Diffusion in correlated random potentials, with applications to DNA

Many biological processes involve one-dimensional diffusion over a correlated inhomogeneous energy landscape with a correlation length ${\xi }_c$. Typical examples are specific protein target location on DNA, nucleosome repositioning, or DNA translocation through a nanopore, in all cases with ${\xi }_c\approx 10\text{nm}$. We investigate such transport processes by the mean first passage time (MFPT) formalism, and find diffusion times which exhibit strong sample to sample fluctuations. For a displacement $N$, the average MFPT is diffusive, while its standard deviation over the ensemble of energy profiles scales as $N^{3∕2}$ with a large prefactor. Fluctuations are thus dominant for displacements smaller than a characteristic $N_c⪢{\xi }_c$: typical values are much less than the mean, and governed by an anomalous diffusion rule. Potential biological consequences of such random walks, composed of rapid scans in the vicinity of favorable energy valleys and occasional jumps to further valleys, is discussed.