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Langmuir 1998, 14, 3062-3066
Characterization of Helical Sense Transition for Poly(β-benzyl L-aspartate) Constrained to the Air-Water Interface S. A. Riou and S. L. Hsu* Polymer Science and Engineering Department, University of Massachusetts, Amherst, Massachusetts 01003
H. D. Stidham Department of Chemistry, University of Massachusetts, Amherst, Massachusetts 01003 Received October 16, 1997. In Final Form: March 9, 1998 From external reflectance infrared spectra directly obtained at the air-water interface, the conformation of poly(β-benzyl L-aspartate) is established in monolayer films. It was found that a right-handed R-helix, 4.013 ω-helix, and a left-handed R-helix can all exist simultaneously at such an interface. The right-handed R-helix forms immediately when the film is cast on a surface of pure water, and these helices persist at all compressions in a Langmuir trough. An alternate conformation similar to the 4.013 ω-helix forms on addition of small amounts of 2-propanol to the aqueous subphase and converts either to left-handed or right-handed R-helices depending on the amount of 2-propanol added.
Introduction Insoluble polypeptide monolayer films at the air-water interface have been used as model compounds for biological systems by a number of authors to investigate chain folding, molecular orientation and domain formation.1-6 Very little is known about the conformational state and stability of these polypeptides confined at this interface, however. Different conformations are stable in different spreading solvents, and the effect of the spreading solvent on the conformation stable in the monolayer film has been investigated.7,8 Other factors that may affect the stability and packing of microstructures in a monolayer include pH, polarity, and temperature of the liquid subphase.9 Finally, it is possible that the conformations of polypeptide chains in the asymmetric environment of the air-water interface may be modified upon slow compression of an initially expanded film. Poly(β-benzyl L-aspartate) and poly(β-methyl L-aspartate) form an unusual left-handed 3.613 R-helix in the crystal and in chloroform or trifluoroethanol solutions,10,11 while the larger aliphatic and para-substituted benzyl esters of poly(L-aspartic acid) form more commonly encountered right-handed R-helices.12,13 The relative * To whom correspondence should be addressed. (1) Malcolm, B. R. Prog. Surf. Membr. Sci. 1973, 7, 183-229. (2) Malcolm, B. R. J. Polym. Sci., Part C 1971, 34, 87-99. (3) Loeb, G. I.; Baier, R. E. J. Colloid Interface Sci. 1968, 27, 38-45. (4) Kawai, T.; Komoto, T.; Kato, S. Makromol. Chem. 1981, 182, 21392149. (5) Cheesman, D. F.; Davies, J. T. In Advances in Protein Chemistry; Anson, M. L., Ed.; Academic Press: New York, 1954; Vol. 9, pp 439501. (6) Watanabe, J.; Okamoto, S.; Satoh, K.; Sakajiri, K.; Furuya, H.; Abe, A. Macromolecules 1996, 29, 7084-7088. (7) Baglioni, P.; Dei, L.; Gabrielli, G.; Innocenti, F. M.; Niccolai, A. Colloid Polym. Sci. 1988, 266, 783-792. (8) Gabrielli, G.; Davidson, A. Prog. Colloid Polym. Sci. 1975, 58, 169-177. (9) Jaffe, J.; Ruysschaert, J.-M.; Hecq, W. Biochim. Biophys. Acta 1970, 207, 11-17. (10) Karlson, R. H.; Norland, K. S.; Fasman, G. D.; Blout, E. R. J. Am. Chem. Soc. 1960, 82, 2268-2275. (11) Bradbury, E. M.; Downie, A. R.; Elliott, A.; Hanby, W. E. Proc. R. Soc. (London) 1960, A259, 110-128.
