Protein Folding Optimization Project Report

Introduction

Protein folding is the process by which a protein acquires its functional three-dimensional structure. In our simplified model, a protein is represented as a chain of beads in 3D space. The total potential energy of the system is given by

$$ U_{\text{total}} = \sum_{i=1}^{n-1} k_b \left(|x_{i+1} - x_i| - b\right)^2 + \sum_{i=1}^{n}\sum_{j=i+1}^{n} 4\epsilon \left[\left(\frac{\sigma}{|x_j-x_i|}\right)^{12} - \left(\frac{\sigma}{|x_j-x_i|}\right)^6\right],$$

where:

• $|x_j-x_i|$ is the Euclidean distance between bead $i$ and bead $j$,
• $k_b$ is the bond stiffness constant and $b$ is the equilibrium bond length,
• $\epsilon$ and $\sigma$ characterize the depth and zero-crossing distance of the Lennard-Jones potential.

Our goal is to optimize the protein configuration by finding a local minimum of this energy function. We use the BFGS optimization algorithm provided by SciPy.

Methodology

See Usage.md for instructions on how to run code.

We developed 3 implementations:

1. Vanilla Python + SciPy:
   10 particles average: 4.22 s

2. Cython + SciPy:
   10 particles average: o.32 s 100 particles average: 399.5216

3. cython self implemented BFGS.