Optimal shapes of compact strings

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Abstract
Optimal geometrical arrangements, such as the stacking of atoms, are of relevance in diverse disciplines1,2,3,4,5. A classic problem is the determination of the optimal arrangement of spheres in three dimensions in order to achieve the highest packing fraction; only recently has it been proved1,2 that the answer for infinite systems is a face-centred-cubic lattice. This simply stated problem has had a profound impact in many areas3,4,5, ranging from the crystallization and melting of atomic systems, to optimal packing of objects and the sub-division of space. Here we study an analogous problem—that of determining the optimal shapes of closely packed compact strings. This problem is a mathematical idealization of situations commonly encountered in biology, chemistry and physics, involving the optimal structure of folded polymeric chains. We find that, in cases where boundary effects6 are not dominant, helices with a particular pitch-radius ratio are selected. Interestingly, the same geometry is observed in helices in naturally occurring proteins.