Molecular orbital calculations on the conformation of polypeptides and proteins

J. theor. Biol. (1970) 26, 321-333

Molecular Orbital Calculations on the Conformation of Polypeptides and Proteins I. Preliminary Investigations and Simple Dipeptides B. MAIGRET,B. PULLMAN AND M. DREYFUS Institut de Biologie Physico-Chimique, 13, rue P. et M. Curie, Paris 5~, France (Received 27 September 1969) The quantum-mechanical method PCILO based on a perturbative configuration interaction using localized orbitals, particularly efficient for the calculation of ground state energies of molecules, is employed for the evaluation of potential energy surfaces and preferred conformations of two dipeptides (glycyl and alanyl residues). It represents an independent procedure for the determination of these properties, omitting the usual partitioning of the potential energy into empirically determined components. In distinction to such empirical calculations, the computations predict as the most stable conformation of the two dipeptides a folded seven-membered hydrogen bonded ring. The prediction is in agreement with very recent experimental findings. The method is rapid enough to be applicable to large polymeric systems.

1. Introduction The methods commonly us6d for computing the energies of polypeptide and protein conformations are essentially empirical (Scheraga & Scott, 1966; Scheraga et al., 1967; Scheraga, 1968; Flory, 1969; Ramachandran & Sasisekharan, 1968; Ramachandran, 1969). They consist of partitioning the potential energy of the system into several discrete contributions, such as electrostatic and non-bonded interactions, barriers to internal rotation around single bonds, hydrogen bonding, etc. The formulas for the contributions are generally deduced from physico-chemical data on model compounds of small molecular weight. An obvious advantage of this method is the relatively short computing time needed for the energy minimization procedure. Coupled with an appropriate statistical treatment, the method is therefore indeed, from the practical point of view, an acceptable tool for discussing the conformation of polypeptide and protein chains. 321

322

B.

MAIGRET,

B.

PULLMAN

AND

M.

DREYFUS

Such a scheme, however, even if frequently fairly successful, cannot be considered because of its empiricism as theoretically satisfactory. The more so as the fundamental formulae and parameters used to define some of the components of the total molecular potential energy are not well established and differ, often appreciably, from one author to another. A more rigorous theoretical investigation, starting from a more fundamental point of view and taking less cognizance of empirical data, seems therefore highly desirable. In particular, time has come for a direct quantum-mechanical treatment of the problem. In fact, such a treatment has now become possible, at least in principle, due essentially to the recent development of methods, which, operating always within the general scheme of the molecular orbital method, deal simultaneously with all valence, a and 7~, or even all (including inner-shells) electrons. Such methods, the most prominent among which are the Extended Hiickel method (Hoffmann, 1963, 1964), the lterative Extended Hiickel method (Caroll, Armstrong & McGlynn, 1966; Rein, Fukuda, Win, Clarke & Harris, 1966), the so-called CNDO/2 method (Pople & Segal, 1965, 1966) and the so-called ab initio procedure (Clementi & Davis, 1966; Clementi, 1968) are therefore able to evaluate the total molecular energy corresponding to any given configuration of the constituent atoms (for a general review see Pullman & Pullman, 1969). In fact, while this work was in preparation, three preliminary quantummechanical calculations pertinent to the problem have been published or announced. They all utilize the simplest of the all valence electrons methods, the Extended Hiickel one, and are devoted to the evaluation of the conformational energy maps for two simple dipeptides (Hoffmann & Imamura, 1969), the conformation of a few amino acids (Kier & George, 1969) and to a comparison of the energy of a pentapeptide s-helix with that of a pentapeptide extended chain (Rossi, David & Schor, 1969, private communication). Of particular interest to our present study are the calculations by Hoffmann & Imamura on dipeptides. The results obtained seem to be, following these authors, quite comparable to those of the previous empirical procedures. They provide thus essentially an independent check on the partitioned energy potential surfaces, and at the same time they demonstrate the possibility of a direct evaluation of the total energy without necessitating the partition, which appears thus as somewhat artificial and arbitrary. Praiseworthy as these attempts are, they suffer from two essential drawbacks. (a) They use the simplest of all valence electron methods, which represents essentially the extension to all valence electrons of the well known Htickel method used abundantly over many years for the study of n-electronic systems. Being a HiJckel type method it neglects to a large extent inter-

