# Topological index

For topological index in mathematics, see Atiyah–Singer index theorem.

In the fields of chemical graph theory, molecular topology, and mathematical chemistry, a topological index also known as a connectivity index is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound.[1] Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariant. Topological indices are used for example in the development of quantitative structure-activity relationships (QSARs) in which the biological activity or other properties of molecules are correlated with their chemical structure.[2]

## Calculation

Topological descriptors are derived from hydrogen-suppressed molecular graphs, in which the atoms are represented by vertices and the bonds by edges. The connections between the atoms can be described by various types of topological matrices (e.g., distance or adjacency matrices), which can be mathematically manipulated so as to derive a single number, usually known as graph invariant, graph-theoretical index or topological index.[3][4] As a result, the topological index can be defined as two-dimensional descriptors that can be easily calculated from the molecular graphs, and do not depend on the way the graph is depicted or labeled and no need of energy minimization of the chemical structure.

## Types

The simplest topological indices do not recognize double bonds and atom types (C, N, O etc.) and ignore hydrogen atoms ("hydrogen suppressed") and defined for connected undirected molecular graphs only.[5] More sophisticated topological indices also take into account the hybridization state of each of the atoms contained in the molecule. The Hosoya index is the first topological index recognized in chemical graph theory, and it is often referred to as "the" topological index.[6] Other examples include the Wiener index, Randić’s molecular connectivity index, Balaban’s J index,[7] and the TAU descriptors.[8][9] The extended topochemical atom (ETA)[10] indices have been developed based on refinement of TAU descriptors.

### Global and local indices

Hosoya index and Wiener index are global (integral) indices to describe entire molecule, Bonchev and Polansky introduced local (differential) index for every atom in a molecule.[5] Another examples of local indices are modifications of Hosoya index.[11]

### Discrimination capability and superindices

A topological index may have the same value for a subset of different molecular graphs, i.e. the index is unable to discriminate the graphs from this subset. The discrimination capability is very important characteristic of topological index. To increase the discrimination capability a few topological indices may be combined to superindex.[12]

### Computational complexity

Computational complexity is another important characteristic of topological index. The Wiener index, Randic’s molecular connectivity index, Balaban’s J index may be calculated by fast algorithms, in contrast to Hosoya index and its modifications for which non-exponential algorithms are unknown.[11]

## Application

### QSAR

QSARs represent predictive models derived from application of statistical tools correlating biological activity (including desirable therapeutic effect and undesirable side effects) of chemicals (drugs/toxicants/environmental pollutants) with descriptors representative of molecular structure and/or properties. QSARs are being applied in many disciplines for example risk assessment, toxicity prediction, and regulatory decisions[13] in addition to drug discovery and lead optimization.[14]

For example, ETA indices have been applied in the development of predictive QSAR/QSPR/QSTR models.[15]