stabilities of the left- and right-handed R-helical forms were estimated by molecular mechanics methods, and differences of only a few tenths of a kilocalorie per mole per residue were reported.14 The determining factor in inducing the left-handed helix appeared to be the contribution of electrostatic energy that arises in the interaction between dipoles associated with the ester and amide groups. The calculation suggested that the interaction was more repulsive in the right-handed than in the lefthanded form, thus destabilizing the right-handed helix.14,15 The similar conformational energies of left- and righthanded helices suggest that these forms may interconvert on heating or on altering the strength of the dipole-dipole interaction. However, when solid films of poly(β-benzyl L-aspartate) were heated to 150 °C, the left-handed structure was irreversibly transformed to a distorted lefthanded 4.013 ω-helix.16 From infrared and electron diffraction measurements, Malcolm reported the presence of right-handed R-helices in dried films of collapsed monolayers of high molecular weight poly(β-benzyl Laspartate) removed from the water surface on which the films were formed.17 The presence of water may weaken the interactions between effective charges on the atoms of the ester and amide groups, thereby destabilizing the left-handed helical structure and favoring the righthanded form. Malcolm’s investigation identified righthanded helices only in the dried films removed from the water surface, and although Malcolm postulated the existence of right-handed helices in the films at the airwater interface, the experiment did not firmly establish the presence of right-handed helices in Langmuir films at (12) Bradbury, E. M.; Carpenter, B. G.; Goldman, H. Biopolymers 1968, 6, 837-850. (13) Goodman, M.; Boardman, F.; Listowsky, I. J. Am. Chem. Soc. 1963, 85, 2491-2497. (14) Yan, J. F.; Vanderkooi, G.; Scheraga, H. A. J. Chem. Phys. 1968, 49, 2713-2726. (15) Ooi, T.; Scott, R. A.; Vanderkooi, G.; Scheraga, H. A. J. Chem. Phys. 1967, 46, 4410-4426. (16) Bradbury, E. M.; Brown, L.; Downie, A. R.; Elliott, A.; Fraser, R. D. B. J. Mol. Biol. 1962, 5, 230-247. (17) Malcolm, B. R. Biopolymers 1970, 9, 911-922.
S0743-7463(97)01132-3 CCC: $15.00 © 1998 American Chemical Society Published on Web 05/05/1998
Helical Transition for Poly(β-benzyl L-aspartate)
Figure 1. Chemical structure of poly(β-benzyl L-aspartate).
surface pressures other than that required to collapse the films. The circumstances in which right-handed R-helices form remained undefined. The presence of different helical forms in polypeptide systems can be detected by using infrared measurements of conformationally sensitive vibrational bands of the amide linkage. First Elliott18 and later Hashimoto19 and Bradbury12,20 showed from transmission infrared spectra of homopolymers and copolymers of L-aspartate esters that there was a correlation between the helix screw sense and the frequencies of the amide I and amide II bands. The ranges of the amide I and amide II bands were respectively 1656-1659 and 1552-1555 cm-1 for the righthanded R-helix, 1664-1668 and 1555-1561 cm-1 for the left-handed R-helix, and 1675 and 1536 cm-1 for the ω-helix of poly(β-benzyl L-aspartate). Recently, external reflectance infrared spectroscopy has proven to be an exceptional technique to acquire structural information at the molecular level even for molecules present at the air-water interface of a Langmuir trough.21-27 Our study demonstrating the coexistence of multiple structures at the airwater interface is reported here. Experimental Section Materials. Poly(β-benzyl L-aspartate) (PBLA) (molecular weight 39 000, degree of polymerization 190 measured by viscosity) was purchased from Sigma and used without further purification (Figure 1). Poly(β-benzyl L-aspartate) solutions were prepared from 99:1 (v/v) mixtures of chloroform (99.0+%, Fisher Scientific) and dichloroacetic acid (99.0+%, Fisher Scientific) at 0.1 and 0.2 mg/mL PBLA. To ensure complete solubilization of the polymer, PBLA was first dissolved in dichloroacetic acid, and then chloroform was added. The solutions were stored at +5 °C and allowed to equilibrate at room temperature before use. Different spreading amounts between 300 and 500 µL for (18) Elliott, A.; Ambrose, E. J. Nature 1950, 165, 921-922. (19) Hashimoto, M.; Arakawa, S. Bull. Chem. Soc. Jpn. 1967, 40, 1698-1701. (20) Bradbury, E. M.; Carpenter, B. G.; Stephens, R. M. Biopolymers 1968, 6, 905-915. (21) Dluhy, R. A.; Cornell, D. G. J. Phys. Chem. 1985, 89, 3195. (22) Dluhy, R. A.; Wright, N. A.; Griffiths, P. R. Appl. Spectrosc. 1988, 42, 138-141. (23) Dluhy, R. A.; Stephens, S. M.; Widayati, S.; Williams, A. D. Spectrochim. Acta Part A 1995, 51, 1413-1447. (24) Gericke, A.; Huehnerfuss, H. J. Phys. Chem. 1993, 97, 1289912908. (25) Gericke, A.; Michailov, A. V.; Huehnerfuss, H. Vibr. Spectrosc. 1993, 4, 335-348. (26) Ren, Y.; Soichet, M. S.; McCarthy, T. J.; Stidham, H. D.; Hsu, S. L. Macromolecules 1995, 28, 358-364. (27) Riou, S. A.; Chien, B. T.; Hsu, S. L.; Stidham, H. D. J. Polym. Sci., Polym. Phys. 1997, 35, 2843-2856.
