Resonant fluctuations of selection pressure exponentially accelerate fitness valley crossing

Two functional protein sequences can sometimes be separated by a fitness valley - a series of low or non-functional intermediate mutations that must be traversed to reach a more optimal or refined function. Time-varying selection pressure modulates evolutionary sampling of such valleys. Yet, how the amplitude and frequency of fluctuating selection influence the rate of protein evolution is poorly understood. Here, we derive a simple equation for the time-dependent probability of crossing a fitness valley as a function of evolutionary parameters: valley width, protein size, mutation rate, and selection pressure. The equation predicts that, under low selection pressure, the valley crossing rate is magnified by a factor that depends exponentially on valley width. However, after a characteristic time set by the evolutionary parameters, the rate rapidly decays. Thus, there is an optimal frequency of selection-pressure fluctuations that maximizes the rate of protein optimization. This result is reminiscent of the resonance frequency in mechanical systems. The equation unites empirical and theoretical results that were previously disconnected, and is consistent with time-dependent in vitro and clinical data. More generally, these results suggest that seasonal and climate oscillations could synchronously drive protein evolution at the resonant frequency across a range of organism hosts and timescales. This theory could also be applied to optimize de novo protein evolution in laboratory directed evolution using time-varying protocols.