A complete analysis of protein binding, including the multiple equilibria below. We write the interaction between a group or free receptor P in a protein and a drug molecule D as
The equilibrium constant, disregarding the difference between activities and concentrations, is
(Or) K [P] [Df] = [PD] … equation(1)
Where, K is the association constant, [P] is the concentration of the protein in terms of free binding sites, [Df] is the concentration of free drug (in moles), sometimes called the ligand, and [PD] is the concentration of the protein–drug complex. The value of K varies with temperature and would be better represented as K(T); [PD], is a bound drug and is sometimes written as [Db] or [D], the free drug, as [Df]. If the total protein concentration is designated as [Pf], we can write
[Pt] = [P] + [PD]
(Or) [P] = [Pt] − [PD] … equation(2)
Substituting the expression for [P] from equation (2) into (1) gives
[PD] = K [Df] ([Pt – [PD]) … equation(3)
[PD] = K [Df ] [PD] … equation(4)
= K [Df ] [Pt]
Let r be the number of moles of drug bound, [PD], per mole of total protein, [Pt]; then
The ratio r can also be expressed in other units, such as milligrams of drug bound, x, per gram of protein (m). Although equation (6) is one form of the Langmuir adsorption isotherm and is quite useful for expressing protein-binding data, it must not be it must not necessarily be that protein binding is an adsorption phenomenon. The equation (6) can be converted to a linear form, convenient for plotting, by inverting it:
If v independent binding sites are available, the expression for r, equation (7), is simply v times that for a single site, or
and equation (8) becomes,
The equation (9) is called a Klotz reciprocal plot. An alternative manner of writing equation (9) is to rearrange it first to
r + rK [Df] = ν K [Df] … equation(10)
r/[Df] = ν K – r K … equation(11)
Data presented according to equation (4) are known as a Scatchard plot.
The binding of bishydroxycoumarin to human serum albumin and the graphical treatment of data using equation (11) heavily weights those experimental points obtained at low concentrations of free drug, D, and may therefore lead to misinterpretations regarding the protein binding behavior at high concentrations of free drug. Equation (11) does not have this disadvantage and is the method of choice for plotting data. Curvature in these plots usually indicates the existence of more than one type of binding site. Equations 3 and 4 cannot be used for the analysis of data if the nature and the amount of protein in the experimental system are unknown. For these situations, Sandberg recommended the use of a slightly modified form of equation (11):
where [Db] is the concentration of the bound drug. The equation (12) is plotted as the ratio [Db]/[Df] versus [Db], and in this way, K is determined from the slope and vK[Pt] is determined from the intercept.
The Scatchard plot yields a straight line when only one class of binding sites is present. Frequently in drug-binding studies, n classes of sites exist, each class I having vi sites with a unique association constant Ki. In such a case, the plot of r/[Df] versus r is not linear but exhibits a curvature that suggests the presence of more than one class of binding sites. The data is analyzed in terms of one class of sites for simplification. The plots at 20 °C and 40 °C clearly show that multiple sites are involved. Blanchard reviewed the case of multiple classes of sites. The equation (8) is then written as
As mentioned earlier, only v and K need to be evaluated when the sites are all of one class. When n classes of sites exist, equation (13) and equation (14) can be written as
The binding constant, Kn, in the term on the right, is small, indicating extremely weak affinity of the drug for the sites, but this class may have many sites and so be considered unsaturated.
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