Effects of symmetry on coupled rotary molecular motors

TL;DR

This work analyzes how symmetry between two coupled rotary molecular motors affects energy transduction under constant and scaling driving schemes. By modeling Fo and F1 as Brownian rotors on coupled periodic landscapes and solving a Fokker-Planck description, the authors quantify local and average fluxes, input/output powers, and slippage, revealing that symmetry match can reduce output power under constant driving and has nuanced effects under scaling driving, including a disruption regime where motors decouple. Across both schemes, output power peaks at intermediate coupling, highlighting the value of flexible coupling for efficient operation. The study combines numerical simulations, barrier- and pathway-based analyses, and barrierless reductions to explain disruption and dominant transport channels, with implications for designing synthetic nanomotors and informing structure-based drug strategies. Overall, symmetry, driving scheme, and coupling strength jointly govern performance, offering design principles for optimized coupled rotary motors in biology and nanotechnology.

Abstract

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.

Figures (14)

• Figure 1: Bacterial ATP synthase structure. Left: molecular structure, adapted from pdb101bioenergy; right: schematic. $\rm{H}^{+}$ crosses the membrane through ${\mathrm{F_{o}}}$, turning the central crankshaft ($\gamma$ and $\epsilon$). Turning of the central crankshaft causes a conformational change in $\mathrm{F_1}$ that results in the chemical reaction of ADP and ${\rm P}_{\rm i}$ to produce ATP. ${\mathrm{F_{o}}}$ is mainly the multi-component C-ring C$_n$, which is composed of $n$ subunits. $\mathrm{F_1}$ has three $\alpha\beta$ subunits across all species.
• Figure 2: a,b) Output power as a function of the number $n_{\mathrm o}$ of ${\mathrm{F_{o}}}$ subunits, for different coupling strengths $\beta E_{\rm c}$. Vertical lines indicate $n_1 = n_{\rm o} = 3$. Star labels disruption. c,d) Output power as a function of the coupling strength, for different numbers $n_{\mathrm o}$ of ${\mathrm{F_{o}}}$ subunits. a,c) Constant driving forces $\beta\mu_{\rm o} = 4$ and $\beta \mu_1 = -2$. b,d) Scaling driving forces $\beta \mu_{\rm o} = 0.5 \times n_{\rm o}$ and $\beta \mu_1 = -\frac{2}{3}\times 3 = -2$. $n_1 =3$ throughout.
• Figure 3: Flux as a function of the number $n_{\mathrm o}$ of ${\mathrm{F_{o}}}$ subunits, for different coupling strengths $\beta E_{\rm c}$, using a scaling driving force. a) Hollow circles: ${\mathrm{F_{o}}}$ flux $J_{\rm o}$; solid squares: $\mathrm{F_1}$ flux $J_1$. Star labels disruption. b) Slippage flux $J_{\rm slip}$. $n_1 =3$ throughout.
• Figure 4: The scaling driving force $\beta \mu_{\rm o}^{*}$ that maximizes output power, as a function of coupling strength $\beta E_{\rm c}$, for various barrier heights. Dashed lines are linear fits. $\beta \mu_1 = -2$ and $n_1 = 3$ throughout.
• Figure 5: Parametric plot of output power and input power with varying $n_{\rm o}$ under a scaling driving force $\mu_{\mathrm o}$, for several coupling strengths $\beta E_{\rm c}$. As a guide to the eye, curves connect points for different $n_{\mathrm o}$. Points with shades of purple represent $n_{\rm o} = 6,\ 12,\ 18$ from light to dark. Star labels disruption. $\beta \mu_1 = -2$ and $n_1 = 3$ throughout.