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Article

An Application of the Eigenproblem for Biochemical Similarity

1
Chemistry Doctoral School, Babeş-Bolyai University, 400084 Cluj, Romania
2
Department of Mathematics and Informatics, University of Petroșani, 332006 Hunedoara, Romania
3
Department of Physics and Chemistry, Technical University of Cluj-Napoca, 400641 Cluj, Romania
*
Authors to whom correspondence should be addressed.
Symmetry 2021, 13(10), 1849; https://doi.org/10.3390/sym13101849
Submission received: 13 August 2021 / Revised: 14 September 2021 / Accepted: 23 September 2021 / Published: 2 October 2021
(This article belongs to the Collection Feature Papers in Chemistry)

Abstract

:
Protein alignment finds its application in refining results of sequence alignment and understanding protein function. A previous study aligned single molecules, making use of the minimization 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 distance 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 distinguished by human observation.

1. Introduction

Just visualizing two simple similar structures leads to an immediate detection of patterns. Similarity is of convenience for humans, but to power automatic decision mechanisms 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 overlapping contiguous stretches of residues to find an initial superposition [6].
DALI [7].
LS-align generates fast and accurate atom-level structural alignments of ligand molecules through an iterative heuristic search of the target function that combines comparisons of inter-atom distance with mass and chemical bonds [8].
MATT uses a fragment-based approach that allows for local flexibility between fragment pairs from two input structures and then a dynamic programming algorithm to assemble these intermediate pairs [9].
TM-align uses the length-independent TM-score as a measure of similarity between two proteins in a dynamic programming approach [10].
Some advances have been made in relation to these algorithms, such as parallel re-implementation of mTM-align/TM-align pm-TM-align [11], parMATT [12], heuristic algorithms, and hierarchical organization mTM-align [13].
The 3D variant of the distance matrix alignment method (DALI) uses rotation and translation in order to achieve a smaller distance between equivalent points in the two molecules [14].
In a previous study, the eigenproblem was employed to achieve the proper alignment of single molecules, or the mirror of the proper alignment, and this can be exploited to reduce the number of rotations for which a scoring function needs to run [15].
The eigenproblem is thus defined in the literature as follows:
Given the quadratic matrix A, of the order n, λ is called the eigenvalue of the matrix A and X 0 its associated eigenvector if the relationship A X = λ X is satisfied. The matrix λ I A is singular (because d e t λ I A = 0 ), where I is the unit matrix of the order n. The solutions of the equation d e t λ I A = 0 represent the eigenvalues of the matrix A.
The determinant d e t λ I A is called the characteristic polynomial (ChP) associated with the matrix A. It has a degree equal to the order of the matrix so that the eigenvalues of the matrix A are its roots.
The eigenproblem in relation to geometrical alignment was stated before in the context of surface analysis [16] and control and can go in another direction in the context of amino acids. A subject of the study is a solution to the eigenproblem of amino acid alignment. The Cartesian system is rotated and eventually translated and reflected until the structure arrives at a position characterized by the highest absolute values of the eigenvalues observed on the Cartesian coordinates.
The aim of this study is to find the best geometric alignment of 20 selected amino acids with regard to each other. An extension to the previous study described by Jäntschi [15] has been elaborated. Sums of the squares of eigenvalues (ST = −2Sx − 2Sy − 2Sz) for all three Cartesian coordinate distance matrices (Dx, Dy, and Dz) are compared. By removing atoms, smaller Dx, Dy, and Dz matrices are obtained and more ST sums are added to the comparison. Percentual similarities are found between these sums. Candidates are aligned by the eigenproblem algorithm, and trilateration is used to attach all previously striped atoms. To verify, a TM-score is run on the resulting full-structure candidates.

