Subknots in ideal knots, random knots, and knotted proteins

Area of Science:

• Computational topology
• Biophysics
• Knot theory

Background:

• Circular knots are complex topological structures found in various scientific domains.
• Understanding the substructure of these knots, known as subknots, is crucial for their characterization.
• Previous methods lacked a systematic way to dissect and identify all subknots within a circular knot.

Purpose of the Study:

• To introduce a novel computational tool, disk matrices, for analyzing subknots in circular configurations.
• To characterize and compare subknots in different knot representations, including energy-minimized and random structures.
• To investigate the presence and distribution of subknots in biological molecules like proteins.

Main Methods:

• Development of disk matrices to encode subchain knotting in circular knots.
• Dissection of circular knots into constituent subknots using a robust closure protocol on linear chains.
• Analysis of subknots in energy-minimized (KnotPlot, ideal geometric) and random knot configurations.
• Comparison of subknots in ideal configurations with classical prime knot diagrams and protein knot structures.

Main Results:

• Disk matrices effectively identify and characterize subknot types within global circular knots.
• Subknot sets from ideal geometric configurations are consistently found across numerous random configurations of the same knot type.
• A comparison between ideal configurations and open protein knots reveals specific subknot dispositions in biological knots.

Conclusions:

• Disk matrices provide a powerful framework for dissecting and understanding the complex topology of circular knots.
• The consistent presence of specific subknots in ideal and random configurations suggests underlying topological constraints.
• This approach offers insights into the topological principles governing protein knot structures.