Energy parameters in polypeptides. 8. Empirical potential energy

Petra Johannesson, Gunnar Lindeberg, Anja Johansson, Gregory V. Nikiforovich, Adolf Gogoll, Barbro Synnergren, Madeleine Le Grèves, Fred Nyberg, Ande...
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United Atom Potentials for Polypeptides and Proteins

(6) A. D. Bangham and R. W. Horne, J . Mol. Biol., 8, 660 (1964). (7)J. Seelig, Q . Rev. Biophys., 10,411 (1977). (6) A. Davidson and B. NordBn, Chem. Phys., 8, 223 (1975). (9) J. H. Davis, K. R. Jeffrey, M. Bloom, M. I. Valic, and T. P. Higgs, Chem. fhys. Lett., 42, 390 (1976). (10) H. J. J. Braddick, “Vibrations, Waves and Diffraction”, McGraw-Hill, New York, 1965. (11) G. R. Fowles, “Introduction to Modern Optics”, 2nd ed., Holt, Rinehari and Winston, New York, 1965. (12) P. Ekwall, Adv. Liq. Cryst., 1, 1 (1975). (13) A. Davidson and B. Norden, Chem. Phys. Lett., 28, 221 (1974). (14) U. Henriksson, T. Klason, L. Cdberg, and J.C. Eriksson, Chem. fhys. Lett., 52, 554 (1977).

The Journal of Physical Chemistry, Vol. 82, No. 24, 1978 2609

(15) J. Breton, M. Michel-Villaz, and G. Paillotin, Biochim. Biophys. Acta, 314, 42 (1973). (16)F. Tjerneld, B. NordBn, H. E. Akerlund, B. Andersson, and P.-A. Albertsson, Spectrosc. Lett., 10, 489 (1977). (17)L. N. M. Duysens, Biochim. Biophys. Acta, 19, l(1956). (16) M. H. Cohen and F. Reif, Solid State fhys., 5, 321 (1957). (19) H.Wennerstrom, G.Lindblom, and 8.Lindman, Chem. Scr., 6, 97 (1974). (20) D. M. Brink and G. R. Satchler, “Angular Momentum”, Oxford University Press, London, 1962. (21) R. G.Barnes, Adv. Nucl. Quadrupole Reson., 1, 335 (1974). (22) This technique was recently used to study diffusion: G. Lindblom and H. Wennerstrom, Biophys. Chem., 6,167 (1977).

Energy Parameters in Polypeptides. 8. Empirical Potential Energy Algorithm for the Conformational Analysis of Large Molecules’ Lawrence G. Dunfield,2’$bAntony W. Burgess,2b-dand Harold A. Scheraga*2b,cse Chemistry Department, Cornell University, Ithaca, New York 14853 and Biophysics Department, Weizmann Institute of Science, Rehovoth, Israel (Received October 27, 1977; Revised Manuscript Received August 25, 1978)

An empirical conformational energy algorithm has been developed for application to polypeptides and proteins. Aliphatic and aromatic CH3, CHz,and CH groups have been represented effectively by several types of single “united atoms”, with the net partial charges taken as the sum of those on the carbon and hydrogen atoms. The parameters for the nonbonded Lennard-Jones 6-12 interactionsof these groups were derived from calculations of crystal structures of hydrocarbons, and were tested, in part, in similar calculations on amino acids. The united atom parameters were then incorporated into an empirical energy algorithm, which treats nonpolar hydrogen atoms implicitly, by using Fourier series of one and two variables (to represent the interactions of all atoms across each bond about which rotation takes place) for 1-4 and 1-5 interactions, respectively. The accuracy of this approach is assessed, and its applicability to large molecules discussed.

Introduction potential energy algorithm, based on the earlier results,13 but which requires less computer time for calculating the Algorithms which approximate the potential energy properties of macromolecules such as proteins. surfaces of peptides have been used widely as techniques In such a potential function, the nonpolar hydrogen for conformational analy~is.~-’~ In all cases, the parameters for these algorithms have been derived in a semiempirical atoms are not included explicitly in the calculation, but manner from an examination of experimental data. From rather they are included implicitly by (i) representing the calculations of crystal structures of small m o l e ~ u l e s , l ~ ~ ~ Jnonpolar ~ - ~ ~ carbon atoms and their attached hydrogen atoms it has been shown that parameters for simulating interas a single g r o ~ p ~ centered J ~ J ~ a~t ~the~ carbon atom molecular interactions can be estimated quantitatively. (“united atoms”), (ii) representing by a one-dimensional Most of these ~ t u d i e s ~ ~ Jwere l J ~ successful J~ in treating Fourier series both the intrinsic torsional potential and nonpolar molecules, and they have been extended only those interactions between atoms whose relative positions recently12J6-18to systems containing heteroatomic interare affected by rotation about only one intervening bond actions similar to those found in proteins and their sub(so-called 1-4 interactions), and (iii) representing by a strates. two-dimensional Fourier series, the interactions between Although empirical intermolecular potential energy atoms (one of which is a nonpolar hydrogen) whose relative functions can be parameterized from calculations of crystal positions depend upon rotation about precisely two bonds structures, their application to the calculation of con(explicit 1-5 interactions). The Lennard-Jones 6-12 informational (intramolecular) energies in molecules is not teraction parameters for the aliphatic and aromatic united simple. In most cases, the calculated intramolecular inatoms were determined by calculations of the crystal teraction energies have to be adjusted empirically to agree structures of n-pentane, n-octane, adamantane, 1-biwith experimental rotational barriers and differences in adamantane, and benzene, and these were subsequently the energies of stable conformers. Recently, an empirical tested on several other crystals. The coefficients for the potential energy function based on calculations of crystal Fourier series mentioned above were derived by using the structures and corrected to give rotational barriers close ECEPP” parameters for nonpolar hydrogens. The success to experiment has been developed for peptides.13 It is of these approximations was evaluated by comparing the designated as ECEPP (empirical conformational energy “total”13 and “united-atom’’ energy surfaces of several program for peptides). This scheme considers all interamino acid residues. A brief description of the applications atomic interactions (including those involving hydrogen of this “united-atom” potential energy function to the atoms) which are separated by a t least one degree of instudy of protein conformations is presented. A Fortran ternal rotational freedom;13the procedure is applicable to IV program which allows the calculation of the conformacromolecules, but requires large amounts of computer mational energy of a protein (using this united atom altime. The purpose of this paper is to present an empirical gorithm) is available.21 This program is designated as 0022-3654/78/2082-2609$01 .OO/O

0 1978 American

Chemical Society

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The Jourqal of Physical Chemistry, Vol. 82, No. 24, 1978

(united atom conformational energy program for peptides).

