Simple Mechanical Equivalents of Stepping Rotary Dynamics in   F$_1$-ATPase
  Two simple (rotator and one-particle) mechanistic models are suggested to describe simultaneously at a minimal level of sophistication two basic functions of F$_1$-ATPase: a motor regime driven by ATP hydrolysis and its inverted function as ATP synthesis. This description is consistent with the so-called rotary binding-change mechanism, a milestone of functioning ATP synthase, and uses a stepping (driving) function associated with two sequences of time instants, at which hydrolysis and synthesis reactions occur. It is useful to analyse experimental data and numerical simulations indeed predict corresponding dynamic behavior. 
A.V. Zolotaryuk, V.N. Ermakov, P.L. Christiansen, and B. Norden and Y.   Zolotaryuk