Subdiffusion and Weak Ergodicity Breaking in the Presence of a Reactive Boundary

We derive the boundary condition for a subdiffusive particle interacting with a reactive boundary with a finite reaction rate. Molecular crowding conditions, that are found to cause subdiffusion of larger molecules in biological cells, are shown to effect long-tailed distributions with an identical exponent for both the unbinding times from the boundary to the bulk and the rebinding times from the bulk. This causes a weak ergodicity breaking: typically, an individual particle either stays bound or remains in the bulk for very long times. We discuss why this may be beneficial for in vivo gene regulation by DNA-binding proteins, whose typical concentrations are nanomolar.

Figure 1

The distribution ${\delta }_{\alpha }$, Eq. (18), for various $\beta$, with $\alpha =0.75$. In all cases, a divergence at $\rho =0$ and 1 is observed. The points are results from a stochastic simulation [21].