Effects of symmetry on coupled rotary molecular motors

As engineering advances toward the nanoscale, understanding design principles for molecular motors becomes increasingly valuable. Many molecular motors consist of coupled components transducing one free-energy source into another. Here, we study the performance of coupled rotary molecular motors with different rotational symmetries under constant and scaling driving forces. Under constant driving and strong coupling, symmetry match between the motors decreases the output power. In contrast, under a scaling driving force, the output power is not sensitive to symmetries. However, driving the upstream motor too strongly reduces the downstream motor’s output power, leading to a perhaps counterintuitive phenomenon we term disruption, in which the two motors become disconnected. Across both driving schemes, output power peaks at intermediate coupling, confirming the value of flexible coupling. Beyond providing insights into biological motors, these findings could inform the future design of synthetic nanomotors and structure-based drugs.

💡 Research Summary

The paper investigates how rotational symmetry and coupling strength affect the performance of two coupled rotary molecular motors, modeled after the biological Fo–F1 ATP synthase system. The authors formulate a stochastic Fokker‑Planck description for the joint probability distribution of the angular coordinates of the upstream motor (Fo) and downstream motor (F1). Each motor experiences a periodic potential proportional to cos (nθ), where n denotes the number of subunits (or energy barriers). A spring‑like coupling potential Vc ∝ cos(θFo − θF1) enforces in‑phase motion, with coupling strength Ec. Driving forces μ0 and μ1 represent free‑energy input per full rotation for Fo and F1, respectively, and can be either (i) constant (fixed free‑energy per rotation) or (ii) scaling with the number of subunits (μ0 ∝ no, μ1 ∝ n1).

Numerical integration of the Fokker‑Planck equation yields steady‑state average fluxes ⟨J0⟩ and ⟨J1⟩, from which input power Po = 2π μ0 ⟨J0⟩ and output power P1 = ‑2π μ1 ⟨J1⟩ are computed. The slippage flux Jslip = ⟨J0⟩‑⟨J1⟩ quantifies how much of the upstream rotation fails to be transmitted downstream.

Key findings:

1. Constant driving, strong coupling – When the number of Fo subunits no equals the fixed three subunits of F1 (symmetry match), both motors face coincident energy barriers simultaneously. This maximizes slippage and reduces ⟨J1⟩, leading to the lowest output power. Thus, symmetry matching is detrimental under tight coupling and fixed driving energy.

2. Intermediate coupling (βEc ≈ 10) – A modest mismatch (no ≠ n1) allows Fo to advance a barrier first, “helping” F1 over its barrier. Output power rises slightly relative to the matched case, but the effect is modest.

3. Scaling driving (μ0 ∝ no) – Output power becomes largely independent of the actual subunit count; the dominant factor is the magnitude of μ0. However, as μ0 grows, it competes with the fixed coupling Ec. Beyond a critical no* (or μ0*), the coupling can no longer enforce synchrony, leading to a phenomenon the authors call disruption: the downstream motor’s output power becomes negative while the upstream motor still consumes positive input power. This reflects a breakdown of energy transduction; each motor rotates according to its own driving force, and slippage spikes dramatically.

4. Coupling strength dependence – In the weak‑coupling limit the two motors act independently, giving negligible output power regardless of symmetry. In the strong‑coupling limit the motors lock together, but symmetry match again reduces performance because of barrier overlap. Across both driving schemes, an intermediate coupling strength maximizes output power, balancing thermal fluctuations (which aid barrier crossing) against mechanical rigidity (which transmits torque).

The authors relate these results to real biological systems. In ATP synthase, the Fo C‑ring subunit number varies from 8 to 17 across species, yet experimental evidence suggests the mechanical coupling between Fo and F1 is relatively invariant. This implies evolution has selected a flexible coupling regime that tolerates symmetry mismatch while preserving efficient energy conversion.

From an engineering perspective, the study offers several design guidelines for synthetic nanomotors: (i) prioritize tuning the coupling strength relative to the driving torque rather than focusing on matching subunit numbers; (ii) avoid excessive driving forces in scaling‑force designs, as they can induce disruption; (iii) aim for intermediate coupling to exploit thermal noise for barrier crossing while maintaining sufficient torque transmission.

In summary, the work demonstrates that rotational symmetry alone does not guarantee optimal performance in coupled rotary motors. Instead, the interplay between symmetry, coupling elasticity, and the nature of the driving force determines output power and the likelihood of disruption. These insights deepen our understanding of natural molecular machines and provide concrete principles for the design of future synthetic nanomechanical devices.