Internal protein motion in a rough model potential

Proteins are understood to exhibit complex internal motions on multiple time scales in their rugged free energy landscapes and often show subdiffusive behavior that significantly influences their biochemical functions. In this study, we employ the fractional Fokker-Planck equation and continuous-time random walk models to investigate the anomalous diffusion of particles within rough confining potentials, drawing inspiration from protein internal dynamics. Our analysis reveals that the dynamics exhibit three distinct regimes: initial free subdiffusion, an intermediate regime where roughness markedly impacts motion, and a long-term thermal plateau due to confinement effects. We derive approximate expressions for the mean displacement and the ensemble-averaged mean squared displacement in the low-roughness limit, revealing good agreement with simulation results. Furthermore, our examination of the ergodic properties of the dynamics indicates that systems with high roughness exhibit enhanced weak ergodicity breaking. As a consequence, the time-averaged mean squared displacement does not reach a plateau but shows a power-law increase in time and individual trajectories intrinsically exhibit an amplitude scatter. In addition, we demonstrate that the mean maximal excursion effectively quantifies the extent of confinement, offering a robust measure for characterizing subdiffusive dynamics in complex systems.