Single molecule tracking of heterogeneous diffusion

The mean square displacement of heterogeneous diffusion obeys the Einstein relation, thereby showing no sign of heterogeneities in the ensemble measurement of the diffusion constant. The signature of spatial heterogeneities appears in the time evolution of the non-Gaussian distribution and in the cross correlation between the square displacements at different times, both available from single molecule diffusional trajectories. As a quantitative measure, the non-Gaussian indicator $g\left(t\right)\mathrm{}$ decays asymptotically to zero according to $1/t$ for finite time correlation, but saturates at a plateau value for power-law correlation. In addition, the joint moment correlation function $f(t,\tau )$ provides a direct probe of the memory effect of the fluctuating rate constant. A two-state diffusion model and a stochastic Gaussian model are constructed to evaluate these quantities and are shown to yield the same result within the second cumulant expansion.