CONFORMATION OF DIPEPTIDES

323

electronic repulsion and correlation effects. Although it has definitely been successful in a number of conformational studies, it stumbled against dif~culties in others. It is obvious that a more refined treatment, carried out, say, at the level of the self-consistent field molecular orbital theory, is more desirable. (b) They consume a prohibitively long computer time. Thus, Hoffmann-Imamura's calculations require one minute per point of the potential surface and are thus much slower than the classical empirical computations. Work carried out in this laboratory during the last two years in particular by Diner, Malrieu and Claverie, with the aim of eliminating these drawbacks resulted successfully in the creation of a new method of calculations labelled PCILO, which seems to be both sufficiently refined and sufficiently rapid to enable successful computations of conformational energies to be carried and in principle for large polymeric systems. The method has been checked on a number of organic molecules for which it predicted the most stable conformation in better agreement with experiment than done by previous calculations.i- In this paper it is used, as a first application to biopolymers, for the problem of the conformational analysis of two simple dipeptides. 2. Method

The designation PCILO stands for Perturbative Configuration Interaction using Localized Orbitals. Details of the method are found in the series of papers by Diner, Malrieu & Claverie (Diner, Malrieu & Claverie, 1969; Malrieu, Claverie & Diner, 1969; Malrieu, Claverie, Diner & Gilbert, 1969). Only its broad principles are outlined here. The method belongs to the all valence electrons procedures studying therefore simultaneously the o" and ~ electrons. Moreover it takes into account explicitly interelectronic repulsions and proposes to go beyond the self-consistent field approximation in the calculation of the ground state energy by incorporating an appreciable fraction of the correlation energy. The method does not require any variational procedure. Its fundamental idea is to choose a set of reasonable bonding and antibonding orbitals localized on the chemical bonds. Such a set may be constructed on a basis of hybridized atomic orbitals [Z~], the bond orbitals being obtained as linear combinations of distinct hybrids taken two by two, each bonding orbital ~ being associated with an orthogonal antibonding orbital ~*: f~)i = C i l ~ i l "~-Ci2zi2 ¢~* = C t 2 x n - Ciixi2 t For example, ethane, hydrogen-peroxide, methyl alcohol, formaldehyde, propionaldehyde, methylamine, propylene, biphenyl, acetonitrile, bicyclobutane, retinal, ,8-ionylidenecrotonic acid, cyclohexane.

324

B. MAIGRET, B. PULLMAN AND M. DREYFUS

A localized orbital representing a lone pair is described by a single hybrid orbital. The bonding orbitals are then used to construct a fully localized Slater determinant. This determinant represents the zeroth order wave function for the ground state of the system. The antibonding orbitals are utilized to build the excited states and the configuration interaction matrix is considered to be constructed on such a basis of configurations. Finally, the lowest eigenvalue and eigenstate, i.e. the energy and the wave function of the ground state of the system are obtained by a RayleighSchr6dinger perturbation expansion truncated after the third order. As a technical simplification, the principal working hypotheses of the CNDO/2 procedure have been retained, in particular the hypothesis of complete neglect of differential overlap as well as the general parametrization of this procedure.t The calculations have been carried out on a CDC 3600 computer. For the two peptides studied below, and which are the same as those investigated by Hoffmann and Imamura, the computing time is a few seconds per point of the potential surface. It thus represents a manifold gain over the preceding authors in spite of the appreciable refinement of the method. 3. Results and Discussion (A) STUDIES ON MODEL COMPOUNDS In connection with our attempted work on the conformation of polypeptides we made a preliminary study of the rotational barriers around single bonds in the three model compounds: acetamide (I), N-methylformamide (II) and N-methylacetamide (III). The rotation around the C-methyl single bond of I and III and around the N-methyl single bond of II and III are expected to provide preliminary information about the hindered rotations around the C~--C ' and C~--N bonds (IV) in a polypeptide chain. As is well known, owing to the planarity of the C'---N semi-double peptide bond, these two

.\

o N

I. Acetamide.

L% N

/o C

II. N-methylformamide.

c\ "

/

N

III. N-methylacetamide.

rotations represent the two degrees of freedom which have to be considered I" This consists in particular in choosing atomic matrix elements empirically, using data on both atomic ionization potentials and electron affinities and omitting certain penetration type terms. The numerical values are those of Table I of Pople & Segal (1966).

