Passage through a sub-diffusing geometrical bottleneck

The usual Kramers theory of reaction rates in condensed media predict the rate to have an inverse dependence on the viscosity of the medium, η. However, experiments on ligand binding to proteins, performed long ago, showed the rate to have η^-ν dependence, with ν in the range of 0.4-0.8. Zwanzig [J. Chem. Phys. 97, 3587 (1992)] suggested a model in which the ligand has to pass through a fluctuating opening to reach the binding site. This fluctuating gate model predicted the rate to be proportional to η^-1/2. More recently, experiments performed by Xie et al. [Phys. Rev. Lett. 93, 180603 (2004)] showed that the distance between two groups in a protein undergoes not normal diffusion, but subdiffusion. Hence, in this paper, we suggest and solve a generalization of the Zwanzig model, viz., passage through an opening, whose size undergoes subdiffusion. Our solution shows that the rate is proportional to η^-ν with ν in the range of 0.5-1, and hence, the subdiffusion model can explain the experimental observations.