Symmetry 2021, 13, 1849. https://doi.org/10.3390/sym13101849 www.mdpi.com/journal/symmetry Article An Application of the Eigenproblem for Biochemical Similarity Dan-Marian Joiţa 1, *, Mihaela Aurelia Tomescu 2 , Donatella Bàlint 1 and Lorentz Jäntschi 1,3, * 1 Chemistry Doctoral School, Babeş-Bolyai University, 400084 Cluj, Romania; balintdonna@gmail.com or donatella.balint@ubbcluj.ro 2 Department of Mathematics and Informatics, University of Petroșani, 332006 Hunedoara, Romania; MihaelaTomescu@upet.ro 3 Department of Physics and Chemistry, Technical University of Cluj-Napoca, 400641 Cluj, Romania * Correspondence: djoita@chem.ubbcluj.ro (D.-M.J.); lorentz.jantschi@chem.utcluj.ro (L.J.) Abstract: Protein alignment finds its application in refining results of sequence alignment and un- derstanding protein function. A previous study aligned single molecules, making use of the mini- mization of sums of the squares of eigenvalues, obtained for the antisymmetric Cartesian coordinate distance matrices Dx and Dy. This is used in our program to search for similarities between amino acids by comparing the sums of the squares of eigenvalues associated with the Dx, Dy, and Dz dis- tance matrices. These matrices are obtained by removing atoms that could lead to low similarity. Candidates are aligned, and trilateration is used to attach all previously striped atoms. A TM-score is the scoring function that chooses the best alignment from supplied candidates. Twenty essential amino acids that take many forms in nature are selected for comparison. The correct alignment is taken into account most of the time by the alignment algorithm. It was numerically detected by the TM-score 70% of the time, on average, and 15% more cases with close scores can be easily distin- guished by human observation. Keywords: eigenproblem; eigenvalues; molecular alignment; orthogonal alignment; biochemical similarity; antisymmetric matrix 1. Introduction Just visualizing two simple similar structures leads to an immediate detection of pat- terns. Similarity is of convenience for humans, but to power automatic decision mecha- nisms for a PC, it must be measurable. It is mostly used for comparing proteins, but the growing number of PDB structures (currently over 180,000) is many orders of magnitude higher than what the human eye can compare. Because of the large number, it takes days even for current programs to search the database for a query structure. A more reasonable time can be achieved by developing new algorithms [1]. Protein alignment finds its application in refining results of sequence alignment and understanding protein function [2,3]. Choosing the alignment that is most geometrically similar is an easier task compared to evaluating its biological significance [4]. The pursuit of the best method is in progress, with multiple programs being developed during the past decades: • CAB-Align uses the residue–residue contact area to identify regions of similarity [5]. • Caretta uses rotation-invariant technique signals of distances derived from overlap- ping contiguous stretches of residues to find an initial superposition [6]. • DALI [7]. • LS-align generates fast and accurate atom-level structural alignments of ligand mol- ecules through an iterative heuristic search of the target function that combines com- parisons of inter-atom distance with mass and chemical bonds [8]. Citation: Joiţa, D.-M.; Tomescu, M. A.; Bàlint, D.; Jäntschi, L. An Application of the Eigenproblem for Biochemical Simi- larity. Symmetry 2021, 13, 1849. https://doi.org/10.3390/sym1310184 9 Academic Editors: Anthony Harriman and Enrico Bodo Received: 13 August 2021 Accepted: 23 September 2021 Published: 2 October 2021 Publisher’s Note: MDPI stays neu- tral with regard to jurisdictional claims in published maps and insti- tutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and con- ditions of the Creative Commons At- tribution (CC BY) license (https://creativecommons.org/li- censes/by/4.0/).