2. Materials and Methods

In [15], it was shown that the Cartesian distance matrix is antisymmetric and therefore its eigenvalues are purely imaginary, as well as the fact that the best alignment of a molecule is obtained for the minimum value of the sum of the squares of eigenvalues of the Cartesian distance matrix.
Thus, the angle of rotation of the structure must be found around an axis for which the minimum of this amount is obtained. One method of finding the angle of rotation around an axis for the best alignment is as follows: in the case of an amino acid with 5 atoms, we note the vertices of the graph corresponding to the organic compound with V i x i , y i , z i ,   i = 1 , 5 ¯ . We want to find the optimal angle of rotation around the Oz axis, for example. The characteristic polynomial associated with the matrix of Cartesian distances on Ox can be approximated in this way:
C h P λ ,   D x = λ 3 λ 2 + j = 1 , 4 ¯ i = 2 , 5 ¯ j < i x i x j 2 ,
which leads to the problem of finding the rotation angle in the xOy plane so as to obtain the maximum value of the sum
S x = j = 1 , 4 ¯ i = 2 , 5 ¯ j < i x i x j 2 .
Because the term x i x j 2 becomes maximum when V j V i , O x = 0 , we calculate the amount S x   using the law of motion of the rotation of a body about a fixed axis:
x i = x i cos φ y i sin φ y i = x i sin φ + y i cos φ
where φ, in turn, takes the value V j V i , O x ;   j = 1 , 4 ¯ ;   i = 2 , 5 ¯ ;   j < i .
Using the interpolation method, we find the value of the angle of rotation around the Oz axis. Similarly, we proceed to find the angle of rotation of the structure around one of the other two axes.
The eigenvalues of the associated Cartesian coordinate distance matrix Dx are always two conjugate purely imaginary solutions: λ 1 2 = λ 2 2 = S x . Sums of the form ST = −2Sx − 2Sy − 2Sz, associated with Dx, Dy, and Dz matrices, are compared in order to find similarities.
Starting from the eigenproblem approach, 20 essential amino acids that take many forms in nature are selected from available databases.
The alignments for these amino acids (downloaded from PubChem), with compound CIDs 750, 5862, 5950, 5951, 5960, 5961, 5962, 6057, 6106, 6137, 6140, 6267, 6274, 6287, 6288, 6305, 6306, 6322, 33032, 145742, are computed. In this example, the one with the fewest heavy atoms is chosen for reference, glycine 00750.sdf. The following tables for the other cases in which the rest of the structures are references can be found in the Supplementary Materials section:
3D structural data for heavy atoms
3D distance matrix for heavy atoms
Table 1, Table 2 and Table 3 depict the Cartesian coordinate distance matrices for heavy atoms. They are antisymmetric, so their eigenvalues, in Table 4, are imaginary.
It can be observed that unlike eigenvalues for a symmetric matrix, we obtain a single pair of complementary imaginary numbers regardless of the number of atoms in the compound. Another good part of this approach is that, as shown in Table 5, the polynomial can be expressed with real-value coefficients as a product of a polynomial of degree 2 and a monomial of degree (n − 2), leading to a faster response from the program.
Making use of the eigenproblem approach (named the OrigEig function), the other amino acids are aligned to glycine. Candidates with a lower number of atoms than the original are processed while searching for ST similarities. The rest of the atoms are later added using a trilateration algorithm found and used from the literature [17]. Some capabilities are added, such as importing original data (*.sdf or *.xyz by the impCart function); performing *.sdf to *.xyz file conversion; removing hydrogen atoms for convenience; and exporting all compared rotated structures as *.xyz (by the writexyz function), a scoring function based on the TM-score and the creation of *.xls files. The code and its explanation can be found in the Supplementary Materials section, and a schematic overview is available in Figure 1.
The requirements for this application are:
The “in” and “results” directories, the former containing an “xyz” directory and the latter containing “aligned,” “rotated,” and “tables” directories
Geometrically optimized amino acid *.xyz or *.sdf files that need to be located in the “in” folder
The name of the file representing the selected reference amino acid or the number associated with the file (1 representing the first file in the “in” directory)
Input variable Num.M, which defines how many extra candidates can be taken into consideration in case Num.low is satisfied by only one candidate
Input variable Num.low, which defines the target percentage differences between ST of two candidates in order to accept and stop searching for candidates with fewer atoms
Input variable Num.low2, which defines the percentage of the maximum found TM-score such that even lower-scored candidates are exported in *.xls tables and *.xyz files
Input variables Num.empi1 through 3 needed by the TM-score or another means of choosing between alignments
After the requirements are met, the original eigenproblem algorithm is run in order to be sure that the starting point of the program is a good initial alignment. Then all possible combinations with a smaller number of atoms are found by eliminating atom by atom in the AllE function. Eigenvalues are found for each combination without rotating the candidates. ST sums are compared until the input variables are satisfied or all combinations with a minimum of three atoms are compared. Candidates are aligned by the original eigenproblem approach, possibly good pi/2 rotations are taken into consideration, and trilateration is run. Since the TM-score compares distances between atoms of molecules, candidates are translated on top of the reference structure. Good final candidates are exported.
The following tables are exported as *.xls files in the “results\tables” directory:
1.
3D structural data for heavy atoms as T1
2.
3D distance matrix for heavy atoms as T2
3.
Cartesian coordinate distance matrices for heavy atoms as T3–T5
4.
Eigenvalues for above Cartesian coordinate distance matrices as T6
5.
Polynomials for the same Cartesian coordinate distance matrices as T7
6.
A table containing data such as Table A1 available in Appendix A, but no images, named Tscore
The following files are exported as *.xyz geometry files:
Initial *.sdf files are converted in the “in\xyz” directory.
In the “results\aligned” directory, the results from the original eigenproblem program are exported.
In the “results\rotated” directory, all *.xyz files related to the Tscore table can be found.

3. Results

Eigenvalues of all combinations of atoms are computed for each structure. The −2Sx, −2Sy, and −2Sz values of Dx, Dy, and Dz matrices for aligned glycine are −73.557, −27.349, and −0.004, respectively; sum ST = −100.91.
Comparing alanine 005950.sdf to glycine, six possible combinations of five atoms can be found, the fifth having the closest sum to −100.91, as seen in Table 6.
All possible candidates are parsed by the moreData function in the search for a lower percentage difference between ST sums (in the indx function). The targeted percentage difference is defined by Num.low. A multiplier is chosen to extend the search range at the cost of time, Num.M, since the best alignment might not necessarily be the one with the lowest difference between sums. In this case, the following three are chosen by the program: 1, 3, and 5.
The eigenproblem approach is used on the chosen candidates to obtain an eigenvalue-wise rotation alignment. It is suggested that compounds are obtained in their correct alignment or in the mirror of the proper alignment [15]. The search is extended to these possible good rotations (by the first “for” instruction of the align function). To obtain the position of the other unmatched unaligned atoms, a trilateration algorithm (receiving data from the rest of the align function) is found and used from the literature [17].
Since one of these rotations should lead to a good superposition of the two amino acids, the mean values on each of the axes are found for selected atoms of both structures. The selection is based on atoms indexed in the candidate search presented in Table 6. Subtracting for each of the axes, the candidate structure is translated on top of glycine (by the trans function).
For the resulting candidate combinations, distances are found between pairs of a number of atoms. A MATLAB function matchpairs is used to find atoms that will be superposed based on a linear assignment problem that allows for minimum-cost solutions. These pairs are introduced into a scoring function chosen from the literature, in this case the geometric part of UniAlign-TMscore [2]. All these are executed by the choice function. One change was made since our chosen structures contain a small number of atoms: the 15 subtraction was set to 0 so that we obtained a positive distance under the square root of the empirical scaling factor for distance normalization, d0. This can be modified in empi3. Other scoring functions may be applied. The best result for alanine is superposed in Figure 2 in tube style, on top of glycine, which is presented in ball-and-stick-with-non-colored-bond style.
The best score for each compared structure is exported to the final results in Table A1 available in Appendix A. Using another parameter (Num.low2), scores close to it are added. Elements selected for candidates with fewer atoms are presented in the table since they help make an easy choice between close scores. A *Tscore.xls file is generated at the end of the choice function.

