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

Wagner A. A. da Rocha, Carlile Lavor, Leo Liberti, Leticia de Melo Costa, Leonardo D. Secchin, Therese E. Malliavin

TL;DR

The paper addresses 3D protein structure determination under uncertain distance data by extending the interval DGP framework with angular and torsion-angle discretization. It introduces two new methods, iABP and iTBP, that transform interval distances into angular constraints, enabling sampling over circular arcs and incorporating torsion signs to enforce chirality and planarity. The authors provide formal foundations, a PDB-based instance-generation pipeline, and comprehensive experiments showing that iABP and especially iTBP improve solution rates, depth, and structural plausibility (lower RMSD variance) compared to the baseline iBP. These advances offer a more robust approach for reconstructing biomolecular conformations from noisy NMR-derived restraints, with potential impact on structural biology workflows and computational protein modeling.

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.

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

TL;DR

The paper addresses 3D protein structure determination under uncertain distance data by extending the interval DGP framework with angular and torsion-angle discretization. It introduces two new methods, iABP and iTBP, that transform interval distances into angular constraints, enabling sampling over circular arcs and incorporating torsion signs to enforce chirality and planarity. The authors provide formal foundations, a PDB-based instance-generation pipeline, and comprehensive experiments showing that iABP and especially iTBP improve solution rates, depth, and structural plausibility (lower RMSD variance) compared to the baseline iBP. These advances offer a more robust approach for reconstructing biomolecular conformations from noisy NMR-derived restraints, with potential impact on structural biology workflows and computational protein modeling.

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.

Paper Structure

This paper contains 9 sections, 10 theorems, 45 equations, 8 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The iDDGP and the iBP Algorithm
  3. A Torsion-Angle-Based Approach for the iDDGP
  4. Generating iDDGP Instances from Protein Structures
  5. Protein Chain Modeling
  6. Construction of a Vertex Order for the iDDGP
  7. Instance Generation
  8. Computational Experiments and Analysis
  9. Conclusions and Future Directions

Key Result

Lemma 1

Let $p, q \in \mathbb{R}_+$ such that $0 < 2q \leq p$. Also, let the functions $\tau^+ : [\sqrt{p - 2 q}, \sqrt{p + 2 q}] \rightarrow [0, \pi]$ and $\tau^- : [\sqrt{p - 2 q}, \sqrt{p + 2 q}] \rightarrow [-\pi,0]$ defined by Then $\tau^\pm$ are continuous, bijective and for an interval $\mathbf{d} = [\underline{d}, \overline{d}] \subset [\sqrt{p - 2 q}, \sqrt{p + 2 q}]$, we have $\tau^+(\mathbf{d}

Figures (8)

  • Figure 1: Illustration of the intersection $S_i^1 \cap S_i^2 \cap \mathbf{S}_i^3$ and its resulting set ${^{(3,2,1)}\mathcal{A}}_i = \mathcal{A}_i^- \cup \mathcal{A}_i^+$.
  • Figure 2: Geometric framework used to describe $x_z$ as a function of $x_u,\ x_v,\ x_w,\ d_{z,u},\ d_{z,v},\ d_{z,w}$, and $\tau_{u,v,w,z}$.
  • Figure 3: View of the plane containing the intersection $S_i^1 \cap S_i^2$, represented as a dashed black circumference, where the angle $\varphi$ denotes the torsion between the planes $\{x_{i_4}, x_{i_2}, x_{i_1}\}$ and $\{x_{i_3}, x_{i_2}, x_{i_1}\}$. In (a), the magenta arcs correspond to the intersection of $\mathbf{S}_i^3$ with the fixed circumference, while in (b) the cyan arcs correspond to the intersection of $\mathbf{S}_i^4$ with the same circumference. A sample of $S_i^1 \cap S_i^2 \cap \mathbf{S}_i^3$ is represented by the set ${^{(i_3,i_2,i_1)}A}_i = \{(x_i^-)_1,\ \dots,\ (x_i^-)_5,\ (x_i^+)_1,\ \dots,\ (x_i^+)_5\}$. Its feasible subset is $A_i = {^{(i_3,i_2,i_1)}A}_i \cap \mathbf{S}_i^4 = \{(x_i^+)_4, \ (x_i^+)_5\}$.
  • Figure 4: View of the plane containing the intersection $S_i^1 \cap S_i^2$, represented as a dashed black circumference, where the angle $\varphi$ denotes the torsion between the planes $\{x_{i_4}, x_{i_2}, x_{i_1}\}$ and $\{x_{i_3}, x_{i_2}, x_{i_1}\}$. In (a), the magenta arcs correspond to the intersection of $\mathbf{S}_i^3$ with this fixed circumference, yielding the set ${^{(3,2,1)}\mathcal{A}}_i$, while the cyan arcs correspond to the intersection of $\mathbf{S}_i^4$ with the same circumference, resulting in ${^{(4,2,1)}\mathcal{A}}_i$. The purple arc represents the refined set $\mathcal{A}_i = {^{(3,2,1)}\mathcal{A}}_i \cap {^{(4,2,1)}\mathcal{A}}_i$. In (b), the sampled points from $\mathcal{A}_i$ are $A_i = \{(\tilde{x}_i^+)_1, \ (\tilde{x}_i^+)_2, \ (\tilde{x}_i^+)_3, \ (\tilde{x}_i^+)_4, \ (\tilde{x}_i^+)_5\}$.
  • Figure 5: Representation of the $i$-th residue of a protein and its local adjacency, highlighting all relevant atoms and the torsion angles $\omega_i,\ \phi_i$, and $\psi_i$.
  • ...and 3 more figures

Theorems & Definitions (21)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 11 more