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

How molecules bind to one another
Quantitative assessment of binding strength
Statistical effects

Let us consider a straight-chain dicarboxylic acid containing two identical binding sites. If the chain connecting the two carboxylate anions is sufficiently long, the carboxyl groups will be far enough apart for Electrostatic Interactions between them to be negligible.

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Each group is characterized by a microscopic binding constant (K*) equal to 5∙10⁴ (which can also be termed the true binding constant, as it pertains to a carboxyl group that does not interact with other groups). Intuitively, it is clear that with respect to proton binding, a solution of the dicarboxylic acid dianion should behave identically to a solution of the R—COO^- monoion at twice the concentration. If this is indeed the case, knowing a single true binding constant is sufficient to characterize both binding sites. Nevertheless, strangely enough, the formation constants K₁ and K₂, corresponding to the binding of the First and Second protons, turn out to be unequal: K₁ = 10∙10⁴; K₂ = 2.5∙10⁴. This fact is due to the so-called statistical effect. In The First stage, a proton can attach to either of the two carboxyl groups, and the resulting molecules are indistinguishable:

If we denote the two forms of PH as A and B [Equation (4-26)] and assume that each of them is related to P by an independent equilibrium step characterized by a microscopic formation constant K*, it follows from Equations (4-23) and (4-24) that

The Nature of this effect is purely probabilistic. It is rooted in the same principle by which a pair of differently colored balls is drawn from an urn containing 50% white and 50% black balls, on average, twice as often as a pair of balls of the same color. In the general case, when a molecule P contains n binding sites, the relationship between the microscopic formation constants K^*_i and the constants K_i characterizing the individual binding steps has the following form [12, 13]:

Using Equations (4-17) and (4-28), it is also easy to show that the following relations hold for identical and independent binding sites:

In this case, all microscopic formation constants turn out to be equal and represent a single true binding constant applicable to all binding sites. Indeed, Equation (4-30) coincides with the analogous equation describing The addition of a single proton (or any other Ligand) to a molecule possessing a single binding site. This is consistent with our earlier intuitive notion that a solution of a substance whose molecules contain n independent binding sites should behave in the exact same manner as an n-times more concentrated solution of a substance with a single site per molecule. Thus, all our calculations merely confirm the Conclusions that logically follow from physical principles. In reality, however, binding sites within a single macromolecule are rarely completely independent; some interaction is almost always present between them, and the equations we derived for determining the constants corresponding to individual Stages of the binding process and the true constants are fully applicable to such cases.