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

How molecules join together
Cooperative conformational changes
Binding equilibria for a dimerizing protein

All the equilibrium processes described in equations (4-42) to (4-48), during which dimers A₂, AB, and B₂ are formed or one or two molecules of X are added to a dimer, are represented schematically in Fig. 4-16. Above each arrow is the microscopic constant characterizing this stage, multiplied by the corresponding

statistical factor. The degree of saturation is given by the following equation:

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Each of the nine terms in the numerator represents the concentration of one of the nine complexes containing X (Fig. 4-16). The fourteen terms in the denominator represent the protein concentration in each of the forms (including those not containing X). The given protein concentrations correspond to the Molecular Weight of the dimers, which is why some terms in the denominator are multiplied by 1/2.

All terms in the numerator and denominator of equation (4-49) can be expressed in terms of [X] using the microscopic constants shown in Fig. 4-16. To do this, one must formulate an equation [similar to equation (4-15)] relating $y$ to [X], K_AX, K_BX, K_t, and the interaction constants K_AA, K_AB, and K_BB. Since such an equation in its general form is very cumbersome, let us consider some special cases where it takes a simpler form.

FIG. 4-16. Possible forms of dimerizing Proteins existing in two conformational states; each protomer has one binding site for Ligand X. Dashed arrows indicate the equilibrium processes considered by Monod, Wyman, and Changeux, while solid arrows indicate those considered by Koshland et al. [61, 62]. Bold arrows refer to the simplest induced-fit model that does not take dimer dissociation into account. (Note that although all arrows point in only one direction, the corresponding processes are reversible.) The values of K_AX and K_BX are assumed to be the same for all subunits regardless of whether they are in the monomeric or dimeric form.

a. Absence of oligomer dissociation

If both constants—K_AA and K_BB—are sufficiently large, dissociation into monomers will not occur. Nevertheless, the transition between Conformations A and B within a dimer or a higher-order oligomer can still take place, and the mathematical relations given in Fig. 4-16 remain applicable. To describe the most common models of oligomeric Enzymes, further simplifications must be introduced.

1. The Monod–Wyman–Changeux model [33]. Assuming that only symmetric dimers are formed, i.e., K_AA and K_BB ≫ K_AB [equation (4-45)], we can restrict our consideration to only those equilibrium processes indicated by dashed arrows in Fig. 4-16. In the absence of ligand X, the ratio [B₂]/[A₂] = const [1/L in the notation of Monod, Wyman, and Changeux; see also equation (4-45)]:

THE POSITION OF equilibrium is influenced by both the association constants K_AA and K_BB and the transition constant K_t. The ratio of [B₂] to [A₂] can be small if K_AA and K_BB are close and K_t is small. If K_t ≃ 1, this ratio can also be small provided that K_AA ≫ K_BB, i.e., when subunits are bound more tightly in the A₂ complex than in B₂. For this case, equation (4-49) simplifies and takes the following form:

Substituting the expression for from (4-50) into (4-51), we obtain

In the case of an oligomer composed of $n$ subunits, Monod et al. make the assumption that all binding sites in a given conformer are independent and equivalent. The equation for derived from (4-29) has the following form in this case:

Initially, we assumed that B₂ binds X more strongly than A₂. Therefore, if the equilibrium in equation (4-50) is strongly shifted toward The formation of B₂ (L is small), The addition of X will not affect the equilibrium between the two conformers, and binding will be noncooperative. Equation (4-53) then reduces to (4-30). If, however, the equilibrium is shifted toward the formation of A₂ (L is large), the addition of X will shift the equilibrium toward the formation of B₂ (which binds X more strongly). Furthermore, since the expression for $y$ (4-52) contains the term K²_BX [X]² in the denominator, binding will begin to exhibit cooperativity. In the limiting case when L is large and K_AX ≃ 0, most terms in equation (4-62) become 0, and it takes the form of equation (4-33) given earlier, which pertains to the case of fully cooperative binding when K = K²_BX L. At other values of K_AX, K_BX, and L, non-complete cooperativity is observed [60].

The Koshland induced-fit model [61, 62]. This model considers only the A₂, ABX, and B₂X₂ forms (bold arrows in Fig. 4-16). The expression for has the following form¹:

If the value of K_AB is small (no "mixed" dimer), this expression also reduces to equation (4-33), which describes fully cooperative binding, with defined in this case as follows:

If, on the other hand, K_AB is much larger than K_AA and K_BB, anticooperativity (negative cooperativity) will occur. In this case, the saturation curve, like the proton binding curve of succinate dianion (Fig. 4-4), will be biphasic.

b. Case where the protein dissociates in one of the conformations

Let us consider the case where the constant K_BB is very small and B₂ readily dissociates into monomers. Then the addition of X will lead to dissociation of the dimer. A well-known example of such a protein is lamprey Hemoglobin, which is a dimer and dissociates into monomers upon oxygen binding [64]. In this case, equation (4-49) reduces to the following form:    

The reader is presented with an interesting opportunity to analyze whether this equation can be used to predict the weak Cooperative binding of oxygen by lamprey hemoglobin.

Let us look once again at the expression for the constant L, which determines the ratio between the amounts of protein in conformations A and B in the absence of a ligand. It follows from equation (4-50) that L can reach large values (with conformer A predominantly present) either when K_t is very small or when К_ВВ≪К_АА. Thus, if K_t≃1 and L is large, this implies that the subunit interaction in B₂ is significantly weaker than in A₂, and it is quite possible that the binding of X will lead to dissociation of the molecule, as is the case with lamprey hemoglobin. Conversely, if K_t is small (meaning that the protein molecules exist predominantly in conformation A due to the greater stability of this form), К_ВВ can be considerably larger than К_АА; if К_АА is sufficiently small at the same time, the A₂ dimer will completely dissociate. Ligand binding will then lead to association and cooperative binding. This can be easily demonstrated by writing out the corresponding terms from equation (4-49).

¹ Sometimes Koshland arbitrarily sets K_АА = 1, assuming that K_ВВ is an interaction constant equal to К_ВВ/К_АА. Although this assumption simplifies the algebraic manipulations, it is valid only for fully associated systems; therefore, in this book we will use the constants defined by equations (4-46) – (4-48).