Appendix V
125
APPENDIX V
Determination of site size and equilibrium constant for complex
formation according to the Schwarz theory
When a ligand interacts with a linear polymer chain, the affinity for free sites could change. This phenomenon is called cooperativity. The cooperativity could be positive, if the ligand affinity for free sites increases as the sites are filled. In the reverse case, one has negative cooperativity. The Schwarz theory (Schwarz, 1970) of cooperative binding is based on two fundamental processes: nucleation and growth.
The occupation of a site modifies the energy of the others two adjacent sites, for these reasons the cooperativity could be described in terms of consecutive base triplets. If D represents a free ligand molecule and d a bound ligand molecule, while u denotes a free site, the overall binding process could be described by two reactions: nucleation and growth.
(nucleation) D + uuu ↔ udu (growth) D + udu ↔ udd
Two equilibrium constants, respectively K* and K, are associated to these reactions.
[
]
[ ] [
D
uuu
]
udu
K
⋅
=
*
(nucleation) (V.1)[
]
[ ] [
D
udu
]
udd
K
⋅
=
(growth) (V.2)The extent of cooperativity is determined by the ratio of the two binding constants, q, which is defined as q = K/K*.
If the cooperativity parameter q is larger than unity, q >1, the cooperativity is positive. On the other hand, q < 1 indicates negative cooperativity, which favours the binding of ligands at isolated sites. q equal to 1 indicates the absence of cooperativity.
Schwarz theory
126
To describe the behaviour of the system at equilibrium the parameter g is introduced as, the number of ligand bound for monomeric unit of polymer. It is important to note that g corresponds to 1/n where n is the site size in the McGhee and von Hippel definition. It is possible to determine the above parameters using γD, the molar fraction of free dye, and CP/CD, the ratio between polymer and dye analytical concentrations.
The treatment makes use of the auxiliary expression:
s = K[D] (V.3)
that is
K = s/γDCD (V.4)
where CD and [D] are the analytical and free dye concentrations respectively. The fraction of bound dye, γd, is defined by the equation
γd = θ g (CP/CD) (V.5)
where θ is the fraction of occupied binding sites (or degree of saturation) and g is the reciprocal site size as defined above. Note that the molar fraction of free dye γD is linked to γd by the equation γD + γd = 1. Then eq. (V.5) becomes:
γD = 1- θ g (CP/CD) (V.6)
If the binding is sufficiently strong and CP/CD sufficiently small, then θ ≅ 1 and eq. (V.6) will represent a linear relationship between γD and CP/CD. Thus a plot of γD vs. CP/CD will provide a straight line whose slope will yield g and therefore the value of site size as 1/g. Note that γD is found from optical measurements (absorbance) as d D d D ε -ε ε ) (A/C − = D γ (V.7)
Appendix V 127 θ = ) (CP/CD g 1 ⋅ −γD (V.8)
The cooperativity parameter (q) could be finally determined using the following relationship proved by Schwarz (Schwarz, 1970):
(
)
2(1
-
2
)
2θ)
θ(1
q
s
1
s
θ
−
=
−
(V.9)The graphical representation of eq. (V.9) yields a straight line with slope equal to q. One can demonstrate that for θ = 0.5, s = 1. The value of γD for θ = 0.5 is obtained by eq. (V.6) and let’s indicate it as γ∗. Then eq. (V.4) is reduced to
K = 1/γ∗CD (V.10)
Schwarz theory
