Dynamics of particles with key-lock interactions

We present a theoretical discussion of particles which interact through the reversible formation of multiple key-lock bridges. Two potential experimental realizations include DNA- grafted particles which interact with a two-dimensional DNA substrate, and particles grafted with antibodies interacting with a protein substrate. We argue there is a percolation transition characterized by the average number of bridges realized between a particle and the substrate. The transition separates a regime in which particles are localized from a diffusive regime where they explore the substrate surface through mutiple breaking and reconnecting of bridges. This diffusion behavior is dispersive, characterized by $\langle r^ {2}(t) \rangle \sim t^{\alpha}$ with $\alpha<1$. The distribution of departure times in a multi-particle system is calculated in two different models which account for the particle dynamics above and below the percolation threshold.