Increases in Receptor Occupancy

In the following discussion, we continue with the simple model for the combination of a ligand with its binding sites that was introduced in Section 1.2.1 (Eq. (1.1)). Assuming as before that the law of mass action applies, the rate at which receptor occupancy (pAR) changes with time should be given by the expression  

In words, this states that the rate of change of occupancy is simply the difference between the rate at which ligand-receptor complexes are formed and the rate at which they break down.  

At first sight, Eq. (1.18) looks difficult to solve because there are no less than four variables pAR, t, [A], and pR. However, we know that pR = (1 - pAR). Also, we will assume, as before, that [A] remains constant that is, so much A is present in relation to the number of binding sites that the combination of some of it with the sites will not appreciably reduce the overall concentration. Hence, only pAR and t remain as variables, and the equation becomes easier to handle. Substituting for pR, we have  

This still looks rather complicated, so we will drop the subscript from pAR and make the following substitutions for the constants in the equation  

This can be rearranged to a standard form that is easily integrated to determine how the occupancy changes with time  

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