATPase mutation sites mapped onto the crystal structure of KaiC 32,33 . Zoomed-in-views of c S258, d F247, e Q361, f N390, g F470, and h phosphorylated S431 and T432 in KaiC. Hydrogen bonds are highlighted by green dotted lines. act at the level of protein dynamics level, i.e., one contribution accelerates but the other decelerates so that they differentially affect the elementary reaction steps to achieve overall compensation. Experimental investigations of the dynamics of temperature-compensated clock proteins are needed to distinguish among these cases.
KaiC, a temperature-compensated clock protein in cyanobacterium Synechococcus elongatus PCC 7942 [11][12][13] , acts as a circadian oscillator with other two clock proteins, KaiA and KaiB 24 . Its most striking feature is that its temperature-compensated circadian rhythm can be reconstructed even in vitro by mixing KaiA, KaiB, and KaiC in the presence of ATP 25 (Fig. 1a). An ATP molecule binds a Walker motif in each of the tandemly duplicated domains called the N-terminal C1 and C-terminal C2 domains (Fig. 1b). These ATP binding events trigger oligomerization of KaiC into a double-ring hexamer. The ATP molecule bound to the C2 domain (C2-ATP) is used mostly as the source of phosphoryl group that is transferred to (auto-kinase) and then removed from (auto-phosphatase) S431 and T432 in KaiC. In the presence of KaiA and KaiB, the status of the dual phosphorylation site alters in a cyclic manner: ST → SpT → pSpT → pST → ST, where S, T, pS, and pT represent S431, T432, phosphorylated S431, and phosphorylated T432, respectively 26,27 . The frequency of this phosphorylation cycle (P-cycle) is proportional to the rate of hydrolysis of C1-bound ATP (C1-ATP) into adenosine diphosphate (ADP) in the absence of KaiA and KaiB ( Fig. 1a) 12,13,17 . More importantly, the ATPase activity of KaiC is perfectly temperature-compensated (Q10 ATP = 1.0) 12 . The in vitro Kai oscillator that consists of temperature-compensated KaiC provides a practical means for studying cross-scale properties of temperature-compensation phenomena at the system, reaction, and dynamics levels (Fig. 1a).
Quasielastic neutron scattering (QENS) is a powerful and direct technique for accessing protein dynamics at the picosecond to nanosecond time scales 28 . Because hydrogen atoms, which constitute up to half of all atoms in proteins, are distributed near uniformly in the three-dimensional structures of proteins, an averaged view of protein dynamics can be extracted from QENS spectra 29 . In this study, we designed a series of KaiC mutants ( Fig. 1b) with temperature-dependent periods ( Fig. 1c-e), inverse temperature-dependent period (Fig. 1f), temperature-compensated but shortened-period (Fig. 1g), or arrhythmicity (Fig. 1h), and measured the temperature sensitivity of their protein dynamics using the neutron spectrometer BL02 (DNA: Biomolecular Dynamics Spectrometer) in the Material and Life Science Experimental Facility of Japan Proton Accelerator Research Complex (MLF/J-PARC) 30 . Our QENS data revealed that the frequency of picosecond to sub-nanosecond incoherent motions is essentially temperature-insensitive in KaiC, regardless of the presence or absence of the temperature-sensitive mutations. The spatiotemporal scale at which cross-scale causality of the temperature sensitivity is established is finite, and extends down to picosecond to sub-nanosecond dynamics only in a very limited part of KaiC, not in its entire part.

Results
Screening and characterization of temperature-sensitive mutants of KaiC. As confirmed by the positions along the horizontal axis in Fig. 2a, the ATPase activity of KaiC WT is as low as 11 d −1 and almost temperature-insensitive (Q10 ATP WT = 0.89 ± 0.10), as previously reported 12 . Consistently with previous studies 11,15,25 , the P-cycle frequency (f P = 24/period) of KaiC WT in the presence of KaiA and KaiB was also temperaturecompensated (Fig. 2b) (Q10 fp WT = 1.08 ± 0.04). Consequently, data points taken at different temperatures for KaiC WT nearly overlapped in the ATPase-f P plot (Fig. 2a). Taking advantage of potential reaction-system correlations, we screened the temperature-sensitive mutants of KaiC using an in vitro ATPasebased screening system 31 . Among a number of KaiC mutants screened for temperature-dependent ATPase activity, four candidates were characterized in detail as they revealed stable but temperature-dependent system-level oscillation ( Fig. 2c-f).
