A model for bacterial flagellar motor: free energy transduction and self-organization of rotational motion

A bacterial flagellar motor is an energy transducing molecular machine which shows some attractive characteristics. First, this motor is driven by a protonmotive force (PMF) across the membrane, two components of which, electric potential delta psi and chemical potential -(2.3RT/F)delta pH, are equivalently transduced to the mechanical work of the motor rotation. Second, a PMF threshold for rotation is observed. Third, this motor can rotate reversibly either counterclockwise (CCW) or clockwise (CW) at almost the same speed. To clarify the osmomechanical coupling of this motor, these characteristics must be explained consistently at the molecular level. In this paper, in order to allow quantitative analyses of the above characteristics, a theoretical model of a bacterial flagellar motor is constructed assuming that the torque generating sites are electrodes which can be charged by protons and that the electrostatic interaction between the electrodes generates the rotation torque. Electrode reaction reasonably derives the equivalence of delta psi and -(2.3RT/F)delta pH. In this model, rates of charging and discharging of protons are influenced by the motor rotation rate, so that the torque generating sites co-operatively work through the motor rotation. We named this kind of co-operativity among them "dynamic co-operativity" in torque generation. This co-operativity causes autocatalytic generation of motor torque and the existence of the rotation threshold. In this model, the appearance of the stable rotational states can be described by phase transition caused by the dynamic co-operativity among torque generating sites. According to this model, the flagellar motor has two stable rotational states corresponding to CCW and CW, which show the same torques. The motor selects one direction from them to rotate, and that is self-organization of rotational motion. Interpretation of the transition between the two stable rotational states as the chemotactic reversals of the flagellar motor is also discussed.