Optimal Control of the F${_1}$-ATPase Molecular Motor

F$_{1}$-ATPase is a rotary molecular motor that \emph{in vivo} is subject to strong nonequilibrium driving forces. There is great interest in understanding the operational principles governing its high efficiency of free-energy transduction. Here we use a near-equilibrium framework to design a non-trivial control protocol to minimize dissipation in rotating F$_{1}$ to synthesize ATP. We find that the designed protocol requires much less work than a naive (constant-velocity) protocol across a wide range of protocol durations. Our analysis points to a possible mechanism for energetically efficient driving of F$_{1}$ \emph{in vivo} and provides insight into free-energy transduction for a broader class of biomolecular and synthetic machines.