Revealing circadian mechanisms of integration and resilience by visualizing clock proteins working in real time

The circadian clock proteins KaiA, KaiB, and KaiC reconstitute a remarkable circa-24 h oscillation of KaiC phosphorylation that persists for many days in vitro. Here we use high-speed atomic force microscopy (HS-AFM) to visualize in real time and quantify the dynamic interactions of KaiA with KaiC on sub-second timescales. KaiA transiently interacts with KaiC, thereby stimulating KaiC autokinase activity. As KaiC becomes progressively more phosphorylated, KaiA’s affinity for KaiC weakens, revealing a feedback of KaiC phosphostatus back onto the KaiA-binding events. These non-equilibrium interactions integrate high-frequency binding and unbinding events, thereby refining the period of the longer term oscillations. Moreover, this differential affinity phenomenon broadens the range of Kai protein stoichiometries that allow rhythmicity, explaining how the oscillation is resilient in an in vivo milieu that includes noise. Therefore, robustness of rhythmicity on a 24-h scale is explainable by molecular events occurring on a scale of sub-seconds.

(b and c) A schematic of the differential affinity of the interaction between KaiA and KaiC (based on van Zon et al, 2007 1 ). b. Phosphorylation and dephosphorylation of KaiC hexamers in complexes with KaiA; j is a state variable for population phosphorylation (j = 0 to 6 in this model). c.
Simulating differential affinity of KaiA for KaiC hexamers as a function of KaiC phosphorylation status. The term a is a measure of the affinity of KaiA interaction with KaiC as assessed by the offrate of KaiA binding; a = 1 means there is no difference in KaiA off-rate as a function of KaiC phospho-status, whereas a > 1 means that KaiA off-rate speeds up with increasing KaiC phosphorylation as a power law in a. ODEs were all rescaled to the initial KaiC concentration ([C 0 ]) and circadian timescale phosphorylation (k ps ), dephosphorylation (k dps ), and transition rates (b 0 , f 6 ) of 1.0 hr -1 were assumed for simplicity. We investigated the system dynamics as both k Af and k j Ab were varied, where the latter rate, k j Ab = k Ab α j with α > 1 (and j = 0 to 6) was varied in simulations by varying α (Supplementary Fig. 1d). A minimum "slow" KaiA-KaiC off-rate of k Ab = 0.1 hr -1 was assumed in Supplementary Fig. 1 and the scaled KaiA on-rate was varied, k A+ ≡ k Af [C 0 ] (hr -1 unit). The initial conditions in simulations were , with the rest of the complexes initially set to zero. These simple PDDA model simulations suggest an optimal range of PDDA (α), where either insufficient PDDA (low α) or excess PDDA (high α) do not allow stable oscillations; furthermore, the amplitude variation as a function of α shows that the system smoothly transitions through a maximum as α increases monotonically. The latter effect of loss of oscillations for high α can be understood qualitatively from the model; for sufficiently large α the system will be unable to phosphorylate the higher states (j = 4 to 6, for example) as the probability of forming higher phosphorylated KaiA-KaiC complexes becomes too low. We expect this result (a limited range of PDDA which improves oscillatory dynamics) to be a general result in more complicated PDDA models which we investigate below.

Simulating site-dependent phosphorylation and de-phosphorylation
To investigate the possible role of PDDA in system dynamics including site-dependence we updated a previously published stochastic matrix model (a matrix of N rows = N hexamers by 6 columns = protomers) to simulate KaiC hexamer dynamics and phosphoform transitions using Monte Carlo for reaction probabilities (per timestep and per molecule). Each monomer can be in one of four phosphostates (U= S/T, T = S/pT, D = pS/pT, S = pS/T) encoded by the labels (0,1,2,3).
Previously published linear interconversion rates (fitted to partial reaction data in the presence or absence of KaiA) between the four phosphoform states (S/T → S/pT, etc) were applied as transition probabilites (per time) to single protomers in simulations and compared with a 4 th order Runge-Kutta solution to the corresponding linear differential equations to confirm the stochastic simulation algorithm (Supplementary Fig. 9a). Previously published linear interconversion rates between phosphoforms were initially tested 2 , Supplementary Table 1.
We define a measure of the degree of synchronization in the hexamer population by counting the number of each phosphoform type in each hexamer and computing the Euclidean "distance" between ",$ . In the linear interconversion simulations (without PDDA) of the phosphorylation and dephosphorylation phases synchronization of the population decreases during the phosphorylation phase and gradually increases again during dephosphorylation ( Supplementary Fig. 9a). In all stochastic simulations we also allowed for the possibility of "monomer exchange" (random shuffling of monomers among hexamers) between any hexamers during any phase [3][4][5] ; the default rate of exchange in simulations was set faster than the fastest transition rate; for the simulations shown in the manuscript a monomer exchange rate of 1.0 hr -1 was used unless stated otherwise.

