Space-filling discrete helices

Abstract

Proteins are linear chain molecules that play a central role in life and health. Protein native state folds are modular assemblies of space-filling building blocks of {\alpha}-helices, \{beta}-sheets and tight turns. Here we deduce the structures of a countable set of space-filling helical forms of a uniform discrete thick string from first principles with no additional input or adjustable parameters. These forms occur in correspondence with the natural numbers, loosely analogous to the energy levels in a Bohr atom. We find the remarkable result that one of these helical forms is an excellent candidate for an {\alpha}-helix through seemingly improbable quantum chemistry coincidences that fit the geometrical requirements. Our work suggests that geometry and chemistry are complementary ways of looking at proteins and suggests a route for developing a unified framework for understanding proteins.