L. G. Dunfield, A. W. Burgess, and H. A. Scheraga

the calculated and observed lattice constants and lattice energy for n - h e ~ a n e The . ~ ~ experimental lattice constants for adamantaneZs and l-biadamantaneZ9were used to Methods obtain the Lennard-Jones 6-12 parameters for the aliNonpolar atoms are defined as aliphatic or aromatic C phatic CH group. The lattice constants of congressane30 and H, and polar atoms are N, 0, and S and H’s bonded were used to verify these parameters. The experimental to them; a carbon atom with no hydrogens bonded to it lattice constants31and lattice energy32of benzene were used is also considered to be polar. to obtain the parameters for aromatic united atoms. The The Lennard-Jones 6-12 nonbonded potential, Um(rii), parameters for the aromatic CH were then tested by for interactions between like united atoms i is comparing the calculated and observed lattice constants33 and lattice energy34of anthracene. The parameters thus UNB(r..)= A”/r..12 - cii/ rii6 (1) obtained were tested further on 13 different crystals by or minimizing the energy of the crystal and comparing the observed and calculated lattice ~0nstants.l~ The molecules in these additional crystals contain both polar and nonwhere rii is the distance between the atom pair. The polar atoms. The partial charges (and the parameters for minimum in Um(rii)is located at Rii, where its value is -eii, the nonbonded potentials) of all polar atoms were taken with e” = (Cii)2/4Au.Both the attractive (CU) and repulsive directly from ECEPP.” The ECEPP charges of nonpolar C (Au) coefficients were determined by calculations of crystal and H were added together to obtain the net charge on a structures. In addition, the resulting attractive coefficients united atom (in molecules that also contained polar atoms). were used to evaluate empirical united-atom polarizaThe accelerated-convergence methodz2was used for the bilities (from the Slater-Kirkwood equation, as described summation of the interactions in the crystals. This was for ECEPP13) so that the attractive coefficient, C’j, for incombined with the cutoff procedure described by Momany teractions between unlike united atoms i and j could be et al.I7 to determine the size of the crystal used for the computed by calculations. r ’ I B. Intramolecular Torsional Parameters. Additional computer time can be saved in conformational analysis by using a Fourier series to represent the torsional energy for rotation about a bond instead of calculating the intrinsic where ai and a j are the calculated group polarizabilities torsional term and all of the 1-4 interactions across the and Ni and Nj are the effective number of electrons for bond separately. However, the united-atom approximation each group.17 The van der Waals radii, Rli, of the united is not a very good one for 1-4 interactions involving a atoms were evaluated from the relationship hydrogen atom. Therefore, it is necessary to include the hydrogen atom 1-4 interactions explicitly in the UNICEPP Rii = (2A”/C”)l/6 (4) algorithm. In such cases, the total ECEPP energy for roThis, in turn, allowed the evaluation of the repulsive tation about a bond is determined both by the pairwise coefficient for interactions between unlike atoms as interatomic nonbonded (or hydrogen bonded) and electrostatic interactions between all atoms across the bond (5) Aij = (CU/2)[(Rii + Rjj)/2I6 and by the intrinsic torsional potential associated with that The application of these parameters to crystals not used bond (this total energy will be referred to as the local in the parameterization provided a test of the intermoenergy surface for a given bond). In ECEPP, the intrinsic lecular interactions in UNICEPP. As indicated above, torsional term pertains only to the dihedral angle o and Fourier series were introduced to represent the close 1-4 most x ’ s . ~ ~ intramolecular interactions between all atom types and Before ECEPP can be used, however, to treat 1-4 inthe similarly close 1-5 intramolecular interactions between teractions, it must be recognized that certain 1-5 and 1-6 nonpolar H’s with other (polar or nonpolar) atoms. interactions involving hydrogens have a very important A. United-Atom Intermolecular Parameters. The basic influence on the local energy surface. Since these interprinciples and description for simulating crystal properties actions will not be present in a united-atom representation may be found e l s e ~ h e r e Those . ~ ~ ~lattice ~ ~ constants which they must be added to the Fourier representation of the are themselves independent were taken as the variables local energy surface. This requires that 1-5 and 1-6 induring energy minimization.17 The crystal lattice binding teractions (which depend on more than one dihedral angle) energy was calculated using energy functions, parameters, be expressed as functions of two and one dihedral angles, and procedures reported by Mommy et al.17except for the respectively. The following steps take this into account. omission of nonpolar hydrogen atoms (united-atom apFirst, if a 1-4 interaction between two aliphatic carbon proximation). In the parameterization, the partial elecatoms was involved in the calculation of a local energy trostatic charge on the united atoms of all hydrocarbons surface, then the 1-6 interactions of all hydrogens on each was assumed to be zero,l0J7and the two parameters, ti, and of these carbons are omitted in the united-atom apRii, of the Lennard-Jones 6-12 potential for each type of proximation. This means that, in the calculation of the united atom (see eq 2) was obtained by minimizing the local energy surface, the two aliphatic carbons should be differences between the observed and calculated lattice treated as united atoms. All other interactions were constants and binding energies. treated with ECEPP parameters. In this way, the interFor each set of Lennard-Jones parameters, the lattice actions involving the hydrogens on these two carbons are constants were altered using the method of until included in the UNICEPP representation without having to the lattice binding energy was a t a minimum. The sum specify the additional dihedral angles that would define of the squares of the relative errors in the calculated lattice constant^^^,^^ and in the binding energies26for n - ~ e n t a n e ~ ~ their positions. Second, in the special case of CH3 groups, they can be and n-octaneZ5was minimized, to obtain the best Lenrotated in the ECEPP algorithm, but such rotations do not nard-Jones 6-12 parameters for the CH2and CH3 groups. occur in UNICEPP since CH3 groups have spherical symThese parameters were tested subsequently by comparing UNICEPP

United Atom Potentials for Polypeptides and Proteins

The Journal of Physical Chemistry, Vol. 82, No. 24, 1978 2611

TABLE I: Parameters for "United-Atom" Lennard-Jones Interaction Energy interactiona

q i , kcal/mol

Rii, A

CH. . .CH CH,. *CH, CH,. * .CH, aromatic CH. .CH

0.13 0.14 0.18 0.12

4.7 5 4.45 4.25 4.20

aliphatic

a See text for derivation of cross terms for interactions between different types of united atoms.

