Pump process of the rotatory molecular motor and its energy efficiency
  The pump process of the ratchet model inspired by the $F_o$ rotatory motor of ATP synthase is investigated. In this model there are two kinds of characteristic time. One is dynamical, the relaxation time of the system. Others are chemical, the chemical reaction rates at which a proton binds to or dissociates from the motor protein. The inequalities between them affect the behavior of the physical quantities, such as the rotation velocity and the proton pumping rates across the membrane. The energy transduction efficiency is calculated and the condition under which the efficiency can become higher is discussed. The proton pumping rate and the efficiency have a peak where a certain set of inequalities between the chemical reaction rates and the reciprocal of the relaxation time holds. The efficiency also has a peak for a certain value of the load. The best efficiency condition for the pump process is consistent with that for the motor process. 
Hiroshi Miki, Masatoshi Sato, and Mahito Kohmoto