4. Discussion

The TM-score can be used to select a best match from all candidates found by the eigenproblem algorithm, as seen in Table A1 and Table A2 of Appendix A. Of the total of 19 amino acids aligned to glycine, 13 results are singular high-confidence alignments, of which 11 give a high TM-score. Another three (cysteine, lysine, and arginine) give two possible good results each, and the TM-score can be used to distinguish the best one.
There are some mismatches made by the program. For example, in the case of glutamine 005961, the best score is found for a four-atom alignment instead of the correct five-atom alignment case number 483. Another difficulty can be observed in the cases of tryptophan 006305 and glutamic acid 033032, where a small score is given to the aligned case numbers 4/115, which are the only ones with elemental similarities, as depicted in Table 7.
Cysteine is the second amino acid taken as a reference for alignment, and all the candidates that our program outputs are depicted in Table A2 of Appendix A. From the total of 19 amino acids aligned to cysteine, six results can be chosen by the highest TM-score, of which two are singular results. Another five give two or more possible good results each, and the TM-score can be used to distinguish the best one.
The following eight mismatches are presented for cysteine, of which the first four are available in Table 8:
In the case of alanine 005950, a small score is given to the aligned case number 269, which is the only one with elemental similarity.
For valine 006287, threonine 006288, and arginine 006322, the best scores are found for candidates with a lower number of aligned atoms. The best candidates with more aligned atoms are 006287-1, 006288-1, and 6322-19.
The outputs for aspartic acid 005960, lysine 005962, histidine 006274, and tryptophan 006305 did not contain the expected alignments.
As stated above, a parameter is introduced such that close scores are not ignored. In this case, a score of 80% of the maximum is accepted for output. This percentage can be indicated in the Num.low2 parameter. This is needed so that the best alignment is given as a result, even though it is not the one with the highest TM-score.
Another easy way to choose from these candidates is to view the chosen elements and eliminate candidates that might have close numerical scores but wrong atom types. Other scoring functions or a combination of such means could lead to even better results.
The use and applicability of the eigenproblem goes beyond the alignment of molecules [15] and biochemical similarity. Recent reports include analysis of regular graphs for their properties, including eigen-spectra and automorphisms [18], molecular topology [19,20,21,22], characteristic equations, principal component decomposition [23], algebraic topology and generalized Bertrand curves [24], treatment of fuzzy decisions [25] and tridiagonal matrices [26], commutator tables, and Laplacian [27], systems of differential [28], and integro-differential [29] equations, while challenging problems appear in polynomial root evaluation [30] and the characteristic equation of a square matrix of a great order [31].

5. Conclusions

An application of the eigenproblem was elaborated, aiming to find the best geometric alignment of selected amino acids with regard to each other.
We can conclude that the best alignment does not obey a strict trend. The close results of the same algorithm can be taken into account. Even after running a score function, we can conclude that the alignment with the highest score is not always the best alignment.
To reduce the number of rotations for which a scoring function is run, the present algorithm needs to be restricted with a few parameters. In addition, a combination of multiple approaches could lead to faster results.
Taking glycine as a reference, 84% of the best alignments can be numerically pointed by a scoring function such as the TM-score, of which 68% are exported as single candidates, meaning that the restrictive parameters are relevant to the present comparison. For cysteine, only 58% can benefit from the presented scoring function. An extensive database would reveal a logical way of choosing them and help training for machine learning.
After running the present algorithm with the other amino acids as a reference, the correct alignment was numerically detected by the TM-score 70% of the time, on average, and 15% more cases with close scores can be easily distinguished by human observation. The present algorithm can be sped up by full vectorization. Machine learning needs to be added to scoring functions as a means to reduce the impact of limited description capabilities and predetermined theory-inspired functional form. These shortcomings can be solved by not imposing a strict algorithm but letting machine learning capture properties that are hard to model because of many unmeasured/unknown/undiscovered quantitative structure–activity relationships (QSAR). Machine learning can assimilate the fast-growing volume of high-quality structural and interaction data found in the literature.

Supplementary Materials

The following archive is available online at https://www.mdpi.com/article/10.3390/sym13101849/s1: *.zip archive, containing results in *.xyz format, *.xls tables, and pictures of a 3D view of alignments for each amino acid taken as a reference.

Author Contributions

Conceptualization, L.J.; software, L.J. and D.-M.J.; data curation, D.B.; writing, D.-M.J. and M.A.T.; supervision, L.J. All authors have read and agreed to the published version of the manuscript.

Funding

The present work received financial support through the project “Entrepreneurship for Innovation through Doctoral and Postdoctoral Research,” POCU/380/6/13/123886, co-financed by the European Social Fund, through the Operational Program for Human Capital 2014-2020. This research was funded by the Technical University of Cluj-Napoca open access publication grant.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are available in the *.zip archive of the Supplementary Materials section of this article.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A

