Biochemistry - The Chemical Reactions of Living Cells, Volume 1 - D. Metzler 1980

How molecules bind to one another
Quantitative evaluation of binding strength
Electrostatic repulsion. Anticooperativity

As noted above, a hypothetical dicarboxylic acid with infinitely separated carboxyl groups, for which $\lg K^* = 4.8$, is characterized by two binding constants; the difference between them (see the first row of Table 4-1) is due to the statistical factor ($\lg 4 = 0.6$). The observed proton-binding constants for the dianions of acids containing 7, 4, 2, and 1 CH₂ groups are listed in Table 4-1. For the acid with the longest chain (azelaic acid), the $\lg K$ values are close to those of the hypothetical acid with an infinitely long chain. However, the closer the carboxyl groups are to each other, the more their behavior deviates from the predictions of the hypothetical model. The first binding constant increases noticeably due to additional electrostatic attraction, whereas the second one decreases¹. The interaction between binding sites causes the difference between the two $\lg K$ values to increase from 0.6 to 2.9. Thus, the binding of the first proton hinders the binding of the second one. Such a negative mutual influence, or anticooperativity, in interactions between binding sites is very common; it always leads to an increased difference between formation constants, which manifests as a "stretching" of the binding curve representing the dependence of $y$ on $\lg X$. Fig. 4-4 compares, as an example, the proton-binding curves for the acetate anion and the succinate dianion.

Class="center">Table 4-1 Proton-Binding Constants for Dicarboxylic Acids³

a From Bell R. P., The Proton in Chemistry, 2nd ed., p. 96. Cornell Univ. Press., Ithaca, New York, 1973.

FIG. 4-4 Proton binding to the acetate anion, succinate dianion, and thiamine anion. The curve corresponding to the binding of a single proton by the acetate anion (dashed line) exhibits the conventional slope for such processes. The binding of two protons by the succinate dianion is an anticooperative process, resulting in a more stretched curve. The binding of two protons by the thiamine anion [Eq. (4-31)] is a fully cooperative process; the curve is steeper.

Anticooperativity in proton binding follows naturally from elementary electrostatic considerations. The corresponding calculations can be performed using Eq. (4-7) if the Dielectric Constant of the medium is known. The difference between the Logarithms of two consecutive constants, $\lg K$, minus 0.6 (the statistical factor) can serve as a measure of METABOLISM/18.html">The Influence of electrostatic forces. For malonic acid, the $\Delta pK$ value is 2.25, and for succinic acid, it is 0.69 (Table 4-1). In 1923, N. Bjerrum suggested that $\Delta pK$ can be directly related to the work required to bring two negative charges together to the intercharge distance in the malonate dianion:

¹ Note that in the case of malonic and succinic acids, the first bound proton may be shared between both carboxyl groups due to hydrogen bonding. In the case of oxalic acid, where the carboxyl groups are directly adjacent to each other, additional factors come into play, which is why its pK values are 4.19 and 1.23.

From Eq. (3-30), we find that $\Delta G$ for the malonate anion is equal to $5.708 \cdot \Delta pK \text{ kJ}\cdot\text{mol}^{-1} = 12.84 \text{ kJ}\cdot\text{mol}^{-1}$. Equating this value to the work $W$ [Eq. (4-7)] and taking the dielectric constant to be 78.5 (the dielectric constant of Water), we obtain an intercharge distance $r$ of 0.138 nm. Because this is significantly smaller than the actual distance between the charges, Westheimer and Kirkwood introduced a correction into the calculation by assuming an internal molecular dielectric constant of 2. By modeling the molecule as a prolate spheroid, they were able to calculate the electrostatic effect on $\mathrm{p}K_a$ with good accuracy [15]. For instance, for the malonate dianion, reference [16] gives a value of $r = 0.41 \text{ nm}$.

The laws of electrostatics have also been successfully applied to interpret protein titration curves, where the net surface positive or negative charge continuously changes with an increasing number of bound protons upon transition from high to low pH values [17].

Electrostatic effects can be transmitted with high efficiency through a system of aromatic rings. This circumstance is undoubtedly crucial for the functioning of biologically active molecules containing heterocyclic aromatic systems. Let us examine The Effect of the protonation state of the pyridine ring nitrogen on the microscopic proton-binding constant of the pyridoxine phenolate anion:

We can see that $\Delta pK = 3.26$, which even exceeds the $\Delta pK$ value for malonic acid.

Electrostatic effects in such aromatic systems are generally well described by the Hammett equation. As an exercise, one can first calculate the four microscopic dissociation constants for 3-hydroxypyridine using the step-specific constants $\mathrm{p}K_a = 4.91$ and $9.62$, as well as the tautomeric ratio [7]

and then calculate these same constants based on the Hammett equation [Eq. (3-66)] and compare the results [18].