Static and dynamic rough energy landscapes can lead to identical diffusivity

Molecules in dense environments, such as biological cells, are subjected to forces that fluctuate both in time and in space. While spatial fluctuations are captured by Lifson-Jackson-Zwanzig's model of "diffusion in a rough potential," and temporal fluctuations are often viewed as leading to additional friction effects, a unified view where the environment fluctuates both in time and in space is currently lacking. Here, we introduce a discrete-state model of a landscape fluctuating both in time and in space. Importantly, the model accounts for the reciprocal interaction of the diffusing particle with the landscape, which alters the landscape dynamics. As a result we find, surprisingly, that many features of the observable dynamics do not depend on the temporal fluctuation timescales and are already captured by the model of diffusion in a rough potential, even though this assumes a static energy landscape. Using this model, we reevaluate results of several experimental studies of protein dynamics and propose more accurate bounds on the inferred energetic roughness scales, which account for landscape dynamics.