New biologically motivated knot table

Area of Science:

• Biophysics
• Computational Biology
• Biochemistry

Background:

• Current knot nomenclature, exemplified by Rolfsen's knot table, lacks the ability to differentiate between mirror-image knots.
• This ambiguity poses challenges in studying knotted biopolymers (DNA, proteins) where chirality is crucial.
• Chirality is a significant property in biological systems, affecting molecular function and interactions.

Purpose of the Study:

• To propose a biologically meaningful knot table that resolves the ambiguity of mirror-image knots.
• To establish a method for selecting a representative from a chiral knot pair based on a quantifiable property.
• To enhance the study of chiral knotted biopolymers by providing unambiguous knot identification.

Main Methods:

• Proposed a novel knot classification system based on the mean writhe of knot realizations.
• Investigated numerical evidence for the invariance of the sign of mean writhe within chiral knot pairs.
• Examined the writhe of minimal or ideal conformations as an alternative method for distinguishing chiral knots.

Main Results:

• Numerical evidence suggests that the sign of the mean writhe is invariant for each knot within a chiral pair.
• The mean writhe, averaged over large ensembles of random knot realizations, provides a consistent chiral indicator.
• The writhe of minimal/ideal conformations also effectively distinguishes chiral knot pairs, aligning with mean writhe results.

Conclusions:

• The proposed mean writhe method offers a biologically relevant way to distinguish between enantiomorphic knots.
• This new nomenclature provides a robust solution for identifying and classifying chiral knots in biopolymers.
• The consistency between mean writhe and minimal conformation writhe strengthens the proposed classification system.