Efficiency at Maximum Power of Interacting Molecular Machines

We investigate the efficiency of systems of molecular motors operating at maximum power. We consider two models of kinesin motors on a microtubule: for both the simplified and the detailed model, we find that the many-body exclusion effect enhances the efficiency at maximum power of the many-motor system, with respect to the single motor case. Remarkably, we find that this effect occurs in a limited region of the system parameters, compatible with the biologically relevant range.

Figure 1

Model I: (a) EMP for a single-state model of kinesin interacting through self-exclusion (solid lines) for two different values of the load distribution factor $\theta$. For comparison, the EMP for noninteracting motors with the same parameter values is shown with dashed lines. Parameter values: $a=8\mathrm{nm}$, ${\omega }_0=1.3\times {10}^{-7}{\mathrm{s}}^{-1}$, $\alpha =5{\mathrm{s}}^{-1}$, $\beta =3{\mathrm{s}}^{-1}$. The values for $\theta$ are taken from 17 and correspond to two independent experiments on the kinesin motor. (b)–(d) $P_{\mathrm{out}}$ as a function of $f$ for three different values of the chemical free energy $\Delta \mu$ and for $\theta =0.3$. The vertical dashed line in (c)–(d) represents the phase boundary between the HD and the MC phase.

Figure 2

Model II: The six-state network model from 17 for a single kinesin motor is adapted to describe kinesin traffic by introducing self-exclusion. States marked by $D$ or $T$ represent a head domain containing bound ADP or ATP, respectively, while an empty circle denotes an empty head domain. The motor uses ATP hydrolysis to perform a forward mechanical step from site $j$ to site $j+1$ through the ${\mathcal{F}}^{+}=|12341⟩$ dicycle, or a backward step to site $j-1$ through ${\mathcal{B}}^{+}=|45614⟩$, if the respective neighboring site is unoccupied. Furthermore, the network contains the futile hydrolysis dicycle $|1234561⟩$.

Figure 3

Model II: (a) The EMP for interacting (solid lines) and noninteracting (dashed lines) kinesin motors as a function of the hydrolysis free energy $\Delta \mu$ for two different parameter sets. Squares: parameters obtained in 17 for the experiment of Visscher et al. [21]. Crosses: as before but with chemical load parameters ${\chi }_{1,2}$ as in 24 (see text). For all curves we take the biological concentrations $[\mathrm{ADP}]=0.07\mathrm{mM}$, $[{\mathrm{P}}_{\mathrm{i}}]=1\mathrm{mM}$ [29] and vary the ATP concentration. Where relevant, we take $\alpha =5{\mathrm{s}}^{-1}$ and $\beta =3{\mathrm{s}}^{-1}$. The biologically relevant regime $\Delta \mu /T\approx 20-25$ is marked with vertical lines. The discontinuity in the EMP for interacting motors at $\Delta \mu \simeq 14.5T$ arises due to a discontinuity in the maximizing force $f^{*}$ as a result of the MC-LD phase transition. (b) $P_{\mathrm{out}}$ as a function of $f$ for the second parameter set and for $\Delta \mu =20T$. As $f$ increases, the motor goes from operating in the HD phase to the LD phase, while the MC phase is entered close to stall force. The maximum power is achieved at the HD-LD boundary. (c) The efficiency $\eta (f)$ corresponding to the parameters as in (b). The shift of $f^{*}$ to higher values results in a boost of the EMP, ${\eta }^{*}$, with respect to the noninteracting case, ${\eta }_0^{*}$.