Cooperativity parameter

The cooperative parameter based on Hill s equation3 was determined and the results are listed in Table 11107 108 The cooperative parameter (r) of the PLL-... 

This hypothesis is further supported by the effect on the absorption behavior of molecular oxygen of additives that destroy the a-helix of PLL. When we added to the PLL system non-coordinative polymers, which do not act as ligands but are able to interact with PLL to form polymer aggregates, the cooperative parameter decreased to unity, as shown in Table 11. In these cases, it was found that the a-heli-cal conformation was made unstable by the formation of polymer aggregates111 ... 

The fact that no change was observed in the visible spectrum after the addition of the polymers showed that the structure of the deoxy-heme complex was not affected thereby. Therefore, the added polymers must cause the decrease in the cooperative interaction through instabilization of the helical structure of PLL. The cooperative parameter was also reduced by the addition of sodium chloride, which destroys the a-helical structure. 

The parameter statistical weight of a terminal unit of a helical sequence. The AG associated with it is the free energy required to create a helix unit located at the boundary between a helical sequence and a random-coil one. Usually, a is called the cooperativity parameter for the formation of helical sequences. The reason will be explained in Section B.3. 

Thus we find that a smaller a induces cooperation of peptide units to create a longer helical sequence, provided the environmental conditions are in favor of helix formation. This is the reason why a is often called the cooperativity parameter for the formation of helical sequences, or a polypeptide with a smaller a is referred to as more cooperative. It is crucial to understand that helix formation in polypeptides is a cooperative phenomenon. 

Another interesting contribution to the study of viscosity behavior in the helix-coil Jransition region is the one due to Hayashi et al. (22) on a PBLA sample (Mw = 23.2 x 104) in m-cresol and a mixture of chloroform and DCA (5.7 voL-% DCA). As mentioned in Chapter B, PBLA undergoes an inverse transition in the chloroform-DCA mixture, while it undergoes a normal transition in m-cresol. Furthermore, its cooperativity parameter is distinctly smaller in the former solvent than in the latter. Thus we may expect that, when compared at the same helical fraction and chain length, the PBLA molecule in the chloroform-DCA mixture assumes a more extended shape and hence a larger intrinsic viscosity than in m-cresol, provided these two solvents have comparable solvent powers for the polymer. The experimental results shown in Fig. 32 are taken to substantiate this prediction, because the approximate agreement of the data points atfN=0 indicates that the two solvents have nearly equal solvent powers for the solute. 

Omura et al. (117) investigated PCBL with m-cresol as solvent. This solvent was chosen because not only was it advantageous for dielectric measurements but also we had found (23) that the cooperativity parameter of the system PCBL-m-cresol was unusually small compared with those of other systems... 

PCBL-ro-cresol is unusually small. Thus, at least qualitatively, the behavior of the experimental data displayed in Fig. 40 conforms to the theoretical prediction. Omura et al. (117) then examined whether these data can be described by Eq. (C-3) (with , a0, and at being replaced by , nc, and h, respectively) together with the cooperativity parameter determined experimentally for the system (21, 23). The curves in Fig. 40 represent the theoretical values obtained with a112 — 0.0027 and /ic = /ih = 6.2 D, and agree fairly well with the experimental points. It is to be noted that the calculated results predict an almost linear variation of 1/2 with f 12 over a broad range despite the fact that the value of c1/2 used for the computations was not altogether negligible. 

It is to be noted that none of these models considers explicitly the cooperativity by a cooperativity parameter or a size limiting step. 

The corresponding plots, modelled for certain values of parameters, are given in Fig. 9.15. It can be seen that, in addition to the inverse S-shaped curve (owing to the cooperativity parameter J > 0) with increasing abruptness, also a systematic shift of the tangentials is obtained (owing to vibrational effects). 

Diporphinatodiiron 5fe(Mi = Mj = Fe(II) in 5) showed the same reduced affinity to CO, whereas it bound two CO molecules due to the diiron structure The CO-binding equilibrium curve for the 5h-imidazole complex appears sigmoidal, while the curves for the 5a-imidazole complex and the imidazole complex of diporphinato-iron 5c (M, — Fe(II), Mj = 2 H in 5) are hyperbolic The cooperative parameter in) was estimated for the CO-binding to 56 to be 3.4 which meant a strong coopera-... 

The number of observables described here is not sufficient to determine uniquely the several parameters of the theory. We can, however, assign p and u on the basis of independent information concerning the conformation of aqueous amylosic chains in the absence of iodine. A realistic model of aqueous amylose (31) discloses that perhaps 25% of an amylose chain in water might be classified as nearly regular helix at any instant, but the chain conformation is extremely labile, and there is no evidence for any conformational cooperativity in the absence of iodine. Hence, the cooperativity parameter u may be set equal to unity, and for convenience we also take p = 1, which implies equal proportions of helix and coil in the absence of iodine. Calculations not reported in detail here reveal that the results described below are quite insensitive to the exact numerical value of p, provided u = 1 and p is of order unity. 

