N

WN = II sy = SN since all s, are identical and equal to s. j = l

All-or-none transition is highly cooperative with N being the cooperative length of a transition.

In the noncooperative model (Figure 9.1), the biomacromolecule converts stepwise from a to b, with an increase in the fraction of b with each transition. Each step j represents a different state j with wj being the statistical weight in which j also represents the total number of residues in b conformation with the probability, s = [b]/[a]. There are N unique combinations, i.e. degeneracy with a single residue b for a chain of N residues, therefore w1 = Ns and wj = gjs. The general form of the partition function for N number of residues is the polynomial expression:

The binomial expansion gives

(1 + s)N = 1 + Ns + [N-(N - 1)/2!]s2 + ... + [{N!/{j!(N - j)!}]sj + ... sN

The expansion of this expression yields the binomial coefficients of the power of s (the degeneracy factors):

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