Noncooperative model

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,... 

The Noncooperative Model, (a = 7 = 1, c= 0). This model applies to assembhes that involve only intemiolecular interactions without any allosteric effect. The occupation of the various binding sites of the receptor is dictated only by statistics. This model is the reference for spotting the presence of allosteric effects in real systems. It also applies to the formation of a given ohgomer in isodesmic polymerizations. This process is exemplified by a monomer A—B that undergoes a reversible polymerization in which all of the stepwise association constants are identical and equal to K. The formation constant of each oligomer (A—B), is given by Eq. [51] in which a = y = 1, c = 0, = 1 and f) = 1 -... 

Noncooperative model, 60 Noncovalent interaction, 79, 81—82, 85 Nonlinear optics (NLO), 216—217 Norrish type 1 photocleavage, 139, 150, 154... 

Figure 26.10 For the noncooperative model, lowering the temperature increases the helicity gradually, not sharply, uh is the number of helical units.

Noncooperative Model Neighboring Units Are Independent of Each Other... 

The noncooperative model predicts that the helicity changes gradually, not cooperatively, with temperature. It proves that independent units are not cooperative. Cooperativity requires some interdependence. Equation (26.12) shows that the noncooperative model fails to predict the dependence of fn on chain length that is indicated by the experiments (see Figure 26.9). 

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. 

When a system behaves according to this model, a nonlinear fit using Eqs. 18,19 and 20 can fit the titration data (qi vs. [X]i, or vs. [X]t/[M]t). Fig. 2 (left panel) shows a typacal ITC profile for the binding of GSNO (12.7 mM) to dimeric wt-hGSTPl-1 (43.7 pM) in phosphate buffer at pH 7.0 and 25°C. Control experiments were also carried out in order to measure the ligand dilution heat. A noncooperative model is imable to fit these calorimetric data properly. 

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...

Calorimetric (DSC) measurements yield thermodynamic properties of duplex melting in these oligonucleotides independent of any assumptions concerning the model of melting, such as a cooperative all-or-none process versus a noncooperative, multiple-stage melting process. Comparison of calorimetric enthalpies with van t Hoff enthalpies obtained either from the manipulation of heat capacity curves outlined in equations (16.19) to (16.22), or from optical or NMR measurements [equations (16.14) to (16.17)] allows conclusions to be drawn concerning the size of the cooperative unit. If the two... 

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. 

Equation 3.34 can be derived from a completely different model. Let us assume that we have two different types of sites on the surface and that these sites are independent, so a molecule that occupies a site of one type does not interact with a neighbor. Adsorption on these two types of sites is noncooperative. Let us assume that adsorption on each site follows a Langmuir adsorption model ... 

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... 

Strictly taken, a prerequisite for the discussion of cooperativity or nonadditivity requires the definition of the additive or noncooperative case [50]. Generally, in the field of intermolecular interaction, the additive model is a model based on the concept of pairwise additive interactions. For atomic clusters per definition, but also for molecular clusters, the use of pairwise additive interactions is almost always used in combination with the assumption of structurally frozen interaction partners. Even in cases of much stronger intermolecular interactions the concept of pair potentials modified to that of effective pair potentials is often used. Most of the molecular dynamics calculations of liquids and molecular solids take advantage of this concept. 

Noncooperative binding of protein nonspecifically to an infinite lattice is described by the McGhee-von Hippel model where... 

---