A hybrid combinatorial-continuous strategy for solving molecular distance geometry problems

TL;DR

This work addresses the Molecular Distance Geometry Problem and its interval variant iDMDGP by introducing a hybrid solver that combines discrete DMDGP-inspired enumeration with continuous optimization. The method builds angle-aware initial conformations, applies sign-consistent discrete improvements, and then refines promising candidates via Spectral Projected Gradient on a nonconvex stress that enforces interval restraints with auxiliary distances in Omega_d. Key contributions include a geometry-aware placement primitive, a multistart framework with RMSD filtering, and a robust SPG refinement that achieves higher problem coverage and lower runtimes on 30 protein instances compared to a MDjeep baseline. The results demonstrate that discrete structure sharply narrows the search while continuous refinement reconciles wide interval data, offering a scalable approach for NMR based structure determination where distance bounds are often broad and heterogeneous.

Abstract

The Molecular Distance Geometry Problem (MDGP) is essential in structural biology, as it seeks to determine three-dimensional protein structures from partial interatomic distances. Its discretizable subclass (DMDGP) admits an exact combinatorial formulation that enables efficient exploration of the search space. However, in practical settings such as Nuclear Magnetic Resonance (NMR) spectroscopy, distances are available only within uncertainty bounds, leading to the interval variant (\emph{i}DMDGP). We propose a hybrid combinatorial--continuous framework for solving the \emph{i}DMDGP. The method combines an enumeration process derived from the DMDGP with a continuous refinement stage that minimizes a nonconvex stress function that penalizes deviations from admissible distance intervals. This integration supports a systematic exploration guided by discrete structure and local optimization. The formulation incorporates torsion-angle intervals and chirality constraints through a refined atom ordering that preserves protein-backbone geometry. Numerical experiments show that the approach efficiently reconstructs geometrically valid conformations even under wide distance bounds, whereas most existing studies assume narrow ones.

Figures (2)

• Figure 1: Enumerative search with exact distances. For $i\ge4$, two symmetric placements correspond to the sign of $\tau_i$.
• Figure 2: Runtime performance profiles among all instances ($x$-axis in $\log_2$ scale).