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MLD50 is the minimum dose that kills 50% of the flies treated. The equation was calculated using a mathematical process called regression (least squares) analysis. The

Figure 5.2 Insecticidal activities of some diethyl-substituted phenyl phosphates. (Results taken from Fukata, T.R. and Metcalf, R.L. (1956) Structure and insecticidal activity of some diethyl-substituted phenyl phosphates. Journal of Agricultural and Food Chemistry 4, 930-935; and Metcalf, R.L. and Fukata, T.R. (1962)

Metasulfapentafluorophenyldiethyl phosphate and metasulfapentafluorophenyl-N-methyl carbamate as insecticides and anticholinesterases. Journal of Economic Entomology 55, 340-341.

Figure 5.2 Insecticidal activities of some diethyl-substituted phenyl phosphates. (Results taken from Fukata, T.R. and Metcalf, R.L. (1956) Structure and insecticidal activity of some diethyl-substituted phenyl phosphates. Journal of Agricultural and Food Chemistry 4, 930-935; and Metcalf, R.L. and Fukata, T.R. (1962)

Metasulfapentafluorophenyldiethyl phosphate and metasulfapentafluorophenyl-N-methyl carbamate as insecticides and anticholinesterases. Journal of Economic Entomology 55, 340-341.

calculation is not difficult, but is rather protracted. It is, therefore, only since the introduction of computers and electronic calculating machines that these so-called regression equations have been used extensively.

A regression equation is a convenient way of quantitatively expressing a correlation, but on its own it does not give as much information as a graph, because the graph indicates how many results were considered, how scattered they were around the best fitting line, and how representative of the results the line is. Additional data should therefore be given, the basic minimum being shown in Equation [5.9], which is laid out in the conventional manner. k1 is log (L/C) = Jfc, + k,a (5.9)

(number of results)

(standard error of the estimate)

(correlation coefficient)

the intercept, and k2 the coefficient for c. The correlation coefficient is a number which varies from zero to 1. The higher the number, the better the correlation. What constitutes a satisfactory correlation coefficient depends on the number of results; the greater the number of results, the lower the acceptable correlation coefficient. In quantitative structure-activity relationships, a figure in excess of 0.9 is aimed for. A useful feature of the correlation-coefficient is that it is the square root of the explained variation; for example, a relationship having a correlation coefficient of 0.990 explains 0.9902x100= 98% of the variation between results.

Additional statistical information, in particular the F value for the equation, is sometimes quoted. The F value is a numerical indicator of whether or not the relationship expressed by the equation is coincidental; the higher the value, the less likely the relationship is due to chance. The format for expressing F distribution and other data is shown in Equation [5.27], and the interpretation explained in Section 5.6.2.5).

The reason for using the logarithm of biological response in Figure 5.2 and Equation [5.8] has thermodynamic origins. The free energy of a transition involving a given molecule is assumed to be the sum of the free energies of its substituent groups. Thus for example, the excess free energy of ionization of p-toluic acid ((5.2) X=p-CH3) over that of benzoic acid is equal to the contribution of the p-methyl group. Equation [5.6] uses log (Kx/Ko) instead of free energy because equilibrium constants are logarithmically related to free energy (AG) through the van't Hoff equation [5.10] in which R is the gas constant and T is the temperature. Log (Kx/Ko) and are therefore also additive. Because of this direct relationship, Hammett's and equivalent equations are said to be linear free energy relationships (LFER). It is therefore logical that the logarithms of biological parameters should also be used in quantitative structure-activity relationships.

Hammett substituent constants can be used only for nuclear aromatic substituents and their effects upon side-chain groups in the meta or para position to them. They cannot be used for ortho substituents because of short-range effects such as steric hindrance and intramolecular hydrogen bonding. Numerous other electronic substituent constants have been introduced since Hammett's original work, many of which have been used in quantitative structure-activity relationships, but only two of them have been used to any great extent. These are the inductive substituent constant and the Taft substituent constant. For information on other constants the reader is referred to the more comprehensive treatises listed at the end of the chapter.

### 5.3.2 Inductive substituent constants

Hammett substituent constants are a measure of both inductive and mesomeric effects. The p-substituent constant (cp) has a greater resonance component than the equivalent meta constant (cm), and the inductive contribution can be calculated from Equation [5.11].

c1 is the inductive substituent constant, and can be used in aliphatic compounds in which the influencing and influenced groups do not form part of a conjugated system. Inductive substituent constants have also been obtained from the dissociation constants of 4-substituted bicyclo(2,2,2)octane carboxylic acids (5.4), and a-substituted acetic acids. A

small selection of inductive substituent constants is given in Table 5.3. More comprehensive lists can be found in the references cited below the Table and at the end of the chapter.

Another set of constants is that devised by Swain and Lupton, who factored the Hammett constant into its inductive and resonance components, and a selection of these is listed in Table 5.3.

5.3.3 Taft's substituent constants

Taft's substituent constants (a*) are a measure of the polar effects of substituents in aliphatic compounds when the group in question does not form part of a conjugated system. They are based on the hydrolysis of esters and are calculated from Equation [5.12], where k represents the rate constant for the hydrolysis of the substituted compound, and k0 that of the methyl derivative.

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