A dynamics theory of association

The previous section worked out the relationship between the statistics of association of a collection of A and B molecules as a function of p, the probability of a single pair of A and B molecules to form a complex in a volume V. We saw that the probability p is related to the experimentally determined dissociation constant Kd via 

The kinetic constants k+ and k-1, which are properties of the molecules A and B and their interaction under specific conditions, are more difficult to determine experimentally than their ratio, which does not require resolving rapid transients to measure. One method to measure rate constants is by stopped-flow experiments, in which small reacting volumes are rapidly mixed and reaction progress followed, usually using some spectrophotometric assay. 

Let r be the distance between the centers of mass of the two molecules, and the interaction energy be U (r). The distance r is known as the reaction coordinate for the association process. The potential U(r) could take the form, for example, of Lennard-Jones 6-12 potential  

As is apparent in the figure, a minimum in energy occurs where the molecules A and B are separated by tq. Conformations in or near this energy correspond well to the AB complex in Equation (10.14). Values of r that are sufficiently greater than ro correspond to the A + B uncomplexed state. To make this definition precise, we note that the probability of the distance between A and B, RAB, is directly related to U(r) following the Boltzmann probability law  

We can gain important insight by rewriting the right-hand side of Equation (10.30) as e F(r)/kBTdr in which 

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