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f(P) represents a mathematical function of partition coefficient.

The random walk involves passage across hydrophilic barriers and lipophilic barriers. Substances with low aqueous solubilities will be impeded (or, if the solubility is sufficiently low, prevented) from crossing hydrophilic barriers and there is a similar connection between low lipid solubilities and ease of crossing lipid barriers. Somewhere between the two extremes, there will be an optimal balance between hydrophilic and lipophilic properties, so that a plot of hydrophilic-lipophilic nature against the likelihood of the molecule completing the random walk would be expected to take the form shown in Figure 5.4. Hydrophilic-lipophilic nature can be expressed in terms of a lipid-water partition coefficient, so that log P, where P is the 1-octanol-water partition coefficient, can be used as the abscissa scale. The plot approximates to a parabola, for which the general equation is y = a + i>x + cx1,

and therefore the random walk can be expressed as log y?„ = k, + log P + kj(log P)\

or alternatively for a group of related compounds as log = + ¿-jTC +A-jTt%

Kvdnophilie !°ípo Hydrophot)ic

Figure 5.4 Parabolic dependency of biological response on octanol-water partition coefficient.

Kvdnophilie !°ípo Hydrophot)ic

Figure 5.4 Parabolic dependency of biological response on octanol-water partition coefficient.

where k\, k2 and k3 represent the relevant coefficients. The complete biological process can therefore be fitted into equations of the form

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