2D 16-mer Polymer Unfolding Simulation

Course: BT2042: Fundamentals of Biophysical Chemistry
Authors: Ritvik Pandey (BS21B028), S Deepak Kumar (BS21B013), Rahul Srinivas (BS21B029)

Overview

This repository contains the simulation code and data analysis for the unfolding of a 2D 16-mer polymer. The simulations start from a fully folded state and explore the unfolding dynamics under various non-covalent interaction energy values (-0.25, -0.75, -1.25, and -2.5 kT).

Methodology & Algorithm

The simulation utilizes a Monte Carlo approach to sample the conformational space of the polymer.

• Residue Mapping: The protein structure is tracked using a 2D NumPy array storing the coordinates of each of the 16 residues.
• Move Sets: * End Moves: Available only for residues 1 and 16. Allows movement to 4 adjacent diagonal coordinates if no overlap occurs and bond length is conserved.
  • Corner Moves: Allows intermediate residues to move diagonally if the previous and next residues are fixed, maintaining structural integrity.
• Metropolis Criterion: Move acceptance is dictated by the energy difference between states ($E_{i+1} - E_i$). Favorable moves are accepted; unfavorable moves are accepted with a probability of $e^{(-\Delta E/kT)}$.
• Simulation Length: 100,000 steps per run, averaged over 50 individual molecules for each energy state.

Key Parameters & Metrics

The script tracks multiple structural and thermodynamic parameters to monitor unfolding:

• Radius of Gyration ($R_g$): Measures the compactness of the polymer configuration around its center of mass.
• End-to-End Distance: The spatial distance between the 1st and 16th residue.
• Interaction Energy: Calculated as $E = N imes E_S$, where $N$ is the number of native non-covalent interactions.
• Microstates: Sampled and identified uniquely based on a combined score of Interaction Energy, $R_g$, and End-to-End distance.
• Free Energy ($\Delta G$): Evaluated using $\Delta G = \Delta U - T\Delta S$, utilizing the number of sampled microstates to approximate entropy.

Key Findings

• Energy Dependence: As interaction energies become more negative (stronger), the polymer requires significantly more turns to unfold. For example, at -0.25 kT the mean unfolding turns is ~279, whereas at -2.50 kT it jumps to ~995.
• Free Energy Profiles: The reaction coordinate ($Q$) was defined as $n/9$ (where $n$ is the number of native interactions). The plotted profiles show distinct energy barriers that shift dynamically based on the assigned $E_S$.
• All Interactions vs. Native Only: When non-covalent interactions are permitted between all non-adjacent residues (not just native ones), the unfolding time drastically increases (e.g., mean turns increased from 375 to 743 at E = -1.25 kT).

Data Exports

The simulation outputs the following data for further analysis:

• Total Energy matrices
• Radius of Gyration arrays
• End-to-End distance tracking
• Turn counts for unfolding events
• Microstate frequency distributions
• Animation files (.mp4) generated via matplotlib.animation