Topological Data Analysis on Proteins

This repository contains a Jupyter Notebook project focused on the application of Topological Data Analysis (TDA) to protein structures, with particular attention to the theoretical foundations and algorithmic implementation of topological methods.

Overview

The aim of this project is to study protein structures using tools from Topological Data Analysis, combining mathematical theory and computational implementation.

Proteins are represented as point clouds derived from atomic coordinates, and their topological properties are analyzed through simplicial complexes and the Mapper algorithm.
The project also includes custom implementations of key algebraic and combinatorial structures used in TDA.

Notebook Content

The core of the project is contained in the notebook:

• Topological_Data_Analysis_on_Proteins.ipynb

The notebook is structured as follows:

1. Installation (Google Colab)

Instructions and dependencies required to run the notebook in a Google Colab environment.

2. Libraries

Import and usage of the main scientific Python libraries used throughout the analysis.

3. Auxiliary Functions

Helper functions used for data manipulation, distance computations, and intermediate steps in the TDA pipeline.

4. Sparse Matrices over 𝑍₂

A custom class for handling sparse matrices over the field 𝑍₂, which are essential for boundary operators in simplicial homology computations.

5. Simplicial Complexes

Implementation of a class representing simplices and simplicial complexes.

A brief theoretical recall is included:

A simplicial complex is a collection of simplices closed under the operation of taking faces.

This section provides the mathematical backbone for the topological constructions used later in the notebook.

6. The Mapper Algorithm

Implementation and explanation of the Mapper algorithm, one of the main tools in Topological Data Analysis.

Mapper is used to:

• Project high-dimensional data to a lower-dimensional filter function
• Construct a simplicial representation of the data
• Reveal topological structures and clusters in protein data

Dataset

Protein data used in this project are stored in the Data/ directory (if present).
The data typically consist of atomic coordinates extracted from protein structures and transformed into point clouds.

Goals of the Project

• Provide a theoretical and practical introduction to TDA applied to proteins
• Implement core TDA structures from scratch for educational purposes
• Apply the Mapper algorithm to protein-related data
• Explore how topology can describe structural properties of proteins