Taking into consideration the limited persistence of the P-cycle for KaiC F247A and KaiC N390A at 318 K ( Fig. 2d and Fig. 2b).
KaiC dynamics detected by QENS. To investigate the temperature dependence of KaiC dynamics, we conducted QENS experiments of KaiC WT at five different temperatures from 283 to 317 K. It must be noted that 293 K is approximately the critical temperature of the in vitro oscillator 34 , below which the P-cycle becomes arrhythmic (zero amplitude). As shown in Fig. 3a, a difference QENS spectrum S(Q, E) of KaiC WT , where Q is the momentum transfer and E is the energy transfer of neutrons, could be obtained at each temperature by subtracting the background spectrum of the D 2 O solvent from the sample spectrum, on the basis of scaling factors calculated from neutron scattering cross-sections. The resultant S(Q, E) for KaiC WT at all the temperatures shared a common feature of temperaturedependent broadening of a narrow elastic peak and a wide quasielastic component derived from global and local motions, respectively (Fig. 3a). To analyze the Q-and temperature dependencies of these two components quantitatively, we attempted to fit the following equation 28 to each S(Q,E) in For the KaiC mutants, QENS experiments were conducted at low (302 K) and high (310 or 313 K) temperatures that were within the physiologically functional range of temperature as the circadian clock. Although most of the KaiC mutants showed similar S(Q,E) to KaiC WT (Fig. 4a, 4c, 4e, and 4f), visually distinguishable differences were observed for KaiC F247A and KaiC N390A (Fig. 4b and d). In order to make more detailed and quantitative comparisons, each S(Q,E) of the KaiC mutants was Dark-colored circles correspond to the mean from independent preparations and measurements (pale-colored squares, triangles, and diamonds). Light blue circles plotted in panel b correspond to the P-cycle of KaiC WT in a D 2 O buffer. h Correlation of Q10 values between ATPase (Q10 ATP ) and P-cycle (Q10 fp ). Error bars correspond to one standard deviation obtained from three independent experiments, except that they are shown as three standard deviations for Q10 fp . analyzed using Eq. (1) in the same way as KaiC WT (panels g-r in Fig. 4). In the following, we describe the global and local motions of KaiC on the basis of temperature dependencies of Γ global (Q) and Γ local (Q), respectively.
Global motions of KaiC. Γ global (Q) provides information on the frequency of the global motions, whose properties are most straightforwardly inspected using the plot of Γ global (Q) vs Q 2 . Linear Q 2 -dependencies of Γ global (Q) were confirmed for KaiC WT at every temperature (Fig. 5a), indicating its boundary-free diffusions including translational and rotational diffusions. Thus, the linear slope in Fig. 5a relates to an apparent boundary-free diffusion coefficient (D global ). The D global for KaiC WT at 302 and 313 K were 3.54 ± 0.09 and 4.48 ± 0.10 × 10 −7 cm 2 s −1 , respectively ( Fig. 5h), which were similar to those (3.75 and 4.97 × 10 −7 cm 2 s −1 ) calculated using the known crystal structure of the KaiC hexamer (Supplementary Note 2).
The temperature dependencies of the D global (Q10 global ) for KaiC WT    temperature and solvent viscosity at T, respectively. Assuming η(302) = 1.016 cP and η(313) = 0.799 cP for the present D 2 O solvent (Supplementary Note 2), the term of T/η(T) increases 1.3fold by raising the temperature form 302 to 313 K. This agreement between Q10 global and the increase ratio of T/η(T) indicates that the temperature dependence of the hydrodynamic radius is rather small. Thus, we suggest that KaiC WT and the KaiC mutants are maintained as intact hexamers during the QENS measurements in the observed temperature range.
In contrast to Q10 global , the absolute D global values of the temperature-sensitive mutants shared a common feature. The D global values estimated at high temperature for KaiC S258A , KaiC F247A , KaiC Q361E , and KaiC N390A were systematically smaller than those of KaiC WT , KaiC F470Y , and KaiC DE (Fig. 5h). In order to make a fair comparison, it is necessary to take into account that the measured temperature of KaiC S258A , KaiC F247A , KaiC Q361E is lower by 3 K than the others. On the basis of the pseudo-linear temperature dependence of the D global values for KaiC WT (Fig. 5h) Fig. 5h). Even after applying these corrections, the D global values tended to be smaller in the temperature-sensitive mutants. These results suggest that the global hexamer structure of the temperature-sensitive mutants may be slightly different from those of KaiC WT and the temperature-compensated KaiC mutants (KaiC F470Y , KaiC DE ).
