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

How molecules join together
Quantitative evaluation of binding strength
Cooperative processes

Can group interactions lead to higher values of the constants characterizing sequential Ligand-binding steps? At first glance, this seems impossible, as it would imply that the true binding constant for the second proton is greater than that for the first. Common sense tells us that the first proton should bind to the site with the higher binding constant, not the lower one. Let us examine, however, the experimental proton-binding curve for the thiamine anion (Fig. 4-4). Compared to the analogous curve for the acetate ion, it is not stretched out at all; on the contrary, it becomes twice as steep. This phenomenon is explained by some striking Features of the Chemical Structure of thiamine (Vitamin B1). Under certain conditions, this vitamin can crystallize as a yellow sodium salt; The structure of the corresponding anion is shown below. Weak proton binding to one of the nitrogen atoms [Equation (4-31)] leads to a decrease in electron density at the adjacent carbon atom, to which the negatively charged sulfur atom attaches, closing the ring of the unstable tricyclic form of thiamine¹.

¹ This issue is discussed in more detail in Ch. 8, Sec. D, as well as in References [19] and [20].

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This cyclic form is stable in methanol and can be isolated in crystalline form. In Water, however, it is highly unstable because the central ring can open in this solvent; subsequent electron displacement (as indicated by the small arrows) toward the nitrogen atom greatly enhances its basicity. A second proton then binds very tightly to this basic nitrogen atom, resulting in The formation of a cation. Thus, the reversal in the order of the binding constants is caused by an intramolecular rearrangement occurring between the binding of the First and Second protons. In this particular case, we essentially cannot measure the sequential binding constants K₁ and K₂ because K₂ is much larger than K₁ (presumably by one or more orders of magnitude). Consequently, the slope of the middle portion of the binding curve (Fig. 4-4) increases twofold (2 × 0.576) within experimental error compared to the corresponding curve for the acetate anion. This curve is accurately described by Equation (4-32), where . A comparison of Equations (4-32) and (4-15) shows how much simpler the former is. This is because, in the presence of cooperativity, the concentration of the product formed upon attachment of the first ligand, PX, is negligibly small.

Proton binding by the thiamine anion is an example of a cooperative process, so named because the attachment of the first proton facilitates the attachment of the second. Cooperative processes are relatively rare in the binding of small molecules, but they are extremely widespread and play a major role in biochemistry. A cooperative binding curve is called sigmoidal (S-shaped) because the plot of versus [X] (the binding isotherm) has an S-shape. A binding process is termed fully cooperative if the possible degree of cooperativity is maximal. This means that the n-th ligand-binding site has practically no affinity for X until the remaining (n − 1) sites are occupied. However, once these sites are occupied, the affinity of the n-th site for X increases so dramatically that only P and PX are present in significant quantities in any equilibrium mixture.

It is easy to show that the fractional saturation for the case of fully cooperative binding on n sites is

Where

The slope in the middle region of the curve plotted in coordinates of is equal to 0.576n, and The change in lg[X] when changes from 0.1 to 0.9 is 1.81/n.

Equation (4-33) can be rewritten as follows:

Taking the logarithm of this expression, we obtain

The straight line representing the plot of versus lg[X] is called the Hill plot, and its slope is equal to n.

Recall that Equation (4-34) was derived for the ideal case of complete cooperativity of ligand binding by all n sites. Nevertheless, biochemists often use Hill plots to analyze processes in which cooperativity is incomplete. In these cases, the experimentally determined slope of the Hill line (n_Hill) will be less than the number of binding sites.

The difference between n_Hill and n is frequently used as a measure of the degree of cooperativity. For fully cooperative processes, the ratio n_Hill/n is 1.00, whereas for cases of incomplete cooperativity, it is less than 1. To determine n_Hill, it is not strictly necessary to construct a Hill plot. Instead, it is sufficient to accurately measure the slope in the middle

portion of the conventional binding curve plotted in coordinates of (or ΔA).

Alternatively, one can determine Δlg[X] corresponding to the change in from 0.1 to 0.9, and then calculate n_Hill from Equation (4-36):

Note that Hill plots are sometimes nonlinear and cannot be used to determine the degree of cooperativity.

A second example of a cooperative process is the reversible Denaturation of folded polypeptide chains. The pH of the solutions of certain Proteins can be brought down to approximately 4 by adding acid without protonating groups "buried" inside the protein globule for which pK > 4. Upon further addition of a small amount of acid, a less basic group becomes protonated, which triggers the unfolding of the polypeptide chain and exposes previously sequestered, more basic groups to protonation. Thus, proton binding in this case is a cooperative process, and, as in thiamine, the cooperativity is driven by a conformational change induced by the protonation of a specific group.

Another type of cooperativity in protein molecules is observed during the reversible conformational transition between an α-Helix and a random coil. If conditions are established under which the helical conformation is more stable, all molecules currently in the random coil state will rapidly adopt the helical form. Similarly, under conditions where the random coil is the more stable conformation, all helices will uncoil and undergo complete conversion into coils. DNA melting (Ch. 2, Sec. 10), like that of any crystal, occurs cooperatively [21]. The formation of a new polynucleotide chain on a complementary template, leading to stacking interactions, can also be a cooperative process. For example, the formation of a polyadenylic acid chain on two polyuridylic acid chains leads to the cooperative formation of a triple-helix complex (Ch. 2, Sec. D.6). The presence of stacking interactions makes helix growth energetically more favorable than the initiation of new helical segments [22]. There is an extensive literature dedicated to The problem of cooperativity, notably works [23–25].