325

C O N F O R M A T I O N OF D I P E P T I D E S

for each peptide residue. Moreover, while rotations ~bi and ~ki around bonds adjacent to the same C7~are strongly interdependent (i.e. a conformation of a given angle q~, may or may not be allowed, depending on the value of ~Ol), the rotations around single bonds of different residues are to a large extent, because of the planarity of the amide groups, independent of each other. Interaction between different peptide units occurs only in a small fraction of their individual angle space. For these reasons conformational studies are often centred on the chain fragment IV. TABLE 1

Rotational barriers in methyl derivatives of formamide Compound

Calculated barrier heights (kcal/mole)

Stable methyl conformation

Formamide

H\ ~ / /

/o

O

N'~I--C ,~ 0'9

Acetamide

H

//~,

14.3

C

Me

~C --H

/,,

H HI

N-Methylformamide

/ 13-1 ~

~__ H/

H

Hf

Me.

N-Methylacetamide

H

o.~ >~__Lc// H/

IH

o

°~c//,o

H

H

c/

~o

\. H

~H

The results summarized in Table 1 indicate that: (a) As concerns the rotations around the C-methyl bond, the barrier obtained has a threefold symmetry, with three equal minima and three equal maxima. The heights are of the order of 0.9 to 1.0 kcal/mole. The minima

326

B. MAIGRET, B. PULLMAN AND M. DREYFUS

(most stable conformations) are obtained when the C-----O bond eclipses a C - - H bond of the methyl group and the maxima when the ~ O bond is staggered with respect to a C - - H bond of the methyl group. (b) Quite similar conclusions are obtained for the rotational barrier around the N-methyl bonds, the same threefold symmetry being observed, with the minima occurring when the N - - H bond eclipses one of the C - - H bonds of the methyl group and the maxima when the N - - H bond is staggered with a C--H bond of the methyl group. It seems particularly worthwhile stressing that this conclusion is supported by recent experimental observation due to Bystrov, Portnova, Tsetlin, Ivanov & Ovchinnikov (1969). The predicted barriers, which are of the order of 0-6 to 0-8 kcal/mole are slightly smaller than those for the C-methyl rotation. Altogether, the calculated absolute values of the barrier heights are probably somewhat too small a result which is a general feature of the CNDO parametrization-~ and which we hope to improve upon in the future (A. Pullman, to be published). On the other hand if, folIowing Scheraga & Scott (1966), we adopt the idea of partitioning the rotational barriers into two principal contributions --intrinsic bond rotation potential and non-bonded interactions between atoms adjacent to that bond--it appears that the intrinsic component of the C~--C ' barrier should have a threefold symmetry with minima at 4 / = 0°, 120° and 240° and maxima at 4/ = 60°, 180° and 300°. This conclusion is identical to that of Scheraga and Flory and we agree therefore with these authors that such a barrier may be represented in empirical computations by an expression of the type ½q/,(1- cos 34/). r-

.7 Hi

_

['-

O/

I

II

I,

,I

/, \

./,,\

o,_1,

cfL

i

-]

I II

"H

i

I

I

,

i H,+,_j I

IV. Standard conventions for studying the conformation of polypeptides (Edsall et aL, 1966 a, b and c). II'

-I

I

limits of a residue

A

[i. ii limits ofapeptideunit t This is due probably to the underestimation of repulsion terms between non-bonded atoms.

CONFORMATION

327

OF D I P E P T I D E S

On the other hand, following our results the same conclusion should be valid for the barrier around the C~----N bond: threefold minima at ~b = 0 ~, 120 ° and 240 ° and three maxima at ~b = 60 °, 180° and 300% This conclusion, however, is the opposite to that of Scheraga & Scott (1967) who suppose the minima in this case to be at ~b = 60°, 180° and 300 ° and uses the expression {ag,(1 + 3 cos ~) to represent the barrier. (c) It may be observed from Table 1 that we have also computed the barriers to rotation around the semi-double peptide bond (angle co following the standard convention). The most significant aspect of the results obtained concerns the influence of the number (and probably the size) of the substituents on the N and C atoms upon the value of the computed barrier: this value decreases substantially with the increase in steric hindrance around the bond. This result is therefore an indication that in polypeptide chains it may be reasonable to consider that distortions from planarity of the peptide bonds are likely to occur to some degree and that the effect of the c0-rotations should therefore be included in studies of the most stable conformations of complex biopolymers. But, it is true, that comparatively with the barriers around single bonds, this one requires probably always an appreciable amount of energy. (B) STUDIES ON DIPEPTIDES For this first study we have selected the two simplest dipcptides, which are, in fact, the same as those studicd by Hoffmann and [mamura: N-acetylN'-mcthylglycylamide (V) and N-acctyl-N'-methylalanylamidc(VI), the more so as very recently, detailed experimental data have been obtained by infrarcd H