Local motions of KaiC. In contrast to Γ global (Q), Γ local (Q) reflects picosecond to sub-nanosecond incoherent local fluctuations of sidechain and main-chain in KaiC. Γ local (Q) of KaiC WT increased asymptotically to approach a plateau at high Q 2 (Fig. 6a). As at high Q, the motions over short distances predominate, the plateau values correspond to the elementary displacements of the local motions. Slight but temperature-dependent increase in the high-Q plateau of KaiC WT suggests that the side-chain and main-chain motions in KaiC WT are accelerated by thermal fluctuations (Fig. 6a).
To determine the temperature sensitivity of the local motions quantitatively, Q 2 -dependencies of Γ local (Q) were analyzed by using a jump-diffusion model 28 Fig. 2a), as previously reported 65 . A shaded bar represents Q10 global WT estimated from the slope of Arrhenius plot using three different temperatures (302, 313, and 317 K) within the physiologically functional range of temperature ( Supplementary Fig. 2a). Other Q10 global presented as black bars were estimated as the ratio of measurement results (R Low and R High ) at two temperatures (T Low and T High ) using Q10 = (R High / R Low ) 10/(THigh-TLow) ; T Low = 302 K, T High = 313 or 310 K. Error bars correspond to the errors associated with the linear regression analysis (a-g).
(1 + D local Q 2 τ) −1 , where D local is the jump-diffusion coefficient and τ is the residence time spent on one site before jumping to others. The model gave reasonable fits to temperature-and Q 2dependent Γ local (Q) of KaiC WT (lines in Fig. 6a). The resultant τ −1 values for KaiC WT were slightly temperature-dependent (Fig. 6h). Arrhenius plot analyses using five different temperatures (283, 293, 302, 313, and 317 K) resulted in the activation energy of 4.0 ± 0.6 kcal mol −1 (Supplementary Note 3), which corresponded to the Q10 value of 1.23 ± 0.04 for the internal motions (Q10 local , hatched bar in Fig. 6h). The estimation of the Q10 local WT value was not seriously affected by limiting the physiologically functional range of temperature (302, 313, and 317 K: Q10 local WT = 1.15 ± 0.11, shaded bar in Fig. 6h, Supplementary Note 3) and by taking a ratio of τ −1 at low (302 K) and high (313 K) temperatures (Q10 local WT = 1.10 ± 0.14, black bar in Fig. 6h). The Q10 values of D local (Q10 D-local ) for KaiC WT varied  Fig. 2b and 2c), as previously reported 65 . A shaded bar represents Q10 local WT or Q10 D-local WT estimated from the slope of Arrhenius plot using three different temperatures (302, 313, and 317 K) within the physiologically functional range of temperature ( Supplementary Fig. 2b and 2c). Other Q10 local presented as black bars were estimated as the ratio of measurement results (R Low and R High ) at two temperatures (T Low and T High ) using Q10 = (R High /R Low ) 10/(THigh-TLow) ; T Low = 302 K, T High = 313 or 310 K. Error bars correspond to the errors associated with the analysis using jump-diffusion model 28 (a-g).
in the range from 1.16 to 1.36 depending on the analysis method (Fig. 6i), probably due to the larger associated errors compared to the τ −1 values.
Γ local (Q) of the KaiC mutants, which revealed the same high-Q saturation as KaiC WT (Fig. 6b-g), could be analyzed reasonably with the jump-diffusion model. Regardless of whether they are temperature-compensated or temperature-dependent, the Q10 local values of the KaiC mutants ranged from 0.97 to 1.20 and were comparable to KaiC WT within error (Fig. 6h). A similar trend was confirmed for Q10 D-local (Fig. 6i). These results suggest that the average behavior of the local fluctuations occurring throughout KaiC, at least on picosecond to sub-nanosecond time scales, is not clearly correlated with the reaction and system levels in terms of temperature dependence.
