Effect of Twisting a Polypeptide on Its Geometry and Electron

Shanish Kumar , Ganesan Mani , Debodyuti Dutta , and Sabyashachi Mishra ... R. Parthasarathi, S. Sundar Raman, V. Subramanian, and T. Ramasami...
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J . Phys. Chem. 1994,98.4413481

4473

Effect of Twisting a Polypeptide on Its Geometry and Electron Distribution P. L. A. Popelier and R. F. W. Bader' Department of Chemistry, McMaster University, Hamilton, Ontario U S 4 M l . Canada Received: November 30, 1993; In Final Form: February 18, 1994"

The geometry of N-formyltriglycine amide, which possess four amidic bonds, is completely optimized at the Hartree-Fock level using the 6-31+G8 basis, starting with a set of geometrical parameters characteristic of the 3.61, a-helix. Optimization results in the partial unfolding of this geometry to one characteristic of the 310 helix. The effect of allowing for nonplanarity of the amide bonds is separately determined in arriving at the final geometry. A hydrogen-bonded a-helix structure is obtained by optimizing the bond lengths and angles of the starting structure with fixed torsion angles. The existence of the hydrogen bond in this molecule and in Cs and C7 structures of blocked N-formylglycine amide is confirmed by the presence of a corresponding bond path in the electron density p, and their properties are characterized in terms of p at the associated bond and ring critical points. The changes in the charge distribution caused by the twisting of the peptide chain are summarized in terms of the atomic properties and properties of the bond critical points. It is shown that charge neutrality of a peptide group, a necessary requirement for its transferability, is preserved in the helical structures. The study indicates that it should be possible to assign characteristic properties to the atoms of a peptide chain by taking into account their dependence on the torsion angles and nonplanarity of the amide bond.

Introduction The possibility of using peptide units defined by the theory of atoms in molecules' to construct a polypeptide or a particular portion of it was recently described.z The manner and reasons underlyingthisapproachcan beillustrated in termsoftheserinyl group illustrated in Figure 1. This group is obtained by separating the hasins of neighboring N and C atoms at their common interatomic surfaces in the two amide bonds that link the group to a peptide chain. The group is represented symbolically by INHCH(CH*OH)C(==O)I, the vertical bars denoting the interatomic surfaces, and graphically by showing the two interatomic surfaces and their intersection with the 0.001 au envelope of the electrondensity.p(r). Thisenvelopedetermins thevander Waals shape of the group and is used to replace those portions of the atomic surfaces that occur infinitely far from the nuclei. The interatomicsurfacesare themselvesdefined in termsofa property of p(r), and thus a group is represented by its distribution of charge in real space, that is, as a bounded space filling object. Suchgroupsaretobelinked together by matchingtheappropriate interatomic surfaces, just as corresponding groups are linked together in an experimental synthesis. Aside from curved, as opposed to flat surfaces between the atoms, this representation of a group bears a striking resemblance in form and function to the space-filling models employed by chemists? Peptide groups serve as building blocks in models used to understand the structure and propcrties of proteins, a model that tacitly assumes the transferability of the individual residues. The electrostatic field generated by the amino acid residues is used in theconstruction of force fieldsin molecular mechanics programs for modeling protein structures and for the determination of electrostatic maps for studies of their solvation and reactivity. Much effort has been expended in obtaining atomic charges and moments for amino acid residues. In most of these studies the moments havebeen arbitrarilydefined in termsofbasis functions used in theexpansionofthemolecularorbitals.c'2 Thedistributed multiple analysis (DMA) of Stonel) has been used to define moments of all naturally occurring amino acids14 and to study their transferahility.'5,'6 Price and Stonet6used blocked alanine and diglycine residues to study the effect of changes in torsion angles @ and $about the N-C, and C,-C' axes respectively, on *Abstract published in Adoonce ACS Abslrocls, April I, 1994.

