Concatenative nonmonotonicity and optimal links in HP protein folding models

TL;DR

This paper investigates the nonmonotonic behavior of optimal folding energies in HP protein models across various lattices, revealing that concatenation can worsen fold quality and uncovering links to knot theory.

Contribution

It proves nonmonotonicity of optimal fold energy under concatenation in multiple lattice types and connects HP models to knot/link theory through proper links.

Findings

Optimal fold energy is nonmonotonic under concatenation.

Proper links can be uniquely optimal folds in certain models.

Connections between HP models and knot theory are established.

Abstract

The hydrophobic-polar (HP) model represents proteins as binary strings embedded in lattices, with fold quality measured by an energy score. We prove that the optimal fold energy is not monotonic under concatenation for several standard lattices, including the 2D and 3D rectangular, hexagonal, and triangular lattices. In other words, concatenating two polymers can produce a fold with strictly worse optimal energy than one of the polymers alone. For closed chains, we show that under the levels-of-hydrophobicity model of Agarwala et al. (1997), proper links can arise as uniquely optimal folds, revealing an unexpected connection between HP models and knot/link theory.