Ht

01

"H -\- o C,_,=T--- .C.;_-T---, Nr--t---I__1o,C / - - / - - ClII ./ II 41J H oi-i

H

i'"

'

C ~ /- - - H

N+~ V. N-acctyI-N'-methylglycylamidc.

.

II0/_1

.2,'

C//ffl-I~

,,,+, cr¢. I \. H/+I

VI. N-acetyl-N'-mcthylalanylamide.

and nuclear magnetic resonance spectroscopy concerning the conformation of these molecules (Koyama & Shimanouchi, 1968; Bystrov et al., 1969) enabling thus a comparison between theory and reality. In these calculations we have adopted the fixed bond lengths and bond

328

B.

MAIGRET,

B.

PULLMAN

AND

M. D R E Y F U S

angles commonly used (Scheraga, 1968)and assumed the peptide bond to be in the planer t r a n s conformation (o9 = 0). The energy changes have been investigated for 20 ° increments in rotations of angles ~b and ~P. (C) RESULTS ON N-ACETYL-N'-METHYLGLYCYLAMIDE, V

The energy map for the ground state of N-acetyl-N'-methylglycylamide is available in Fig. 1. The most stable conformation obtained corresponds

900

~

"

45 °

0*

451

90*

155"

80*

225*

270*

f

515"

~60"

¢ FIG. 1. Calculated energy curves for the ground state of N-methyl-N'-acetylglycylamide. [The contours of the map are labelled in kcal/mole relative to a zero of energy at the absolute minimum of the calculation (A).] The terminal methyl groups have the most stable conformations: C~ -1--O~-1 bond eclipsed with respect to the C~_ 1--H bonds Nt+x--H~+l bond eclipsed with respect to the C~+x---H bonds.

to a seven-membered hydrogen-bonded ring [Fig. 2(a)] (q~ = 100°, ~b = 220 ° and the symmetric position ~b = 260 °, ~O = 140°). This folded form of the molecule is stabilized by an intramolecular hydrogen bond between the Hi+l hydrogen and the O~_ 1 oxygen [Fig. 2(a)]. This leads to an eclipsed position of the C~--O~ and the Nr---H i bonds with respect to the CT--H bond. The situation is similar to that of the model molecules studied in the preceding section. The effect of the terminal methyl groups on the energy map may be specified. The most stable positions of these groups are the

329

CONFORMATION OF DIPEPTIDE$

o,C 1

~

C?

N~(

(o)

Oi- 1

i

(b)

FIG. 2 (a) Preferred conformation of N-acetyl-N'-methylglycylamide (seven-membered ring with intramolecular H-bond). (b) Secondary energy minimum for the conformation of the glycyl residue.

same as those found in N-methylacetamide, i.e. when the N~+I--Hi+ 1 bond is eclipsed by one of the C~+ 1--H bonds, and when the C i_ l~-----Oi_ x bond is eclipsed by one of the C7-x--H bonds. The barrier heights are, however, slightly lower than in the corresponding model compounds. Secondary, local minima appear 2 kcal/mole above the absolute preceding minimum in the regions ~b = 0 ,~k = 0 and (k = 180°, ~k = 90° (and in the symmetrical areas). The first of those secondary minima corresponds to the fully extended form of the molecule. The stabilization of this form is due to favourable positions for rotational conformations about CT--NI and C7--C~ bonds (Ni--Hi cis with respect to one of the C7--C~ bonds, C;~---Ot cis with respect to C~--N~ bond), favourable orientation of the electrical dipoles, and small repulsive interactions. The other secondary minimum corresponds to a minimum of the repulsive part of the energy along the line ~b = 180° [Fig. 2(b)]. (D) RESULTS ON N-ACETY'L-N'-METHYLALANYLAMIDE, VI

For this compound two independent computations have been performed with the side chain methyl group either staggered or eclipsed with respect to the main backbone chain. As shown on the corresponding energy maps [Figs 3(a) and (b)], the most stable form taken up by this compound is, following our calculations, also a folded one with a stabilizing intramolecular hydrogen bond between the Hi+ 1 hydrogen and the Ol-1 oxygen [Fig. 4(a)], the corresponding rotational angles being ~b = 260° and ~ = 140°. It may T.B.