Aligned amino acids are superposed in Table A1 in tube style on top of glycine, which is presented in ball-and-stick-with-non-colored-bond style. Each amino acid is taken as a reference and presented in its own *Tscore.xls file in the Supplementary Materials section. Three-dimensional renders can be found in the pictures folders of the archive.
Table A1. All structures aligned to glycine; their candidate indexes as exported by the program in *.xyz format; TM-scores; selected elements; and −2Sx, −2Sy, and −2Sz.
Table A1. All structures aligned to glycine; their candidate indexes as exported by the program in *.xyz format; TM-scores; selected elements; and −2Sx, −2Sy, and −2Sz.
3D Views of AlignmentAligned Structure and IndexTM-ScoreSelected Atoms from 000750−2Sx, −2Sy, and −2Sz of the Reference CandidateSelected Atoms from the Aligned Structure −2Sx, −2Sy, and −2Sz of the Aligned Candidate
Symmetry 13 01849 i001005862-130.49953OONCC−0.004
−27.3493
−73.5567
OONCC−13.611
−25.8092
−61.4329
Symmetry 13 01849 i002005862-140.59034OONCC−0.004
−27.3493
−73.5567
OONCC−13.611
−25.8092
−61.4329
Symmetry 13 01849 i003005950-30.97619OONCC−0.004
−27.3493
−73.5567
OONCC−4.4817
−70.1638
−28.7497
Symmetry 13 01849 i004005951-410.80012OONCC−0.004
−27.3493
−73.5567
OONCC−69.7742
−4.2881
−26.9964
Symmetry 13 01849 i005005960-370.86944OONCC−0.004
−27.3493
−73.5567
OONCC−71.2314
−27.4581
−2.0944
Symmetry 13 01849 i006005961-330.84632OONCC−0.004
−27.3493
−73.5567
OONCC−71.2576
−3.3832
−27.3196
Symmetry 13 01849 i007005961-1700.95817OOCC−0.0011
−20.6582
−23.8791
OCNC−24.0884
−0.0004
−21.1849
Symmetry 13 01849 i008005961-1730.93357OOCC−0.0011
−20.6582
−23.8791
NCOC−0.0004
−24.0884
−21.1849
Symmetry 13 01849 i009005961-1980.79095OOCC−0.0011
−20.6582
−23.8791
OCOC−0.2581
−24.3734
−20.2611
Symmetry 13 01849 i010005961-2830.82694OONCC−0.0022
−14.8281
−54.3193
OONCC−53.2066
−1.5086
−15.0372
Symmetry 13 01849 i011005961-4830.85761OONCC−0.0032
−4.3992
−57.2565
OONCC−54.6736
−2.475
−21.8424
Symmetry 13 01849 i012005962-120.75429OONCC−0.004
−27.3493
−73.5567
CNOCC−70.2261
−29.0349
−4.2049
Symmetry 13 01849 i013005962-400.91715OONCC−0.004
−27.3493
−73.5567
OONCC−1.5282
−26.4302
−73.3027
Symmetry 13 01849 i014006057-400.86353OONCC−0.004
−27.3493
−73.5567
OONCC−2.3691
−27.3223
−70.9619
Symmetry 13 01849 i015006106-20.6392OONCC−0.004
−27.3493
−73.5567
OONCC−8.1806
−61.8107
−30.7736
Symmetry 13 01849 i016006137-400.86184OONCC−0.004
−27.3493
−73.5567
OONCC−2.2896
−27.3359
−71.1183
Symmetry 13 01849 i017006140-400.86912OONCC−0.004
−27.3493
−73.5567
OONCC−2.1499
−27.3655
−71.1355
Symmetry 13 01849 i018006267-20.86616OONCC−0.004
−27.3493
−73.5567
OONCC−7.4136
−64.3092
−29.0284
Symmetry 13 01849 i019006274-410.83006OONCC−0.004
−27.3493
−73.5567
OONCC−69.8017
−3.9136
−26.9388
Symmetry 13 01849 i020006287-150.69499OONCC−0.004
−27.3493
−73.5567
OONCC−68.0382
−8.7019
−24.3289
Symmetry 13 01849 i021006288-650.92522OONCC−0.004
−27.3493
−73.5567
OONCC−72.4925
−27.4791
−1.3521
Symmetry 13 01849 i022006305-10.80948OONCC−0.004
−27.3493
−73.5567
CCCCC−0.0317
−25.8926
−74.5664
Symmetry 13 01849 i023006305-40.66756OONCC−0.004
−27.3493
−73.5567
OONCC−0.0317
−74.5664
−25.8926
Symmetry 13 01849 i024006305-420.7386OONCC−0.004
−27.3493
−73.5567
CCNCC−1.1175
−25.7209
−72.1151
Symmetry 13 01849 i025006306-110.82228OONCC−0.004
−27.3493
−73.5567
OONCC−70.3452
−26.3299
−4.0766
Symmetry 13 01849 i026006322-140.97621OONCC−0.004
−27.3493
−73.5567
NNCNC−73.6395
−27.5846
−0.0063
Symmetry 13 01849 i027006322-930.99322OONCC−0.004
−27.3493
−73.5567
OONCC−72.695
−0.0803
−27.8005
Symmetry 13 01849 i028033032-690.96492OONCC−0.004
−27.3493
−73.5567
OOCCC−79.1546
−0.3715
−26.9955
Symmetry 13 01849 i029033032-1150.93477OONCC−0.004
−27.3493
−73.5567
OONCC−73.0115
−27.7341
−1.111
Symmetry 13 01849 i030145742-40.92766OONCC−0.004
−27.3493
−73.5567
OONCC−6.2518
−65.5464
−27.6594
Table A2. All structures aligned to cysteine; their candidate indexes as exported by the program in *.xyz format; TM-scores; selected elements; and −2Sx, −2Sy, and −2Sz.
Table A2. All structures aligned to cysteine; their candidate indexes as exported by the program in *.xyz format; TM-scores; selected elements; and −2Sx, −2Sy, and −2Sz.
3D Views of AlignmentAligned Structure and IndexTM-ScoreSelected Atoms from 000750−2Sx, −2Sy, and −2Sz of the Reference CandidateSelected Atoms from the Aligned Structure−2Sx, −2Sy, and −2Sz of the Aligned Candidate
Symmetry 13 01849 i031000750-60.48834OONCC−13.611
−25.8092
−61.4329
OONCC−73.5567
−27.3493
−0.004
Symmetry 13 01849 i032000750-80.42207OONCC−13.611
−25.8092
−61.4329
OONCC−73.5567
−0.004
−27.3493
Symmetry 13 01849 i033000750-100.44278ONCCC−13.611
−25.8092
−61.4329
ONCOC−0.004
−27.3493
−73.5567
Symmetry 13 01849 i034000750-110.44133OCCC−13.611
−25.8092
−61.4329
OCNC−73.5567
−27.3493
−0.004
