Determination of the equilibrium constant for complex formation when the ligand is a polynucleotide
The interaction between a ligand site, S, and a dye, D, to form the DS complex can be expressed by the reaction (II.1)
D + S ' DS (II.1)
whose equilibrium constant is
K = [DS] (II.2)
where [DS], [D] and [S] represent the equilibrium concentrations of DS, D and S respectively, S concentration being expressed in base pairs.
The following relationship also applyies
CD = [D] + [DS] (II.3)
where CD is the overall concentration of the dye.
If the Lambert & Beer law applies, for a wavelength where the free and bound dye only absorb and for a 1cm path length cell, the overall absorbance is given by the equation
A = εD [D] + εDS [DS] (II.4)
Introduction of the eq. (II.3) into (II.2) yields
1 = D− (II.5)
that can be rewritten as
1 1 [DS]
CD = + (II.6)
Determination of equilibrium constant
Introducing (II.3) into (II.4) one obtains
A - εD CD = (εDS - εD) [DS] (II.7)
If we now define
∆A = A - εDCD (II.8)
∆ε = εDS - εD (II.9)
eq. (II.7) becomes
[DS] =∆A (II.10)
Introducing (I.10) into (I.6) and rearranging we finally obtain the relationship
A plot of CD/∆A vs. 1/[S] is a straight line whose slope and intercept are equal to 1/∆εK and 1/∆ε respectively. Therefore, K is obtained as the intercept/slope ratio, whereas ∆ε is the intercept reciprocal.
The reciprocal of eq. (II.11) yields eq. (II.12)
where under conditions of polymer excess [S] ≅ CP.
Under conditions of polymer excess (CP>10CD) the mass conservation for the polymer sites holds in the ordinary way
CP = [S] + [DS] (II.13)
117 where CP is the total polymer concentration expressed in molarity of base pairs. If the equation eq. (II.13) holds, an alternative equation could be derived. From (II.2), (II.3) and (II.13) we have
(C K [DS]
P − −
that, taking into account relationship (II.10) can be rewritten as eq. (II.15)
∆ε ) C (C
C P D
⎛ + (II.15)
Eq. (II.15) enables K and ∆ε to be obtained in an iterative way. That is, disregarding the ∆A/∆ε2 term on a first approximation, ∆ε can be calculated from the reciprocal of the slope of the straight line fitting the experimental CPCD/∆A vs.
(CP+CD). This ∆ε value will be used to re-evaluate the (CPCD/∆A + ∆A/∆ε2) term and so on, until convergence is reached.
It should be remarked that the free site concentration is a function of the saturation degree and therefore the approximation [S] ≅ CP not always applies.
Therefore, the “simple” equations above do not apply anymore as they are but, a relationship between the free site concentration [S] and the total polymer concentration CP is needed. Such a relationship was found by McGhee and Von Hippel (McGhee and von Hippel, 1974). On the basis of a statistical approach, the above authors defined the f(r) function (eq. II.16).
[ ]n1 P
1)r (n 1
f(r) 1 =
= − − (II.16)
This function takes into account both the saturation degree of the polynucleotide r = [DS]/CP and the size of the polynucleotide binding sites n, defined as the number of repetitive units of the polymer involved in the binding of one dye molecule under saturation conditions. The n value can be obtained both by low ionic strength titrations, where complex formation is quantitative, or by means of
Determination of equilibrium constant
the Scatchard analysis of the data (see Appendix III). The r value is directly obtained from the experimental data as
From (II.16) it turns out that
[S] =P (II.18)
and therefore eq. (II.11) becomes
D = + (II.19)
A first estimate of ∆ε from the amplitude of the binding isotherm will enable us to obtain approximate values of r and f(r). f(r) will be used for fitting the data according to eq. (II.19), which provides a better value of ∆ε, that will be used to re- estimate r and f(r). The procedure is repeated until convergence is reached (three iterations only are usually sufficient) and the equilibrium constant K is obtained as the ratio intercept/slope of the line fitting data points.
Fluorescence data are processed using the same procedure, just replacing ∆A and
∆ε by the analogous ∆F = F – F° and ∆ϕ = ϕDS - ϕD quantities respectively.