Langmuir, Vol. 14, No. 11, 1998 3063 both polymer solutions were used for monolayer formation. No significant differences in the resulting isotherms and infrared data were observed. Distilled water was further treated by filtration through a Milli-Q purification system to yield deionized water of nominal resistivity 18.2 MΩ cm-1. Instrumentation and Methods. Infrared transmission measurements were performed using a Bruker Model 113 FT-IR spectrometer equipped with a liquid nitrogen-cooled wide-band mercury-cadmium-telluride (MCT) detector. Films were cast onto calcium fluoride windows and infrared spectra were recorded in the mid-infrared region by coadding 512 scans with a resolution of 2 cm-1. Temperature-dependent infrared spectra were obtained using a home-built heating cell ((1 °C) controlled by a Watlow temperature controller and a copper-constantan thermocouple. External reflectance infrared spectra were obtained in conjunction with surface pressure, using an apparatus described earlier.27 A Langmuir trough equipped with a Cahn electrobalance coupled to a Fourier transform infrared spectrometer has been constructed in our laboratory. The monolayer films were spread on the surface of water using a Hamilton microsyringe. Particular care was necessary to spread a film on the aqueous subphase when 2-propanol was added. To ensure complete evaporation of the solvent and for film to reach equilibration, sufficient relaxation time (∼1 h) was allowed before starting the measurements. Characteristic infrared bands of the solvents used were monitored until they disappeared completely. Typically, 30 min was needed after the spreading step for solvents to evaporate. Monolayer films were then compressed and external reflectance infrared spectra acquired as a function of molecular area and surface pressure. The surface pressure measurements ((0.2 mN m-1) were obtained by using the Wilhelmy plate attached to the modified Cahn electrobalance. The surface area values were calculated to within (0.5 Å2/residue. The isotherms were recorded by compressing continuously the monolayer films at the compression rate of 3.0 cm2 min-1. The temperature of the liquid subphase was maintained constant at 20 ( 0.2 °C using a refrigerated bath circulator. The infrared incident beam was brought to a focus at the air-liquid interface at an angle of incidence of 30 ( 4° and detected by a narrow-band mercury-cadmiumtelluride (MCT) detector cooled with liquid nitrogen. The external reflection-absorption infrared spectra were obtained by typically coadding 512 scans with a resolution of 4 cm-1. The surface pressure for values below ∼40 mN m-1 remained constant during the short acquisition time of the infrared data (∼4 min). A gain of 8 was employed to compensate for the weak reflectance signal. The conformationally sensitive amide I, amide II, and carbonyl ester stretching vibrations in the 1400-1800 cm-1 region were analyzed. The interfering water bands could be reduced and, in some cases, almost totally compensated by carefully regulating the humidity level in the compartment enclosing the Langmuir trough.
Results and Discussion When poly(β-benzyl L-aspartate) films were cast from a 99:1 (v/v) chloroform/dichloroacetic acid solution on a CaF2 plate, the transmission infrared spectra obtained at 20 °C showed bands arising in the NH stretch, ester carbonyl stretch, amide I, and amide II regions at 3301, 1735, 1666, and 1558 cm-1, respectively (Figure 2). These frequencies characterize the left-handed R-helix. When the sample and substrate were heated to 120 °C, the frequencies of the amide I and amide II bands were observed at 1674 and 1536 cm-1, respectively (Figure 2). The frequency shifts for all four bands showed a sigmoidal transition region with an inflection point near 90 °C, in good agreement with the NMR results28 and the differential thermal analysis results29 that indicated transi(28) Happey, F.; Jones, D. W.; Watson, B. M. Biopolymers 1971, 10, 2039-2048. (29) Watson, B. M.; Green, D. B.; Happey, F. Nature 1966, 211, 13941395.