While the mutations had small effects on the temperature dependence of the local dynamics, the absolute values of τ −1 were markedly different from KaiC WT in several temperature-sensitive mutants. The τ −1 values for KaiC F247A were 0.14 ± 0.01 ps −1 at low and high temperatures and were reduced~50-70% relative to KaiC WT (Fig. 6h). Conversely, the local motions in KaiC N390A were 30-40% faster (0.29 ± 0.01-0.33 ± 0.02 ps −1 ) than KaiC WT (Fig. 6h). D local was essentially unaffected by temperature change or mutations (Fig. 6i), except in the case of KaiC F247A , where a large error prevented us from identifying clear increase in D local .
Fractional change in apparently immobile atoms in the temperature-dependent ATPase mutants of KaiC. EISF is defined as the ratio of the elastic peak intensity to the sum of elastic and quasielastic scattering intensities as in Eq. (1), providing information on the geometry of molecular motions and the fraction of mobile and immobile atoms on the time scale (~55 ps) of the spectrometer. As shown in Fig. 7a-g, EISF of KaiC WT and the mutants were plotted against Q and then fitted using the following equation, assuming diffusion in an ensemble of spheres whose radii (a) follow a lognormal distribution 37 : where p is the fraction of atoms whose motions are outside the current instrumental energy window and therefore appear immobile; (1 − p) correspond to the fraction of atoms diffusing within the sphere ensemble; c is the median of the distribution; s is the variance in the natural logarithmic space; and j 1 denotes the spherical Bessel function of the first kind of order.
The results of EISF analysis support the current observations on the local dynamics. Consistent with the Q10 local WT value of 1.15 ± 0.11, the immobile fraction of KaiC WT was maintained at 0.6 in the physiologically functional range of temperature (Fig. 7h). In KaiC F470Y and KaiC DE , the immobile fractions were nearly unaffected by temperature change or mutations within the experimental error. On the other hand, the immobile fraction of KaiC F247A increased 25% as the temperature increased, despite its nearly temperature-compensated τ −1 value (Q10 local F247A = 0.97 ± 0.16). Interestingly, a reverse trend was confirmed in KaiC N390A . At the same time, all the temperature-sensitive mutants of KaiC exhibited smaller immobile fractions than KaiC WT at 302 K. This result that the temperature-dependent mutations shifted certain motions from immobile to mobile fractions (Fig. 7h) is directly or indirectly related to the systematic decrease in the D global values (Fig. 5h).
In KaiC WT and KaiC F470Y , atoms diffusing within a radius of 3-6 Å (Fig. 7i) constituted a major mobile component. By contrast, the mobile fractions of the temperature-sensitive mutants and KaiC DE were distributed mainly in reduced radii of 1-2 Å relative to KaiC WT . Thus, KaiC S258A , KaiC F247A , KaiC Q361E , and KaiC N390A are the temperature-dependent ATPase mutants with the reductions in D global (Fig. 5h), immobile fraction p (Fig. 7h), and amplitudes of local motions (Fig. 7i).

Discussion
Over the past decades, chronobiologists have sought a reasonable model that explains the three physiological properties of the circadian clock systems: self-sustained oscillation, temperature compensation, and synchronization [2][3][4] . The circadian clock of cyanobacteria is an ideal experimental system for this purpose, as its physiological properties can be studied in relation to the physicochemical properties of clock-related components at the molecular and atomic scales.
Among the three Kai proteins, KaiC is the core of the cyanobacterial clock system. In the presence of both KaiA and KaiB, KaiC exhibits a phosphorylation rhythm (Fig. 2b) whose frequency (f P ) is correlated with the ATPase activity of KaiC alone (Fig. 2a). For example, when the ATPase activity of KaiC doubles as a result of amino acid substitutions, the frequencies of both the in vitro systemscale and the cellular-scale rhythms also double ( Fig. 1a) 12,17,25 .
This causal relationship, in which properties are transferred through upward causation from bottom to top in the spatiotemporal hierarchy, was also confirmed for temperature compensation from the reaction to the cellular scale (Fig. 1a). The ATPase activity of KaiC WT is temperature-compensated; probably because of this, both the in vitro and in vivo rhythms 13 are also temperature-independent. This phenomenological interpretation is further supported by the four examples of temperature-sensitive mutants of KaiC. In contrast to KaiC WT (Q10 ATP WT = 0.89 ± 0.10), the f P values for KaiC S258A (Q10 ATP S258A = 1.73 ± 0.22), KaiC F247A (Q10 ATP F247A = 1.47 ± 0.05), and KaiC Q361E (Q10 ATP Q361E = 1.70 ± 0.15) increased in a temperature-dependent manner (Fig. 2c-e), and the f P value for KaiC N390A (Q10 ATP N390A = 0.47 ± 0.07) obeyed the inverse temperature dependence (Fig. 2f). The well correlated Q10 values between the ATPase activity and f P values in the four mutants (Fig. 2h) clearly demonstrate that the reaction-tosystem scale causal relationship arises from the temperaturecompensated ATPase activity of KaiC.