Here, W is a cooperation parameter, kT/e = 25 mV, A(p is the fraction of membrane voltage measured between adjacent positions of the gating particle, K = const. 6 is analogous to the variable m and gives the fraction of displaced... 

Helical growth (HG) occurs when a sequence of linear steps according to MSOA leads to a critical concentration C at which cooperative helical growth begins (Fig. 2b). Because more bonds per unimer are expected in the helical than in the linear chain, the sudden increase of DP at C > C is described by a constant K] > K and by the familiar Zimm-Bragg cooperativity parameter."." The original theory was developed by Oosawa for the nucleus schematized in Fig. lb and applied to the G- -F transformation of actin. A recent extension by van der Schoot and coworkers,"" applied to helical assemblies of discotic molecules, does not require the specification of the critical nucleus. 

Small values of the cooperativity parameter, E, favor sharp helix-coil transitions which necessarily have very few segments. In terms of these parameters, the free energy per macromolecule may be simply written as... 

In Figure 1, we sketch the persistence length as a function of s for several values of E. For small values of the cooperativity parameter it is too costly to have very many domain walls and, thus, the system tends to a situation of a completely random coil or a rigid helix depending on whether s is smaller or larger than unity. 

The principal goal of this work is to pursue the concept of induced rigidity. Our current investigations include extensions to the cases of arbitrary values of the cooperativity parameter 1 > E > 0, more accurate treatments of the rod-rod excluded volume interaction,— and the inclusion of some attractive dispersion forces. While we refer to the phase transition as lyotropic, in fact we do expect that it also occurs as a function of temperature (as well as pH, ionic strength, etc.) because of the dependence of s on these parameters. We must further emphasize the semi-quantitative nature of our results. Indeed under most actual circumstances, s and c will not be completely independent variables. Nevertheless we believe that the physical picture presented here has some relation to reality. 

Figure 1.16(a) shows the test calculation to see how the coil-globule transition becomes sharper with cooperativity. The DP is fixed at n = 100 and the cooperativity parameter is varied from curve to curve. We can see clearly that the transition becomes sharper with a. The broken lines show the fraction of the hydrated parts. 

The calculation was done as a test case by assuming that all parameters are symmetric and with p = l. The cooperativity parameter a varies from curve to curve. We can clearly see that the coverage takes a minimum value at Xm = 0.5 (stoichiometric concentration) as a result of the competition, so that the end-to-end distance also takes a minimum value at x =0.5. As cooperativity becomes stronger, the depression of the end-to-end distance becomes narrower and deeper. In a real mixture, the association constant and cooperativity parameter are different for water and methanol, so that we expect asymmetric behavior with respect to the molar fraction. 

Figure 6.13(a) draws the spinodal curves for different cooperative parameters a with other parameters fixed. The bottom part of the miscibility square becomes flatter with decreasing a. In the calculation, the usual miscibility domes with UCST appear at low temperatures, but these are not observable in the experiments because the water freezes. For polymer concentrations higher than (/>=0.5, our theoretical description becomes poor because of the depletion of water molecules the number of water molecules becomes insufficient to cover the polymers. 

Figure 6.13(b) shows the dehydration curves. The coverage 0 of a polymer chain by H-bonded water molecules is plotted against the temperature. The cooperative parameter... 

Fig. 6.16 Second virial coefficient A2 plotted against temperature. The cooperative parameter cr is varied from curve to curve. (Reprinted with permission from Ref. [55].)...

Figure 6.16 plots the second virial coefficient A2 as a function of the temperature. The cooperative parameter a is varied from curve to curve. There are in principle three theta temperatures where A2 (6.91) vanishes. The one lying in the middle temperature is the relevant theta temperature (inverted theta temperature) to which the observed LCST approaches for infinite molecular weight. With an increase in coop-erativity, the dehydration becomes sharper, so that the (negative) slope of A2 becomes larger. 

The first equation of (10.156) ensures that a chain doesn t form helices by itself. The second equation is described by the weight 02 for the initiation (nucleation) of a double helix, i.e., the probability for an arbitrarily chosen pair of monomers on different chains to start winding. This is the counterpart of single-chain helix nucleation. The cooperativity parameter a is expected to be small, but <72 can be of order unity if there is no strict restriction on monomer conformation in starting chain winding. 

Two cooperativity parameters are necessary a for helix formation, and 02 for helix association. We can study the weak association case (ca/ h << 1), and the strong... 

---