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New Aspects of Analyzing Amyloid Fibrils

Xiaoxi Lin, Yunpeng Zi, Fengling Li, Jingyan Li

TL;DR

The paper addresses how to quantify amyloid fibril structure beyond RMSD by combining discrete-curve geometry with topological data analysis. It represents peptide chains as discrete curves in $\mathbb{R}^3$ and introduces the $N$-truncated hop distance along with discrete curvature and discrete torsion (built from cross ratios and osculating circles) to capture local and layer-to-layer interactions. Topological methods, including persistent homology and Vietoris-Rips complexes, provide multiscale descriptors via persistent diagrams, enabling more sensitive discrimination of fibril polymorphs. Application to transthyretin (TTR) ATTR fibrils shows torsion anomalies at carbonyl carbons correlating with hydrogen-bond networks, and results suggest TDA offers advantages for structural classification and drug-design insight.

Abstract

This is a summary of mathematical tools we used in research of analyzing the structure of proteins with amyloid form \cite{xi2024Top}. We defined several geometry indicators on the discrete curve namely the hop distance, the discrete curvature and the discrete torsion. Then, we used these indicators to analyze the structure of amyloid fibrils by regarding its peptide chains as discrete curves in $\Rds^3$. We gave examples to show that these indicators give novel insights in the characterization analysis of the structure of amyloid fibrils, for example the discrete torsion can detect the hydrogen bonds interactions between layers of amyloid fibril. {Moreover,} the topological tool performs better than the root mean square deviation (RMSD) in quantifying the difference of the structure of amyloid fibrils, etc.

New Aspects of Analyzing Amyloid Fibrils

TL;DR

The paper addresses how to quantify amyloid fibril structure beyond RMSD by combining discrete-curve geometry with topological data analysis. It represents peptide chains as discrete curves in and introduces the -truncated hop distance along with discrete curvature and discrete torsion (built from cross ratios and osculating circles) to capture local and layer-to-layer interactions. Topological methods, including persistent homology and Vietoris-Rips complexes, provide multiscale descriptors via persistent diagrams, enabling more sensitive discrimination of fibril polymorphs. Application to transthyretin (TTR) ATTR fibrils shows torsion anomalies at carbonyl carbons correlating with hydrogen-bond networks, and results suggest TDA offers advantages for structural classification and drug-design insight.

Abstract

This is a summary of mathematical tools we used in research of analyzing the structure of proteins with amyloid form \cite{xi2024Top}. We defined several geometry indicators on the discrete curve namely the hop distance, the discrete curvature and the discrete torsion. Then, we used these indicators to analyze the structure of amyloid fibrils by regarding its peptide chains as discrete curves in . We gave examples to show that these indicators give novel insights in the characterization analysis of the structure of amyloid fibrils, for example the discrete torsion can detect the hydrogen bonds interactions between layers of amyloid fibril. {Moreover,} the topological tool performs better than the root mean square deviation (RMSD) in quantifying the difference of the structure of amyloid fibrils, etc.
Paper Structure (19 sections, 7 theorems, 27 equations, 10 figures, 3 tables, 2 algorithms)

This paper contains 19 sections, 7 theorems, 27 equations, 10 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Primary Structures of Proteins
  3. Geometry in Biology
  4. Discrete Curve Model
  5. Topological Model
  6. Arrangement of the Paper
  7. Discrete Curve and Hop Distance
  8. Hop distance
  9. Applications to Analysis of the Structure of Amyloid fibrils
  10. Discrete Curvatures and Discrete Torsions
  11. Curvatures and Torsions for Smooth Curves
  12. Basics in quaternions
  13. Curvatures and Torsions for Discrete Curves
  14. Applications to Analysis of the Structure of Amyloid fibrils
  15. Topological Data Analysis
  16. ...and 4 more sections

Key Result

Lemma 3.1

Let $a,b,c,d,d'\in \mathds{H}$ be four quaternions, then we have

Figures (10)

  • Figure 1: A peptide chain. The backbone of this peptide chain is the set $\{\dots, \text{C}_{\alpha_{i-1}}, \text{C}_{i-1}, \text{O}_{i-1}, \text{N}_{i}, \text{C}_{\alpha_{i}}, \text{C}_{i}, \text{O}_{i}, \dots\}.$ Blue rectangles display the peptide bonds. Here, $\text{res}_{i}$ represents the $i$-th residue of the peptide should be given in the caption.
  • Figure 2: Convert peptide chains into discrete curves. (A) The top is the cartoon representation of chain A of TTR V30M mutant (PDB entry: V30Mn), and the bottom is its corresponding discrete curve. (B) The top is the cartoon representation of chain A of one amyloid form of V30Mn (V30Ma-dimer), and the bottom is its corresponding discrete curve. Red balls represent the $\text{C}_\alpha$ atoms.
  • Figure 3: The truncated hop distance between chain A of V30Mn and V30Ma-dimer. The entries represented by the rows and columns are the corresponding residue numbers. Here, we only considered the truncated hop distance of AAs of the same sequence fragment.
  • Figure 4: Structural analysis based on the binary matrices with 25Å cutoff distance. (A) Binary matrices constructed by setting the cutoff distance to 25Å. (B) The sequence fragments corresponding to the first and second brighter regions and their hydrogen bond interaction. (C) The shape of gate of chain A of ATTR.
  • Figure 5: The computation of discrete curvature and discrete torsion where the inserting points were marked with green and the orange circle is the osculating circle at $\gamma_i$.
  • ...and 5 more figures

Theorems & Definitions (26)

  • Definition 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Remark 3.3
  • proof
  • Lemma 3.4
  • Remark 3.5
  • Corollary 3.6
  • proof
  • ...and 16 more