A Novel Branch-and-Prune Algorithmic Framework for the 3D Interval Discretizable Distance Geometry Problem: An Approach Based on Torsion Angles of Molecular Structures

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.