Paper detail

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

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.