Potential of Average Force between the Subunits

Consider now the work required to bring the two subunits from fixed positions at infinite separation i = 00 to the final position R = d, where d is the distance between the subunits in the final state of our model. This work is denoted AA(oo - d) and is given by the ratio of the corresponding PFs in the initial and final states. 

In most of this chapter we shall discuss the binding properties of the entire polymer in its final form. In particular, the interaction energies Eap will be constants. However, for the sake of the following derivation, we view each of the interaction energies Eap as a function of the intersubunit distance R. In this case (3,3.17) is written, for any R, as 

Note that Epp(R) is a function of R, but the probability distribution of states used in the average (3.3.19) corresponds to R= oo. Physically we can think of a system for which we have frozen in the equilibrium L HSit R oo. We bring the two subunits from infinity to R but keep the mole fraction of states and that correspond to 

-dEap R)/dR is the direct force operating between the two subunits in states a, at a distance R. The sum on the rhs of (3.3.20) is over all possible states of the (empty) subunits, with the distribution x ap R) at that particular R (note that each of xap R) as defined in (3.3.3) is a function of R through Qap)- This justified the reference to A (oc i ) as the potential of average force between the two subunits. 

It should be noted that (3.3.20) is an average over the various forces this is different from the force defined by the average interaction energy. The reason, as we shall soon see, is that the averaging and the differentiation processes do not commute. 

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