Simulating KaiA-KaiC reactions and PDDA
Previous fits to partial reaction data parameterized the effect of varying KaiA concentration on the phosphoform transition rates whereas we wish to directly investigate the effect of PDDA of KaiA interacting with KaiC. We assume that these previously estimated effective phospho-transition rates (with rates fitted to the linear model above) are physically a result of (differential) on-off kinetics of KaiA to KaiC as suggested by the experimental data.
KaiA on-off dynamics occur much faster (timescale s -1 ) than the effective phosphoform transition rates (timescale hr -1 ) so that simulating KaiA-KaiC reactions in "pseudo" steady state should be a reasonable approximation where N(T) is the number of S/pT protomers in the hexamers, etc., and γ i > 0 are relative weights for altering the dissociation probability of the KaiA-KaiC complex for each phosphoform type (increased phosphorylation of a hexamer lowers the probability of complex formation), k d0 is the dissociation constant for completely unphosphorylated KaiC, and c > 0 is a scaling factor parameterizing the overall strength of PDDA (c = 0 corresponds to "non-PDDA" in the main manuscript figures) . Relative weights were selected based on the measured relative off-rates (t bound ) from the experimental data in the ratio γ 1 : γ 2 : γ 3 = 1.0 : 3.9 : 2.3 (Fig. 3). The model assumes the dissociation rate(s) are inversely proportional to the measured t bound values, so we chose ratios suggested by Fig. 3.
To be consistent with the apparent high affinity measured in the mutant phospho-mimic of the dephosphorylated state (KaiC-AA, Fig. 3 Supplementary Fig. 9b); the simulated population average affinity varies by approximately a factor of 3 for the standard parameters given above ( Supplementary Fig. 11b,d). The partial reaction simulations indicate that the inclusion of PDDA increases the relative fraction of pS/T states at the expense of pS/pT states which would be expected from a decreased affinity of KaiA for more highly phosphorylated hexamers. The transition rates assumed for simulation figures in the manuscript are given below (Supplementary Table 2), with a physical interpretation that these rates reflect probabilities (per time) of the phospho-transitions when bound-state KaiA-KaiC complexes transiently form. Since PDDA results in dephosphorylation transitions the rates are modified from the original published values. We also considered additional non-PDDA controls (c = 0, varying k d0 ) to determine the dynamical effect of PDDA, in which non-PDDA can promote the pS/pT → pS/T transition ( Supplementary Figs. 10b, 11f, 11h).

Simulating KaiB-KaiC reactions and KaiA sequestration
PDDA of KaiA for KaiC by itself is insufficient for generating oscillations in population phosphorylation as the KaiA-KaiC partial reactions do not appear to oscillate (experimentally) for any choice of initial KaiA and KaiC concentrations. KaiB seems to be required to produce sustained population dephosphorylation by sequestration of KaiA in stable complexes, thereby removing "active" KaiA. Based on previous experiments and models 2,6-12 , we consider in simulations two simple "switch" based models for KaiB binding: (A) a pS/T-based switch and (B) a dual (pS/T and pS/pT) switch. In Model A, a minimum number of pS/T protomers per hexamer is required to initiate KaiB binding to an individual hexamer ( Supplementary Fig. 11c-f). In Model B, a minimum number of pS/T or pS/pT protomers per hexamer is required to initiate KaiB binding ( Supplementary Fig. 11 a-b, g, h). Model B was used in the simulations in Figures 6 and 7 because we are persuaded that there are effective allosteric switch-like mechanisms for KaiB binding 6,9 . Provided this condition is met, KaiB-KaiC complexes form at some rate, k +B . Each of these KaiB-KaiC complexes is assumed to rapidly sequester, on average, n seq free KaiA 13 , so that where (BC) indicates the number of KaiB-bound KaiC. Given a KaiB-KaiC complex we allow a general (slow) off-rate for de-sequestration of KaiA and release of KaiB, k-B. This rate constant needs to be sufficiently slow so that sequestration is effective, yet the inclusion of such a rate allows a "small" fraction of free KaiA during the dephosphorylation phase, which is consistent with the experimental observations of Clodong and co-authors and Kageyama and co-authors 3,14 . When a given hexamer dephosphorylates, we assume the rate of KaiB-KaiC complex dissociation (k relax ) occurs on a similar timescale (~1.0 hr -1 ) to the fastest dephosphorylation reactions -this transition is implemented with a similar simple threshold based on the number of pS/T protomers per hexamer falling at or below some minimum value, B-,thresh . We use the following parameters summarized in Supplementary Table 3 to simulate the KaiB-KaiC and KaiA-KaiB-KaiC interactions:  Fig. 11, panels b & d).