21

330

B.

MAIGRET,

B.

++0o

PULLMAN

AND

M.

DREYFUS

++

+1 0°

45 °

90 °

155 °

180 °

225 °

270 °

315 °

360 °

¢ (o)

225°

~56 ~

45 ° .

0°

45 °

90 °

135 °

180 °

225" J,

(b)

FiG. 3.

270 °

515 °

560 °

CONFORMATION

OF D I P E P T I D E S

331

be underlined that this form is indeed very close to the seven-membered ring found experimentally by Bystrov et al. (1969). The C'~------O~bond and the Nf---H~ bond are then eclipsed by the CTL--H bond. Two secondary minima appear again 1 kcal/mole above the deepest minimum. The first one (4, = 100°, $ = 220 °) corresponds closely to Mizushima's conformation (Mizushima, Shimanouchi, Tsuboi & Azakawa, 1957) as shown in Fig. 4(b). The second one corresponds to the totally extended form of the molecule (4, = 0 °, ~, = o°).

°,C

,

i

NI+ 1

'

~141

(o)

C~-1

~...JOi_

-~OiNi, c7(

,J

~

X

-1

1

(b)

FzG. 4. (a) The most stable conformation of N-acetyl-N'-methylalanylmide. (b) The stable form of Mizushima for N-acetyl-N'-methylalanylamide. The effect of the rotation of the terminal methyl groups leads to the same conclusions as in the case of N-acetyl-N'-methylglycylamide: the C'~_1~--~O~-t bond is cis with respect to one of the C ~ _ I - - H bonds and the N~+I--H~+t bonds is cis with respect to one of the C~+ t - - H bonds. The rotation of the side chain methyl group leads to a stable conformation when the C~--C~, CT--N~ and C~--H bonds are staggered with respect to the C ~ - - H bonds.

FIG. 3. (a) Calculated energy curves for the ground state of the N-acetyl-N'-methylalanylamide, with the Ca methyl staggered with respect to the C-ffi bonds. [The contours are labelled in kcal/mole relative to a zero of energy at the absolute minimum of the calculation (A).] The terminal methyl groups have the most stable conformations: C;-x---O~-i bond eclipsed with respect to the C~- t - - H bonds Nl+t--H~+t bond eclipsed with respect to the C~+x--H bonds. (b) Calculated energy curves for the ground state of the N-acetyl-N'-methyl-alanylamide, with the Ca methyl eclipsed with respect to the C~ bonds. [The contours are labelled in kcal/mole relative to a zero of energy at the absolute minimum of the calculation (A).] The terminal methyl groups have the most stable conformations: C~_1--O,_~ bond eclipsed with respect to C~-x--H bonds Na+x--H~+x bond eclipsed with respect to C~l+~--H bonds.

332

n.

MAIGRET,

B.

PULLMAN

AND

M.

