On the Maximum Span of Fixed-Angle Chains

TL;DR

This paper investigates the computational complexity of maximizing the span of fixed-angle chains in 2D and 3D, providing polynomial-time solutions for specific cases relevant to protein modeling.

Contribution

It identifies special cases of fixed-angle chains where the maximum span problem is solvable in polynomial time, including flat configurations and chains with specific angle constraints.

Findings

Maximum 3D span with equal lengths and angles is achieved in a flat configuration.

Maximum flat span for simple chains with equal angles can be computed in linear time.

Maximum 3D span with 90-degree angles can be found in quadratic time.

Abstract

Soss proved that it is NP-hard to find the maximum 2D span of a fixed-angle polygonal chain: the largest distance achievable between the endpoints in a planar embedding. These fixed-angle chains can serve as models of protein backbones. The corresponding problem in 3D is open. We show that three special cases of particular relevance to the protein model are solvable in polynomial time. When all link lengths and all angles are equal, the maximum 3D span is achieved in a flat configuration and can be computed in constant time. When all angles are equal and the chain is simple (non-self-crossing), the maximum flat span can be found in linear time. In 3D, when all angles are equal to 90 deg (but the link lengths arbitrary), the maximum 3D span is in general nonplanar but can be found in quadratic time.