Inferring Subsystem Efficiencies in Bipartite Molecular Machines

Molecular machines composed of coupled subsystems transduce free energy between different external reservoirs, in the process internally transducing energy and information. While subsystem efficiencies of these molecular machines have been measured in isolation, less is known about how they behave in their natural setting when coupled together and acting in concert. Here, we derive upper and lower bounds on the subsystem efficiencies of a bipartite molecular machine. We demonstrate their utility by estimating the efficiencies of the $F_o$ and $F_1$ subunits of ATP synthase and that of kinesin pulling a diffusive cargo.

Figure 1

Steady-state energy and information flows for a bipartite molecular machine. The $X$ and $Y$ subsystems can exchange work and heat with external reservoirs, and energy and information with each other. Arrows indicate positive flows under our sign convention.

Figure 2

Inferred subsystem efficiencies ${\eta }_o$ for $F_o$ (the $Y$ subsystem) and ${\eta }_1$ for $F_1$ (the $X$ subsystem) in $F_o{F}_1$-ATP synthase. Red horizontal and vertical lines: minimum possible values of the lower bounds, Eq. (18), on ${\eta }_1$ and ${\eta }_o$, respectively, that are consistent with the estimated parameter ranges detailed in the main text. Green horizontal and vertical lines: maximum possible values of the two upper bounds, Eq. (18). Red and green dashed curves: minimum and maximum possible values of the product ${\eta }_o{\eta }_1={\eta }_T$ given our parameter estimates. Purple region: efficiencies satisfying all bounds.

Figure 3

Upper and lower bounds, Eq. (19), on the motor efficiency computed from experimental data for a single transport motor (the $Y$ subsystem) pulling a diffusive cargo (the $X$ subsystem) [36]. For each paired velocity and cargo diameter from Ref. [36], red and green points respectively denote lower and upper bounds [Eq. (19)] on the motor efficiency, with error bars showing experimental error propagated through Eq. (19). Purple dashed curve: nonlinear best-fit curve separating the upper and lower bounds computed using a nonlinear support vector machine [40]. Purple shaded region: corresponding confidence interval. Blue dot-dashed line: theoretical prediction from Ref. [20].