Symmetry 13 01849 i035005950-1480.50246OONCC−8.3671
−29.9695
−62.0051
OOCCC−71.7263
−5.4464
−25.4841
Symmetry 13 01849 i036005950-1640.5433OOCCC−8.3671
−29.9695
−62.0051
OOCCC−71.7263
−25.4841
−5.4464
Symmetry 13 01849 i037005950-2480.60276OONCC−8.3671
−29.9695
−62.0051
OOCCC−71.7263
−5.4464
−25.4841
Symmetry 13 01849 i038005950-2690.56223OONCC−8.3671
−29.9695
−62.0051
OONCC−0.31
−73.3751
−27.8286
Symmetry 13 01849 i039005951-160.78486SOONCCC−37.2279
−107.1892
−159.5151
OOONCCC−147.0913
−24.8576
−106.5052
Symmetry 13 01849 i040005960-10.42025SOONCCC−37.2279
−107.1892
−159.5151
NOCOCCC−11.3045
−110.1124
−182.4483
Symmetry 13 01849 i041005960-60.4143SONCCC−37.2279
−107.1892
−159.5151
OCNCCC−11.3045
−182.4483
−110.1124
Symmetry 13 01849 i042005960-70.3736SOONCCC−37.2279
−107.1892
−159.5151
NOOOCCC−182.4483
−110.1124
−11.3045
Symmetry 13 01849 i043005960-100.42411SOOCCC−37.2279
−107.1892
−159.5151
CNCCOC−11.3045
−182.4483
−110.1124
Symmetry 13 01849 i044005960-120.40875SOONCCC−37.2279
−107.1892
−159.5151
OOOCCCC−182.4483
−110.1124
−11.3045
Symmetry 13 01849 i045005960-130.39022OONCC−37.2279
−107.1892
−159.5151
OCCCC−110.1124
−11.3045
−182.4483
Symmetry 13 01849 i046005960-140.39347SOONCCC−37.2279
−107.1892
−159.5151
ONCOCCC−11.3045
−110.1124
−182.4483
Symmetry 13 01849 i047005960-150.45748SOONCCC−37.2279
−107.1892
−159.5151
CCNOCOC−11.3045
−110.1124
−182.4483
Symmetry 13 01849 i048005960-240.42411SOOCCC−37.2279
−107.1892
−159.5151
CNCCOC−110.1124
−11.3045
−182.4483
Symmetry 13 01849 i049005960-250.37188SOOCCC−37.2279
−107.1892
−159.5151
CONCOC−182.4483
−110.1124
−11.3045
Symmetry 13 01849 i050005960-260.39022OONCC−37.2279
−107.1892
−159.5151
OCCCC−11.3045
−110.1124
−182.4483
Symmetry 13 01849 i051005960-270.40875SOONCCC−37.2279
−107.1892
−159.5151
OOOCCCC−24.9542
−207.716
−70.9892
Symmetry 13 01849 i052005960-280.39485SOONCCC−37.2279
−107.1892
−159.5151
NOOOCCC−207.716
−24.9542
−70.9892
Symmetry 13 01849 i053005960-310.3717SOONCCC−37.2279
−107.1892
−159.5151
OCCOCOC−24.9542
−207.716
−70.9892
Symmetry 13 01849 i054005960-350.42601SOCCC−37.2279
−107.1892
−159.5151
OCCOC−24.9542
−207.716
−70.9892
Symmetry 13 01849 i055005960-370.42553SOONCCC−37.2279
−107.1892
−159.5151
OOOCCCC−207.716
−70.9892
−24.9542
Symmetry 13 01849 i056005960-430.43719SOOCCC−37.2279
−107.1892
−159.5151
OOCOCC−207.716
−70.9892
−24.9542
Symmetry 13 01849 i057005960-490.42601SOCCC−37.2279
−107.1892
−159.5151
OCCOC−70.9892
−24.9542
−207.716
Symmetry 13 01849 i058005960-500.46341SOOCCC−37.2279
−107.1892
−159.5151
COCOCC−207.716
−70.9892
−24.9542
Symmetry 13 01849 i059005960-520.42553SOONCCC−37.2279
−107.1892
−159.5151
OOOCCCC−207.716
−70.9892
−24.9542
Symmetry 13 01849 i060005961-140.61685SCCC−31.8416
−89.3906
−126.7171
OCCC−1.4071
−33.3867
−210.7718
Symmetry 13 01849 i061005961-1550.55368SOOCCC−5.2465
−88.3648
−122.3962
COOCNC−118.8959
−26.4831
−65.5527
Symmetry 13 01849 i062005961-1790.5482SOONCC−28.5971
−38.2796
−135.5565
OCCNCC−14.3175
−66.8735
−113.4675
Symmetry 13 01849 i063005961-2300.59214SOOCCC−28.5971
−38.2796
−135.5565
COOCNC−118.8959
−26.4831
−65.5527
Symmetry 13 01849 i064005961-2390.55869SOOCCC−28.5971
−38.2796
−135.5565
OCNCOC−118.8959
−65.5527
−26.4831
Symmetry 13 01849 i065005961-2420.56355SNCCC−28.9822
−86.0507
−136.5352
CNCOC−210.7718
−1.4071
−33.3867
Symmetry 13 01849 i066005962-10.43119SONCCC−37.2279
−107.1892
−159.5151
OCNCOC−8.8452
−80.0543
−237.5683
Symmetry 13 01849 i067005962-20.39314SONCCC−31.7674
−35.7196
−105.9193
OCNCOC−8.8452
−237.5683
−80.0543
Symmetry 13 01849 i068005962-30.41372SONCCC−37.2279
−107.1892
−159.5151
CNOCOC−237.5683
−8.8452
−80.0543
Symmetry 13 01849 i069005962-150.44908SOONCCC−37.2279
−107.1892
−159.5151
COONCCC−8.8452
−80.0543
−237.5683
Symmetry 13 01849 i070005962-180.40717SOOCCC−37.2279
−107.1892
−159.5151
ONCCCC−237.5683
−80.0543
−8.8452
Symmetry 13 01849 i071005962-230.44908SOONCCC−37.2279
−107.1892
−159.5151
COONCCC−237.5683
−8.8452
−80.0543
Symmetry 13 01849 i072005962-250.4643SOOCCC−37.2279
−107.1892
−159.5151
COOCCC−237.5683
−80.0543
−8.8452
Symmetry 13 01849 i073005962-300.40299OONCC−31.7674
−35.7196
−105.9193
CNOCC−7.4698
−103.8043
−62.1387
Symmetry 13 01849 i074005962-320.39753SONCCC−31.7674
−35.7196
−105.9193
OONCCC−103.8043
−62.1387
−7.4698
Symmetry 13 01849 i075005962-340.44778OONCC−31.7674
−35.7196
−105.9193
OONCC−103.8043
−7.4698
−62.1387
Symmetry 13 01849 i076005962-390.41984SOOCCC−31.7674
−35.7196
−105.9193
COOCNC−7.4698
−62.1387
−103.8043
Symmetry 13 01849 i077005962-400.46743SONCCC−31.7674
−35.7196
−105.9193
COOCNC−7.4698
−62.1387
−103.8043
Symmetry 13 01849 i078005962-460.38057SOOCCC−31.7674
−35.7196
−105.9193
OCNCOC−7.4698
−62.1387
−103.8043
Symmetry 13 01849 i079005962-480.46743SONCCC−31.7674
−35.7196
−105.9193
COOCNC−103.8043
−7.4698
−62.1387
Symmetry 13 01849 i080006057-160.76753SOONCCC−37.2279
−107.1892
−159.5151
COONCCC−173.9182
