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We present a simple stochastical model for motor molecules that cooperate in large groups. This model could apply for actin-myosin motors in muscles and for motility assays with a high concentration of motor molecules. We calculate the dependence of the velocity on the applied force as a function of ATP concentration and show the existence of a dynamical phase transition allowing for spontaneous directed motion even if the system is spatially symmetric. In the symmetric case, the problem is isomorphous to a paramagnet-ferromagnet transition, in the asymmetric case to a liquid-vapor transition.
Jülicher et al. (Mon,) studied this question.