Passage through fluctuating geometrical bottlenecks. Subdiffusive dynamics of the opening -- exact solution

The usual Kramers theory of reaction rates in a condensed medium predict the rate to have an $\eta^{-1}$ dependence, $\eta$ being the viscosity of the medium. However, experiments on ligand binding to proteins performed long ago, showed the rate to have $\eta^{-\nu}$ dependence, with $\nu$ in the range $0.4-0.8$. Zwanzig {\it (Journal of Chemical Physics 97, 3587 (1992))} suggested a model, in which the ligand has to pass through a fluctuating opening to bind. Thus fluctuating gate model predicted the rate to be proportional to $\eta^{-1/2}$. Experiments performed by Xie et. al. ({\it Physical Review Letters 93, 1 (2004)}) showed that the distance between two groups in a protein undergoes subdiffusion. Hence in this paper, we suggest and solve a generalisation of the Zwanzig model, viz., passage through a gate that undergoes subdiffusion. Our solution shows that the rate is proportional to $\eta^{-\nu}$ with $\nu$ in the range $0.5-1$, and hence the model can explain the experimental observations.