Figure 1. Illustration of 1-4 type interactions in a portion of a polypeptide chain.

metry. However, in the ECEPP algorithm, the value of the dihedral angle for the methyl rotation will influence the local energy surface for rotation about the bond adjacent to the one about which the methyl group rotates. This difficulty was circumvented by setting the dihedral angle for rotation of the methyl group a t 60°, and explicitly including the 1-5 interactions of the methyl hydrogens with atoms other than aliphatic carbons (using ECEPP parameters for type 1 hydrogens) in the calculation of the local energy surface for the adjacent dihedral angle. The restriction to interactions with atoms other than aliphatic carbons was made because the procedure mentioned in the first step already has included the interaction of methyl groups with other united-atom types. Third, the explicit 1-5 interactions involving nonpolar (non-methyl) hydrogens were treated separately by using the ECEPP parameters for the nonpolar hydrogens and representing such local energy surfaces (that are functions of two dihedral angles) by a two-dimensional Fourier series. Thus, two adjacent dihedral angles must be specified in order to compute the energies involved in such 1-5 interactions. With the inclusion of all appropriate interactions, as indicated above, each of the one-dimensional local energy surfaces was then approximated as a function of the dihedral angle for rotation about the given bond, by a Fourier series Elocal= A. C(Akcos h0 + Bk sin h0) (6)

+

k

where the A's and B's are Fourier coefficients, and 0 is the dihedral angle. The two-dimensional local energy surfaces were approximated in a similar manner by fitting a two-dimensional Fourier series for each pair of adjacent dihedral angles

+ bkncos k0 sin ny -t Ckn sin h0 cos ny + dknsin k0 sin ny)

E(0,y) = C(akn cos h0 cos ny k,n

(7) where 8,y are the dihedral angles and akn,bkn,Ckn, and dkn are the Fourier coefficients. Figure 1 provides illustrations of the interactions that are included in typical local energy surfaces. For example, consider the 1-4 interactions that are involved across the N-C" bond (viz., those between the pairs of atoms 1 , 3 ; 1, 4; 1, 5; 2, 3; 2,4; and 2,5). For these pairs of atoms only, the electrostatic and nonbonded energy terms35(ECEPP13) were calculated, and summed at each value of the dihedral angle 4, in 10' increments in 4. This is the procedure followed for all dihedral angles. The interactions of the nonpolar hydrogens, 6 and 7 , with atoms 1 and 2 of Figure 1 are clearly important. Here, the 1-5 interactions of H6 and H7 with C'l and Hz depend

explicitly on two dihedral angles. These important interactions are included by calculating the ECEPP energy surface for these interactions at 9' intervals of both dihedral angles in order to obtain the correct two-dimensional local energy surface. Fourier series were calculated in an identical manner for all pairs of adjacent dihedral angles that define explicit 1-5 interactions of nonpolar hydrogen atoms. Thus, in this united atom algorithm, the total conformational energy of a peptide consists of the one-dimensional Fourier torsional term to represent the local energy surface around each single bond (which replaces the intrinsic torsionals and the explicit treatment of 1-4 interactions in E C E P P ~the ~ ) ,two-dimensional Fourier terms to represent the interactions of a nonpolar hydrogen with atoms that are separated from it by two bonds about which rotation can take place, plus the hydrogen bonding, Lennard-Jones nonbonded and electrostatic interactions between all pairs of atoms (other than nonpolar hydrogens) separated by two or more degrees of internal rotational freedom. C. Residue Geometries and Partial Charges. The molecular geometries for all heavy atoms and hydrogen atoms attached to polar atoms were taken from E C E P P . ~ ~ All of the protons attached to C atoms were omitted, and the partial charge associated with each of these was added to the atom to which it was attached. In most cases, the partial charges on united atoms net to almost zero but, when the united atom is next to a polar group such as carboxyl or amine, there is a small residual charge on the united atom which reflects charge migration along the side chain.18 The nonpolar C atoms were then assigned the appropriate united-atom Lennard-Jones parameters determined from the crystal structure calculations described in Methods section A.

Results and Discussion A. Intermolecular United-Atom Parameters. The united-atom parameters derived from the crystal packing studies on n-pentane, n-octane, adamantane, l-biadamantane, and benzene are given in Table I. It is interesting to note that the calculated value of Rii (4.75 A) for the CH group is larger than Rii (4.45 A) for the methylene group which in turn is larger than that for the methyl group (Rii = 4.25 A). Previous united atom p o t e n t i a l ~ ~ Jconsidered ~ J ~ ~ ~ ~the value of Rii for all the united atoms to be the same. The distance of approach of the methyl groups in the n-octaneZ5crystal is much closer than any of the methylene-methylene contacts, but not so for pentane.24 If the same value of Rii is used for both CH3 and CH2 groups, it is not possible to obtain a reasonable fit to the experimental lattice constants for n-pentane even when using different t parameters for each group. A qualitative understanding of the larger effective sizes of the CH and CH2 groups can be achieved by considering the directions of interactions involving these groups. The effective size of a CH,, CH2, or CH group manifests itself in the directions that would normally be occupied by the protons; in other directions (e.g., toward

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The Journal of Physical Chemistry, Vol. 82, No. 24, 1978

L. G. Dunfield, A. W. Burgess, and H. A. Scheraga

TABLE 11: Comparison of Results of Calculations on Crystals of Aliphatic and Aromatic Hydrocarbons Using Empirical Potentials

molecule n-pentane

n-hexane

n-octane

benzene

ref 24 (expt) 17 19 9 1O d 20 this work 27 (expt) 17 19 9 20 this work 25 (expt) 17 19 9 1oe 20 this work 31 (expt)

space group Pbcn

3.2.1

,I7

I1

anthracene

19 9 1O d this work 34 (expt) 17 19 9 1O d this work

P2Ja

adaman tane

1- biadaman tane

congressane

9 this work 29 (expt) 19 9 this work 30 (expt) 19 9 this work

cell sizea

pi

pa3

a

b

Aa,b

Ab,b

a

4.10 0.16 -0.01 -0.31 0.03 -0.50 0.17 4.17 0.15 -0.04 -0.36 -0.55 0.15 4.16 0.10 -0.10 -0.38 -0.15 -0.58 0.13 7.39 -0.11 -0.22 -0.49 -0.23 0.04 8.44 0.18 -0.13 -0.30 -0.11 0.18 9.45 -0.59 -0.91 -0.11 6.53 -0.34 -0.60 -0.01 10.11 0.06 -0.27 -0.06