0022-3654/94/2098-4413$04.50/0

Figurel. SerinylgrouplNHCH(CH2OH)C(=O)Iasiepresentedby the intcrsection of its van der Waals surface with the -C(=O)l interatomic surface on the top left and the INH- surface in ccnter at bottom. The interatomic surfacts are characteristic of the amidic surface found in a polypeptide.

the computed moments, all other geometrical parameters being held fixed. They found a marked dependence of the DMA's on the torsion angles, but in agreement with the results obtained by Chang and Bader,2 a high degree of transferability was found from the single to the dipeptide residue when the torsion angles are held fixed. It was demonstrated early on1' that the multiple moments up to and including the quadrupole, determined by the charge distributionsofthe individualatoms in a molecule, provided a fast converging expansion to the electrostatic fields used in the prediction of the relative approach geometries of two interacting 0 1994 American Chemical Society

4474 The Journal of Physical Chemistry, Vol. 98, No. 16, 1994

molecules. The advantage of using atomic moments defined by theory is their high degree of transferability. Thus, it is possible to construct the electrostatic fields generated by large molecules such as the nucleic acid bases by expressing them in terms of the characteristic moments of the appropriate groups taken from a standard bank of “transferable atom equivalents”..’* The remarkable similarity in the electrostatic potential fields generated by the glycyl group in N-formylglycinamideand the central such group in triglycyl and by the carboxylic acid group in di- and triglycine has been dem0n~trated.l~ Bader and Chang2 studied the transferability of geometrical parameters, of atomic moments, energies, and volumes, and of the critical point data needed to summarize the distribution of the electron density and its associated Laplacian V2p(r) for the glycine and alanyl residues in systems with geometries optimized at the 4-3 1G level, followed by single-point calculations using the 4-3 1G**set. The groups studied were the singly blocked terminal glycyl and alanine groups with -NHI blocked with formyl and -O=)CJ blocked with amino, the doubly blocked or internally bound glycyl group G” INHCHzC(=O)(, all dipeptide combinations of these groups, and the tripeptide, triglycyl. The torsion angles 4, JI, and w, the latter being the torsion angle for the amidic bond, all equal 180° in the di- and tripeptides and deviate by less than a degree from these values in the mixed dipeptides formed from glycine and alanine. The initial studies were thus precluded from considering the consequences of the twistingof the peptidechain on the transferabilityof the properties of the peptide groups. This is the primary purpose of the present work, which reports on the geometriesand propertiesof the blocked triglycyl peptide, HC(=O)JG”IG”IG”(NHz (I). With four amide bonds, this molecule is the simplest of the polypeptides capable of possessing a 13-membered hydrogen bond and of exhibiting the 3.613 structure of the a-helix. Transferability of atomic properties requires transferability of the related geometrical parameters. Thus, is is essential to determine whether or not these parameters are indeed transferable from one system to another, requiring complete geometry optimizations of all the systems studied. The average change in bond length and bond angle among the various combinations of the residues in the planar peptides studied by Bader and ChangZ was found to be 0.001 A and 0.17O, with maximum deviations of 0.004 A and O S 0 , respectively. An illustration of the transferability of bond lengths and of the associated atomic property is provided by the lengths predicted for the amide bonds formed by linking the complementary -HNI and I C ( 4 ) interatomic surfaces. The resulting length is given by the sum of the bonded radii of the N and C atoms, rb(N) and b(C), the distance of a nucleus from the bond critical point. The error in the predicted lengths of the amide bond in the synthesized planar di- and tripeptides was 0.002 A or less with the single exception of G”INHCHzCOzH, for which it equaled 0.005 A. The properties of a bond are succinctly summarized in terms of the values of pb, vzpband e, the electron density, its Laplacian, and the bond ellipticity, respectively,evaluated at the bond critical point, the point where p(r) attains its minimum value along the associated bond path. These values were found to change by such small amounts for a particular bond in a given group throughout the series of molecules,2as to demonstrate that the bonds and their associated charge distributions are transferable from one peptide interactionto another without significant change in their physical or chemical characteristics. The transferability of thegeometrical parameters and of the critical point data implies a corresponding transferability of the distribution of electronic charge over each atomic basin, up to their interatomic surfaces. Thus the atomic properties, including their multipole moments, energies, and volumes, exhibit relatively small changes within the set of molecules for a given atom in a given functional group. For example, the average change in a population is 0.OOSe. Even

