Computational Complexity of Flattening Fixed-Angle Orthogonal Chains

Abstract

Planar/flat configurations of fixed-angle chains and trees are well studied in the context of polymer science, molecular biology, and puzzles. In this paper, we focus on a simple type of fixed-angle linkage: every edge has unit length (equilateral), and each joint has a fixed angle of $90^\circ$ (orthogonal) or $180^\circ$ (straight). When the linkage forms a path (open chain), it always has a planar configuration, namely the zig-zag which alternating the $90^\circ$ angles between left and right turns. But when the linkage forms a cycle (closed chain), or is forced to lie in a box of fixed size, we prove that the flattening problem -- deciding whether there is a planar noncrossing configuration -- is strongly NP-complete. Back to open chains, we turn to the Hydrophobic-Hydrophilic (HP) model of protein folding, where each vertex is labeled H or P, and the goal is to find a folding that maximizes the number of H-H adjacencies. In the well-studied HP model, the joint angles are not fixed. We introduce and analyze the fixed-angle HP model, which is motivated by real-world proteins. We prove strong NP-completeness of finding a planar noncrossing configuration of a fixed-angle orthogonal equilateral open chain with the most H--H adjacencies, even if the chain has only two H vertices. (Effectively, this lets us force the chain to be closed.)