The nonlinear chemo-mechanic coupled dynamics of the F 1 -ATPase molecular motor

The ATP synthase consists of two opposing rotary motors, F₀ and F₁, coupled to each other. When the F₁ motor is not coupled to the F₀ motor, it can work in the direction hydrolyzing ATP, as a nanomotor called F₁-ATPase. It has been reported that the stiffness of the protein varies nonlinearly with increasing load. The nonlinearity has an important effect on the rotating rate of the F₁-ATPase. Here, considering the nonlinearity of the γ shaft stiffness for the F₁-ATPase, a nonlinear chemo-mechanical coupled dynamic model of F₁ motor is proposed. Nonlinear vibration frequencies of the γ shaft and their changes along with the system parameters are investigated. The nonlinear stochastic response of the elastic γ shaft to thermal excitation is analyzed. The results show that the stiffness nonlinearity of the γ shaft causes an increase of the vibration frequency for the F₁ motor, which increases the motor’s rotation rate. When the concentration of ATP is relatively high and the load torque is small, the effects of the stiffness nonlinearity on the rotating rates of the F₁ motor are obvious and should be considered. These results are useful for improving calculation of the rotating rate for the F₁ motor and provide insight about the stochastic wave mechanics of F₁-ATPase.