Optimizing Efficiency and Motility of a Polyvalent Molecular Motor

Abstract

Molecular motors play a vital role in the transport of material within the cell. A family of motors of growing interest are burnt bridge ratchets (BBRs). BBRs rectify spatial fluctuations into directed motion by creating and destroying motor-substrate bonds. It has been shown that the motility of a BBR can be optimized as a function of the system parameters. However, the amount of energy input required to generate such motion and the resulting efficiency has been less well characterized. Here, using a deterministic model, we calculate the efficiency of a particular type of BBR, namely a polyvalent hub interacting with a surface of substrate. We find that there is an optimal burn rate and substrate concentration that leads to optimal efficiency. Additionally, the substrate turnover rate has important implications on motor efficiency. We also consider the effects of force-dependent unbinding on the efficiency and find that under certain conditions the motor works more efficiently when bond breaking is included. Our results provide guidance for how to optimize the efficiency of BBRs.

Keywords: burnt bridge ratchet; computational model; molecular motor.

Figure A1

Velocity scaling with consumption rate. Velocity as a function of consumption rate for different $a_{tot}$. Crosses represent simulated results. Lines represent a linear fitting. At substrate concentrations of less than 0.1 the hub was stalled leading to velocities below computational tolerance.

Figure 1

Model schematic. (A) The motor hub (green) interacts with surface-bound substrate (blue) forming bound complex (orange) which is eventually burnt and freed to the buffer (red). (B) Example concentration surface-bound substrate and bound complex and the relative location of the hub at a single point in time. (C) Schematic of state transitions and associated rates. The bound complex to surface-bound state transition initially ignored.

Figure 2

Time dependence of hub velocity and substrate consumption rate. (A) Substrate concentration profiles and the hub position at different points in time. (B) The velocity of the motor as a function of time at different burn rates, ${\nu }^{\prime }$. There is an initial transient period where the hub accelerates and attains peak speed. For some burn rates, the hub eventually achieves a constant steady-state velocity. Whereas at very high or very low burn rates, there is a steady slowing down of the hub with time. (C) The consumption rate, b, as a function of time at different burn rates. The substrate consumption rate peaks and reaches steady state at similar times to the velocity. In all of the above calculations, $a_{tot}=1$, ${\omega }^{\prime }=0$, $k_{off}^{\prime }=K=0.01$.

Figure 3

Dependence of steady-state velocity and consumption rate on system parameters. (A) Steady-state velocity as a function of ${\nu }^{\prime }$ at different values of $a_{tot}$. At fixed $a_{tot}$ there is an optimal ${\nu }^{\prime }$ that leads to the largest hub speed. (B) Steady-state velocity as a function of $a_{tot}$ at different ${\nu }^{\prime }$. At fixed ${\nu }^{\prime }$, the steady-state velocity increases with $a_{tot}$, but eventually saturates. (C) Steady-state substrate consumption rate as a function of ${\nu }^{\prime }$ at fixed $a_{tot}$. As with velocity, the consumption rate peaks at a given burn rate. (D) Steady-state consumption rate as a function of $a_{tot}$ for fixed ${\nu }^{\prime }$. At a fixed burn rate, the consumption rate increases without bounds with increasing $a_{tot}$. In all of the above calculations, ${\omega }^{\prime }=0$, $k_{off}^{\prime }=K=0.01$.

Figure 4

Dependence of efficiency on system parameters with no complex unbinding. (A) Stokes efficiency of system as a function of burn rate, ${\nu }^{\prime }$ at different substrate concentrations $a_{tot}$. Given $a_{tot}$, ${\nu }^{\prime }$ may be adjusted to provide maximum efficiency. Optimal efficiency coincides with ${\nu }^{\prime }$ that maximizes velocity. (B) Efficiency as a function of $a_{tot}$ at different burn rates, ${\nu }^{\prime }$. (C) Heat map of the efficiency as a function of $a_{tot}$ and ${\nu }^{\prime }$. There is an optimal $a_{tot}$ and ${\nu }^{\prime }$ that lead to the highest efficiency given a fixed amount of input chemical energy, $\Delta g$. In all of the above calculations, ${\omega }^{\prime }=0$, $k_{off}^{\prime }=K=0.01$, $\Delta g=15$.

Figure 5

Effects of rebinding on efficiency. Efficiency heat maps for different $k_{off}^{\prime }$. In all of the above calculations, ${\omega }^{\prime }=0$, $\Delta g=15$.

Figure 6

Effects of complex unbinding on efficiency. (A) The normalized velocity as a function of ${\omega }^{\prime }$. Velocities normalized by the maximum velocity for a given $a_{tot}$. (B) The normalized consumption rate as a function of ${\omega }^{\prime }$. (In A,B some curves approach 1 slowly). (C) The absolute efficiency as a function of ${\omega }^{\prime }$ at different $a_{tot}$. At lower substrate concentrations the maximal efficiency occurs when there is no complex unbinding (i.e., ${\omega }^{\prime }=0$). At higher substrate concentration (i.e., $a_{tot}=10$) the optimal efficiency occurs at a non-zero unbinding rate. For calculations in (A–C), $\delta =0$. (D) The efficiency as a function of ${\omega }^{\prime }$ at different $\delta$. For larger $\delta$ the optimal ${\omega }^{\prime }$ is greater. $\delta$ makes no significant difference in the global maximal efficiency of the motor. For calculations in D, $a_{tot}=10$. In all of the above calculations, ${\nu }^{\prime }=1$, $k_{off}^{\prime }=K=0.01$, $\Delta g=15$.