A hybrid phase-synchronization framework for rotary motors: Discrete dynamics in ATP synthase and continuous dynamics in the bacterial flagellar motor

ATP synthase functions as a dual-rotor molecular motor, with the F${}_0$ and F${}_1$ units stepping in mismatched increments, yet achieving near 100% chemomechanical efficiency of the F${}_1$ motor under near-reversible conditions. This raises the question of how stable phase synchronization is maintained despite such symmetry mismatch. We address this problem by modeling ATP synthase as a driven oscillator system in which the central elastic stalk acts as a torsional filter, transmitting and modulating torque. We propose a hybrid synchronization model that integrates continuous and discrete dynamics, governed by a torsional energy-dependent mixing parameter that determines interpolation between limits. The resulting single hybrid phase synchronization equation captures both gradual continuous phase drift and discrete pulsed entrainment. This framework reproduces key experimental features, including stable synchronization, intermittent slip events in ATP synthase, and recovery dynamics under varying loads, and offers testable predictions. In the discrete limit, the model specializes to a van Slooten-type pulse map that accords with the well-established 120° stepping of F${}_1$-ATPase; in the continuous limit, it reduces to an Adler-type equation appropriate for the near-constant-torque behavior of the bacterial flagellar motor. This framing unifies two historically separate descriptions without requiring a literal mode change within a single molecule and clarifies how elastic energy can interpolate between limits via the mixing parameter $\sigma \left(E\right)$. The hybrid model proposes that ATP synthase and the bacterial flagellar motor exploit elastic filtering and energy-regulated regime interpolation between limits to achieve robust rotational coordination, providing new insights into the dynamics of biological rotary motors.