C

A c , ~

a

a

9.04 -0.14 -0.46 -0.92 -0.44 -1.26 -0.23 4.70 -0.14 -0.24 -0.53 -0.72 -0.03 4.75 -0.16 -0.30 -0.56 -0.29 -0.75 -0.09 9.42 0.01 -0.47 -0.72 -0.15 -0.24 6.00 -0.11 -0.21 -0.38 0.07 -0.08 9.45 -0.55 -0.91 -0.11 6.58 -0.39 -0.63 -0.06 10.11 0.06 -0.27 -0.08

14.70 0.13 0.06 -0.24 0.14 -0.47 -0.06 8.57 0.05 0.27 0.06 -0.14 0.22 11.00 0.03 0.41 0.06 1.09 -0.12 0.37 6.81 -0.05 -0.02 -0.26 0.03 0.25 11.12 0.10 -0.32 -0.48 -0.40 -0.20 9.45 -0.58 -0.91 -0.11 10.46 -0.23 -0.47 0.09 10.11 0.06 -0.27 -0.03

CY

AcY,~

deg 90 0 0 0 0 0 0

96.6 0.2 -9.2 -6.7 -8.8 -4.4 94.8 0.4 -2.5 -6.8 5.8 -6.7 -1.7 90 0 0 0 0 0 90 0 0 0 0 0 90 0 0 0 87.5 2.41 2.07 3.2 90 0 0 0

'

Ap,

Y A Y , ~

deg

deg

90 0 0 0 0 0 0 87.2 0.4 4.4 3.3 2.4 0.4 84.5 0.1 -1.2 2.7 13.5 2.0 -1.5 90 0 0 0

90 0 0 0

0 0

125.6 0.0 -1.3 -0.8 -1.2 -1.1 90 0 0 0 104.6 -0.35 - 0.64 0.3 90 0 0 0

0 0 0

105.0 -1.4 10.2 9.4 8.6 10.7 105.1 -1.7 9.3 9.3 10.9 8.6 10.0 90 0 0 0 0 0 90 0 0 0 0 0 90 0 0 0 119.9 -1.10 -0.88 -0.9 90 0 0 0

lattice binding energy, kcal/mol -11.2c -9.9 -7.6 -7.0 -18.6 -4.8 -10.4 -13.2' -12.0 -9.5 -8.9 -6.3 -12.6 -17.2' -15.7 -12.9 -12.0 -18.5 -8.7 -17.8 -11.3? -9.5 -14.4 -13.6 -11.4 -11.3 -22.6 t o -24.5' -19.4 -28.1 -27.2 -24.0 -23.4 -13.4 -11.9 -23.7 -23.8 -21.3 -40.9 -13.0 -11.0 -35.9

a The cell size is given as the number of unit cells in each direction along the a, b, and c axes, in addition to the cell containing the asymmetric unit. The A ' s are defined as A1 = l c d c d - lobsd, where 1 = a, b, c, CY, p , or y. Reference 26 and 36. d This result was reported for the full-atom potential in ref 10, Le., H atoms were considered explicitly. e This result was reported for the united atom potential in ref 10. f Reference 32. g Reference 33.

'

the C" atom in C"-CH3), the large van der Waals radius of the C" atom prevents the close approach of another atom to the CH, (or CH,, or CH) group along the direction of the C"-CH3 bond. When another atom approaches a methyl or methylene group, the contact distance is determined mainly by the positions of the methyl or methylene protons. The methyl protons can rotate away from the approaching atom, allowing it to pack more closely to the methyl group. In addition, the presence of the heavy atoms bonded to the CH, or CH group serve to direct the approach along the C-H bond directions, causing the van der Waals force of the hydrogen to be felt sooner than for close approach to a CH3 group. When an atom approaches a CH2 or CH group, the protons cannot adjust their orientation without changing the conformation of the heavy atoms in the molecule, and this will often prevent atoms from packing as closely to CH, or CH groups as they do to a CH, group. In Table 11, the results of the calculations on crystals of both aliphatic and aromatic hydrocarbons are presented. Those quoted for Warme and Scheraga,lg Gibson and

S ~ h e r a g aand , ~ Pletnev et aL20were calculated here from the united atom parameters given by each of these authors. The results quoted for Momany et and for Ferro and Hermanslo were taken directly from their papers; their calculations10J7considered hydrogen atoms explicitly, but have been included so that a direct comparison of the united atom approximation can be made with these potentials. The united atom parameters obtained in this study are able to simulate the experimental lattice constants almost as well as the treatment of Momany et al.17 and are better than the calculations presented by Ferro and Hermans for n-pentanelO (Table 11). For n-pentane, n-hexane, and n-octane, all the calculated lattice constants (except the c axis of n-octane) are within 0.25 A of the experimental values. The lattice binding energies obtained for these molecules are actually closer to the value estimated from e ~ p e r i m e n t ~than ~ t , ~the values calculated by Momany et al.,I7 and all are within 1 kcal/mol of the experimental binding energies. The other united atom p o t e n t i a l ~ ~fail J ~ to J ~reproduce ~~~ the experimental results in several cases. The reason for this is that the Len-

The Journal of Physical Chemistry, Vol. 82, No. 24, 1978 2613

United Atom Potentials for Polypeptides and Proteins

TABLE 111: Results of Calculations on Crystals of Several Polar Molecules Using United-Atom Approach

space group

molecule a-glycine p-glycine y-glycine glycylglycine L -alanine

~4hreonine acetyl-L -prolinamide glycyl-L-asparagine succinic acid succinamide a - D -glucose

L-tyrosine ethyl ester IV-acetyl-a-D-glucosamine

exptla calcd exptlb calcd exptlC calcd exptld calcd exptle calcd exptlf calcd exptlg calcd exptlh calcd, exptl' calcd exptlj calcd exptlk calcd exptll calcd exptlm calcd

cell size 2,1,2 2,2,2 2,2,2 2,2,2 2,1,2 1,2,2 2,1,2 3,1,1 2,2,2 2,2,2

1,1,2 2,1,2 1,2,1

a,

a

5.10 5.08 5.08 5.08 7.04 7.09 9.43 9.46 6.03 5.96 13.61 13.48 9.74 9.80 4.81 4.78 5.13 5.09 6.93 6.83 10.36 10.22 12.79 12.71 11.25 11.41

b,

C,

a,

a

a

deg

11.97 12.01 6.27 6.25 7.04 6.89 9.56 9.64 12.34 12.56 7.74 7.87 13.20 13.15 12.85 12.80 8.88 8.33 7.99 8.04 14.84 14.82 16.98 16.95 4.82 4.87

