Diffusion-limited formation of internal loops in polymer chains

The speed of diffusional motion of a polypeptide places an upper limit on the speed of protein folding: folding often requires two distant portions of the polypeptide chain to diffuse into contact. Although several studies have investigated the speed at which external (i.e. end-to-end) loops form in polypeptide chains, the more realistic case of internal loops (i.e. connecting two interior points) involves additional excluded volume and potentially slower dynamics. We have used a simple statistical approach to estimate this effect. We generate model chains in continuous, three-dimensional space, where the hard-sphere, excluded volume interaction represents the only deviation from ideal chain behavior. This yields the probability distribution for the distance between particular sites in the chain. We then use a first-passage-time approach (Szabo, Schulten, and Schulten 1980) to estimate the rate of contact formation under various combinations of loop lengths, ``tail'' lengths, and excluded volume. We find that the addition of just a few extra links to an end-to-end loop significantly reduces the speed of loop formation.