Design principles and optimal performance for molecular motors under realistic constraints

The performance of a molecular motor, characterized by its power output and energy efficiency, is investigated in the motor design space spanned by the stepping rate function and the motor-track interaction potential. Analytic results and simulations show that a gating mechanism that restricts forward stepping in a narrow window in configuration space is needed for generating high power at physiologically relevant loads. By deriving general thermodynamics laws for nonequilibrium motors, we find that the maximum torque (force) at stall is less than its theoretical limit for any realistic motor-track interactions due to speed fluctuations. Our study reveals a tradeoff for the motor-track interaction: while a strong interaction generates a high power output for forward steps, it also leads to a higher probability of wasteful spontaneous back steps. Our analysis and simulations show that this tradeoff sets a fundamental limit to the maximum motor efficiency in the presence of spontaneous back steps, i.e., loose-coupling. Balancing this tradeoff leads to an optimal design of the motor-track interaction for achieving a maximum efficiency close to 1 for realistic motors that are not perfectly coupled with the energy source. Comparison with existing data and suggestions for future experiments are discussed.

Figure 1

Illustration of the minimal motor model. (a) The motor is described by its motion (red arrows) in physical space (angle $\theta$) along the interaction potential $V\left(\theta \right)$. The gray box highlights the forward and backward stepping transitions, represented by the solid and dotted green arrows respectively, between two adjacent potentials (black and blue lines) shifted by half a period ${\theta }_0$. A V-shaped potential is shown with its minimum at $(1+\varepsilon ){\theta }_0$ and depth $V_d$. An energy barrier $V_B\left(\theta \right)$ is added to prevent back flow. (b) There are two types of chemical transitions highlighted in the gray box in (a): the PMF-coupled transitions (red arrowed lines) and the spontaneous transitions (gray arrowed lines). Detailed balance is broken in the reaction loop which leads to a dissipative reaction cycle. Forward and backward reactions between state-1 to state-2 are represented by solid and dotted lines, respectively. (c) The chemical transitions shown in the chemical conformation space. The ratio between the total forward rate ($k_{+}$) and the total backward rate ($k_{-}$) depends on the energy gap $E_g$, which is the difference between the effective driving energy $G_0$ and potential gain $\mathrm{\Delta }V\left(\theta \right)$.

Figure 2

The motor probability distribution and the gating effect on the torque-speed curve. (a) The stator-rotor interaction potential $V\left(\theta \right)$ (blue line); and the forward stepping rate $k_{+}\left(\theta \right)$ (green line). A positive torque ${\tau }_{+}$ is generated when $\theta <{\theta }_m$ and a negative torque $-{\tau }_{-}$ is generated when $\theta >{\theta }_m$. (b) The steady-state distribution $P_s\left(\theta \right)$ at three representative loads: high load $\xi =1$ (red), medium load $\xi =0.1$ (blue), and low load $\xi =0.01$ (green) for $k_g=5\times {10}^5$ with $V\left(\theta \right)$ and $k_{+}\left(\theta \right)$ given in (a). Note that the peak of $P_s\left(\theta \right)$ at the bottom of potential ${\theta }_m$, indicated by the arrows in both (a) and (b), increases as the load ($\xi$) decreases. (c) The torque-speed curves for different values of the gating strength $k_g$. The concavity increases with the gating strength $k_g$. (d) The torque-speed curves for different values of the distance ${\mathrm{\Delta }}_g\equiv {\theta }_m-({\theta }_0+{\theta }_{\epsilon })$ between the gate and the potential minimum. The concavity increases with ${\mathrm{\Delta }}_g$. The square symbols in both (c) and (d) represent data from [36] (pH = 7.0, ${\left[\text{Na}\right]}_{\text{ex}}=30\text{mM}$).

Figure 3

Maximum torque ${\tau }_m$ and the energy dissipations. (a) ${\tau }_m$ depends nonmonotonically on the potential gradient ${\tau }_{+}$ for different values of $\varepsilon =0.2,0.5,0.8$. The experimentally measured ${\tau }_m$ and its corresponding ${\tau }_{+}$ is marked by the star. (b) Fractions of energy dissipation due to torque (speed) fluctuations $f_{\text{mech}}$ (red line) and entropy production in chemical reactions $f_{\text{chem}}$ (blue line) versus load ($\xi$). $f_{\text{mech}}$ dominates at high loads, $f_{\text{chem}}$ dominates at low loads. The total dissipation $(f_{\text{mech}}+f_{\text{chem}})$ is shown as the green line. (c) The experimentally measured speed distribution when hook-only motors were attached to a large 1 $\mu \text{m}$ polystyrene bead with an estimated high load of $\xi \approx 8$ (see Ref. [36] for details). The variance around the first peak ${\sigma }_{\omega }$ corresponds to the speed fluctuation for motors with a single stator. The fractional dissipation due to speed fluctuation at $\xi =8$ can be estimated $f_{\text{mech}}\approx 0.16$ (marked as a star) from (c). The original data are from Ref. [36].

