A combinatorial theory of macromolecular association

Let us first consider a simple system containing one protein A molecule and one protein B molecule confined in a volume V. Due to intra-molecular interaction there are two possibilities for the system A and B are separated, or A and B are associated in a complex AB. There are two possible states of the system A + B and 

corresponding to the left and right sides of Equation (10.14). If the probability of complex existing (the relative amount of time the system is found in the AB state) is p, then the Boltzmann probability law says that the energy difference between the two possible states is —ksT In As we shall see, the probability p is a function of V, the volume of the system. We define Ka = which will be shown to be related to (but not equal to) the equilibrium constant for the reaction. 

we consider a container with volume held constant containing a number of A and B molecules. We define n and n°B to be the total numbers of A and B (including molecules found in complexed and uncomplexed states) in the system and to be the number of AB complexes. By elementary combinatorics, the number of ways of choosing l A molecules out of the total set of n°A is ( / ).5 The number of ways of choosing l B molecules out of the total set of n°B is ( / ). Since there are number of ways to pair the A and B molecules, the total number of ways we can form complexes is ( / ) ( /)T  

The energy associated with AB complexes (compared to the reference state of all in the uncomplexed state) is —IksT In KLl. Thus the Boltzmann weight for an individual state with complexes is 

If we count all of the independent ways of combining A and B molecules into l complexes, then the non-normalized probability of AB complexes existing is 

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