Noncooperative transition model

Figure 8.15. Bis for the model of Section 8.8, with parameters given in Eq. (8.8.10), and m = 10. The curves from left to right correspond to increasing values of Xg = 0,5,10,15,..., 40. Note the transition from the noncooperative BI for = 0 to a highly (positive and homotropic) cooperative curve for...

It may be mentioned here that a recent study (Vasconcelos 1996) of a simple noncooperative (one-block) model of stick-slip motion (described by eqn (4.2) with / o = 0 or eqn (4.4) with k = 0) shows discontinuous velocity-dependent transition in the block displacement, for generic velocity-dependent friction forces. Naive generalisation of this observation for the coupled Burridge-Knopoff model would indicate a possible absence of criticality in the model. 

Finally, we have not observed a spontaneous transition from a low flux to a high flux state (Figure 24.7 A) with our previous MR-based membranes [3,5]. The fact whether this transition is observed depends on the feed concentration suggests that the transition is a transport-related phenomenon. It is possible that this transition relates to the concept of cooperative (high flux) vs. noncooperative (low flux) dehybridization (Figure 24.9), but further studies, both experimental and modeling, will be required before a definitive mechanism for this transition can be proposed. 

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 wy being the statistical weight in which j also represents the total number of residues in b conformation with the probability, s = [h]/[a]. There are N unique combinations, i.e. degeneracy with a single residue b for a chain of N residues, therefore wi = Ns and Wy = gys. The general form of the partition function for N number of residues is the polynomial expression ... 

Most transitions in the secondary structures of biomacromolecules fall somewhere between the cooperative none-or-all and the noncooperative models. Many of these transitions can be described by the zipper model, which dissects the structural transition of a polymeric chain into a number of discrete steps (Figure 9.1). The model is a special case of the cooperative structural transition of biomacromolecules. In the zipper model, the initiation of the transition is harder than extension (propagation) and therefore low probability. This initiation step is of high energy and provides a nucleation point for the transition. The subsequent extension steps occur by a series of lower energy and consequently higher probability. 

The partition function for the zipper model is derived from the basic relationships of the noncooperative model. The only difference is the statistical weight for the first step Wi that must include a nucleation parameter, a to represent the probability (lower probability therefore a < 1) for initiating the transition. Therefore the statistical weight for the state / = 1 is w, = as. Two possibilities exist for the next step of the transition. In one case,... 

This is equivalent to the degeneracy of each state in the noncooperative structural transition for the two-state model. The probability of having j forward steps and therefore N - y backward steps, P, is... 

Figure 26.11(b) shows that the probability density function for this model has two peaks, separated by a trough. At low temperatures, all molecules are fully helical. At high temperatures, the molecules in coil conformations substantially outnumber the molecules in helical states. At the midpoint of the transition, the helicity is 1 /2, not because the individual molecules are half helix and coil but because half of the molecules are all-helix, and half of them are all-coil. In the two-state model no molecule is in an intermediate state. In this regard, the two-state model differs markedly from the noncooperative model. 

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