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

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.