In the present study, we utilized QENS technique to test how far down the microscopic spatiotemporal scale the causality of the temperature sensitivity can be established. The Q10 global values were estimated to be~1.2 for KaiC WT and the mutants (Fig. 5h), confirming that the global motions of KaiC are accelerated by the temperature dependence of T/η(T) according to the Stokes-Einstein diffusion law. The limited temperature dependence was also observed for the parameters, such as τ −1 and D local , which characterize the local dynamics of KaiC. Even though the P-cycle and ATPase activity were accelerated in a temperature-dependent manner (Q10 ATP and Q10 fp as high as 1.7) in the three mutants (KaiC S258A , KaiC F247A , KaiC Q361E ) and decelerated in the inverse temperature-dependent manner (Q10 ATP and Q10 fp as low as 0.5) in one mutant (KaiC N390A ) (Fig. 2h), the Q10 local and Q10 D-local values for all the four temperature-sensitive mutants monotonously resulted in 1.0-1.2 ( Fig. 6h and i). The similar temperature dependence was also confirmed for the local dynamics of the temperature-compensated but short-period (KaiC F470Y ) and arrhythmic (KaiC DE ) mutants. Our current QENS observations indicate that the spatiotemporal scale at which causality of the temperature sensitivity is established is finite. Given that QENS detects the average image of the fluctuations of all the hydrogen atoms scattered in the protein molecule, we can conclude that causality is established down to the local dynamics only in a very limited part of KaiC (inset in Fig. 1a).
It is worth discussing our observations in the light of previous QENS studies on proteins other than clock-related proteins. To date, the QENS method has been used in many studies to characterize protein dynamics of small single-domain proteins 38,39 to larger and more complex molecular systems [40][41][42][43] , as well as to largescale conformational changes such as those that occur upon ligand binding 35,44,45 , pressurization 46-48 , unfolding 37,49-53 , and fibrillization 36,54 . The results reported to date for various protein samples indicate that the τ values are mostly distributed in the range of 1-20 ps (τ −1 = 0.05-1.0 ps −1 ) (see Grimaldo et al 29 . and references therein), although attention must be paid to the differences in measurement temperature, energy resolution, and analysis methods. The τ value for KaiC WT , 3.7-8.0 ps (τ −1 = 0.13-0.27 ps −1 ), is included in the above range, indicating that the internal motion of KaiC WT is neither exceptionally slow nor too fast.
More detailed comparisons of internal motions are possible by making reference for QENS data for Hb 35 and αSyn 36 , which were acquired at the same beamline with the same energy resolution (12 μeV/55 ps). As shown in Fig. 8  CO-bound Hb (CO-Hb), a fibril state of αSyn (fib-αSyn), and a monomeric but intrinsically disordered form of αSyn (mon-αSyn). This result points to the intrinsically low energy barrier for the local motions independent of protein folds or functions, supporting the present interpretation that the temperature sensitivity of the Kai oscillator does not correlate well with that of the picosecond to sub-nanosecond local dynamics. In the measured range of temperature, however, the τ −1 values for KaiC WT are 30-50% smaller than those of the four examples. Considering the insensitivity of the internal dynamics even upon an R(CO)-to-T(deoxy) allosteric transition of Hb (Fig. 8), we speculate that the slower jump-diffusion frequency of KaiC WT may be related to its function and structure as the circadian pacemaker. In fact, the τ −1 values for KaiC F247A and KaiC N390A were 50-70% reduced and 30-40% enhanced, respectively, relative to KaiC WT (Figs. 6h and 8), although not for all temperature-sensitive mutants.
In contrast to the featureless temperature dependence of the local motions, all the temperature-sensitive mutants shared the reduced global motions including the translational and rotational diffusions relative to KaiC WT and temperature-insensitive mutants (KaiC F470Y and KaiC DE ) (Fig. 5h). Two possibilities can be considered for the reduced D global .