5.46 5.54 5.38 5.57 5.48 5.57 7.83 7.68 5.78 5.85 5.14 5.11 7.17 7.28 13.52 13.37 7.62 7.63 9.88 9.95 4.97 5.34 5.28 5.35 9.72 9.74

90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90

90 90

P, deg 111.7 109.9 113.2 111.7 90 90 124.9 125.5 90 90 90 90 90 90

90 90 133.6 134.6 102.5 108.9 90 90

90 90 113.7 112.7

a Reference 3 9 . . Reference 4 0 , Reference 37. Reference 43. e Reference 47. f Reference 48. Reference 46. Reference 38. J Reference 45. Reference 41. Reference 50. Reference 49.

7,

deg 90 90 90 90 120 120 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 g

lattice binding energy, kcal/mol -27.6 -27.1 -27.2 -40.0 -28.4 -34.2 -21.4 -44.0 -30.0 -23.4 -24.0 -42.69 -34.9

Reference 44.

nard-Jones Rii parameters for these p ~ t e n t i a l s were ~ ~ ~ ~ ~anthracene. '~~~~ Lattice constants for adamantane, l-biadamantane, and obtained from a qualitative inspection of crystal structures. congressane, as measured experimentally and as calculated The parameters of Warme and Scheragalg give lattice with the united-atom potential of this paper, are presented constants for the aliphatic hydrocarbons which are within in Table 11. Although they are constituted from bridged 0.6 b, of the experimental results (Table 11);however, the molecules, these crystals have close to ideal geometry and calculated lattice binding energies are approximately 4 are relatively strain-free. The near-spherical shape of kcal/mol more positive than the corresponding experiadamantane guarantees close contacts of CH and CH2 mental values. Gibson and Scheragag chose a value of Rii groups. Congressane and l-biadamantane form crystals of 3.90 A, which causes deviations of up to 0.9 b, for the with a large number of close contacts and, thus, serve as calculated lattice constants of the aliphatic hydrocarbons a good test of the united-atom approximation, with cal(Table 11). The set of united-atom parameters of Pletnev culated lattice constants within 5% of the corresponding et a1.20 was derived from the early potential function experimental values. described by Scott and Scheraga.l The potentials deThese initial results with hydrocarbons indicate that a scribed by Scott and Scheragal appear to underestimate'l united-atom model is able to give a reasonable reprethe effective sizes of atoms, and this result is reflected in sentation of intermolecular interactions between hydrothe parameters obtained by Pletnev et alaz0which give carbons. In order to test the united-atom parameters on calculated lattice constants which are considerably smaller than the experimental results (see Table 11). polar molecules such as amino acids and carbohydrates, calculations of crystal structures were performed on 13 The united-atom representation of the aromatic CH group, used here, leads to calculated values of the crystal additional molecules (which contain aliphatic CH, CHz, and CH3 and aromatic CH united atoms). The results of lattice constants and energies of benzene and anthracene these calculations are given in Table 111. The calculated which are all within 4% of the corresponding experimental values (Table 11),the largest deviation being 0.25 b,. This lattice constants, using the united atom approximation were usually within 0.2 8, of the experimental value. In is better than the results obtained by Ferro and Hermanslo (e.g., Ac = -0.4 for anthracene) where the hydrogen atoms only two cases was the deviation greater than 0.3 A (the were treated explicitly. Although the potential described calculated length of the c axis of a-D-glucose was 0.37 b, by Momany et al.17 led to calculated lattice constants that larger than e ~ p e r i m e n tand , ~ ~the b axis for succinic acid were within 2% of the experimental values, the calculated was 0.55 b, smaller than e ~ p e r i m e n t ~A~ )comparison . of the calculated results for a-glycine, &glycine, and y-glycine lattice energy for benzene was 15% more positive than the with the corresponding experimental result^^^,^^ shows experimental value. The united-atom parameters of agreement to within 0.15 b, and 2'. The hydrogen bonding Warme and Scheragalg lead to a deviation in one aromatic lattice constant of nearly 0.5 A, and the calculated lattice arrangements used by Momany et al,17J8were used for all energies for benzene and anthracene are 25% more of the crystals studied here (except for a-D-glucose where the neutron diffraction results*l were used). The calcunegative than the experimental values (Table 11). The lations using the united atom potential for a-, p-, and results obtained with the Gibson and Scheragag unitedatom parameters for the aromatic CH group also lead to y-glycine give more negative lattice binding energies than large deviations between the calculated and experimental the full atom potential;18 this reflects the closer fit to the lattice constants and binding energies of benzene and experimental lattice binding energies, obtained by the