Popelier and Bader the atomic energies, which vary from 23 X lo3for C to 76 X 103 kcal mol-’ for 0,change on average by only 3 kcal mol-’ (0.01%) or less on transfer. The largest variation in bond and atomic properties are found for the N and C atoms of the peptide bonds, as these are the atoms whose immediate neighbors change on transfer of the group. Equally transferable are the shapes of the groups and their associated volumes, as determined by the 0.001 au density envelope. The local charge concentrations and depletions defined by the topology of Vzp(r) correctly predict the sites of electrophilic and nucleophilic attack, respectively.’ Just as molecular similarityis defined and determined by a comparison of the properties of the atoms, as defined by the topology of p(r), molecular complementarity is contained within the topology of VZp(r). Complementarity is achieved by complementingthe local charge concentrations and depletions on one molecule with the respective local charge depletionsand concentrationson the other, the positions of the maxima and minima being determined by corresponding critical points in Vzp. This information can be displayed visually in terms of a molecule’s *reactive surface”, the surfacedefmed by the zero-valued envelopeof Vzpwhich separates the regions of charge concentration from those of depletion. Such surfaces have been illustrated for creatine and its analogue inhibitor ~arbamoylsarcosine.’~ One finds2that the positions as well as the magnitudes of Vzp at the relevant critical points in the peptide atoms change by only small amounts when a group is transferred from a doubly blocked peptide group to di- and tripeptides. Thus, it has been demonstrated that groups, like tho serinyl group shown in Figure 1, can be linked at the C-N interatomic surface of their amide bonds to provide a prediction of the properties of any portion of a planar polypeptide. The linking is accomplishedZby aligning, in an antiparallel manner, the bond path vector originating at the critical point in the amidic surface ofthe Catomand directed at theC nucleus with thecorresponding vector originating in the amidic surface of the N atom. As discussed by Chang and BaderY2 this results in a matching of the two surfaces and of the density contours across the single resulting surface out to values of p = 0.02 au. Ignoring the small errors in p in the outer reaches of the amidic surface, one can in this manner construct visual displays of the molecule’s van der Waals shape, of its electrostaticfield, and of its reactive surfaceas defined by the zero envelope of the Laplacian of the electron density. Constructing a polypeptide in the manner enables one to predict all of its properties for a given conformation, for all properties are defined for each individual group. The dependence of the group properties upon the torsion angles 4. +, and w is the focus of this study. Charge neutrality for a peptide group used as a building block in the constructionof a polypeptide is a necessityif one is to avoid an accompanying buildup of charge. This property was shown to be exhibited by theglycylgroup G” in its planar conformations.2 The present study demonstrates that charge neutrality is independent of the torsion angles. Another way of viewing the requirement of charge neutrality is through the use of Gauss’ theorem which states that the net charge of a group equals the flux in the electrostatic field through its interatomic surfaces. Hence, the requirement of charge neutrality for a peptide group insures that the flux of the electric field out of the -CHR-HN( surface of one group will equal the flux into the IC(=O)-CHRsurface of the group it is linked to. Thus, the two surfaces will be electrostaticallymatched to yield a -HN(C(=O)- interatomic surface with a zero net flux in the electrostatic field, thereby resulting in the good degree of matching noted above. The geometry of the a-helix is critically dependent upon the presence of hydrogen bonds forming 13-memberedrings, and the presence of hydrogen bonding in five- and seven-memberedrings of single peptides has been surmised. The presence of a bond path linking two nuclei, as defined by a trajectory of Vp, provides

Geometry of N-Formyltriglycine Amide

The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 4415

the necessary and sufficient conditions for two atoms be bonded toone another.’ Thus, the topology of theelectron densityenables one to unambiguously determine the presence of a hydrogen bond and to quantify the resulting interaction. A bond defined in this manner is subject to experimental verification. Bond paths with the characteristics anticipated for a hydrogen bond have been observed in the experimetnally determined charge distributions of crystalline amino acids.20 The presence of hydrogen bonds in the C5 and C7 structures of blocked glycine and in the a-helix conformation of blocked triglycine is demonstrated, and their properties are determined.

