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Drug Design 2 - Applied Bioinformatics Group

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Wiener Index<br />

• In 1947, H. Wiener observed that the boiling<br />

point of alkanes can be predicted based on their<br />

topology<br />

• For n-­‐alkanes this is trivial based on chain length<br />

alone, but not for branched alkanes<br />

• For example, different nonanes have boiling<br />

points between 122°C (2,2,4,4-­‐tetramethyl-­‐<br />

pentane) and 150°C (n-­‐nonane)<br />

• Wiener proposed to model the boiling point as<br />

a linear funcDon of two variables, p and w:<br />

T B = a p + b w + c<br />

with constants a, b, c<br />

• Based on this simple model, he could predict<br />

boiling points with an accuracy of about 1 K<br />

Wiener Index<br />

• p is called polarity number, a 1D<br />

descriptor, coun%ng the number of<br />

carbon-­‐carbon pairs separated by<br />

exactly three bonds (C—C—C—C)<br />

• W, the path number, is a topological<br />

descriptor and is also called Wiener<br />

index<br />

• w is the sum of lengths of all<br />

pairwise paths in the molecular<br />

graph<br />

• In acyclic alkanes this corresponds to<br />

the product of all atoms to the lem<br />

and to the right of any bond<br />

Wiener Index<br />

T B / °C<br />

250<br />

200<br />

150<br />

100<br />

50<br />

0<br />

4 6 8<br />

Chain Kettenlänge length<br />

10 12<br />

Boiling points of n-alkanes<br />

3<br />

1<br />

2<br />

4<br />

5<br />

D 1,2 = 1, D 3,5 = 3, ...<br />

w = (1·4) + (4·1) + (3·2) + (4·1)<br />

= 18<br />

• There is a large number of generaliza%ons of the Wiener index to<br />

make it applicable for cyclic structures as well<br />

• A widely used defini%on is half the sum of the entries of the<br />

distance matrix of the molecular graph (= sum of all minimal pair-­‐<br />

wise paths)<br />

• Wiener index is omen used if the degree of branchedness is<br />

relevant for a property<br />

• It is also relevant for predic%ng pharmacological ac%vi%es, e.g.,<br />

there are models for carboanhydrase II inhibitor ac%vity based on<br />

the Wiener index<br />

4<br />

4<br />

6<br />

4<br />

Saxena, Khadikar, Acta Pharm. 49 (1999) 171-179

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