An Angle-Based Algorithmic Framework for the Interval Discretizable Distance Geometry Problem

TL;DR

This paper introduces two novel angle-based algorithmic frameworks, iABP and iTBP, for solving the interval Discretizable Distance Geometry Problem, improving efficiency and solution quality in protein structure determination from NMR data.

Contribution

The work develops and formalizes two new algorithms, iABP and iTBP, that convert interval distances into angular constraints, enhancing solution feasibility and biological relevance in protein structure modeling.

Findings

Both iABP and iTBP outperform iBP in solution rate and efficiency.

iTBP produces solutions with lower RMSD variance, better reflecting biological structures.

Algorithms are validated on PDB-derived instances, demonstrating practical effectiveness.

Abstract

Distance Geometry plays a central role in determining protein structures from Nuclear Magnetic Resonance (NMR) data, a task known as the Molecular Distance Geometry Problem (MDGP). A subclass of this problem, the Discretizable Distance Geometry Problem (DDGP), allows a recursive solution via the combinatorial Branch-and-Prune (BP) algorithm by exploiting specific vertex orderings in protein backbones. To accommodate the inherent uncertainty in NMR data, the interval Branch-and-Prune (\textit{i}BP) algorithm was introduced, incorporating interval distance constraints through uniform sampling. In this work, we propose two new algorithmic frameworks for solving the three-dimensional interval DDGP (\textit{i}DDGP): the interval Angular Branch-and-Prune (\textit{i}ABP), and its extension, the interval Torsion-angle Branch-and-Prune (\textit{i}TBP). These methods convert interval distances into angular constraints, enabling structured sampling over circular arcs. The \textit{i}ABP method guarantees feasibility by construction and removes the need for explicit constraint checking. The \textit{i}TBP algorithm further incorporates known torsion angle intervals, enforcing local chirality and planarity conditions critical for protein geometry. We present formal mathematical foundations for both methods and a systematic strategy for generating biologically meaningful \textit{i}DDGP instances from the Protein Data Bank (PDB) structures. Computational experiments demonstrate that both \textit{i}ABP and \textit{i}TBP consistently outperform \textit{i}BP in terms of solution rate and computational efficiency. In particular, \textit{i}TBP yields solutions with lower RMSD variance relative to the original PDB structures, better reflecting biologically plausible conformations.