A thermodynamically consistent model of the post-translational Kai circadian clock

<div><p>The principal pacemaker of the circadian clock of the cyanobacterium <i>S. elongatus</i> is a protein phosphorylation cycle consisting of three proteins, KaiA, KaiB and KaiC. KaiC forms a homohexamer, with each monomer consisting of two domains, CI and CII. Both domains can bind and hydrolyze ATP, but only the CII domain can be phosphorylated, at two residues, in a well-defined sequence. While this system has been studied extensively, how the clock is driven thermodynamically has remained elusive. Inspired by recent experimental observations and building on ideas from previous mathematical models, we present a new, thermodynamically consistent, statistical-mechanical model of the clock. At its heart are two main ideas: <i>i</i>) ATP hydrolysis in the CI domain provides the thermodynamic driving force for the clock, switching KaiC between an active conformational state in which its phosphorylation level tends to rise and an inactive one in which it tends to fall; <i>ii</i>) phosphorylation of the CII domain provides the timer for the hydrolysis in the CI domain. The model also naturally explains how KaiA, by acting as a nucleotide exchange factor, can stimulate phosphorylation of KaiC, and how the differential affinity of KaiA for the different KaiC phosphoforms generates the characteristic temporal order of KaiC phosphorylation. As the phosphorylation level in the CII domain rises, the release of ADP from CI slows down, making the inactive conformational state of KaiC more stable. In the inactive state, KaiC binds KaiB, which not only stabilizes this state further, but also leads to the sequestration of KaiA, and hence to KaiC dephosphorylation. Using a dedicated kinetic Monte Carlo algorithm, which makes it possible to efficiently simulate this system consisting of more than a billion reactions, we show that the model can describe a wealth of experimental data.</p></div

Probability density of the time, per period of the phosphorylation cycle, the CII domain of a KaiC hexamer is bound to KaiA, Δ<i>t</i>_CII·KaiA.

<p>(A), or the CI domain has at least one KaiA dimer bound, Δ<i>t</i>_CI·KaiA (B). We compare situations with our standard value for the KaiA dissociation rate, (orange solid line) with a much lower value (blue dotted line). (A) For the CII domain, both values of the KaiA dissociation rate show unimodal distributions of the time KaiA is bound. This indicates that, during a period, KaiA binds to all hexamers in the ensemble equally likely. Of all hexamers, only 1% do not bind KaiA to CII at any time during the period. (B) For the CI domain, the time KaiA is sequestered by KaiC is roughly exponentially distributed, and is the same for both KaiA dissociation rates from the CII domain. Here, 30% of the hexamers do not sequester KaiA at all during a period, indicating many hexamers do not sequester KaiA during a cycle.</p

Model parameters relating to the CI domain are introduced in the theory section on the power cycle and their values are motivated in the results section.

<p>Energies are given in units of kT, were k is Boltzmann’s constant and T the temperature. Note that the rates of binding and unbinding of KaiA are for the CI domain. For rates relating to the CII domain, see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005415#pcbi.1005415.t002" target="_blank">Table 2</a>.</p

Parameters used in the alternative model without hydrolysis in the CII domain that are different from the values in our original model in Tables 2 and 3.

<p>Parameters values are listed for both the active and inactive conformations if they have been changed for either conformation.</p

Hydrolysis of ATP in the CI domain, and the subsequent binding of KaiB and KaiA, forms a hysteresis loop in the free-energy difference between the active and inactive state.

<p>Starting in the active state with no ADP bound to CI, the free-energy difference linearly decreases as ATP is hydrolyzed to ADP, as indicated by the green line. The number of ADP in CI is set by the competition between a fixed hydrolysis rate and a variable ADP off-rate, which is set by the phosphorylation state of the CII domains of the whole hexamer. Phosphorylation of the T site initially enhances the ADP off-rate, but the subsequent phosphorylation of the S-site will decrease the dissociation of ADP. The antagonistic effect of the cyclically phosphorylated T and S sites on the ADP dissociation rate causes a sharp transition in the CI binding pocket from ATP dominated to ADP dominated [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005415#pcbi.1005415.ref024" target="_blank">24</a>]. When there are 5 or 6 ADP nucleotides in the CI domain, the hexamer will flip to the inactive state, increasing the affinity for KaiB, which will then slowly bind inactive KaiC. When 6 KaiB monomers are bound to the hexamer, up to six free KaiA dimers will be sequestered from solution. This complex of KaiB and KaiA on the CI domain stabilizes the inactive state of KaiC further (red line). When all KaiA is sequestered, the dephosphorylation of the S-site in the CII domain will increase the nucleotide exchange rate in the CI domain, decreasing the ADP level and increasing the free-energy difference between the active and inactive state. This allows the hexamer to flip back to the active state. In the active state, KaiA and KaiB dissociate from KaiC, because they have a low affinity for CI in the active conformation.</p

Hydrolysis in the CII domain, without KaiA sequestration by the CI domain, is sufficient to generate the ordered phosphorylation of the T and S sites in a solution of KaiC and KaiA.

<p>In all panels, the bulk ATP fraction is 50%. (A-D) Heatmaps of the probability, , for a single hexamer, of having <i>n</i>_T phosphorylated threonine sites and <i>n</i>_S phosphorylated serine sites. Arrows indicate the net flux through a state, where the length is proportional to the magnitude of the flux. (A) KaiC and KaiA, with hydrolysis of ATP in both the CI and CII domains, shows a clear ordered cycle in state space. (B) Without hydrolysis in both the CI and CII domains, all the equations in our model obey detailed balance, and the fluxes in state space disappear. (C) When we remove hydrolysis in the CI domain, the fluxes in phosphorylation state space are little affected compared to panel A. (D) When there is hydrolysis in both domains, but no differential affinity of the CII domain for KaiA, the hydrolysis cycle in the CII domain does not couple to the phosphorylation cycle, and there is no ordered phosphorylation.</p

KaiA regulates the fraction of ATP in the CII binding pockets by increasing the nucleotide dissociation rates, in combination with the ATPase activity in the CII domain.

