Abstract
In contrast to the standard construction of Voronoi regions, in which the boundaries between different regions are at equal distance from the given points, we consider the construction of modified Voronoi regions obtained by giving greater weights to spots reported to have higher abundance. Specifically we are interested in applying this approach to 2-D proteomics maps and their numerical characterization. As will be seen, the boundaries of the weighted Voronoi regions are sensitive to the relative abundances of the protein spots and thus the abundances of protein spots, the z component of the (x, y, z) triplet, are automatically incorporated in the numerical analysis of the adjacency matrix, rather than used to augment the adjacency matrix as non-zero diagonal matrix elements. The outlined approach is general and it may be of interest for numerical analyses of other maps that are defined by triplets (x, y, z) as input information.
Similar content being viewed by others
References
Randić M.: Quantitative Characterization of Proteomics Maps by Matrix Invariants, (ed.). by Conn., P.M. Handbook of Proteomics Methods, pp. 429–450. Humana Press, Inc., Totowa, NJ (2003)
Randić M., Zupan J., Balaban A.T., Vikić-Topić D., Plavšić D.: Graphical representation of proteins. Chem. Rev. 111, 790–862 (2011)
González-Díaz H., González-Díaz Y., Santana L., Ubeira F.M., Uriarte E.: Proteomics, networks and connectivity indices. Proteomics 8, 750–778 (2008)
Randić M., Orel R.: On numerical characterization of proteomics maps based on partitioning of 2-D maps into Voronoi regions. J. Math. Chem. 49, 1759–1768 (2011)
Randić M., Kleiner A.F., DeAlba L.M.: Distance/distance matrices. J. Chem. Inf. Comput. Sci. 34, 277–286 (1994)
Randić M.: Novel graph theoretical approach to heteroatom in quantitative structure-activity relationship. Intel. Lab. Syst. 10, 213–227 (1991)
Randić M.: Topological Indices, (ed.) by Schleyer, P.V.R., Allinger, N.L., Clark, T., Gasteiger, J., Kollman, P.A., Schaefer, H.F., Schreiner, P.R. The Encyclopedia of Computational Chemistry, pp. 3018–3032. Wiley, Chichester (1998)
Devillers J., Balaban A.T.: Topological Indices and Related Descriptors in QSAR and QSPR. CRC Press, Boca Raton, FL (2000)
Anderson N.L., Esquer-Blasco R., Richardson F., Foxworthy P., Eacho P.: The effects of peroxisome proliferations on protein abundances in mouse liver. Toxicol. Appl. Pharmacol. 137, 75–89 (1996)
Randić M., Novič M., Vračko M.: On characterization of dose variations of 2-D proteomics maps by matrix invariants. J. Proteome Res. 1, 217–226 (2002)
Randić M., Estrada E.: Order from chaos: observing hormesis at the proteome level. J. Proteome Res. 4, 2133–2136 (2005)
Randić M.: On graphical and numerical characterization of proteomics maps. J. Chem. Inf. Comput. Sci. 41, 1330–1338 (2001)
Randić M., Zupan J., Novič M.: On 3-D graphical representation of proteomics maps and their numerical characterization. J. Chem. Inf. Comput. Sci. 41, 1339–1344 (2001)
Randić M., Witzmann F., Vračko M., Basak S.C.: On characterization of proteomivcs maps and chemically induced chanes in proteomes using matrix invariants: application to peroxisome proliferators. Med. Chem. Res. 10, 456–479 (2001)
Randić M., Zupan J., Novič M., Gute B.D., Basak S.C.: Novel matrix invariants for characterization of changes of proteoics maps. SAR QSAR Environ. Res. 13, 689–703 (2002)
Randić M., Lerš N., Vukičević D., Plavšić D., Gute B.D., Basak S.C.: Canonical labeling of proteome maps. J. Proteome Res. 4, 1347–1352 (2005)
Randić M.: A graph theoretical characterization of proteomics maps. Int. J. Quantum Chem. 90, 848–858 (2002)
Randić M., Novič M., Vračko M., Plavšić D.: Study of proteome maps using partial ordering. J. Theor. Biol. 266, 21–28 (2010)
Bajzer Ž., Randić M., Plavšić D., Basak S.C.: Novel map descriptors for characterization of toxic effects in proteomics maps. J. Mol. Graph. Model. 22, 1–9 (2003)
Randić M., Lerš N., Plavšić D., Basak S.C.: Characterization of 2-D proteome maps based on the nearest neighborhoods of spots. Croat. Chem. Acta 77, 345–351 (2004)
Randić M., Novič M., Vračko M.: Novel characterization of proteomics maps by sequential neighborhood of protein spots. J. Chem. Inf. Model. 45, 1205–1213 (2005)
Randić M., Lerš N., Plavšić D., Basak S.C.: On invariants of a 2-D proteome map derived from neighborhood graphs. J. Proteome Res. 3, 778–785 (2004)
Randić M., Orel R.: On numerical characterization of proteomics maps based on partitioning of 2-D maps into Voronoi regions. J. Math. Chem. 49, 1759–1768 (2011)
Voronoi G.: Nouvelles applications des paramètres continus à à la théorie des forms quadratiques. J. Reine Angewandte Math. 133, 97–178 (1907)
Delaunay B: Sur la sphère vide. Izv. Akad. Nauk SSSR, Otdel. Mat. Estest. Nauk 7, 793–800 (1934)
Kowalski B.R., Bender C.F.: Pattern recognition. Powerful approach to interpreting chemical data. J. Am. Cem. Soc. 94, 5632–5639 (1972)
Ash P.F., Bolker E.D.: Generalized Dirichlet tessellations. Geometrie Dedicata 20, 209–243 (1986)
L. Mu, WVD2009 (Multiplicatively weighted Voronoi diagram, computer program) http://www.ggy.uga.edu/people/faculty/mu/ (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Orel, R., Randić, M. On characterizing proteomics maps by using weighted Voronoi maps. J Math Chem 50, 2689–2702 (2012). https://doi.org/10.1007/s10910-012-0058-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-012-0058-y