Before revision:  We unexpectedly find that symmetric sawtooth potentials maximize time asymmetry, while earlier work examining ratchet models of molecular machines required asymmetric sawtooth potentials -- a distinction related to differences between autonomous and deterministic driving .
After revision:  We unexpectedly find that symmetric sawtooth potentials maximize time asymmetry, while earlier work examining ratchet models of molecular machines required asymmetric sawtooth potentials to achieve directed behavior. This distinction arises because deterministically driven machines are externally provided with direction, while autonomous machines must generate directed behavior .