2614

The Journal of Physical Chemistry, Vol. 82, No. 24, 1978

united-atom potential for n-pentane and n-octane [i.e., the potential of Momany et al.17 leads to more positive values for the lattice binding energies of aliphatic hydrocarbons (see Table II)]. Also the use of accelerated convergence sums22leads to a y-glycine lattice binding energy of the correct magnitude; while the slow convergence due to hexagonal symmetry led Momany et a1.18 to an incorrect binding energy. This behavior had also been explained by Derissen et al.42as due to the nonvanishing terms in the multipole expansion of noncentrosymmetric crystals. For the three polymorphic forms of glycine, and glycylglycine, the calculated and e ~ p e r i m e n t a lattice l ~ ~ constants agree within 3%. A residue in which the methylene group is particularly important is proline, Le., the methylene groups of the pyrrolidine ring are very important (stereochemically) in protein structures; the interactions of these methylene groups are accounted for correctly in UNICEPP since the calculated lattice constants of crystalline acetyl proline amide are within 3% of the experimental ~ a l u e s . 4 ~ Some of the specific directionality of the protons of the methylene groups is lost by using the united-atom approximation. This is evident from a comparison of the united-atom results for succinic acid and succinamide with the full-atom results17 for these molecules. With the full-atom potentials, all of the calculated lattice constanW for these molecules are within 3% of the experimental value^;^^,^^ however, with the united atom potential, the discrepancy was within 2% for 6 of the lattice constants but within 6% for the other two lattice constants. Thus, there is a small but detectable difference in the directionality between the united-atom potential and the full-atom potentials. When there are strong hydrogen bonding networks within the crystal such as glycyl-Lasparagine,& the united-atom potential yields calculated lattice constants which are almost identical with the full-atom calculations. However, the lattice energy tends to be approximately 10% more negative for the unitedatom calculations than for the full-atom results, as a result of the improved fit to the lattice energies of pentane and octane. Two crystals where the methyl group is important for the molecular packing are L-alanine and L-threonine. The united-atom parameters give calculated cell lattice constants for these two molecules which are only 1% different from the experimental value^.^'^^^ The smaller methyl radius (4.25 A) appears to be an effective representation for this group. The amino acid crystals considered above do not provide a stringent test of the aliphatic CH united atom because it is not involved in close contacts with atoms on other molecules within the crystal. On the other hand, a-D-glucose has five aliphatic CH groups, and in its crystal lattice there are several contacts involving these groups. There is good agreement (-3%) between the calculated and observed lattice constants for C Y - D - ~ ~ U C Oindicating S~,~~ that the radius assigned to the aliphatic CH group is a reasonable value. N-Acetyl-a-D-glucosamine has both the geometric feature of a ring of united groups and a bulky polar side chain. The united-atom potential handles these different interactions well, with agreement of the calculated lattice constants to within 1%of the experimental values.49 The aromatic united atom (CH) parameters also give good results of the crystal calculations on L-tyrosine ethyl ester. The average deviation for the three calculated lattice constants from their experimental values50 is 1%. B. Intramolecular United-Atom Parameters. If we consider the amino acid residues treated in ECEPP,13 together with the N-acetyl, ”-methyl, NH2, and COOH end

L. G. Dunfield, A. W. Burgess, and H. A. Scheraga T

A

i

0

d

60

I20

180

60

120

180

i

Flgure 2. The $,$ conformational energy surface for N-acetylN’-methylglycinamide as Calculated by the ECEFP’~ and UNICEPP potentials:

(A) lowest energy contour 1 kcal/mol, contour levels 2 kcal/mol for ECEPP; (6) same as for A, but using UNICEPP.

groups, there are over 157 different types of single bonds and 126 paired bonds for which united-atom Fourier series have to be developed. These include the rotations about the N-C* bond ($), C“-C’ bond ($), and C’-N bond ( 0 ) which make up the polypeptide backbone as well as the rotations around the different side-chain bonds (x’s). The coefficients for the terms of all Fourier approximations to the local energy surfaces for each of these bonds are listed in the input data for the Fortran programs developed to use these potentials for calculations on proteins.21 The resulting Fourier series agree to within 0.08 kcal with the local energy surfaces calculated from the ECEPP ~ o t e n t i a 1 . lIt~ is these local energy surfaces that are the most important factor in determining the location of local conformational minima in all but the most sterically crowded minima. Thus, the representation of the energy surfaces by accurate Fourier series results in very good agreement between the united-atom potential presented here and the full-atom potential,13provided that there are no van der Waals overlaps. C. Comparison of Full-Atom13 and United-Atom Amino Acid Residue Energy Surfaces. The united atom approximation appears to fit both the intermolecular interaction properties (see section A of Results) and the local interactions around each single bond (see section B of Results). A further important test is to compare the results of the ECEPP and UNICEPP potentials for different molecules. Several examples of two-dimensional energy surfaces of the N-acetyl-N’-methylamidesof single residues and dipeptides have been chosen to illustrate different interatomic interactions. These surfaces are displayed as two-dimensional contour plots so that the characteristics of the surfaces can be appreciated visually. $,$ maps for N-acetyl-N’-methylglycinamide, computed with both potentials, are given in Figure 2. The boundaries of the low energy regions are almost identical for both potentials, as are the positions and relative energies of the local minima (Table IV). The two-dimensional contour plots for the ECEPP and UNICEPP potentials of this molecule suggest that the shapes of the energy surfaces in the low energy regions of the $,$ surface are very similar. The results for (p,$ energy maps of Nacetyl-N’-methylalaninamide using both potentials are displayed in Figure 3. The characteristics of both the full atom and united atom $,J/maps for the alanine residue are almost identical; the only differences are in the higher energy contours. The positions of the minima (Table IV) and their relative energies indicate that the united-atom approximation is a sufficiently accurate representation of the ECEPP p0tentia1.l~ A larger molecule was used to investigate an energy surface involving proline residues, viz., N-acety1-N’methylprolylprolinamide. When the two dihedral angles

The Journal of Physical Chemistry, Vol. 82,No.

United Atom Potentials for Polypeptides and Proteins

24, 1978 2615

TABLE IV: Positions of Minima on Energy Surfaces for ECEPPI3 (I) and UNICEPP (11) Empirical Potential Energies position of local min, deg molecule

I

I1

I

I1

-83,76 -180,180 -173, 62 -72, -53 -83,79 -154,153 -150, 72 -74, -45 -158, -58 54,57 -172,179 -172,64 -66,179 -66,64 167,171, 78 164, -7,156

-83,76 -180,180 -174,60 -71, -52 .-83, 8 1 -152,147 -143, 82 -72, -44 -157, -58 55,57 -172,179 -172, 64 -67, 179 -67, 64 157, 169, 79 152, -6, 137

0.0 0.8 1.0 1.2 0.0 0.4 0.7 1.1 1.6 2.3 0.0 0.2 0.2 0.4 0.0 2.4'

0.0 0.9 1.1 1.2 0.0 0.7 1.0 1.2 2.0 3.5 0.0 0.1 0.1 0.2 0.0 2. 5'

variables

N-acetyl-N'-methylglycinamide

fb, i a

N-acetyl-N'-methylalaninamide

fb, i

N-ace tyl-N' -methyllysinamide

XI,

x4

-85, )I = 85, x z = x 3 = 180, x 5 = 60 @ =

N-acetyl-N'-methylprolylprolinamide

$

,, w ,, $

conformational energy,b kcal/mol

a The @,I) map is symmetrical for a glycine residue, and only half the minima are reported here. Conformational energy for each molecule is expressed relative t o the lowest energy conformation. This energy difference between two dissimilar conformations should not be confused with the energy difference of the cis-trans isomers of N-acetyl-W-methylprolinamide.51