The first is the deformation into less compact hexamers by the temperature-sensitive mutations. According to the crystallographic structure of KaiC 32,33 , every temperature-sensitive mutation is mapped onto its domain-domain interfaces. S258 is located in the outer radius side of the C2 ring (Fig. 1b); the side chain is hydrogen bonded to that of R326 in the neighboring C2 domain (Fig. 1c). This inter-C2-domain hydrogen bond is thus disrupted in KaiC S258A . F247 and Q361 neighbor each other in the C1-C2 interface (Fig. 1b), which is stabilized by a limited number of interdomain contacts with lower packing density than the C1-C1 and C2-C2 interfaces (Supplementary Note 4), exemplified by the nonbonded interactions associated with F247 and Q361 (Fig. 1b). The plane of the phenyl ring of F247 in the C1 domain lies nearly parallel to the C1-C2 interface (Fig. 1d), filling the loosely packed boundary through a hydrophobic interaction with L360 in the C2 domain. The F247A substitution is thus interpreted as the mutation that makes the intrinsically low-density interface even looser. Although the C1-C2 packing density will be little affected by the replacement of glutamine to glutamate in KaiC Q361E (Fig. 1b), a potential hydrogen bond between Q361 and either T238 or N245 in the C1 domain should be modulated or even disrupted (Fig. 1e). The N390A substitution disrupts an inter-C2domain hydrogen bond between the side chain of N390 and the main-chain carbonyl oxygen atom of G386 (Fig. 1f). Considering these structural features common to the temperature-sensitive mutations, we speculate that the looser inter-domain interactions have resulted in the loss of compactness of the hexamer structure.
The second is the reduction of an additional contribution from internal large-scale motions. Previous QENS study on the R(CO)to-T(deoxy) allosteric transition of Hb shows a certain similarity to our results 35 ; D global is notably increased upon the T(deoxy)-to-R(CO) transition while keeping τ −1 nearly unaffected. Authors of that study suggested that internal large-scale relative motions of two αβ dimers could partially contribute to D global in CO-Hb 35 . If this is the case, the decrease in D global is interpreted as a result of the suppression of domains movements within the KaiC hexamer by the temperature-sensitive mutations. In fact, several lines of evidence have pointed out the dynamic properties of inter-ring stacking of KaiC during the circadian cycle 55 .
Since the current temperature-sensitive mutations will not make the inter-domain interactions within the KaiC hexamer more stable, we believe that the contribution of the first case is greater than that of the second. This interpretation is also consistent with the decrease in the immobile fraction of the temperature-sensitive mutations (Fig. 7h). As shown in Fig. 2c-f, the temperature-sensitive mutations resulted in a preferential shortening and elongation of the dephosphorylation process in the P-cycle. The dephosphorylation process is also the timing of the ATPase activation in KaiC 12 . Taken together, these results suggest that delicate but close inter-domain interactions are important in regulating the temperature dependence of ATPase and dephosphorylation reactions in KaiC. This view is also supported by the fact that the KaiC hexamer undergoes a quaternary structure change in concert with ATPase and dephosphorylation events 33 .
Several studies have suggested that fluctuations and structural polymorphs of the clock proteins play important roles in the circadian clock systems. In mammalian systems, CKIδ-dependent phosphorylation is one of the key reactions that regulates period length and its temperature sensitivity 16 . In CKIδ, the temperature dependence of substrate affinity is compensated by the opposing temperature dependence of product affinity 18 . On the basis of molecular dynamics (MD) simulation, the authors of that study proposed that the temperature dependence of the amplitude of particular local fluctuations is reversed in the substrate-and product-bound forms, and that this reversal is one of the origins of the biochemical opposition. Structural polymorphs of the clock proteins, which have been confirmed by crystallography, NMR, and MD simulation, also play important roles in the substrate selectivity of CKIδ 56 and the interaction between CRY1/2 and the CLOCK-BMAL1 complex 57 . In the cyanobacterial system, the rhythmic stacking/unstacking of C1-and C2-rings is coupled to the P-cycle of KaiC 55 . The MD simulation of KaiC suggests a large-scale conformational change during ADP release from the C2 domain 58 . These previous observations are mainly based on the MD simulations and experiments on nanosecond to microsecond time scales, which are much slower than the time scales targeted by the current QENS study.
