Hourglass model for a protein-based circadian oscillator

Abstract

Many organisms possess internal biochemical clocks, known as circadian oscillators, which allow them to regulate their biological activity with a 24-hour period. It was recently discovered that the circadian oscillator of photosynthetic cyanobacteria is able to function in a test tube with only three proteins, KaiA, KaiB, and KaiC, and ATP. Biochemical events are intrinsically stochastic, and this tends to desynchronize oscillating protein populations. We propose that stability of the Kai-protein oscillator relies on active synchronization by (i) monomer exchange between KaiC hexamers during the day, and (ii) formation of clusters of KaiC hexamers at night. Our results highlight the importance of collective assembly or disassembly of proteins in biochemical networks, and may help guide design of novel protein-based oscillators.

FIG. 1

Schematic of model circadian oscillator in cyanobacteria, including proposed KaiC-monomer exchange during the day and cluster formation at night. KaiC monomers (small circles) self-associate in the presence of ATP to form hexamers. During daylight hours, KaiA (not shown) accelerates phosphorylation of KaiC, which can have 0 (white), 1 (gray), or 2 (black) phosphoryl groups. During this period, KaiC-monomer exchange helps synchronize hexamer phosphorylation levels. Fully phosphorylated KaiC hexamers (hexagons) can complex with KaiA and KaiB (not shown), allowing KaiC to form clusters of N_c hexamers (N_c = 8 shown). During the night, KaiC in clusters dephosphorylates (lighter shaded hexagons). When a cluster’s phosphorylation level reaches P_min, the cluster breaks apart, freeing KaiC hexamers to begin the day cycle again.

FIG. 2

Model circadian oscillations showing fraction of phosphorylated KaiC (solid curves) and fraction of KaiC in clusters (dashed curves) from Eqs.(1)–(5). In (a)–(d) N_o = 6, N_c = 12, γ = 0.129h⁻¹, the total number of oligomers, O₀ = 500, and with initial conditions M₀(t = 0) = (0.7)2 N_oO₀, and M_2P(t = 0) = 0. (a) Wild-type behavior, with parameters α/γ = 8.0, $\beta O_0^{N_c-1}/\gamma =4.88\times {10}^{14}$, P_min = 14. (b) Long-period mutant with parameters as in (a) except for a lower phosphorylation rate α/γ = 6.0. (c) Short-period mutant with parameters as in (a) except for a higher dephosphorylation rate γ′/γ = 1.1. (d) Short-period mutant with parameters as in (a) except for earlier cluster opening, P_min = 29.

FIG. 3

Oscillation phase diagram for different sized oligomers N_o and different sized clusters N_c. White squares are values of N_o and N_c for which Eqs.(1)–(5) produced a limit-cycle oscillator for some choice of parameters. For other N_o and N_c values, gray squares indicate that oscillations occurred in a stochastic simulation with O₀ = 500 oligomers. Black squares indicate no oscillations. Inset: Oscillations of KaiC phosphorylated fraction for a stochastic simulation with 500 KaiC hexamers, N_o = 6, N_c = 6, α/γ = 9.5, $\beta O_0^{N_c-1}/\gamma =3.125\times {10}^7$, and P_min = 14.