−101.9463
−23.1192
Symmetry 13 01849 i081006057-1440.62652SOONCCC−28.5971
−38.2796
−135.5565
COONCCC−13.4258
−40.6418
−148.5831
Symmetry 13 01849 i082006106-160.54749SOOCCC−37.2279
−107.1892
−159.5151
COOCCC−28.1552
−71.8562
−208.2835
Symmetry 13 01849 i083006106-830.68249SOONCCC−37.2279
−107.1892
−159.5151
COONCCC−37.4924
−148.3439
−113.3827
Symmetry 13 01849 i084006137-180.8267SOONCCC−37.2279
−107.1892
−159.5151
COONCCC−156.8033
−107.15
−27.0805
Symmetry 13 01849 i085006140-50.65026SOONCCC−11.1453
−70.4461
−121.9608
COONCCC−9.3713
−53.8297
−141.0272
Symmetry 13 01849 i086006140-280.76831SOONCCC−37.2279
−107.1892
−159.5151
COONCCC−173.9288
−101.9683
−23.0865
Symmetry 13 01849 i087006140-2280.62808SOONCCC−28.5971
−38.2796
−135.5565
COONCCC−108.7941
−87.3884
−5.489
Symmetry 13 01849 i088006140-2560.62895SOONCCC−28.5971
−38.2796
−135.5565
COONCCC−13.4284
−40.6224
−148.603
Symmetry 13 01849 i089006140-4530.69151SOONCCC−31.8416
−89.3906
−126.7171
COONCCC−148.4493
−83.8248
−16.8891
Symmetry 13 01849 i090006267-1500.75426SOOCCC−28.5971
−38.2796
−135.5565
OONCCC−114.7445
−56.4402
−32.1581
Symmetry 13 01849 i091006267-2700.64267OONCCC−30.6696
−83.1536
−110.9209
CNOCOC−157.5302
−52.8112
−14.0738
Symmetry 13 01849 i092006274-110.52022SOOCC−30.6696
−83.1536
−110.9209
CCNCC−29.2987
−34.082
−161.888
Symmetry 13 01849 i093006274-200.51619OCCC−37.2279
−107.1892
−159.5151
CCCN−8.53
−238.2463
−49.0387
Symmetry 13 01849 i094006274-440.47052OONCC−31.7674
−35.7196
−105.9193
OONCC−104.752
−19.0687
−49.267
Symmetry 13 01849 i095006274-750.48041SOOCCC−11.1453
−70.4461
−121.9608
CNOCCC−27.4473
−30.9096
−145.1067
Symmetry 13 01849 i096006274-780.58417SOOCCC−11.1453
−70.4461
−121.9608
CNCCCC−145.1067
−30.9096
−27.4473
Symmetry 13 01849 i097006274-830.48041SOOCCC−11.1453
−70.4461
−121.9608
CNOCCC−145.1067
−27.4473
−30.9096
Symmetry 13 01849 i098006274-1400.47356SOONCC−28.9822
−86.0507
−136.5352
NCOCCC−7.9674
−210.2593
−33.7955
Symmetry 13 01849 i099006274-1510.5215SOOCCC−28.9822
−86.0507
−136.5352
CNCCCC−210.2593
−7.9674
−33.7955
Symmetry 13 01849 i100006274-1700.48165OCCC−28.9822
−86.0507
−136.5352
CCCN−6.8009
−203.291
−41.7099
Symmetry 13 01849 i101006274-1750.48518SOOCCC−28.9822
−86.0507
−136.5352
CCCCNC−6.8009
−41.7099
−203.291
Symmetry 13 01849 i102006274-1840.48165OCCC−28.9822
−86.0507
−136.5352
CCCN−41.7099
−6.8009
−203.291
Symmetry 13 01849 i103006274-1880.47675SOONCCC−28.9822
−86.0507
−136.5352
NOOCCCC−175.3227
−13.4702
−62.6855
Symmetry 13 01849 i104006274-1950.54066SOCCC−28.9822
−86.0507
−136.5352
ONCCC−13.4702
−175.3227
−62.6855
Symmetry 13 01849 i105006274-2530.51888SOOCCC−28.9822
−86.0507
−136.5352
CCCCCC−170.4509
−72.8278
−7.6721
Symmetry 13 01849 i106006274-2660.49887OONCC−30.6696
−83.1536
−110.9209
NCCCC−29.2987
−161.888
−34.082
Symmetry 13 01849 i107006274-2700.47705OONC−30.6696
−83.1536
−110.9209
CNCC−29.2987
−161.888
−34.082
Symmetry 13 01849 i108006274-2750.4984SOOCCC−30.6696
−83.1536
−110.9209
NCCNCC−29.2987
−34.082
−161.888
Symmetry 13 01849 i109006274-2840.47705OONC−30.6696
−83.1536
−110.9209
CNCC−34.082
−29.2987
−161.888
Symmetry 13 01849 i110006274-2910.47128SOONCC−31.8416
−89.3906
−126.7171
OCNCCC−9.7892
−202.8613
−35.0825
Symmetry 13 01849 i111006274-3030.47216SOOCCC−31.8416
−89.3906
−126.7171
CCCCCC−202.8613
−35.0825
−9.7892
Symmetry 13 01849 i112006287-10.44763SOONCCC−37.2279
−107.1892
−159.5151
COONCCC−28.463
−162.9622
−112.2806
Symmetry 13 01849 i113006287-100.50113ONCCC−37.2279
−107.1892
−159.5151
COCNC−28.463
−112.2806
−162.9622
Symmetry 13 01849 i114006287-140.43465SOONCCC−37.2279
−107.1892
−159.5151
OCCNCCC−28.463
−112.2806
−162.9622
Symmetry 13 01849 i115006287-200.43271SOOCCC−37.2279
−107.1892
−159.5151
COOCCC−28.463
−112.2806
−162.9622
Symmetry 13 01849 i116006287-220.43465SOONCCC−37.2279
−107.1892
−159.5151
OCCNCCC−162.9622
−28.463
−112.2806
Symmetry 13 01849 i117006287-240.46687SOOCCC−37.2279
−107.1892
−159.5151
OCCCCC−162.9622
−112.2806
−28.463
Symmetry 13 01849 i118006288-10.48391SOONCCC−37.2279
−107.1892
−159.5151
OOONCCC−43.0635
−66.5743
−180.8565
Symmetry 13 01849 i119006288-30.45585SOONCCC−11.1453
−70.4461
−121.9608
OCCNOCC−43.0635
−180.8565
−66.5743
Symmetry 13 01849 i120006288-60.45633OONCCC−37.2279
−107.1892
−159.5151
CNOCCC−43.0635
−180.8565
−66.5743
Symmetry 13 01849 i121006288-130.51827SONCCC−37.2279
−107.1892
−159.5151
NCOCCC−180.8565
−66.5743
−43.0635
Symmetry 13 01849 i122006288-260.5201SOOCCC−37.2279
−107.1892
−159.5151
OCOCCC−180.8565
−66.5743
−43.0635
Symmetry 13 01849 i123006288-280.51827SONCCC−37.2279
−107.1892
−159.5151
NCOCCC−8.4024
−103.7817
−61.1853
Symmetry 13 01849 i124006288-370.42733SONCCC−31.7674
−35.7196
−105.9193
OONCOC−8.4024
−61.1853
−103.7817
Symmetry 13 01849 i125006288-720.42211SOOCCC−11.1453
−70.4461
−121.9608
OCNCOC−29.1037
−62.1843
−111.9187
Symmetry 13 01849 i126006305-10.50837SONCCC−37.2279