The mechanism by which a single amino acid substitution can exert a notable effect on reaction-or even system-level temperature compensation deserves further investigation. There is growing experimental evidence that ordinary enzymes actively utilize internal motions, thereby increasing the efficiency of Fig. 8 Temperature dependences of jump-diffusion frequency (τ −1 ) for KaiC, human hemoglobin (Hb), and α-synuclein (αSyn). Dashed and solid red lines represent the linear fits of the Arrhenius plot for KaiC WT using entire (283-317 K) and physiologically functional range (302-317 K) of temperature, respectively. Quasielastic neutron scattering (QENS) data of Hb and αSyn are taken from previous studies 35,36 ; activation energies of deoxygenated Hb (deoxy-Hb), CO-bound Hb (CO-Hb), fibrillized αSyn (fib-αSyn), and monomeric αSyn (mon-αSyn) are 5.2 ± 0.3, 5.2 ± 0.3, 3.7 ± 0.7, and 2.6 ± 1.9 kcal mol −1 , respectively. Values given near fitting lines represent the Q10 local values estimated for 303 K.
overall catalytic reactions [59][60][61] . Because functionally relevant fluctuations often refer to collective motions of atoms on slower time scales (μs to ms), care must be taken in discussing those motions in relation to the incoherent and fast (ps) dynamics detected in this study. However, several studies have demonstrated a linkage between ps-ns fluctuations and slower motions associated with catalytic reactions [62][63][64] . Thus, it is necessary to experimentally verify the possibility that the causality of temperature sensitivity in KaiC is still established on limited time scales longer than nanoseconds.

Methods
Expression and purification of Kai proteins. Glutathione S-transferase (GST)tagged versions of Kai proteins were constructed in pGEX-6P-1. Each Kai GST-fusion protein was expressed in E. coli BL21(DE3) and purified as reported previously 11,26 .
In vitro rhythm assay. P-cycles of KaiC WT and its mutants (0.2 mg/ml) were initiated by addition of KaiA (0.04 mg/ml) and KaiB (0.04 mg/ml) in H1-buffer at 303, 308, 313, and 318 K 25 . For KaiC F247A and KaiC Q361E , whose P-cycle were unstable or unobvious at the normal amount of KaiA, KaiA was added at a 2-fold higher concentration (0.08 mg/ml). Aliquots taken from the incubated samples were subjected to SDS-PAGE analysis. The relative abundances of four phosphorylation states of KaiC were quantified by densitometric image analysis of gel bands using the LOUPE software 65 .
Sample preparation for QENS. Every sample was prepared on site immediately before QENS measurements. KaiC WT and its mutants were purified on a gel- QENS experiments. QENS data were recorded using the near-backscattering spectrometer installed at beamline BL02 (DNA) in the Material and Life Science Experimental Facility of Japan Proton Accelerator Research Complex (MLF/J-PARC), Tokai, Ibaraki, Japan 30 . QENS spectra were recorded over an energy transfer range from −0.5 to 1.5 meV with energy resolution (12 μeV) enabling us to access motions faster than~55 ps. QENS data were collected at 283, 293, 302, 310, 313, and 317 K with exposure times of 6-10 h for each condition (520−615 kW). The obtained S(Q,E) were corrected for detector efficiency using a vanadium standard, and intensities were normalized as relative intensities using the standard after subtracting the contributions of the empty cell. The background spectrum of the D2-buffer was subtracted from each sample spectrum on the basis of scaling factors calculated from neutron scattering cross-sections, as reported previously 35,36 . The temperature was controlled by an LS350 (Lakeshore) with He conductance gas through a GM refrigerator and heat transfer from a cartridge heater installed in a copper block at the top of a sample cell, while monitoring the temperature at the bottom of the cell.
Estimation of the Q10 values. Datasets with more than two temperature points were subjected to Arrhenius plot analysis, and the resulted activation energy (E a ) was converted to the Q10 value at 303 K using exp{(E a /R)(10/T/(T + 10))}, where T and R represent temperature and the gas constant, respectively 65 . Other Q10 values were presented as the ratio of measurement results (R 1 and R 2 ) at two temperatures (T 1 and T 2 ) using Q10 = (R 2 /R 1 ) 10/(T2-T1) .
Calculation of D global . The D global values of KaiC WT were simulated at 302 and 313 K using a previously reported method 39 , the crystal structure (2GBL) 32 , and HYDROPRO 66 (Supplementary Note 2).