## References

1. ^ Hendrik Timmerman; Todeschini, Roberto; Viviana Consonni; Raimund Mannhold; Hugo Kubinyi (2002). Handbook of Molecular Descriptors. Weinheim: Wiley-VCH. ISBN 3-527-29913-0.
2. ^ Hall, Lowell H.; Kier, Lemont B. (1976). Molecular connectivity in chemistry and drug research. Boston: Academic Press. ISBN 0-12-406560-0.
3. ^ González-Díaz H, Vilar S, Santana L, Uriarte E (2007). "Medicinal chemistry and bioinformatics--current trends in drugs discovery with networks topological indices". Current Topics in Medicinal Chemistry. 7 (10): 1015–29. doi:10.2174/156802607780906771. PMID 17508935.
4. ^ González-Díaz H, González-Díaz Y, Santana L, Ubeira FM, Uriarte E (February 2008). "Proteomics, networks and connectivity indices". Proteomics. 8 (4): 750–78. doi:10.1002/pmic.200700638. PMID 18297652.
5. ^ a b King, R. Bruce (1983). Chemical applications of topology and graph theory: a collection of papers from a symposium held at the University of Georgia, Athens, Georgia, U. S. A., 18–22 April 1983. Amsterdam: Elsevier. ISBN 0-444-42244-7.
6. ^ Hosoya, Haruo (1971). "Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons". Bulletin of the Chemical Society of Japan. 44 (9): 2332–2339. doi:10.1246/bcsj.44.2332..
7. ^ Katritzky AR, Karelson M, Petrukhin R (2002). "Topological Descriptors". University of Florida. Retrieved 2009-05-06.
8. ^ Pal DK, Sengupta C, De AU (1988). "A new topochemical descriptor (TAU) in molecular connectivity concept: Part I--Aliphatic compounds". Indian J. Chem. 27B: 734–739.
9. ^ Pal DK, Sengupta C, De AU (1989). "Introduction of A Novel Topochemical Index and Exploitation of Group Connectivity Concept to Achieve Predictability in QSAR and RDD". Indian J. Chem. 28B: 261–267.
10. ^ Roy K, Ghosh G (2003). "Extended Topochemical Atom (ETA) Indices in the Valence Electron Mobile (VEM) Environment as Tools". Internet Electronic Journal of Molecular Design. 2: 599–620.
11. ^ a b Trofimov MI (1991). "An optimization of the procedure for the calculation of Hosoya's index". Journal of Mathematical Chemistry. 8 (1): 327–332. doi:10.1007/BF01166946.
12. ^ Bonchev D, Mekenyan O, Trinajstić N (1981). "Isomer discrimination by topological information approach". Journal of Computational Chemistry. 2 (2): 127. doi:10.1002/jcc.540020202.
13. ^ Tong W, Hong H, Xie Q, Shi L, Fang H, Perkins R (April 2005). "Assessing QSAR Limitations – A Regulatory Perspective". Current Computer-Aided Drug Design. 1 (2): 195–205. doi:10.2174/1573409053585663. Archived from the original on 2010-06-20.
14. ^ Dearden JC (2003). "In silico prediction of drug toxicity". Journal of Computer-aided Molecular Design. 17 (2–4): 119–27. Bibcode:2003JCAMD..17..119D. doi:10.1023/A:1025361621494. PMID 13677480.
15. ^ Roy K, Ghosh G (2004). "QSTR with extended topochemical atom indices. 2. Fish toxicity of substituted benzenes". Journal of Chemical Information and Computer Sciences. 44 (2): 559–67. doi:10.1021/ci0342066. PMID 15032536.; Roy K, Ghosh G (February 2005). "QSTR with extended topochemical atom indices. Part 5: Modeling of the acute toxicity of phenylsulfonyl carboxylates to Vibrio fischeri using genetic function approximation". Bioorganic & Medicinal Chemistry. 13 (4): 1185–94. doi:10.1016/j.bmc.2004.11.014. PMID 15670927.; Roy K, Ghosh G (February 2006). "QSTR with extended topochemical atom (ETA) indices. VI. Acute toxicity of benzene derivatives to tadpoles (Rana japonica)". Journal of Molecular Modeling. 12 (3): 306–16. doi:10.1007/s00894-005-0033-7. PMID 16249936.; Roy K, Sanyal I, Roy PP (December 2006). "QSPR of the bioconcentration factors of non-ionic organic compounds in fish using extended topochemical atom (ETA) indices". SAR and QSAR in Environmental Research. 17 (6): 563–82. doi:10.1080/10629360601033499. PMID 17162387.; Roy K, Ghosh G (November 2007). "QSTR with extended topochemical atom (ETA) indices. 9. Comparative QSAR for the toxicity of diverse functional organic compounds to Chlorella vulgaris using chemometric tools". Chemosphere. 70 (1): 1–12. Bibcode:2007Chmsp..70....1R. doi:10.1016/j.chemosphere.2007.07.037. PMID 17765287.