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The bracket polynomial of bonded knots and applications to entangles proteins

Boštjan Gabrovšek, Matic Simonič

TL;DR

This work models protein structures with intramolecular bonds as bonded knots and develops a polynomial invariant framework based on skein theory. It constructs the framed bonded Kauffman bracket skein module $\\mathcal{K}^\text{fr}$, proves it is freely generated by the basis $\\mathcal{B}^\text{fr}=\\{\\Theta^{m}H^{n}\\}$, and defines a normalized invariant $\\llbracket K \\rrbracket \\in \\mathbb{Z}[A^{\pm1},\\Theta,\\H]$. In the topological case, the module collapses to powers of $\\Theta$ with $\\H=(-A^2-A^{-2})\\Theta$, simplifying computations. An explicit example for the TRTX-Tp1a toxin demonstrates the computation workflow and yields a closed-form topological invariant $\\,[K]_{\\mathcal{B}^\text{top}} = \frac{A^4}{(1+A^4)^2}\\Theta^3$.

Abstract

We model proteins with intramolecular bonds, such as disulfide bridges, using the concept of bonded knots -- closed loops in three-dimensional space equipped with additional bonds that connect different segments of the knot. We extend the Kauffman bracket polynomial (which is closely related to the Jones polynomial) to bonded knots through the introduction of the bonded version of the Kauffman bracket skein module. We show that this module is infinitely generated and torsion-free for both the rigid and topological case of bonded knots, providing an invariant of such structures.

The bracket polynomial of bonded knots and applications to entangles proteins

TL;DR

This work models protein structures with intramolecular bonds as bonded knots and develops a polynomial invariant framework based on skein theory. It constructs the framed bonded Kauffman bracket skein module , proves it is freely generated by the basis , and defines a normalized invariant . In the topological case, the module collapses to powers of with , simplifying computations. An explicit example for the TRTX-Tp1a toxin demonstrates the computation workflow and yields a closed-form topological invariant .

Abstract

We model proteins with intramolecular bonds, such as disulfide bridges, using the concept of bonded knots -- closed loops in three-dimensional space equipped with additional bonds that connect different segments of the knot. We extend the Kauffman bracket polynomial (which is closely related to the Jones polynomial) to bonded knots through the introduction of the bonded version of the Kauffman bracket skein module. We show that this module is infinitely generated and torsion-free for both the rigid and topological case of bonded knots, providing an invariant of such structures.

Paper Structure

This paper contains 8 sections, 6 theorems, 62 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. The bonded Kauffman bracket skein module
  4. The generating set
  5. Freeness
  6. Normalization of the framed bonded KBSM
  7. The case of topological bonded knots
  8. An example

Key Result

Theorem 2.1

Two bonded links are ambient isotopic if and only if their diagrams are related by a finite sequence of (Reidemeister) moves I - V depicted in fig:reid.

Figures (9)

  • Figure 1: The bonded structure of a protein. Right: the 3D ribbon model of the toxin FS2 isolated from black mamba (Dendroaspis polylepis) venom (PDB 1TFS), left: the associated bonded knot diagram closed with a direct segment. The PDB code refers to the protein structure entry in the Protein Data Bank burley2019rcsb.
  • Figure 2: Examples of entangles structures in proteins. From left to right: an open knot (PDB 7ECD), a cystine knot (PDB 2MXM), where a bond threads through a loop created by the other two bonds, a deterministic $\theta$-curve (PDB 1AOC), a deterministic link (PDB 2LFK) and a lasso, where a tail pierces the minimal surface enclosed with a bond (PDB 1EWS). The last four structures are bonded structures (the protain contains disulfide bridges). Some examples are taken from sulkowska2020folding.
  • Figure 3: Reidemesiter moves for bonded links. Although not indicated in the figures, arcs in moves I, II, III and the free arcs in IV and IV' can be either link arcs or (colored) bonds.
  • Figure 4: The move RV (a rigid-vertex versions of the move V).
  • Figure 5: Left: a rigid vertex of a framed bonded link, right: framing markers.
  • ...and 4 more figures

Theorems & Definitions (14)

  • Theorem 2.1: Kauffman1989gabrovvsek2021invariant
  • Definition 2.1
  • Theorem 2.2: viro2007quantumBao20
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.1
  • Lemma 3.1
  • proof
  • Definition 3.3
  • Lemma 3.2
  • ...and 4 more