rbLEC - restricted backbone Local Euler Characteristic - from CATH database
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<p><strong>Author: Rodrigo A. Moreira (C) 2023<br>
https://orcid.org/0000-0002-7605-8722<br>
LICENSE: CC BY-NC-ND 4.0 (https://creativecommons.org/licenses/by-nc-nd/4.0/)</strong></p>
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<p><strong>rbLEC - Local Euler Charactersitics - from CATH database</strong></p>
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<p><strong>A. rbLEC NETWORK</strong></p>
<p> [I] The networks for each PDB[1] structure is defined by the PDB atoms N,CA,C of each residue as nodes of a graph G.<br>
[II] An edge of G is set if the distance between two atom in [I] is greater than 2.0 Angstrons.<br>
[III] The graph G is defined in the files with extensions ".network_backboneRE_heavy_gt2"</p>
<p>Equation (1) [6,7]<br>
\begin{equation}<br>
\chi = \sum_{k=1}^{N} \kappa_k = \sum_{k=1}^{N} \underbrace{ \left(1 + \sum_{l=1}^{\infty} (-1)^{l} \frac{v_{l-1}}{l+1} \right)_{k}}_{\kappa_k}<br>
\end{equation}</p>
<p>Equation (2)<br>
\begin{equation}<br>
LEC = \sum_{m \in R} \kappa_m = \kappa_{N} + \kappa_{CA} + \kappa_{C}<br>
\end{equation}</p>
<p><strong>B. FILENAME EXTENSIONS</strong></p>
<p> <strong> B.1</strong> Basic files</p>
<p>".fixed"<br>
PDB file after use of pdbfixer[2] in structures from CATH database.</p>
<p>".dssp"<br>
Output of DSSP[3] software</p>
<p>".stride"<br>
Output of STRIDE[4] software</p>
<p> <strong>B.2</strong> Data files</p>
<p>".network_backboneRE_heavy_gt2" - Generate by <strong>D.2</strong> below.<br>
Describe the network graph, as described in <strong>A.</strong> above.</p>
<p>".knill_curvature" - Generate by <strong>D.1</strong> below.<br>
Contain the filtration of kappas for each vertice of the network.</p>
<p>".residues_curvature" - Generate by <strong>D.1</strong> below.<br>
They are the filtration of LEC, Equation (2) above, for each residue, namely summation of 3 kappas from respective '.knill_curvature', correspoings to PDB atoms N,CA and C, describe in <strong>A.</strong> above.</p>
<p>".label" - Generated by <strong>D.3 </strong>below<br>
Extra file for easier assesment of structures. They have the same information about LEC as described in respective ".residue_curvature" file extensions, but merge also the information from ".dssp" and ".stride" classes as well as residue name and residue ID for each molecule.<br>
Format of columns:<br>
cutoff resname resid DSSP_class STRIDE_class LEC</p>
<p><strong>C. FOLDERS</strong></p>
<p> CATH_FIXED (after uncompress cath_fixed.tar.xz, approximately 13GB)<br>
contains the fixed PDBs and LECs from CATH[5] database</p>
<p><strong>D. SOFTWARE</strong><br>
<strong>D.1</strong> lec.py: compute the kappas in Equation (1) above.<br>
Example usage:<br>
$ python3 lec.py CATH_FIXED/2x0qA02/2x0qA02<br>
It will create the files with extension ".kappas" and ".relec", which reproduces the respectively the files with extension "<strong>.knill_curvature</strong>" and "<strong>.residue_curvature</strong>".</p>
<p> <strong> D.2</strong> pdb2network.lua: creates rbLEC network file (number of nodes and edges list) from PDB to be used as input by lec.py.<br>
Example usage:<br>
$ lua pdb2rbLEC.lua CATH_FIXED/2x0qA02/2x0qA02.fixed<br>
Output reproduces the file CATH_FIXED/2x0qA02/2x0qA02.pdb.<strong>network_backboneRE_heavy_gt2</strong></p>
<p> <strong>D.3</strong> label.lua: create files with extension '*.label' from files '*.pdb.stride', '*.pdb.dssp' and '*.pdb.network_backboneRE_heavy_gt2.residues_curvature.<br>
Example usage:<br>
$ lua label.lua CATH_FIXED/2x0qA02/2x0qA02.pdb<br>
Output reproduces the file CATH_FIXED/2x0qA02/2x0qA02.pdb.<strong>network_backboneRE_heavy_gt2.residues_curvature.label</strong></p>
<p><strong>REFERENCES</strong><br>
[1] Herman, H., Westbrook, J., Feng, Z., Gilliland, G., Bhat, T., Weissig, H., Shindyalov, I., & Bourne, P. (2000). The protein data bank. Nucleic acids research, 28, 235–42.<br>
[2] Eastman, P., Swails, J., Chodera, J., McGibbon, R., Zhao, Y., Beauchamp, K., Wang, L.P., Simmonett, A., Harrigan, M., Stern, C., & others (2017). OpenMM 7: Rapid development of high performance algorithms for molecular dynamics. PLoS computational biology, 13(7), e1005659.<br>
[3] Kabsch, W., & Sander, C. (1983). Dictionary of protein secondary structure: pattern recognition of hydrogen-bonded and geometrical features. Biopolymers: Original Research on Biomolecules, 22(12), 2577–2637.<br>
[4] Frishman, D., & Argos, P. (1995). Knowledge-based protein secondary structure assignment. Proteins: Structure, Function, and Bioinformatics, 23(4), 566–579.<br>
[5] Knudsen, M., & Wiuf, C. (2010). The CATH database. Human genomics, 4(3), 1–6.<br>
[6] Levitt, N. (1992). The Euler characteristic is the unique locally determined numerical homotopy invariant of finite complexes. Discrete & computational geometry, 7, 59–67.<br>
[7] Knill, O. (2011). A graph theoretical Gauss-Bonnet-Chern theorem. arXiv preprint arXiv:1111.5395.</p>
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