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where n is refractive index, M is relative molecular mass and p is density. Its units are those of molar volume, and thus it may be regarded as a measure of bulk, although the constitutive component of polarisability is also present, which leads some workers to regard it as a measure of weak electronic interactions. Although it is measured or calculated as a whole molecule parameter, it is strictly additive, and hence substituent MR values are available (see Hansch, C. and Leo, A. (1979) Substituent Constants for Correlation Analysis in Chemistry and Biology, New York: John Wiley.)

5.4.2 Whole molecule parameters

5.4.2.1 Relative molecular mass (RMM)

RMM or molecular weight is perhaps the simplest of all steric parameters; it is certainly the easiest to calculate, and has been widely used in QSAR. It may be noted that it is often used, together sometimes with log P, in correlating penetration rates through membranes, which suggests a size restriction on molecules passing through pores.

Figure 5.3 Sterimol parameters.

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Figure 5.3 Sterimol parameters.

5.4.2.2 Molecular volume

Molar volume is defined as the volume occupied by one mole of a pure compound; it has an electronic component, since strong intermolecular attraction can hold molecules more closely together. Because of this, it requires experimental measurement.

Molecular volume can be calculated by summing the van der Waals volumes of the constituent atoms (obtained from the van der Waals radii). A more rigorous method is to use a computer program that rolls a water molecule over the molecular surface defined by the van der Waals radii, to give a cavity surface volume.

Perhaps the simplest way of calculating molecular volume accurately is to use the McGowan and Mellors characteristic volume method, which simply sums atomic and bond contributions as follows: C 16.35, H 8.71, O 12.43, N 14.39, F 10.48, Cl 20.95, Br 26.21, I 34.53, S 22.91, P 24.87; for each bond, irrespective of type, subtract 6.56. Thus for NH2COCH3 the value is 2x16.35+12.43+14.39+5x8.71-8x6.56= 50.59 cm3 mol-1.

5.4.2.3 Surface area

The computer program mentioned in Section 5.4.2.2, that rolls a water molecule over the molecular surface, can be used to generate an accessible surface area, a parameter widely used in QSAR studies, since it is molecular surfaces that come into contact with solvent and receptor. It can be particularly effective if the contribution of hydrophobic and hydrophilic surface areas can be distinguished.

5.4.2.4 The kappa index

Introduced by Kier in 1985, this parameter is derived from the number of 2-bond fragments (e.g. C-C-C) in a non-hydrogen molecular skeleton. Linear molecules tend to have higher kappa values, as is shown by the values for the isomeric hexanes:

n-Hexane 5.000

2-Methylpentane 3.200

3-Methylpentane 3.200 2,3-Dimethylbutane 2.222 2,2-Dimethylbutane 1.633

The kappa index is thus a shape parameter. It is readily calculated, and requires no experimental measurement.

5.4.2.5 Minimal steric difference

This parameter assesses the difference between molecules in terms of the parts which do not overlap when one chemical formula is placed on top of the other. If for example, piperidine (5.6) is compared with pyrrolidine (5.7), the methylene group, surrounded by the dotted circle, will determine the MSD, since this is the only portion which does not overlap. The rules of the calculation are as follows:

(i) hydrogen atoms are ignored,

(ii) elements in the second period of the Periodic Table have a weighting of

(iii) elements in the third period have a weighting of 1.5 (e.g. S),

(iv) elements in higher periods have a weighting of 2 (e.g. Br).

Thus the MSD between piperidine and pyrrolidone is 1, and that between pyrrolidine and indole (5.8) is 4.

5.4.2.6 Molecular shape analysis

Devised by Hopfinger in 1980, this method uses a similar principle to that involved in the minimal steric difference method. It uses the concept of common overlap volume between a reference compound (usually the most active) and the other compounds in a series; this common overlap volume is considered to represent the steric requirements of the receptor. The method in effect uses the steric similarity of a set of molecules as a parameter.

5.4.2.7 Molecular similarity

A third parameter using the concept of similarity has been developed by Richards, and is incorporated in the TSAR (Tools for Structure-Activity Relationships) software. Based on equations derived by Carbo and by Hodgkin, it allows comparison of the similarity of a set of molecules to a standard (e.g. the most active in a series) on the basis of either electrostatic potential or steric parameters. Although quite new in concept, it is already finding wide application in QSAR analysis. A number of other methods of determining molecular similarity have recently been developed, and are finding use in, for example, the searching of data-bases for the screening of compounds for specified types of drug action.

5.4.2.8 3-D parameters

Most of the classical QSAR parameters, with the exception of those that model shape, take no account of conformation or of the fact that most molecules are three-dimensional. Nevertheless, since a significant contribution to a molecule's biological activity arises from its fit and binding to a receptor, molecular three-dimensionality is clearly important. In recent years, therefore, much effort has gone into the examination and development of parameters that reflect that three-dimensionality. Such parameters can be as simple as inter-atomic distances or torsion angles or as complex as the distribution of electrostatic potential around a molecule. One approach that has aroused much interest is that known as CoMFA (comparative molecular field analysis). This involves firstly superimposing the molecules to be studied, within a three-dimensional grid or lattice. This is a simple procedure for most congeneric series, whereby the common features of the molecules can readily be superimposed. For non-congeneric series, with no obvious common features, alignment is much more difficult and more subjective. Hence most CoMFA studies to date have been concerned with congeneric series. A probe atom is then placed at each lattice point in turn, and the steric (Lennard-Jones) and electrostatic (Coulombic) fields exerted by each molecule at each lattice point are then calculated. This results in a large number of data-points, and partial least squares (PLS) statistics is used to determine the minimal set of data-points necessary to distinguish the set of compounds according to their biological activities. The PLS model then has to be cross-validated, for example by the leave-one-out method. If necessary and appropriate, re-alignment of the most poorly predicted compounds can then be carried out, and the above steps repeated. The contoured QSAR coefficients can then be displayed to allow visualisation of regions where electrostatic and/or steric fields have the greatest effect on activity.

5.5 TOPOLOGICAL PARAMETERS

Graph theory is that branch of chemistry dealing with molecular topology, since a molecular structure is described as a graph. Graph theory is particularly concerned with the way atoms are connected in a molecule, and many attempts have been made to relate topology to molecular properties.

Of these approaches, the most successful is undoubtedly that of Kier and Hall, who developed a series of topological parameters called molecular connectivities (mx) from an original concept of Randic. The superscript m denotes the order of the parameter. Zero order connectivity (0x) is the simplest and is defined by Equation [5.20], where 5i is a number assigned to each non-hydrogen atom, reflecting the number of non-hydrogen atoms bonded to it. Thus for 1-butane (5.9),

= ! (because €, is attached to Q nnty). = 2 (because C„ is attached to C„ and

The first order connectivity (1x) is derived for each bond by calculating the product of the numbers associated with the two atoms of the bond. The reciprocal of the square root of this number is the bond value. Bond values are summed to give the first order connectivity for the molecule, so that the value of 1-butane is,

The 1% value for 2-butane is similarly calculated to be 1.732.

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