3064 Langmuir, Vol. 14, No. 11, 1998
Figure 2. Raising the temperature from 20 to 120 °C (bold) results in frequency changes in the transmission infrared spectrum of a film of poly(β-benzyl L-aspartate) cast on a calcium fluoride window. The changes are irreversible on cooling the plate to room temperature.
Riou et al.
was essentially no effect of surface pressure on conformation, as all frequencies remained in the range that characterizes the right-handed R-helix (Figure 3B). This not only confirms Malcolm’s conjecture that the structure of the collapsed film is related to that of the monolayer1 but also shows that the right-handed helical conformation is formed immediately after spreading the polypeptide solution on the water surface independently of the surface pressure. These observations were independent of the polymer molecular weight since poly(β-benzyl L-aspartate) samples with viscosity average molecular weights of 14 100 and 59 000 displayed essentially the same monolayer characteristics. It is tempting to speculate that the right-handedness arises due to the presence of water, which would act to decrease the total Coulombic energy of interaction Uel which is expressed as15
Uel )
Figure 3. (A) Surface pressure-area isotherm for a monolayer film of poly(β-benzyl L-aspartate) spread on a water surface at 23 Å2/residue from chloroform solution containing 1 vol % dichloroacetic acid. (B) External reflection infrared spectra of the poly(β-benzyl L-aspartate) monolayer on the water surface at a variety of surface pressures and areas per residue.
tion from the R-helix to the ω-helix, which these authors reported at 90 and 95 °C, respectively. Monolayers of poly(β-benzyl L-aspartate) on the surface of water were spread from a 99:1 (v/v) chloroform/ dichloroacetic acid solution at an initial surface area of ∼23 Å2/residue. After the solvent had evaporated, the monolayers were slowly compressed to the surface areas designated in Figure 3A. The surface pressure-area isotherms presented the usual monolayer-to-bilayer plateau transition characteristic of the R-helical structure.1,17 The limiting area per residue A0 of ∼20.3 Å2/residue estimated by extrapolation to zero surface pressure was in good agreement with electron diffraction data and consistent with the presence of R-helices.17,30,31 External reflection infrared spectra for the monolayers were obtained in situ at the air-water interface and surface pressure-area measurements were performed simultaneously. The slight frequency changes showed that there (30) Malcolm, B. R. Proc. R. Soc. London, Ser. A. 1968, 305, 363385. (31) Birdi, K. S. Lipid and Biopolymer Monolayers at Liquid Interfaces; Plenum Press: New York and London, 1989; pp 173-177.
qiqj
∑Drij
(1)
where D is the dielectric constant, qi and qj are interacting charges, and rij is the distance between them. However, as Ooi et al.15 remark, a reliable estimate of the magnitude of the dielectric constant D is difficult to make on the atomic scale, for when interacting charges are close enough together that there are no intervening solvent molecules or other atoms of the polymer chain, the dielectric constant should be that of free space (except for the reaction field of the environment). However, at greater distances of separation, D should increase, and if water molecules were to intrude between interacting charges, there is the possibility of a really substantial decrease in the contribution of the Coulombic energy to the total. When monolayers were spread on an aqueous subphase at pH 2 or on pure water, similar surface pressure-area isotherms were obtained, and external reflectance infrared spectra recorded on the different substrates showed only vibrational bands characteristic of the right-handed R-helix. However, when the substrate was maintained at a pH of 12 and the surface film was compressed at the usual slow rate about 1 h after spreading, neither surface pressure rise nor characteristic vibrational bands of the polymer were observed. This indicates that the polymer film dissolved into the aqueous subphase by the time the film compression was started. When the film compression was performed right after spreading and a rapid compression rate of 20.0 cm2 min-1 was used to produce a substantial initial surface pressure, the surface pressure was found to decrease slowly with time after completion of the compression, reaching a zero value after about 12 h had elapsed. External reflectance infrared spectra (Figure 4) of the polymer film at the air-liquid interface were obtained at different values of surface pressure. These values remain essentially constant during the short acquisition time of the infrared data (∼4 min). The intensities of vibrational bands at 1661 and 1554 cm-1 were found to decrease with decreasing surface pressure, but no new bands appeared (Figure 4). The ester carbonyl stretch at 1742 cm-1 broadened and decreased in intensity relative to the 1661 cm-1 band, suggesting that the ester was saponified by the basic substrate and the polymer dissolved as the esters converted to carboxylate. Possibly in solution the polypeptide was in random coil conformation, but the spectroscopic evidence is that it remained organized in a right-handed R-helix while at the surface of the basic solution. External reflection infrared spectroscopy does not penetrate into the bulk solution deeply enough to allow any statement to be made concerning the conformation of the polypeptide in solution.
