Page 137 - Thermodynamics of Biochemical Reactions

134 Chapter 7 Thermodynamics of the Binding of Ligands by Proteins

H 7.6 EFFECT OF pH ON PROTEIN-LIGAND EQUILIBRIA

When the binding of a ligand by a protein is accompanied by the production or
consumption of hydrogen ions, the apparent dissociation constant K ' for thc
protein-ligand complex will be a function of the pH. The apparent dissociation
constant is defined by

(7.6-1)

The abbreviations for reactants represent sums of species at a specified pH, and
the expression for the equilibrium constant is written in term of concentrations
because K' is taken to be a function of ionic strength as well as pH. The pH
dependence of K' can be written in terms of the binding polynomials of the three
reactants (see equation 1.4-8):

(7.6-2)

where Krcf = [P][L]/[PL] is written in terms of species for a reaction that is
independent of pH. The binding polynomials in equation 7.6-2 include all weak
acid groups in the three reactants.
If the ligand does not have pKs in the pH range studied. P(L,,,) = 1. In this
case hydrogen ions are produced or consumed when there are acid groups in the
binding site that have pKs in the pH range studied and the pKs of these groups
are changed by the binding of the ligand. If the various acid dissociations of the
protein are independent, the binding polynomials are written as products of terms
of the form (1 + 10-PHH+pK1 + ~o-PH+PK~ -, when there are 11, groups with
)
(1
)
'I1
pK, and n2 groups with pK,. However, it is not necessary to go this far in making
assumptions. If the acid groups in the binding site are independent of the acid
groups in the rest of the protein molecule, the binding polynomial for the protein
is given by (Alberty, 2000d)
P(P,ot) = '(Pnonsitc)'(Psite) (7.6-3)
where P(P,,,,,it,) is the binding polynomial for the acid groups outside of the
binding site and P(P,,,,) is the binding polynomial for the acid groups in the
unoccupied binding site. In this equation the acid groups in the binding site are
defined as the acid groups in the protein that undergo a shift in pK when the
ligand is bound. This major step in the treatment of the effect of the binding of a
ligand by a protein is possible if the binding of the ligand changes the pKs of only
some of the acid dissociations of the protein.
The corresponding binding polynomial for the protein with the binding site
occupied (PL,,,) is given by
P(PLtot) = P(Pnonsitc)P(PLsite) (7.6-4)
where P(P,,,,nsite) is the same polynomial that is in equation 7.6-3 and P(PL,,,,)
has the same number of terms as P(PSi,J. The nonsite groups can interact with
each other, the site groups in the protein can interact with each other, and the
site groups in the protein-ligand complex can interact with each other.
Substituting equations 7.6-3 and 7.6-4 in 7.6-2 when the ligand does not have
pKs in the pH range considered yields a simpler equation for the dependence of
K' on pH:

(7.6-5)

This equation is important because it shows that the pH dependence of K' is
determined entirely by the pKs of the acid groups in the binding site when it is
unoccupied and in the binding site when it is occupied by ligand. If the number