Principal component analysis on a torus: Theory and application to protein dynamics.

Area of Science:

• Computational Chemistry
• Data Science
• Biophysics

Background:

• High-dimensional circular data presents challenges for traditional dimensionality reduction.
• Principal Component Analysis (PCA) is a common technique, but its application to circular data requires adaptation.
• Analyzing molecular dynamics, especially dihedral angles, benefits from robust dimensionality reduction.

Purpose of the Study:

• To develop a novel dimensionality reduction method for high-dimensional circular data.
• To adapt Principal Component Analysis (PCA) for data on a torus.
• To demonstrate the method's utility in analyzing biomolecular dynamics and free energy landscapes.

Main Methods:

• Developed a dimensionality reduction method based on PCA of data points on a torus.
• Introduced geometrical PCA with torus distance measures and subspace projection.
• Implemented a maximal gap shifting technique to minimize projection error for periodic data.

Main Results:

• Successfully applied the method to molecular dynamics simulations of Aib9 and villin headpiece.
• Demonstrated robust construction of metastable states and accurate low-dimensional reaction coordinates.
• Provided direct interpretation of covariances and principal components for angular variables.

Conclusions:

• The developed method offers a robust approach for dimensionality reduction of circular data.
• Maximal gap shifting is a general technique applicable to various dimensionality reduction methods.
• The approach enhances the analysis of biomolecular dynamics and free energy landscapes.