Page 15 - Thermodynamics of Biochemical Reactions

8 Introduction to Apparent Equilibrium Constants

The numbers of hydrogen ions bound that are calculated using this equation are
based on the arbitrary convention of not counting the additional 12 hydrogen
atoms in ATP. Thus, the average number RE, of hydrogen ions bound by ATP is
given by

CH'1 , 2CH'I2
(1.3-7)

At very high pH, the binding of H+ approaches zero, and below pH 4 it
approaches 2.
In dealing with binding, it is convenient to use the concept of a binding
polynomial (Wyman 1948, 1964, 1965, 1975; Edsall and Wyman, 1958; Hermans
and Scheraga, 1961; Schellman, 1975, 1976; Wyman and Gill, 1990). The poly-
nomial in the denominator of equation 1.3-7 is referred to as the binding
polynomial P. It is actually a kind of partition function because it gives the
partition of a reactant between the various species that make it up. The binding
polynomial for the binding of hydrogen ions by ATP is given by

(1.3-8)

The average binding of hydrogen ions is given by
-1 dlnP
-
N [H'] dP - dlnP - ____ (1.3-9)
-
w-
P dLH'] - dln[H+] ln(10) dpH
Equation 1.3-7 is readily obtained from equation 1.3-8 by use of equation 1.3-9.
Substituting the values of the two acid dissociation constants of ATP at
298.15 K, 1 bar, and I = 0.25M from Table 1.2 into equation 1.3-7 or 1.3-9 yields
the plot of Nt, versus pH that is shown in Fig. 1.3.
Figure 1.3 shows that the acid titration curve for a weak acid can be
calculated from its pKs, and this raises the question as to how the pKs can be
calculated from the titration curve. This can be done by first integrating equation
1.3-9 to obtain the natural logarithm of the binding potential P:

NH d[H']
JCH'I = i dln P = In P + const. (1.3- 10)
or
-ln(10) N,pH = dlnP = 1nP + const. (1.3-1 1)

. . . . . .? -..
I . . . . . . . . . . . . . .-:- . . . pb,
4 5 6 7 8 9
Figure 1.3 Binding of hydrogen ions by ATP at 298.15K and I = 0.25 M (see
Problem 1.2).