Optimal Discretization in Hour-Glass Molecular Clocks Driven by Oscillating Free Energy

Abstract

Hour-glass clocks do not free-run; they keep time by riding an external rhythm. Motivated by the primordial KaiBC system in cyanobacteria, we study a driven, finite-state molecular clock that advances through a small number of biochemical states under an intrinsic driving energy and a rotating energy landscape set by day-night metabolism. In the continuum limit, coherence is maximized at a resonant operating point where the intrinsic drift matches the driving frequency. In realistic clocks with a finite number of states, discreteness matters: as the rotating landscape sweeps over a lattice of states, it generates a small and high frequency vibration of the collective phase that makes timing inaccurate. Combining the resonant cost with this discreteness penalty yields a trade-off in the number of states: few states are energetically cheap but noisy; many states are precise but costly. The optimum lies at moderate discretization (typically five to fifteen states) and an environmental coupling that is strong enough for responsiveness yet weak enough to avoid large discrete-state vibrations. These design rules rationalize why KaiC's hexameric architecture falls near the predicted optimum and suggest a general principle for hour-glass clocks across organisms.

Figures (3)

• Figure 1: Schematic illustration of the model and the resonant condition. (a) The system consists of $N$ states, representing the phase $n2\pi/N$, arranged on a ring with periodic boundary conditions. The forward and backward reactions are driven by both an intrinsic force and a time-dependent external potential. (b) Steady-state distribution in the rotating frame $\theta=\phi-\omega t$ under the continuous limit. The figure shows the distribution for different values of the parameter $\alpha=(\omega-\Omega^{'}E_i)/\Omega^{'}$. The distribution is narrowest at resonance ($\alpha=0$). The green dashed line represents the approximate theoretical prediction. (c) The order parameter for different values of $\alpha$, plotted on the complex plane. (d) Analytical and numerical results for the dissipation as a function of $E_o$ for various intrinsic driving strengths $E_i$ (with $N=500$). As $E_o$ increases, the dissipation for different $E_i$ values converges to the dissipation at resonance.
• Figure 2: Dissipation and phase vibration of the hour-glass clock. (a) Time-averaged collective amplitude $\left<R\right>$ as a function of intrinsic driving $E_i$ and step size $\Delta\phi$. The asterisks mark the optimal driving $E_i^{\ast}$ that yields the maximum amplitude for a fixed number of states $N$. The solid curves represent the theoretical predictions for $E_i^{\ast}$. Since $R$ exhibits small vibration over time, we plot its time-averaged value, $\left<R\right>$. (b) Relation between the optimal intrinsic driving $E_i^{\ast}$ and the step size $\Delta\phi$ for different partitions $I$. The case $I=1$ yields the minimum $E_i^{\ast}$. Parameters used:$\Omega'=0.055, \omega=2\pi/24$ (c) Time evolution of the collective amplitude $R$ and collective phase $\Psi$ for a small number of states $N=4$ and a large external potential $E_o=10$. Both the amplitude and phase vibrate with a period of $T/N$. When $E_o/N$ is large, the total phase $\omega t +\Psi$ is strongly pinned to the discrete phases $n2\pi/N$ of the clock, leading to large vibration in $\Psi$. (d) Time evolution of different Fourier modes in the rotating reference frame $\tilde{p}(f)$ for $N=6$. The amplitude of mode $n$ is identical to that of mode $N-n$. Modes closer to $N/2$ exhibit a smaller amplitude.
• Figure 3: Trade-off between driving cost and phase vibration controlled by the external signal strength $E_o$ and the clock's resolution $\Delta\phi$. Simulation results are numerically obtained at the resonant $E_i$ for each set of parameters ($\Omega'=0.055, \omega=2\pi/24$). (a) Both the collective amplitude $\left<R\right>$ and phase vibration increase with $E_o$. A larger number of states $N$ (higher resolution) is required to suppress this vibration. The black dashed line represents the approximate theoretical value. (b) However, a larger $N$ leads to higher dissipation. Thus, $N$ must be chosen within an optimal range to suppress phase vibration in an energy-efficient way. (c) Contour plots of phase vibration $\Delta\Psi$, collective amplitude $\left<R\right>$, and negative dissipation rate $-\dot{W}$ in the $E_o$-$\Delta\phi$ parameter space. Balancing the dissipation and phase vibration, an optimal state number is $N=5\textendash15$. Within this range, maximizing $E_o$ emerges as a viable optimization strategy.