<p>A) Without KaiA bound to CII, the nucleotide dissociation rates are identically zero, and there is no nucleotide exchange with the bulk. Since ATP is hydrolyzed at a constant rate, , eventually all the binding pockets will be occupied by ADP. B) When KaiA is bound to the CII domain, it increases the dissociation rates of ADP and ATP, and , respectively, while leaving the affinities for ATP and ADP unchanged. Now, ADP in the CII domain is replaced by ATP at a rate that is faster than that at which ATP is hydrolyzed; this indeed increases the fraction of ATP in the binding pockets. For simplicity, we assumed equal diffusion limited association rates for ATP and ADP, such that the probability of binding ATP after ADP has dissociated is equal to <i>α</i>_ATP. The rate of the reverse pathway is proportional to the bulk ADP fraction, 1 − <i>α</i>_ATP.</p

Overview of the model and of how a hexamer progresses through its cycle.

<p>(A) KaiC hexamer with one monomer highlighted. Each monomer consists of a CI and a CII domain. Both domains can bind nucleotides, but only the CII domain can be phosphorylated, at a serine and a threonine residue. In the cartoon shown, CI has ATP bound while CII has ADP bound; in CII, the threonine residue is phosphorylated whereas the serine residue is not. Arrows indicate allowed reactions: ATP bound to either domain can be hydrolyzed to ADP, and bound ADP and ATP can be exchanged. The serine and threonine residues in the CII domain can be phosphorylated through phospho-transfer from a bound ATP and de-phosphorylated through phospho-transfer from the residue to a bound ADP. Panels B-G show a top view of the hexamer of panel A folded open for reasons of clarity; in each monomer, the CI domain is towards the center and the CII domain towards the outside. They illustrate how the hexamer traverses a typical phosphorylation cycle in a buffer with ATP. Starting with an unphosphorylated hexamer in the active state, the nucleotide exchange rate of the binding pockets of the CI domain with the bulk will be high such that they have predominantly ATP bound. A single KaiA dimer can bind to the CII domain, which enhances the nucleotide exchange rate in all the CII binding pockets such that the probability that ATP is bound increases. ATP in CII drives phosphorylation, where first the T sites and later the S sites are phosphorylated (C). When more serine sites than threonine sites are phosphorylated (D), the nucleotide exchange rate in the CI domain will drop and any remaining ATP in the binding pockets of CI will be hydrolyzed so that most CI domains have ADP bound. ADP in the CI domain will stabilize the inactive state, causing the hexamer to change conformation and allowing for KaiB monomers to slowly bind the CI domain (E). When six KaiB monomers are bound, the hexamer will sequester 6 KaiA dimers (F). Without KaiA available to bind to the CII domain, the nucleotide exchange rate in the binding pocket of the CII domain is low, and all ATP is hydrolyzed to ADP, allowing the hexamer to dephosphorylate. When no serine sites are phosphorylated, the nucleotide exchange rate in the binding pocket of the CI domain increases again, releasing the ADP and replacing it with ATP (G). The hexamer then flips back to the active state and releases the sequestered KaiA and KaiB.</p

Measured ATPase activity in ADP molecules produced per KaiC monomer per day (24 hours), under different conditions.

<p>First rows show results for parameters given in Tables <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005415#pcbi.1005415.t002" target="_blank">2</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005415#pcbi.1005415.t003" target="_blank">3</a>. The last four rows show results for alternative models. When , the ATP hydrolysis in the CII domain is blocked when KaiA is bound to CII. When, , the CII domain has a higher relative affinity for ADP when the monomer is in the D state. The experimental values for the combined ATPase activity of the CI and CII domain, given in the last column, are taken from [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005415#pcbi.1005415.ref025" target="_blank">25</a>], and are shown for comparison.</p

Model correctly predicts the ATP fractions in KaiC nucleotide binding pockets and the phase difference between this fraction and the phosphorylation level.

<p>Figures A-D show the fraction ATP/(ATP+ADP) in the nucleotide binding pockets of the CI domain (orange), the CII domain (green) and their sum (blue), in a 100% ATP solution, for different scenarios: KaiC initially unphosphorylated, no KaiA and KaiB present (A), KaiC initially phosphorylated, no KaiA and KaiB (B), initially unphosphorylated KaiC + KaiA (C) and KaiC + KaiA + KaiB (D). (A) The ATP levels drop monotonically due to the slow hydrolysis in both the CI and CII domain. (B) ATP fractions in dephosphorylating KaiC shows a clear trough in the ATP fraction of the CI domain due to the peak in the number of monomers in the S state (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005415#pcbi.1005415.g006" target="_blank">Fig 6A</a>), which temporarily decreases the ADP-off-rate in the CI domain. (C) For a system with KaiC and KaiA, the ATP fraction is higher in the CII domain and lower in the CI domain. KaiA increases the nucleotide exchange rate in the CII domain, and the resultant high phosphorylation level decreases the ADP dissociation rate in CI, which decreases the ATP fraction in this domain. (D) Full system with KaiA, KaiB and KaiC shows oscillations in the ATP fractions of both the CI and CII domains. (E) The phase difference and amplitude of the ATP fraction (% ATP, blue line) and the phosphorylation level (% phosphorylated monomers, black line) agree well with experimental results in [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005415#pcbi.1005415.ref039" target="_blank">39</a>]. (F) ATPase levels of the KaiC domains show oscillations proportional to the ATP fractions in the respective domains.</p