DREYFUS

The rotational barrier in the ease of the absolute minima obtained [~b = 260 °, ~O = 140° for both maps [Fig. 3(a) and (b)]] is 2.0 kcaI/mole. All the points of the potential map corresponding to the eclipsed form are less stable than those of the staggered form. This situation is in agreement with the hypothesis used in classical calculations. 4. Conclusion The comparison of our results with previous computations, both quantummechanical (Hoffmann & Imamura, 1969) and empirical (for a review see Ramachandran, 1969) on the glycyl and alanyl residues reveals significant differences. This concerns, in particular, the predicted deepest minimum of the potential surface which following our calculations occurs in or near regions "sterically not allowed" and corresponds to "folded" forms (q~ = 100 °, ~O = 220 ° and ~b = 260 °, q; = 140°). In fact, what happens in this conformation is an atomic overlap between the proton at N~-I and the O~+a, these atoms being only 1.96 A apart. Now, this distance is the usual one for N - - H . . . £ ~ C hydrogen bonds. The possibility of such a bond has been excluded by previous workers, on the basis of an unfavourable orientation of the N H and CO interacting groups. Consequently they have used a non-bonded potential between these two groups instead of a hydrogenbonding one and the conformation came out unstable. The situation, due possibly to the overestimation of the directional character of the hydrogenbond (see Donohue, 1968) is an illustration o f the danger of empirical, additivity type computations. It may be remarked that "other" energy contributions favour the folded form: the N - - C-'~ and C" C' bond torsional potentials are approximately at a minimum and no overlap occurs between any other pair of distant atoms. In any ease, the preferential stability of the folded form has, as already said, been demonstrated experimentally recently for the alanyl residue (Bystrov et al., 1969) and confirms thus the advantage of our unprejudiced mode of approach. This work was supported by grant No. 67-00-532 of the D616gation G6n6rale la Recherche Scientifique et Technique (Comit6 de Biologic Moleculaire). REFERENCES BYSTROV,V. F., PORTNOVA,S. L., TSETLn,t, V. I., IVANOV,V. T. & OVCHINNIKOV,Y. A. (1969). Tetrahedron, 25, 493. CARROLL,D. G., AmMS'n).ONG,A. T. & McGLYNN,S. P. (1966). J. chem. Phys. 44, 1865. C~Mm,n'i, E. (1968). Chem. Rev. 68, 341. CLEMENTI,E. & DAVIS,D. R. (1966). J. comput. Phys. 1, 223. DINER, S., MALRIEU,J. P., CLAVERIE)P. (1969). Theor. chim. Acta, 13, 1. DoNomm, J. (1968). In "Structural Chemistry and Molecular Biology", p. 443 (A. Rich and N. Davidson, eds.). San Francisco and London: Freeman and Co.

CONFORMATION

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EDSALL, J. T., FLORY, P. J., KENDREW,J. C., LIQUORI,A. M., NEMETHY,G., RAMAC~DRAN, G. N. ~ SCHERAGA,H. A. (1966a). J. molec. Biol. 15, 399. EDSALL, J. T., FLORY, P. J., KENDREW,J. C., LIQUORI,A. M., NEMETHY,G., RAMACHAN= DRAN, G. N. & SCHERAGA,H. A. (1966b). J. biol. Chem. 241, 1004. EDSALL) J. T., FLORY, P. J., KENDREW,J. C., LIQUORI,A. M., NEMETHY,G., RAMACHAN= DRAN, G. N. ~ SCHERAGA,H. A. (1966c). Biopolymers, 4, 121. FLORY, J. P. (1969). In "Statistical Mechanics of Chain Molecules." New York: Interscience. HOFFMANN,R. (1963). J. chem. Phys. 39, 1397. HOFFMANN,R. (1964). J. chem. Phys. 40, 2745. HOFFMANN,R. & IMAMURA,A. (1969). Biopolymers, 7, 207. KIER, L. B. & GEORGE,J. M. (1969). Theor. chim. Acta 14, 258. KOVAMA,Y. & SHIMANOUCHLT. (1968). Biopolymers, 6, I037. MALRIEU,J. P., CLAVERIE,P. & DINER, S. (1969). Theor. chim. Acta, 13, 18. MALRIEU, J. P., CLAVERIE,P., DINER, S. & GILBERT, M. (1969). Theor. chim. Acta (in the press). MIZUSHIMA,S., SHIMANOUCH/,T., TSUBOI, M. & AZAKAWA,T. (1957). J. Am. chem. Soc. 79, 5357. POPLE, J. A. t~ SEGAL,G. A. 0965). J. chenL Phys. 43, S136. POPLE, J. A. & SEGALG. A. (1966). J. chem. Phys. 44, 3289. PULLMAN,B. & PULLMAN,A. (1969). Progr. Nucleic acid Res. molec. Biol. 9, 327. RAMACHANDRAN,G. N. & SASISEKHARAN,V. (1968). Adv. Protein Chem. 23, 283. RAMACHANDRAN,G. N. (1969). Int. J. Prot. Res. 1, 5. REIN, R., FUKUDA,N., WIN, H., CLARKE,G. E. & HARRIS, F. E. (1966). J. chem. Phys. 45, 4743. SCHERAGA,H. A. (1968). Adv. phys. erR. Chem. 6, 103. SCHERAGA,H. A. & ScoTT, R. A. (1966). J. chem. Phys. 45, 2091. SCHERAGA, H. A., SCOTT,R. A., VANDERKOOI,G., LEACH, S. J., GIBSON,K. O., Oo1, T. & NEMETHY, G. (1967). In "Conformation of Biopolymers", p. 43. New York: Interscience.