It Takes Two to Make a Thing Go Right: Boosting Current in Coupled Motors

TL;DR

The study tackles boosting current in catalysis-driven synthetic motors operating under loose mechanochemical coupling by coupling two motors on offset tracks. It employs two complementary models—an explicit particle-based MD framework and a coarse-grained jump-diffusion model—to show that mechanical coupling can produce a real, single-digit boost in current $j$ by elevating activity $\mathcal{A}$, even as the bias $\Gamma$ is partially diminished; raising the fuel concentration helps recover the lost bias. A key insight is a two-step design rule: first increase activity via coupling (e.g., offset geometry with intermediate coupling), then restore directionality by stronger driving (higher fuel or $k_\mathrm{attach}^{\mathrm{far}}$). The results imply that cooperative coupling can turn slow, loosely coupled motors into faster collective machines, though achieving orders-of-magnitude boosts would require new architectures that move beyond discrete, rare hops between fixed binding sites. Overall, the work provides mechanistic and design principles for constructing cooperative molecular machines with enhanced current under nonequilibrium fueling.

Abstract

Catalysis-driven synthetic molecular motors operate in a loose mechanochemical coupling regime, one in which a decomposition of a fuel molecule does not reliably produce a forward step. In that regime, stochastic backward steps can significantly degrade the motor's current, prompting us to ask whether mechanically coupling multiple such motors can boost their averaged current. By simulating rotaxane-based motors with two classes of models--particle-based nonequilibrium molecular dynamics and jump-diffusion models--we show that current boosts are physically achievable. Our observed boosts, which amplify current by single-digit factors, emerge when coupling between motors can increase the activity, speeding up the rate of both forward and backward steps. In doing so, the bias for preferring forward steps actually degrades, but the lost bias can be largely recovered by raising the fuel concentration, demonstrating a general design strategy: amplify activity through coupling and restore bias through stronger driving.

Figures (12)

• Figure 1: Current boosts in coupled catalysis-driven linear motors.(a) The system consists of a ring (green) diffusing along a track (black) with four equally spaced binding sites (orange) and adjacent catalytic sites (white). The ring preferentially binds to the binding sites, while white catalytic sites catalyze the decomposition of full tetrahedral clusters (FTC) into empty tetrahedral clusters (ETC) and a central particle (C), which obstructs the ring's movement when attached to the catalytic site. The track is infinitely extended using periodic boundary conditions. A nonequilibrium steady state is maintained by applying a high chemical potential for FTC ($\mu_\mathrm{FTC}$) and low chemical potential for waste products ($\mu_\mathrm{ETC}$ and $\mu_\mathrm{C}$), driving net rightward motion of the ring. (b) Two identical motors are arranged on parallel tracks in an offset configuration. The centers of mass of the two rings are coupled by a harmonic potential with spring constant $\kappa$. (c) When a single motor is operated under the nonequilibrium conditions, it generates an average shuttling-ring current $j$ indicated by the gray line. A boost above this uncoupled value is not possible when tracks are aligned but is possible for the offset configuration provided the coupling is strong enough but not too strong. Means and error bars are collected from 100 independent simulations for each motor, all simulated with equal driving force $\Delta \mu = \mu_{\rm FTC} - \mu_{\rm ETC} - \mu_{\rm C}$ and plotted with reduced units defined in the main text. Simulation methodology is discussed in Methods. Numerical details about interaction strengths and driving forces are provided in the SI.
• Figure 2: Current decomposition into activity and bias.(a) For an offset motor configuration, current $j$ compared to its single-motor uncoupled value $j_0$ rises then falls as coupling between two motors is strengthened. This effect is productively rationalized by decomposing the coupled current into contributions from an activity ($\mathcal{A}$) and a bias ($\mathit\Gamma$) relative to the uncoupled values ($\mathcal{A}_0$ and $\mathit\Gamma_0$) then observing that the amplification of activity exceeds the loss of bias. (b) With no fuel and no coupling, the ring has symmetric jumps regulated by a $h = 6.3$$k_{\rm B} T$ free energy barrier that the ring must jump over to get to a neighboring binding site. See SI Sec. 5 for free energy calculations of the MD model that generate the barrier-hopping cartoon. Adding fuel introduces a directional bias because the shuttling ring at a binding site sterically occludes (green shading) fuel from adding blocking groups at the nearby catalytic site. (c) Coupling two motors with offset tracks and an intermediate value of $\kappa$ yields shallower free-energy basins at the binding sites. More rapid escapes from those basins give an increase in activity, but they also increase the time the ring spends away from its binding sites, thereby decreasing the kinetic asymmetry that generates bias.
• Figure 3: Current is not boosted when the shuttling ring is weakly attracted to binding sites. When the uncoupled motor had a $6.3 k_{\rm B} T$ deep attractive well holding the shuttling ring to each binding site, Fig. \ref{['fig:fig2']} showed that the coupling-induced boost in activity ($\mathcal{A}/\mathcal{A}_0$) could be greater than the corresponding loss in bias ($\mathit \Gamma/\mathit \Gamma _0$). When the binding sites are less attractive, a current boost was not observed. Very weak binding of shuttling ring to the binding sites ($h = 2.3 k_{\rm B} T$) allows for such rapid hops that coupling cannot increase activity, much less current. Being coupled to another ring can boost the activity when $h = 4.5 k_{\rm B} T$, but this boost is not as great as the loss in bias. For these rotaxanes to get a boost, it is important that the well depth is sufficiently deep that the ring spends an overwhelming fraction of the time at a binding site.
• Figure 4: One-dimensional jump-diffusion model to study the effect of fuel concentration.(a) The ring (green circle) diffuses along a one-dimensional potential fitted to MD simulations with parameter $h$ setting the barrier height. Motors on parallel offset tracks are coupled by a harmonic potential. Blocking groups (red circle) are added or removed through a jump process with attachment and detachment rates, $k_\mathrm{attach}$ (red) and $k_\mathrm{detach}$ (blue). $k_\mathrm{attach}$ depends on the distance $d$ between the ring and the catalytic site (white), modeled by a function that switches from zero at short distances to $k_\mathrm{attach}^\mathrm{far}$ at long distances. (b) The coupling-induced current amplification $j/j_0$ at coupling strength $\kappa = 0.2k_\mathrm{B}T / \sigma^2$ as a function of $k_\mathrm{attach}^\mathrm{far}$ with the $h$ parameter set to match the MD barrier heights of Figs. \ref{['fig:fig2']} and \ref{['fig:fig3']}. The gray line highlights the corresponding $k_\mathrm{attach}^\mathrm{far}$ simulated in the MD simulation. Increasing $k_{\rm attach}^{\rm far}$ can generate a current enhancement even for small well depths. (c) Deeper wells consistently yield larger activity boosts across $k_\mathrm{attach}^\mathrm{far}$. (d) The current boost emerges because increasing $k_\mathrm{attach}^\mathrm{far}$ restores the bias towards its uncoupled value while largely preserving the coupling-boost activity boost. Means and error bars are collected from 100 independent simulations for each motor.
• Figure S1: Aligned configuration deepens the free energy and never boosts current.(a) Schematics of motors coupled in aligned-track configuration with coupling strength $\kappa$. (b) By tuning the pair potential between green shuttling ring particles and orange binding site particles, deeper wells can be induced. Columns, labeled by the binding-well depth $h$ for the uncoupled motor (black line), show how these wells grow monotonically deeper with increasing coupling $\kappa$. (c) Higher energy barriers prolong the escape time from the wells, leading to a monotonic decrease in activity with increasing $\kappa$. Although the bias increases, it does not compensate for the loss in activity.