Figure 4

Power and efficiency of the motor. The dependence of (a) power and (b) efficiency (for $\kappa =0.5$) on energy gap $E_g$ and load $\xi$. Both power and efficiency peak at an intermediate load and $E_g$, labeled by the blue dot (for power) and the black star (for efficiency). The red arrow indicates the $E_g/E_0\approx 0.29$ estimated from experiments [36], which is close to the optimal $E_g/E_0$ ratios for maximum power (blue dot) and maximum efficiency (black star). (c) Efficiency $\mathrm{\Lambda }$ versus normalized speed for different values of $\kappa$. $\mathrm{\Lambda }$ vanishes at $\overline{\omega }=0$ for all values of $\kappa <1$. (d) The efficiency of the motor working at maximum power (blue line) is comparable to the (global) maximum efficiency ${\mathrm{\Lambda }}^{*}$ (black line), with their ratio (red line) $\sim 80\%$ for a wide range of $E_0$.

Figure 5

Maximum efficiency and optimal design of the motor. (a) The scaling relationship, $1-{\mathrm{\Lambda }}^{*}\sim ln\left(E_0\right)/E_0$, holds for different values of $\varepsilon$ and $\kappa <1$. (b) The optimal energy gap $E{*}_g$ for achieving the maximum efficiency shown in (a). (c) The prefactor $C_e$ in the scaling relation, Eq. (13), decreases with increasing $\kappa$ for different values of $\varepsilon$. (d) $1-{\mathrm{\Lambda }}^{*}$ versus $ln\left(E_0\right)/E_0$ for randomly chosen motor designs. Each point corresponds to a random stepping rate profile (see Appendix pp2 for details) with $\varepsilon =0.5$ and $\kappa =0.5$. All points lie above the envelop line of $1-{\mathrm{\Lambda }}^{*}\sim ln\left(E_0\right)/E_0$.

Figure 6

The thermodynamic efficiency ${\mathrm{\Lambda }}_T$ and its limit. (a) The dependence of ${\mathrm{\Lambda }}_T$ on the energy gap $E_g$ and the external torque ${\tau }_{\text{ext}}$. The load $\xi =0.1$ is fixed. Other parameters are the same as those in Fig. 4. The optimal ${\mathrm{\Lambda }}_T$ occurs at a finite $E_g$ and an intermediate ${\tau }_{\text{ext}}$. (b) The optimal ${\mathrm{\Lambda }}_T^{*}$ depends on the driving energy $E_0$ approximately following the same relation given in Eq. (13) as the optimal Stokes efficiency.

Figure 7

Illustration of transitions between states in different energy landscapes shifted by the IMF ($G_0$). The black line represents the state 1 with potential $V_1\left(\theta \right)=V\left(\theta \right)$. The blue lines represent the two adjacent states ($2^{\prime }$ and 2) with potentials $V_2\left(\theta \right)=V(\theta +{\theta }_0)$ shifted by $G_0$ and $-G_0$ for state $2^{\prime }$ and state 2, respectively, ${\theta }_0$ is the half period. The green arrowed lines represent the transitions between the states with the transition rates labeled (see Appendix pp1 for details). Due to symmetry, the transition rates ($k_{2^{\prime }1}$ and $k_{{12}^{\prime }}$) between states $2^{\prime }$ and 1 are the same as the transition rates ($k_{12}$ and $k_{21}$) between states 1 and 2 with the angle $\theta$ shifted by ${\theta }_0$.

Figure 8

The distribution $P_s\left(\theta \right)$ for different values of $E_g$ (by varying $V_d$ and fixing $\varepsilon =0.5$). The functions $V\left(\theta \right)$ and $k_{+}\left(\theta \right)$ are the same as shown in Fig. 1 in the main text. The probability in the negative torque regime $P_{-}\equiv {\int }_{{\theta }_m}^{2{\theta }_0}{P}_s\left(\theta \right)d\theta$ decreases with increasing $E_g$ and thus the probability in the positive toque regimes $P_{+}=1-P_{-}$ decreases with decreasing $E_g$.

Figure 9

The results for quadratic potentials. (a) The shape of a quadratic potential, expression given in Eq. (D1), in which the minimum location at ${\theta }_m=(1+\varepsilon ){\theta }_0$ and the depth of the potential $V_d$ are varied to optimize motor performance. (b) The maximum torque versus $1-E_g/G_0=V_d(1+2\varepsilon )/\left[G_0{(1+\varepsilon )}^2\right]$. (c) $1-{\mathrm{\Lambda }}^{*}$ versus $\text{ln}\left(E_0\right)/E_0$, where ${\mathrm{\Lambda }}^{*}$ is the maximum efficiency. (d) The optimal $E_g^{*}$ (for achieving the maximum efficiency) versus $ln{E}_0$.