Non-equilibrium transition state rate theory

Abstract

Transition state or Kramers' rate theory has been used to quantify the kinetic speed of many chemical, physical and biological equilibrium processes successfully. For non-equilibrium systems, the analytical quantification of the kinetic rate is still challenging. We developed a new transition state or Kramers' rate theory for general non-equilibrium stochastic systems with finite fluctuations. We illustrated that the non-equilibrium rate is mainly determined by the exponential factor as the weight action measured from the basin of attraction to the “saddle” or more accurately “global maximum” point on the optimal path rather than the saddle point of the underlying landscape as in the conventional transition state or Kramers' rate formula for equilibrium systems. Furthermore, the pre-factor of the non-equilibrium rate is determined by the fluctuations around the basin of attraction and “saddle” point along the optimal paths. We apply our theory for non-equilibrium rate to fate decisions in stem cell differentiation. The dominant kinetic paths between stem and differentiated cell basins are irreversible and do not follow the gradient path along the landscape. This reflects that the dynamics of non-equilibrium systems is not only determined by the landscape gradient but also the curl flux, suggesting experiments to test theoretical predictions. We calculated the transition rate between cell fates. The predictions are in good agreements with stochastic simulations. Our general rate and path formula can be applied to other non-equilibrium systems.