Diffusion of a Heteropolymer in a Multi-Interface Medium

We consider a heteropolymer, consisting of an i.i.d. concatenation of hydrophilic and hydrophobic monomers, in the presence of water and oil arranged in alternating layers. The heteropolymer is modelled by a directed path ( $\left( {i,S_i } \right)_{i \in \mathbb{N}_0 }$ , where the vertical component lives on $\mathbb{Z}$ , and the layers are horizontal with equal width. The path measure for the vertical component is given by that of simple random walk multiplied by an exponential weight factor that favors matches and disfavors mismatches between the monomers and the medium. We study the vertical motion of the heteropolymer as a function of its total length n when the width of the layers is d _n and the parameters in the exponential weight factor are such that the heteropolymer tends to stay close to an interface (“localized regime”). In the limit as n→∞ and under the condition that lim _n→∞ d _n /log log n=∞ and lim _n→∞ d _n /log n=0, we show that the vertical motion is a diffusive hopping between neighboring interfaces on a time scale expχd _n (1+o(1))], where χ is computed explicitly in terms of a variational problem. An analysis of this variational problem sheds light on the optimal hopping strategy.