Intrachain loops in polymers: Effects of excluded volume

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Abstract
We develop a theory for the formation of loops and intrachain contacts in polymer molecules which are subject to excluded volume. We use two methods: (i) exhaustive simulations of chain conformations on two‐dimensional square lattices, and (ii) the Edwards path integral approach. The predictions are compared to those of the Jacobson–Stockmayer theory, which neglects excluded volume. Our results show that the cyclization probability in two dimensions depends on loop length to a power between −1.6 to −2.4, in contrast to the prediction of Jacobson–Stockmayer of a power of −1. In addition, the cyclization probability depends on the position in the chain, and end effects are significant. A principal result of the present work is the development of ‘‘topological’’ correlation functions among multiple loops in a chain. If two loops are far apart along the chain, they act independently, but as they approach each other, or if they are interlinked, then one can strongly hinder or enhance the likelihood of another. For these situations, the path integral theory is an improvement over Jacobson–Stockmayer, but misses some important features of short‐ranged packing effects. A particularly striking conclusion is that in the presence of the most probable first loop, the formation of a second contact is strongly preferred to be in either of only two possible conformations: a helix or an antiparallel sheet. This suggests that the basis for secondary structure formation in globular proteins may be packing and conformational freedom, rather than hydrogen bonding or other specific interactions.