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

How molecules join together
Cooperative conformational changes
Higher-order oligomers

Mathematical analysis of binding curves for oligomers with a subunit number >2 is more complex, yet numerical Methods can be employed. When using equations from the literature to describe a given system, caution must be exercised to ensure they are truly applicable to the situation at hand. Let us consider two tetrameric structures:

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The Standard Free energy of formation for the isologous square (also shown in Fig. 4-9, B) is determined by the energy of the aj and bk interactions.

Thus, according to Cornish-Bowden and Koshland [62], the following relations hold for the tetramer as a whole:

Since standard free energy values are additive, the formation constant of the tetramer will be equal to the product of the formation constants corresponding to the individual interactions; for instance, K_ajAА represents the formation constant of the dimer containing a single aj bond.

In the isologous tetrahedron (Fig. 4-9, C), a third set of pairwise interactions—cl—must be taken into account. (However, unlike constants of the K_ajAА and K_bkAA types, this third interaction constant is a dimensionless quantity). The heterologous square, by contrast, is characterized by a single interaction constant.

Let us now examine the binding of a single molecule X to an isologous tetramer, accompanied by a conformational transition in one of the subunits:

This process results in a slight alteration of one aj interaction and one bk interaction. The Equilibrium Constant for the binding of X to the tetramer is given by the following equation:

The appearance of the statistical factor 4 in the equation is due to the fact that the A₃BX complex can be formed in four ways. When a second molecule of X binds, the following geometric arrangements of subunits are possible:

Each of these forms can be generated in two ways. It is straightforward to express the microscopic constants for the binding of the second molecule of X as a sum of three terms. Clearly, the three binding variants for the second molecule of X are not equally probable, since the constants corresponding to the aj and bk interactions differ. Consequently, the oligomer will preferentially bind Ligand X via one specific pathway.

The heterologous tetramer A₂B₂X₂ can exist in two distinct geometric forms. Here again, the various packing arrangements are not equally probable; the likelihood of their realization depends on the specific values of the interaction constants¹.

All of the above may seem like a lengthy and somewhat superfluous exercise, yet it lays the essential groundwork required to solve concrete problems. It should be borne in mind, however, that applying mathematical models inevitably requires certain simplifications. Real Proteins can exist in more than two stable Conformations [11], and their external surfaces feature numerous sites with the potential to bind A wide variety of molecules—both large and small. Occupancy of virtually any of these sites can modulate the Functional Properties of the protein.

¹ Note that in the heterologous tetramer A₄, only a single type of subunit-subunit interaction is present. However, as soon as one molecule of X binds and a single subunit undergoes a transition to conformation B, two variants of aj interactions immediately arise. (In the first, group a is in conformer A, while In the second, it is in conformer B). Since the numbers of these interaction types are equal, they can be treated together.