The Lennard-Jones Potential Minimization Problem for Prion AGAAAAGA Amyloid Fibril Molecular Modeling

TL;DR

This paper addresses the complex Lennard-Jones potential minimization problem in molecular modeling, particularly for prion amyloid fibrils, by leveraging specific PDB structures to find global minima despite the problem's nonconvexity and numerous local minima.

Contribution

It introduces a novel approach to solving the Lennard-Jones minimization problem in amyloid fibril modeling using molecular data from PDB files.

Findings

Successful molecular models built for prion amyloid fibrils.

Effective minimization of Lennard-Jones potential in complex molecular structures.

Demonstrated approach can handle large numbers of atoms with many local minima.

Abstract

The simplified Lennard-Jones (LJ) potential minimization problem is minimize f(x)=4\sum_{i=1}^N \sum_{j=1,j<i}^N (\tau_{ij}^{-6} -\tau_{ij}^{-3}) subject to x\in \mathbb{R}^n, where $\tau_{ij}=(x_{3i-2}-x_{3j-2})^2 +(x_{3i-1}-x_{3j-1})^2+(x_{3i} -x_{3j})^2$, $(x_{3i-2},x_{3i-1},x_{3i})$ is the coordinates of atom $i$ in $\mathbb{R}^3$, $i,j=1,2,...,N(\geq 2 \quad \text{integer})$, $n=3N$ and $N$ is the whole number of atoms. The nonconvexity of the objective function and the huge number of local minima, which is growing exponentially with $N$, interest many mathematical optimization experts. The global minimizer should be just at the point of the bottom of the LJ potential well. Based on this point, this paper tackles this problem illuminated by amyloid fibril molecular model building. The 3nhc.pdb, 3nve.pdb, 3nvf.pdb, 3nvg.pdb and 3nvh.pdb of PDB bank are used for the successful molecular modeling.