TABLE V: Comparison of Number of Dihedral Angles and Pairwise Interactions for ECEPPI3 (I) and UNICEPP (11) Potential Energy Calculations no. of molecule

no. of amino acid residues

I

1 1 54 97 129 2 12

19 22 802 1398 1951 3 241

N-acetyl-W-methylglycinamide N-ace tyl-N'-methylalaninamide rubredoxin a-chymotrypsin (C chain) lysozyme papain

t

-,eF-%==% 6 ? : zc

i

atoms

I1

,,~~, F?,

!c

,

-180

-120

-60

0

60

pairwise interactions

I20

11 12 506 866 1255 2 039

j 180

4

4,$ conformational energy surface for N-acetylN'-methylalaninamide as calculated by the ECEPP'~ and UNICEPP potentials: (A) lowest energy contour 1 kcal/mol, contour levels 2 kcal/mol for ECEPP; (B) same as for A, but using UNICEPP. Flgure 3. The

between the proline residues (q1 and wl) are varied, interactions occur between the methylene groups of the two proline residues. The united-atom potential represents the &,wl energy surface (not shown here) of this proline dipeptide adequately, and some minima on the energy surfaces for the two potentials are listed in Table IV. N-Acetyl-N'-methyllysinamide served for an investigation of the effectiveness of the united-atom approximation for representing the interaction of polar groups separated by a long hydrocarbon chain. The ECEPP and UNICEPP results agree well as regards the location and energy of the local minima of the most stable conformers. The magnitude of the error in the total energy resulting from the united-atom approximation was determined by comparing the ECEPP and UNICEPP potentials for all of the residues discussed by Momany et and was found to be less than 0.5 kcal on the average, provided that no bad steric overlaps occurred.

dihedral angles

I

I1

I

I1

123 174 318 427 971 975 1895 964 5 239 535

20 24 125 493 370 677 781 329 2 068 563

6 7 298 532 749 1203

4 4 278 473 688 1099

Thus, the united-atom potential reliably represents the interactions that are present in the low-energy conformations of amino acids. The agreement with the previously published potential13 for polypeptides is excellent, but since fewer pairwise atomic interactions are involved, the united atom potential presented in this work is a more efficient algorithm. D. Applicability of United-Atom Potentials to Conformational Energy Calculations on Polypeptides and Proteins. It was stated earlier that the united-atom potentials were developed because of the need to simplify conformational energy calculations (Le., to reduce the required computer time) on protein molecules. This problem arises because there is a large number of bonds about which rotation can take place, and a large number of pair interactions which have to be considered in order to calculate the energy of a single conformation. As the number of variables increases, it becomes more and more expensive to select a reasonable number of starting points and to carry out the necessary energy calculations to find the local minimum nearest to each starting point. Although empirical conforrhational energy calculations use simple analytical functions to evaluate the interatomic interactions, in molecules as large as proteins there are so many of these terms to evaluate that it becomes increasingly difficult to estimate the conformational energy in a reasonable amount of computer time. A comparison of the number of single bonds (about which rotation can take place) and the total number of pairwise interactions which need to be evaluated when calculating the conformational energy is given for several molecules in Table V. The data in Table V can be used to calculate the ratio

2616

The Journal of Physical Chemistry, Vol. 82, No. 24, 1978

of the number of pairwise interactions in a UNICEPP computation to that number in an ECEPP computation. The ratio is close to 0.40 for all of the molecules in Table V, with the exception of the first two, which contain only one full amino acid residue. A was made of the computer execution time requirements of the two types of computation, using program UNICEPP~~ for “unitedatom,” and ECEPP13 for “full-atom’’ calculations. In both cases, it was found that the time required for the computation of the conformational energy of a polypeptide from the generated atomic coordinates was consumed mainly by the subprogram which computes the electrostatic and nonbonded interaction energies for each pairwise interaction. It would be expected that the ratio of times for energy computation by the two programs would be close to the ratio of the number of pairwise interactions, and this appears to be the case. For the sequence of bovine pancreatic trypsin inhibitor (55 amino acid residues), the ratio of pairwise interactions is 0.40, and the ratio of energy computation times (UNICEPP to ECEPP) was found to be 0.42. For this protein, UNICEPP is faster than ECEPP in energy computation by a factor of approximately 2.5. The same factor of rate improvement was found for a molecule with three amino acid residues, and is expected to be applicable to proteins in general, independent of chain, length. In the case of proteins, it is not possible to locate the positions of protons in a molecular structure determination using X-ray crystallography. Thus, an energy algorithm which introduces as few protons as possible can be more easily applied directly to protein structures from X-ray crystallography. At the present time, these united-atom potentials have been used for two main purposes: (i) to generate polypeptide chains using a standard geometry13and to estimate the energy of a variety of conformations by minimization with respect to the dihedral angles for rotation about single bonds, and (ii) to investigate the conformational energy surface of the molecular structure reported from experiment. Computer programs, based on those used for the full-atom potentials,13 have been developed for both of these purposes. It is possible to use these programs to adjust the positions of residues in a protein so as to minimize their conformational energy; computations with a small protein (bovine pancreatic trypsin inhibitor) indicate that these techniques are valuable tools for directing the refinement of protein coordinates with respect to their stereochemical characteristic^.^^ The united atom potential also appears to be a useful tool for theoretical studies on the folding of polypeptide chains from possible starting conformations (such as those which might arise from empirical predictive algorithms) toward a compact globular structure.

Acknowledgment. The authors are grateful for the help and suggestions provided by Drs. G. I?. Endres, M. R. Pincus, and R. A. Nemenoff. Note Added in Proof: The ECEPP parameters of the nonbonded potential for aromatic carbon atoms have been revised recently,” providing better agreement between the calculated and observed lattice energies of benzene and anthracene.

