Two Different Binding Sites, a Single Ligand

The differential-equation system governing the dynamics of these new variables can then be written as 

From the theory of differential equations (Zill 2008), the general solution for (7.40) is 

Consider the case of a receptor R with two different binding sites for the same ligand X. This system is a particular instance of the one previously studied. Consider that Y molecules are the same as X molecules, and that states (AT,0) and (0, Y) are equal. From the above discussion, the present system can be summarized by means of the following chemical reactions  

The reaction in (7.42) results from lumping together the reactions in (7.1) and (7.2). The factor 2 in the forward reaction rate accounts for the fact that, when the receptor molecule is empty, each ligand molecule has two binding sites available. Similarly, the reaction in (7.43) results from lumping together the reactions in (7.3) and (7.4), while the factor 2 in the backward reaction rate takes into account that there two possible ways in which a fully occupied receptor can turn into a molecule with only one occupied site. 

Under the supposition that the system consists of receptor molecules that are bound and unbound by a constant number nx of X ligands, its state can be determined by the counts of free and bound-by-one-ligand receptors nR,nRx). The number of completely occupied receptors can be calculated as  

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Two Different Binding Sites, a Single Ligand in which hrxx is as defined in (7.44), while... 

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