Appendix III
119
APPENDIX III
Determination of the equilibrium constant for complex formation and of the site size using the Scatchard and McGhee and von Hippel equations
The interaction between a dye molecule D and a free site on the polymer can be expressed by the equation (III.1)
D + S ' DS (III.1)
If CP is the polynucleotide concentration in base pairs and B is the number of binding sites for every base pair, the total site concentration is
[S]0 = BCP (III.2)
Following the Scatchard hypothesis, the sites, independent one to each other, are saturated in such a way that the mass balance in its classical from still applies, that is
[S]0 = [DS] + [S] (III.3)
From (III.2) and (III.3) we have
[DS] + [S] = BCP (III.4)
Assuming that free sites [S] and bound sites [DS] are linked by eq. (III.4), then an equilibrium constant KSC can be introduced, which is represented by eq. (III.5).
[S]
[D]
K
SC[DS]
= ⋅
(III.5)Introducing now the r parameter, already defined in Appendix II
Scatchard equation
120 CP
r = [DS] (III.6)
eq. (III.4) becomes
rCP + [S] = BCP (III.7)
and, therefore
[S] = CP (B - r) (III.8)
Finally, introducing (III.6) and (III.8) in (III.5) and rearranging, one obtains eq.
(III.9)
[ ]
D K B K r rSC SC −
= (III.9)
known as the Scatchard equation (Scatchard, 1949).
A plot of r/[D] vs. r is linear, with slope equal to -KSC and intercept on the x-axis equal to B. According to the Scatchard argument, the reciprocal of B yields the site size.
It can be demonstrated that for isolate binding of the dye, r →0, KSCB = K, with K equilibrium constant of the binding defined in Appendix II.
Actually, the linearity supposed by eq. (III.9) is rarely fully obeyed. This might happen only in case a single class of independent sites is present on the polymer, that moreover, have to be saturated in ordered way, leaving no empty spaces between an occupied site and the following.
However, if the site size is higher than unity (2 or higher), gap that cannot be occupied are formed and the Scatchard equation does not anymore represent a correct model for the equilibrium. Even in the absence of cooperativity the slope of the Scatchard plot is no longer constant, but a function of polynucleotide saturation producing curved Scatchard plot at high dye loadings (Fig. III.1).
Appendix III
121 Fig. III.1. Scatchard plot displaying a positive deviation from linearity for high values of r.
This behaviour was rationalised by McGhee and Von Hippel through rigorous mathematical models that introduce correcting factors into the Scatchard equation, on the basis of cooperativity and probabilistic factors (McGhee and von Hippel, 1974).
They demonstrated that, in the absence of cooperativity, the Scatchard plot should display a positive deviation from linearity at the end of the titration curve, for high values of r. Owing to this phenomenon, due to site overlapping; the intercept on the x-axis is larger than B (Fig. III.1). Its value, 1/n, is related to B through the relationship n = [1+(1/B)]/2 (McGhee and von Hippel, 1974).
The equation derived by McGhee and von Hippel to account for deviation of Scatchard plots from linearity is eq. (III.10)
1) - (n n
1)r]
(n [(1
nr) - K (1
r
[D]=
− −
(III.10)Scatchard equation
122