Helical Transition for Poly(β-benzyl L-aspartate)
Figure 4. External reflection infrared spectra of a poly(βbenzyl L-aspartate) monolayer spread on a water surface of pH 12 that was rapidly compressed at a rate of 20 cm2 min-1 obtained as a function of decreasing surface pressure: (1) 32 mN m-1; (2) 22 mN m-1; (3) 16 mN m-1; (4) 9.7 mN m-1; (5) 6.8 mN m-1; (6) 3.6 mN m-1; and (7) 1.5 mN m-1.
Figure 5. (A) Surface pressure-area isotherms for poly(βbenzyl L-aspartate) films spread on water containing small concentrations of 2-propanol (% v/v): (1) 0%; (2) 0.4%; (3) 0.5%; (4) 0.85%; (5) 1.0%. (B) External reflection infrared spectra obtained at 40 mN m-1 from these films.
When monolayers were spread on an aqueous subphase that contained a small amount of 2-propanol (up to 1% (v/v)), the height of the plateau and the area at which first rise in surface pressure occurred were both reduced, as shown in Figure 5A. When the monolayer was spread on a surface of water containing 1% (v/v) 2-propanol, the plateau entirely disappeared and the measured crosssectional area was extremely low, suggesting the disappearance of helices from the surface. A similar contraction of the cross-sectional area of the polymer adsorbed at the interface was reported by Malcolm.1,17,32 If the polymer is in solution, the Gibbs adsorption isotherm requires both 2-propanol and polymer to be present at the surface in concentrations that exceed the bulk concentration of either,33,34 and the low value of the limiting area may perhaps be attributed to solution of the polymer in the 2-propanol-enriched surface region. However, in the surface of the 1% (v/v) 2-propanol subphase the amide I (32) Malcolm, B. R. In Applied Chemistry at Protein Interfaces; American Chemical Society: Washington, DC, 1973; pp 338-359. (33) Gibbs, J. W. The Collected Works of J. Willard Gibbs, Ph.D.; Yale University Press: New Haven, 1948; Vol. 1, p 231. (34) Adamson, A. W. Physical Chemistry of Surfaces, 5th ed.; New York, 1990.
Langmuir, Vol. 14, No. 11, 1998 3065
and amide II frequencies were 1666 and 1558 cm-1, respectively, characteristic of the left-handed R-helix. The much smaller minimum cross-sectional area and the observed R-helical frequencies may imply that the helices were intact in the surface of 1% (v/v) 2-propanol/water but largely submerged in the mixed solvent, perhaps with the helix axis inclined at some angle to the surface normal rather than parallel to the surface. Figure 5B shows that the frequencies obtained for a monolayer spread on pure water are consistent with a right-handed R-helix, whereas for the monolayer spread on 1% (v/v) 2-propanol, the frequencies are characteristic of the left-handed form. The frequencies observed when the monolayer was spread on 0.5% (v/v) 2-propanol show the presence of both left-handed and right-handed helical forms, and some form quite different from both with bands at 1733, 1675, and 1539 cm-1. All these features are in good agreement with the frequencies of the ω-helix that results from heat treatment of either left- or right-handed helices of poly(β-benzyl L-aspartate). Furthermore, infrared reflection spectra of monolayers spread on subphases of intermediate 2-propanol concentrations show a progressive transition from one conformation to the other. When the 2-propanol concentration was 0.4% (v/v), the amide I band was considerably broadened, and the composite band contains contributions from at least three components at the characteristic frequencies for the righthanded and left-handed R-helices and ω-helices. When the 2-propanol concentration was increased to 0.85% (v/ v), the frequencies of the bands correspond largely to those of the left-handed R-helix with some contribution from the bands of the ω-helix. Thus, as the 2-propanol concentration was increased in the aqueous subphase, the infrared reflection spectra show a progression from the right-handed R-helix through an intermediate form with frequencies similar to those of the ω-helix to the left-handed R-helix. The amide I frequency of the ω-helix is higher than that of either the right- or left-handed R-helix. Since hydrogen bonding usually lowers the frequency of a carbonyl stretch, this suggests that the carbonyl in the ω-helix is involved in hydrogen bonding that is substantially weaker than that found in the right- or left-handed R-helix. As the helical sense converts from right- to lefthanded, it is highly probable that the mechanism involves the breaking and re-forming of hydrogen bonds as the amide groups rotate about the bond to the