A grammatical theory for the conformational changes of simple helix bundles

Polymers, including biomolecules such as proteins, have two particularly important types of single-molecule transitions: a helix-coil transition, driven by interactions that are local in the chain, and a collapse transition, driven by nonlocal interactions. A long-standing challenge of polymer statistical mechanics has been to deal with both types of transition in a single theoretical framework. The simplest paradigmatic problem would be a theory of helix-bundle folding. Here, we show how the machinery of formal grammars, originally developed in the context of linguistic analysis and programming-language compilation, provides a simple and general way to combine the Zimm-Bragg model of α-helices with the model of Chen and Dill for nonlocal interactions in antiparallel polymeric systems. We use a well-known construction in the theory of formal grammars to give the statistical mechanical partition function for two-helix bundles. Predictions are shown to be quite good in comparison to exact enumerations within a lattice model.