Biological molecular machines convert free energy between different forms in cells, often at high efficiency. Optimal control theory provides a framework to elucidate design principles governing energetically efficient driving. Here, we use linear-response theory to design efficient protocols exercising dynamic control of trap center and stiffness in a model of driven ${\mathrm{F}}_1$-ATPase. We find that the key design principles of an efficient protocol can be satisfied either by dynamic control of both parameters or by dynamic control of a single parameter and a good static choice for the second. These results illustrate that accessing a new degree of dynamic control provides varying performance improvements in different systems.

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• Fluctuations & noise
• Nonequilibrium statistical mechanics
• Stochastic thermodynamics
• Biomolecules
• Brownian ratchet
• Langevin equation
• Linear response theory