Supplementary
Therefore, for comparisons with non-PDDA controls we also considered both a locked high affinity constant of KaiA for KaiC, k d (k d = 0.01) and an intermediate affinity k d (k d = 0.25) for the non-PDDA controls in Supplementary Figs. 10b, 11fh, 12bc, 13bd.

Simulation of the KaiA-KaiB-KaiC Oscillatory system, Oscillatory parameter space, and Effect of PDDA
A simulation is specified by choosing the number of hexamers (N hex = 1000 hexamers is typically used), the initial ratio of KaiA dimer to KaiC hexamer (A 2 /C 6 ) , the phosphoform transition rates above, the KaiA-KaiC complex formation parameters, and the KaiB-KaiC parameters listed above (we assumed non-limiting KaiB). A typical simulation output for N = 1000 hexamers, initial A 2 /C 6 = 1.15, +/-PDDA indicates the onset of oscillations from KaiA (autocatalytic) sequestration by KaiB-KaiC complexes (k +B = 0.5 hr -1 shown) in both models, Supplementary Fig.  11a,c. An additional non-PDDA control (c = 0, k d0 = 0.25) was also used to determine the dynamical effect of PDDA, in which we allow non-PDDA to promote the pS/pT → pS/T transition in both models ( Supplementary Fig. 11f,h). The general qualitative modeling results were checked for a range of PDDA strengths (parameter c). For varying KaiA concentrations ( Supplementary Fig. 12a-c) and varying KaiB effective on-rates ( Supplementary Fig. 13a-d), oscillatory period estimates were based on peaks and troughs from smoothed total percent phosphorylation simulation output data for each simulation run; the existence or non-existence of oscillations from such a simulation run was typically based on ~200 h simulated time course intervals. The resulting time course simulations were visually inspected to confirm the absence or presence of noticeable oscillations for Fig. 6 in the main manuscript. Simulations were repeated using varying initial random seeds to check that stochastic effects did not change the general qualitative conclusions from comparison of simulations +/-PDDA.
PDDA enhances resilience whether Model A or Model B switching is considered. As discussed in the main text, PDDA in Model B enhances resilience to noise that is due to fluctuations in KaiA:KaiC stoichiometry. In Model A, without PDDA the transition from the phosphorylation phase to the dephosphorylation phase (pS/pT→pS/T) is disfavored because there is no inactivation of KaiA (by sequestration of KaiA) until pS/T becomes dominant, and until that happens, KaiA continues to stimulate the phosphorylation of KaiC to the pS/pT state. However, with PDDA the transition can be promoted by the reduced affinity of KaiA to hyperphosphorylated (pS/pT) KaiC. The pS/T-only switch (Model A) has been used to simulate the cyanobacterial clock mechanism in oft-cited models (e.g., Rust et al. 2007 2 ), and in this case +PDDA enables oscillations of an appropriate period over a broader range of KaiB on-rate (e.g., compare Supplementary Fig. 13c), assuming the standard concentrations of KaiA and KaiC (1.3 and 3.4 µM, respectively). This phenomenon is potentially important because the demonstration of slow "fold-switching" of KaiB introduces a much slower effective K +B term for KaiB binding to KaiC pS/T than is possible to accommodate in previous models unless an unrealistic "hard" KaiA sequestration is invoked 10 . Conversely, PDDA can generate stable oscillations for a wide range of physiologically reasonable effective KaiB/KaiC association rates, and therefore, whichever model is used for the KaiC phosphostatus that promotes KaiB binding, PDDA plays a crucial role to enhance resilience.