Introduction, Notation, and Some Historical Perspectives

Perhaps the simplest two-site cooperative systems are small molecules having two binding sites for protons, such as dicarboxylic acids and diamines. Despite their molecular simplicity, most of these molecules do not conform with the modelistic assumptions made in this chapter. Therefore, their theoretical treatment is much more intricate. The main reasons for this are (1) there is, in general, a continuous range of macrostates (2) the direct and indirect correlations are both strong and intertwined, so that factorization of the correlation function is impossible. In addition, as with any real biochemical system, the solvent can have a major effect on the binding properties of these molecules. 

The literature on dissociation (or ionization) constants of these and related compounds is immense. We present in this section only a few examples to illustrate various aspects of cooperativity in these systems. 

As a rule, we shall use only intrinsic binding (or association) constants. The relevant experimental data are usually reported in terms of either the thermodynamic dissociation or association constant. The general relation between intrinsic and thermodynamic constants has been discussed in Section 2.3. It will be repeated below for the special cases of this section. 

One should be careful to distinguish k and fe from ka and kb for systems with different binding sites. Both ka and kh are for the first site. See Subsection 4.8.3. 

Note that the terms macroscopic and microscopic constants do not imply that these quantities measure macroscopic or microscopic quantities, respectively. Here, in the macroscopic view we have simply grouped the two microscopic species (0, 1) and (1, 0) into one species denoted by (1). Both of these constants can be macroscopic or microscopic, depending on whether we study the binding per molecule or per one mole of molecules. 

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