References and Notes (1) This wok was supported by research grants from the National Science Foundation (PCM75-08691) and from the National Institute of General Medicial Sciences of the National Institutes of Health, U S . Public Health Service (GM-14312). (2) (a) National Research Council of Canada Postdoctoral Fellow,

L. G. Dunfield, A. W. Burgess, and H. A. Scheraga 1976-1977; (b) Cornell University; (c) Weizmann Institute: (d) Recipient of an Australian American Education Foundation Travel Grant, 1972-1974; (e) To whom requests for reprints should be addressed at Cornell University. (3) G. N. Ramachandran, C. Ramakrishnan, and V. Sasisekharan, J . Mol. Biol., 7, 95 (1963). (4) G. Ndmethy and H. A. Scheraga, Biopolymers, 3, 155 (1965). (5) D. A. Brant and P. J. Flory, J. Am. Chem. Soc., 87, 663, 2791 (1965). (6) S. J. Leach, G. Ndmethy, and H. A. Scheraga, Biopolymers, 4, 369 (1966). (7) R. A. Scott and H. A. Scheraga, J . Chem. fhys., 45, 2091 (1966). (8) T. Ooi, R. A. Scott, G. Vanderkooi, and H. A. Scheraga, J. Chem. fhys., 46, 4410 (1967). (9) K. D. Gibson and H. A. Scheraga, froc. Natl. Acad. Sci. U.S.A ., 58, 420 (1967). (10) D. R. Ferro and J. Hermans,Jr., in “Liquid Crystals and Ordered Fluids”, J. F. Johnson and R. S. Porter, Ed., Plenum Press, New York, 1970, p 259. (11) A. I.Kitaigorodskii, “Advances in Structure Research by Diffraction Methods”, Brunswick, Vol. 3, R. Brill and R. Mason, Ed., Braunschweig, Vieweg, 1970, p 173. (12) A. T. Hagler, E. Huler, and S. Lifson, J. Am. Chem. Soc., 96, 5319 (1974). (13) F. A. Momany, R. F. McGuire, A. W. Burgess, and H. A. Scheraga, J. feys. Chem., 79, 2361 (1975). (14) D. E. Williams, J . Chem. Phys., 45, 3770 (1966). (15) D. E. Williams, J . Chem. Phys., 47, 4680 (1967). (16) H. A. Scheraga, in Nobel Symposium on “Symmetry and Function of Biological Systems at the Macromolecular Level”, A. Engstrom and B. Strandberg, Ed., Almqvist and Wiksell, Stockholm, 1969, p 43. (17) F. A. Momany, L. M. Carruthers, R. F. McGuire, and H. A. Scheraga, J . fhys. Chem., 78 1595 (1974). (18) F. A. Momany, L. M. Carruthers, and H. A. Scheraga, J. fhys. Chem., 78, 1621 (1974). (19) P. K. Warme and H. A. Scheraga, J. Comput. Phys., 12, 49 (1973). (20) V. Z. Pletnev, E. M. Popov, and F. A. Kadymova, Theor. Chim. Acta, 35, 93 (1974). (21) This computer program is available on magnetic tape from the Quantum Chemistry Program Exchange. Write to Quantum Chemistry Program Exchange, Chemistry Department, Room 204, Indiana University, Bloomington, Ind. 47401 for standard program request sheets, and then order No. QCPE 361. (22) D. E. Williams, Acta Crystallogr., Sect. A , 27, 452 (1971). (23) M. J. D. Powell, Comput. J., 7, 155 (1964). (24) N. Norman and H. Mathisen, Acta Chem. Scand., 18, 353 (1964). (25) N. Norman and H. Mathisen, Acta Chem. Scand., 15, 1747 (1961). (26) L. L. Shipman, A. W. Burgess, and H. A. Scheraga, J . fhys. Chem., 80, 52 (1976). (27) N. Norman and H. Mathisen, Acta Chem. Scand., 15, 1755 (1961). (28) W. Nowacki, Helv. Chem. Acta, 28, 1233 (1945). (29) R. A. Alden, J. Kraut, and T. G. Taylor, J . Am. Chem. SOC.,90, 74 (1968). (30) I.L. Karle and J. Karle, J . Am. Chem. Soc., 87, 918 (1965). (31) G. E. Bacon, N. A. Curry, and S. A. Wilson, Roc. R . Soc. London, Ser. A, 279, 98 (1964). (32) A. Bondi, J . Chem. Eng. Data, 8, 371 (1963). (33) G. Milazzo, Ann. Chim. (Rome), 46, 1105 (1956). (34) R. Mason, Acta Crystallogr., 17, 547 (1964). (35) There is no intrinsic torsional potential for variation of 4 or In ECEPP.” (36) A. Warshel and S. Lifson, J. Chem. fhys., 53, 582 (1970). (37) Y. Iitaka, froc. Jpn. Acad., 30, 109 (1954). (38) J. S. Broadlev. D.W. J. Cruickshank. J. D. Morrison. J. M. Robertson. F. R. S. Shearer, and H. M. M. Shearer, froc. R. SOC.London, Ser. A, 251, 441 (1959). (39) R. E. Marsh, Acta Crystallogr., 11, 654 (1958). (40) Y. Iltaka, Acta Crystallogr., 13, 35 (1960). (41) G. M. Brown and H. A. Levv. Science. 147, 1038 (1965). (42) J. L. Derissen, P. H. Smit, and J. Voogd, J . fhys. Chem., E l , 1474 ( 1977). (43) H. C. Freeman, G. L. Paul, and T, M. Sabine, Acta Crystallogr., Sect. B , 26, 925 (1970). (44) T. Matsuzaki and Y. Iitaka, Acta Ctystallogr., Sect. B , 27, 507 (1971). (45) D.R. Davies and R. A. Pasternak, Acta Crystallogr., 9, 334 (1956). (46) R. A. Pasternak, L. Katz, and R. B. Corey, Acta Crystallogr., 7, 225 (1954). (47) H. J. Simpson, Jr., and R. E. Marsh, Acta Crysfalbgr., 20, 550 (1966). (46) D. P. Shoemaker, J. Donohue, V. Schomaker, and R. B. Corey, J. Am. Chem. Soc., 72, 2328 (1950). (49) L. N. Johnson, Acta Crystallogr., 21, 885 (1966). (50) A. F. Pieret, F. Durant, M. Griffe, G. Germain, and T. Debaerdemaeker, Acta Crystallogr., Sect. 6, 26, 2117 (1970). (51) S. S. Zimmerman and H. A. Scheraga, Macromolecules, 9, 408 (1976). (52) G. F. Endres, unpublished results in this laboratory. (53) M. K. Swenson, A. W. Burgess, and H. A. Scheraga in Symposium on Frontiers in Physico-Chemical Biology”, B. Pullman, Ed., Paris, 1977, in press. (54) D. J. Sandman, A. J. Epstein, J. S. Chickos, J. Ketchum, J. S. Fu, and H. A. Scheraga, J . Chem. fhys., in press.