−107.1892
−159.5151
CCCCCC−1.9654
−82.3993
−214.7367
Symmetry 13 01849 i127006305-20.56881SONCCC−37.2279
−107.1892
−159.5151
CCNCCC−3.6968
−75.0753
−224.1486
Symmetry 13 01849 i128006305-50.48382SONCCC−37.2279
−107.1892
−159.5151
CCNCCC−2.4671
−82.2841
−211.504
Symmetry 13 01849 i129006305-90.54483SOONCCC−37.2279
−107.1892
−159.5151
COONCCC−1.9654
−214.7367
−82.3993
Symmetry 13 01849 i130006305-230.47679SONCCC−37.2279
−107.1892
−159.5151
CNCCCC−214.7367
−1.9654
−82.3993
Symmetry 13 01849 i131006305-350.51784SONCCC−37.2279
−107.1892
−159.5151
CNCCCC−224.1486
−3.6968
−75.0753
Symmetry 13 01849 i132006305-360.50682OOCCC−37.2279
−107.1892
−159.5151
CNCCC−75.0753
−224.1486
−3.6968
Symmetry 13 01849 i133006305-440.51678SOONCCC−37.2279
−107.1892
−159.5151
CCCCCCC−224.1486
−75.0753
−3.6968
Symmetry 13 01849 i134006305-490.50682OOCCC−37.2279
−107.1892
−159.5151
CNCCC−75.0753
−3.6968
−224.1486
Symmetry 13 01849 i135006305-590.51678SOONCCC−37.2279
−107.1892
−159.5151
CCCCCCC−1.8573
−208.8667
−86.2881
Symmetry 13 01849 i136006305-600.53173SONCCC−37.2279
−107.1892
−159.5151
CNCCCC−208.8667
−1.8573
−86.2881
Symmetry 13 01849 i137006305-880.49077SONCCC−37.2279
−107.1892
−159.5151
NCCCCC−0.5256
−221.8958
−77.1639
Symmetry 13 01849 i138006305-930.4984SONCCC−37.2279
−107.1892
−159.5151
CCCCCC−0.5256
−77.1639
−221.8958
Symmetry 13 01849 i139006305-970.56351SOONCCC−37.2279
−107.1892
−159.5151
CCCCCNC−0.5256
−77.1639
−221.8958
Symmetry 13 01849 i140006305-980.56197SONCCC−37.2279
−107.1892
−159.5151
CNCCCC−221.8958
−0.5256
−77.1639
Symmetry 13 01849 i141006305-1050.56351SOONCCC−37.2279
−107.1892
−159.5151
CCCCCNC−221.8958
−0.5256
−77.1639
Symmetry 13 01849 i142006305-1100.56356SONCCC−37.2279
−107.1892
−159.5151
CNCCCC−211.504
−2.4671
−82.2841
Symmetry 13 01849 i143006305-1380.50276SOONCCC−37.2279
−107.1892
−159.5151
NCOCCCC−1.4469
−210.87
−83.7564
Symmetry 13 01849 i144006305-1630.59418SONCCC−37.2279
−107.1892
−159.5151
CCCCNC−0.0058
−216.4546
−93.9712
Symmetry 13 01849 i145006305-1860.49257OOCCC−37.2279
−107.1892
−159.5151
CCCCC−48.5216
−239.5349
−13.6686
Symmetry 13 01849 i146006305-1880.48355SONCCC−37.2279
−107.1892
−159.5151
CNCCCC−13.6686
−239.5349
−48.5216
Symmetry 13 01849 i147006305-1990.49257OOCCC−37.2279
−107.1892
−159.5151
CCCCC−48.5216
−13.6686
−239.5349
Symmetry 13 01849 i148006305-2000.47666SOOCCC−37.2279
−107.1892
−159.5151
CCNCCC−239.5349
−48.5216
−13.6686
Symmetry 13 01849 i149006306-260.59343SOONCCC−37.2279
−107.1892
−159.5151
OCCNCCC−187.941
−40.4448
−79.4912
Symmetry 13 01849 i150006306-1060.68164SOCCC−5.2465
−88.3648
−122.3962
NCCCC−20.4327
−39.2382
−155.9536
Symmetry 13 01849 i151006306-1410.57733OONCCC−5.2465
−88.3648
−122.3962
CNCCCC−18.3555
−160.9526
−36.9999
Symmetry 13 01849 i152006306-1750.54571SOOCCC−28.5971
−38.2796
−135.5565
COOCNC−28.3736
−66.0039
−109.0218
Symmetry 13 01849 i153006306-1810.54665SOOCCC−28.5971
−38.2796
−135.5565
OCNCOC−28.3736
−66.0039
−109.0218
Symmetry 13 01849 i154006306-1830.54571SOOCCC−28.5971
−38.2796
−135.5565
COOCNC−109.0218
−28.3736
−66.0039
Symmetry 13 01849 i155006306-2260.56045SOONCCC−28.9822
−86.0507
−136.5352
OCCNCCC−152.9247
−33.7788
−64.0308
Symmetry 13 01849 i156006322-20.55916SOOCCC−28.5971
−38.2796
−135.5565
COOCNC−27.3384
−62.6263
−111.6846
Symmetry 13 01849 i157006322-190.4956SOONCCC−37.2279
−107.1892
−159.5151
COONCCC−158.5429
−38.7577
−80.7785
Symmetry 13 01849 i158006322-420.5614SOOCCC−28.5971
−38.2796
−135.5565
OCNCOC−27.3384
−62.6263
−111.6846
Symmetry 13 01849 i159006322-440.46472SOOCCC−28.5971
−38.2796
−135.5565
NOOCCC−111.6846
−27.3384
−62.6263
Symmetry 13 01849 i160006322-560.49042ONCCC−28.9822
−86.0507
−136.5352
CNCNC−221.7222
−6.933
−23.1061
Symmetry 13 01849 i161006322-600.4524SNCC−28.9822
−86.0507
−136.5352
NCNC−221.7222
−23.1061
−6.933
Symmetry 13 01849 i162006322-690.51145SOOCCC−28.9822
−86.0507
−136.5352
NCCNCC−221.7222
−6.933
−23.1061
Symmetry 13 01849 i163006322-900.4613SOONCCC−28.9822
−86.0507
−136.5352
COCNCCC−204.5901
−27.1111
−18.7144
Symmetry 13 01849 i164033032-130.56115SOONCCC−37.2279
−107.1892
−159.5151
COONCCC−157.6594
−87.1404
−38.0414
Symmetry 13 01849 i165033032-660.45269SOONCCC−31.8416
−89.3906
−126.7171
COONCCC−12.4748
−35.4453
−199.7397
Symmetry 13 01849 i166033032-690.51437SOCCC−31.8416
−89.3906
−126.7171
OCCOC−199.7397
−35.4453
−12.4748
Symmetry 13 01849 i167033032-740.45269SOONCCC−31.8416
−89.3906
−126.7171
COONCCC−199.7397
−12.4748
−35.4453
Symmetry 13 01849 i168145742-210.59575SOOCCC−37.2279
−107.1892
−159.5151
COOCNC−24.7574
−57.3704
−227.05
Symmetry 13 01849 i169145742-280.53132SONCCC−11.1453
−70.4461
−121.9608
CCCCCN−145.192
−12.4766
−45.8437
Symmetry 13 01849 i170145742-290.52833OONCCC−11.1453
−70.4461
−121.9608
OCNCCC−45.8437
−145.192
−12.4766
Symmetry 13 01849 i171145742-430.48734SOCCC−11.1453
−70.4461
−121.9608
CCNCC−145.192
−45.8437
−12.4766

