# Two Methods for Superposing the Structures of Like-Molecule Assemblies: Application to Peptide and Protein Oligomers and Aggregates

## TL;DR

This paper introduces two efficient algorithms for aligning structures of assemblies made of identical molecules, like peptides and proteins, without needing to check all possible arrangements.

## Contribution

The novel contribution is two new algorithms (LMADA and LMAGDA) that efficiently superimpose like-molecule assemblies with lower computational cost than exhaustive methods.

## Key findings

- LMADA achieves lower RMSD compared to exhaustive permutation methods.
- LMAGDA provides better alignment of geometrically matching sections of assemblies.
- Both algorithms scale with N² computational cost instead of N!.

## Abstract

Two algorithms are proposed for the superposition of assemblies of like molecules (e.g., peptide and proteins homooligomers and homoaggregates), which do not require examining all permutations of the molecules. Both start from searching the mutual orientation of the two assemblies over a grid of quaternion components for the sub-optimal mapping and orientation of the molecules of the second to those of the first assembly. The first one, termed Like-Molecule Assembly Distance Alignment (LMADA), uses Singular Value Decomposition to superpose the two assemblies, given the sub-optimal mapping. The second one, termed Like-Molecule Assembly Gaussian Distance Alignment (LMAGDA), minimizes the negative of the logarithm of the sum of the Gaussian terms in the distances between the corresponding atoms/sites of all pairs of molecules of the two assemblies in quaternion components, starting from those estimated in the first stage. Both algorithms yield as good or nearly as good superposition, in terms of root mean square deviation (RMSD), as examining all permutations to find the lowest RMSD. LMADA results in lower RMSDs, while LMAGDA in a better alignment of the geometrically matching sections of the assemblies. The costs of the proposed algorithms scale only with N2, N being the number of molecules in the assembly, as opposed to N! when examining all permutations.