  Microscopic machines utilize free energy to create and maintain
out-of-equilibrium organization in virtually all living things. Often this
takes the form of converting the free energy stored in nonequilibrium chemical
potential differences into useful work, via a series of reactions involving the
binding, chemical catalysis, and unbinding of small molecules. Such chemical
reactions occur on timescales much faster than the protein conformational
rearrangements they induce. Here, we derive the energetic cost for driving a
system out of equilibrium via a series of such effectively instantaneous (and
hence discrete) perturbations. This analysis significantly generalizes
previously established results, and provides insight into qualitative, as well
as quantitative, aspects of finite-time, minimum-dissipation discrete control
protocols. We compare our theoretical formalism to an exactly solvable model
system and also demonstrate the dissipation reduction achievable in a simple
multistable model for a discretely driven molecular machine.