References

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Figure 1. A schematic overview of the algorithm.
Figure 1. A schematic overview of the algorithm.
Symmetry 13 01849 g001
Figure 2. A 3D view of the best alignment of alanine to glycine.
Figure 2. A 3D view of the best alignment of alanine to glycine.
Symmetry 13 01849 g002
Table 1. First Cartesian coordinate (x) distance matrix for glycine (heavy atoms).
Table 1. First Cartesian coordinate (x) distance matrix for glycine (heavy atoms).
Dx12345
100.0100.0010.0180.022
2−0.0100−0.0080.0080.013
3−0.0010.00800.0170.021
4−0.018−0.008−0.01700.004
5−0.022−0.013−0.021−0.0040
Table 2. Second Cartesian coordinate (y) distance matrix for glycine (heavy atoms).
Table 2. Second Cartesian coordinate (y) distance matrix for glycine (heavy atoms).
Dy12345
101.9510.738−0.1300.726
2−1.9510−1.212−2.080−1.224
3−0.7381.2120−0.868−0.012
40.1302.0800.86800.856
5−0.7261.2240.012−0.8560
Table 3. Third Cartesian coordinate (z) distance matrix for glycine (heavy atoms).
Table 3. Third Cartesian coordinate (z) distance matrix for glycine (heavy atoms).
Dz12345
10−1.165−3.549−2.383−1.146
21.1650−2.384−1.2180.019
33.5492.38401.1662.403
42.3831.218−1.16601.236
51.146−0.019−2.403−1.2360
Table 4. Eigenvalues for glycine (heavy atoms).
Table 4. Eigenvalues for glycine (heavy atoms).
x₁x₂x₃x₄x₅
[Dx]6.065i−6.065i000
[Dy]3.698i−3.698i000
[Dz]0.044i−0.044i000
Table 5. The polynomials of [Dx], [Dy], and [Dz] for glycine (heavy atoms).
Table 5. The polynomials of [Dx], [Dy], and [Dz] for glycine (heavy atoms).
Matrix (A)|λ·I−A| Polynomial
[Dx]λ3∙(λ2 + 36.7783)
[Dy]λ3∙(λ2 + 13.6746)
[Dz]λ3∙(λ2 + 0.0019791)
Table 6. All combinations of five atoms in the case of alanine and their ST sums.
Table 6. All combinations of five atoms in the case of alanine and their ST sums.
Possible Atom ChoicesST
1O2N3C4C5C6−103.395
2O1N3C4C5C6−107.168
3O1O2C4C5C6−102.657
4O1O2N3C5C6−134.779
5O1O2N3C4C6−101.514
6O1O2N3C4C5−136.012
Table 7. 3D views of the problematic choice of alignments for glycine.
Table 7. 3D views of the problematic choice of alignments for glycine.
3D Views of AlignmentAligned Structure and IndexTM-ScoreSelected Atoms from 000750Selected Atoms from the Aligned Structure
Symmetry 13 01849 i172006305-10.80948OONCCCCCCC
Symmetry 13 01849 i173006305-40.66756OONCCOONCC
Symmetry 13 01849 i174006305-420.7386OONCCCCNCC
Symmetry 13 01849 i175033032-690.96492OONCCOOCCC
Symmetry 13 01849 i176033032-1150.93477OONCCOONCC
Table 8. 3D views of the problematic choice of alignments for cysteine.
Table 8. 3D views of the problematic choice of alignments for cysteine.
3D Views of AlignmentAligned Structure and IndexTM-ScoreSelected Atoms from 000750Selected Atoms from the Aligned Structure
Symmetry 13 01849 i177005950-2480.60276OONCCOOCCC
Symmetry 13 01849 i178005950-2690.56223OONCCOONCC
Symmetry 13 01849 i179006287-10.44763SOONCCCCOONCCC
Symmetry 13 01849 i180006287-100.50113ONCCCCOCNC
Symmetry 13 01849 i181006288-10.48391SOONCCCOOONCCC
Symmetry 13 01849 i182006288-130.51827SONCCCNCOCCC
Symmetry 13 01849 i183006288-260.5201SOOCCCOCOCCC
Symmetry 13 01849 i184006288-280.51827SONCCCNCOCCC
Symmetry 13 01849 i185006322-190.4956SOONCCCCOONCCC
Symmetry 13 01849 i186006322-420.5614SOOCCCOCNCOC
Symmetry 13 01849 i187006322-690.51145SOOCCCNCCNCC
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Joiţa, D.-M.; Tomescu, M.A.; Bàlint, D.; Jäntschi, L. An Application of the Eigenproblem for Biochemical Similarity. Symmetry 2021, 13, 1849. https://doi.org/10.3390/sym13101849

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Joiţa D-M, Tomescu MA, Bàlint D, Jäntschi L. An Application of the Eigenproblem for Biochemical Similarity. Symmetry. 2021; 13(10):1849. https://doi.org/10.3390/sym13101849

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Joiţa, Dan-Marian, Mihaela Aurelia Tomescu, Donatella Bàlint, and Lorentz Jäntschi. 2021. "An Application of the Eigenproblem for Biochemical Similarity" Symmetry 13, no. 10: 1849. https://doi.org/10.3390/sym13101849

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