R-carbon.17 However, if the mechanism of conversion of right- to lefthanded helicity were by propagation of a short length of open helix along the helix axis as Malcolm suggested, it would be surprising that the intensity of the 1675 cm-1 band is strong enough to be observable. In fact, when the aqueous subphase contains 0.5% (v/v) 2-propanol, the 1675 cm-1 band entirely dominates the amide I region and is far more intense than the 1660 or 1666 cm-1 features, suggesting that many of the amide groups are present in this sample as components of the more weakly hydrogenbonded intermediate structure. Somewhat smaller numbers of amide groups were evidently present in this film in both left- and right-handed R-helices. The strong intensity of the 1675 cm-1 band relative to the intensities of the 1660 and 1666 cm-1 bands in the spectrum obtained for 0.5% (v/v) 2-propanol concentration is especially interesting. This spectrum shows that the transition from one helicity to the other proceeds through a region of weak hydrogen bonding, and further, the broad bandwidth suggests that the structure is disordered. The spectrum thus shows that the transition is very similar to the random coil-helix transition undergone by many
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Riou et al.
helical structures under conditions of changing pH or temperature.35-37 The strong intensities of the 1675, 1666, and 1660 cm-1 bands in the reflection spectrum of the same film for 0.4% (v/v) 2-propanol concentration may be explained as follows. The mean number of coil-helix transition boundaries 〈n〉 along a polymer chain involved in helix-coil transitions and divided into N equal segments is independent of leftor right-handed helicity at a boundary and is expressed as
〈n〉 )
∂ ln Z ∂ ln σ
(2)
where Z is the partition function for helix and coil and σ is a measure of the cooperative effect involved in forming helix from coil. A segment cannot be in a helix all by itself. At least several others must already be laid down, and the parameter σ requires several previous segments in sequence along the chain to be in helical form. A very small value (e.g., 10-4) of σ relatively disfavors the conversion of coil to helix. In terms of a function F defined as
F)
4s(1 - σ) (1 + s)2
(3)
the mean number of transition boundaries 〈n〉 along the polymer chain is given by the theory as
〈n〉 )
2sσN (1 + s)2(1 + x1 - F)x1 - F
(4)
where s ranges between zero and infinity and provides a measure of the energy change that accompanies a change from coil to helix. An s > 1 favors helical form, while an
s < 1 favors the coil. When s is very large, cooperative effects are favored, leading to very few very long helices along the chain, while a very small s implies that cooperative effects are overcome and there are very few very short helices along the chain. Near s ) 1, the function 〈n〉 peaks sharply with a width that grows as σ increases, which implies a very large number of helices of various intermediate lengths interspersed with random coils of similar segmental lengths. Thus, the strong intensity of the 1675 cm-1 band is required by the theory, as is the substantial intensity of the 1660 and 1666 cm-1 bands. It is perhaps worth remarking that the presence of very few very short coils when s is very large, or very few very short lengths of coil when s is very small, corresponds to the survival of a few holes in a crystal annealed near the absolute zero of temperature. The entropy decrease that accompanies removal of the last few holes, or the last few coils, or the last few helices, is extremely large, though much larger in crystals than in a polymer chain. The polymer chain is a small system in comparison with a crystal, and the entropy decrease, though large, is not nearly as large as is the case for the removal of holes from a crystal. Nonetheless, it may be sufficient to provide a nucleus for the cooperatively formed helices when s increases from very small values or to provide an initial site for coil formation when s is declining from very large values. Acknowledgment. This work was supported by the National Science Foundation Materials Research Laboratory at the University of Massachusetts and a grant from American Chemical Society, Petroleum Research Fund, #30573-AC7. LA971132H (35) Zimm, B. H.; Bragg, J. K. J. Chem. Phys. 1958, 28, 1246. (36) Reiss, H.; McQuarrie, D. A.; McTaigue, J. P.; Cohen, E. R. J. Chem. Phys. 1966, 44, 4567. (37) Poland, D.; Scheraga, H. A. Theory of Helix-Coil Transitions in Biopolymers; Academic Press: New York, 1970.