Geometry Optimizations SCF Cdculations. In recent years, ab initio calculations have been performed on blocked single amino residues which serve as models of d i p e p t i d e ~ ~ . ~and l - ~on ~ o l i g o p e p t i d e ~ . ~ -These ~~-~~ calculations have served to determine the geometries of the most stable forms and the degree of transferability of the geometrical parameters from a blocked monopeptide to a dipeptide and on to peptides of increasing length, the blocked forms of the pentapeptides of glycine and alanine having been recently in~estigated.~~ All molecules were fully optimized at the 6-31+G* level using Gaussian 90,31 the choice of the basis set being determined in a benchmark paper by Head-Gordon et a1.22 The minimum-energy geometry of N-formylglycine amide (I) was determined together with the minimum-energy geometries corresponding to the C5 and C7 hydrogen-bonded forms of N-formylglycine amide, HC(=O)~NHCHZC(==O)INHZ,structures II and III, respectively, and the transition state for their interconversion,N.The presence of a transition state was established by computing the Hessian for the energy, as obtained from analytical second-derivative calculations to ensure the existence of a single imaginary frequency. The results for N-formylglycine amide agree with those previously obtained by Head-Gordon et a1.22 The charge density and atomic properties were obtained from singledeterminant SCF wave functions using 6-31 l++G**.32 This basis set with the extra set of diffuse functions has been used to prove the existence of an intramolecular C-H-O hydrogen bond in creatine and carbamoyl~arcosine.~~ The combined level of theory for all calculations is denoted 6-311++G**//6-31+G*. Direct SCF was used in all cases.34 The geometries were determined by first optimizing using 3-21G, the planarity constraint on the peptide groups being removed in the final stage using the 6-31+G* basis. Thus, the effects of assuming planar peptide groups on the energy, bond lengths, and angles and the atomic properties have been separately determined for each molecule. While the relaxation of this constraint is found to have a minimal effect on the energy, the atomic properties are affected because of the small accompanying rearrangement of the electron density. The geometries of all structures are given in Table 1. Geometries and Structures of N-Formylglycine Amide. The global energy minimum for blocked glycine II is the planar molecule with qj = fi = 180O. The other minimum-energy geometry of almost equal energy, hE = 0.83 kcal mol-’, is the twisted structure Ill (Table 2). These molecules are labeled C5 and C7, respectively, because of the presumed presence of hydrogen bonds forming a five- and seven-membered ring, respectively. A single transition-state structure IV,hE = 2.4 kcal mol-’ relative to Cs, links II and HI on a relatively flat potential energy surface. The deviations from peptide planarity are small for III but quite pronounced for the transition-state structure N,but the corresponding changes in energy relative to the related planar structure are remarkably small in all cases (Table 2). Our results for the relative energies of III and IV are in agreement with those reported by Head-Gordon et aLZ2for blocked glycine using the 6-31 +G* basis. Peters and Peters have used the minimal STO-3G basis to map out regions of the potential energy surfaces for di-, tri-, and

TABLE 1: Comparison of Bond Lengths (A) Anglea (deg) between Six Glycyl Residues’ C5 C7 Cs-77 G”i INHCH2C(O=)I C=O 1.2014 1.2020 1.2006 1.2039 C,-C 1.5189 1.5277 1.5263 1.5224 CcN 1.4358 1.4511 1.4430 1.4509 N-H 0.9973 0.9954 0.9959 0.9962 C,-H 1.0852 1.0822 1.0790 1.0813 C,-H’b 1.0852 1.0787 1.0824 1.0823 NC,C 109.43 113.25 110.49 115.59 C,CO 122.15 120.96 120.53 119.13 CNC, 121.69 123.21 120.99 121.01 C,CN 114.92 115.39 116.02 117.85 HNC, 117.29 118.39 117.73 118.07 HNC 121.02 118.39 117.97 117.33 NC,H 110.75 108.58 108.55 111.09 NC,H’ 110.75 108.65 108.80 108.22 NC,CO 0 -113.9 -55.9 159.9 HNC,C 0 93.2 73.8 133.2

+4

180 180

-85.3 66.5

NIC C=O H-C OCN HCN

1.3415 1.2001 1.0888 124.52 113.29

1.3430 1.2017 1.0879 124.54 113.49

ClN N-H N-H” CNH CNH’

1.3479 1.3454 0.9959 0.9957 0.9942 0.9970 118.73 118.57 122.68 121.86

-85.3 126.7

-68.7 -20.9

and Bond G”2

G”3

1.2025 1.5269 1.4459 0.9935 1.0800 1.0818 116.43 118.20 121.71 117.77 119.31 118.80 110.17 108.47 170.5 104.3 -70.9 -9.8

1.2038 1.5252 1.4435 0.9967 1.0795 1.0818 116.02 118.90 122.37 117.84 117.75 119.82 109.33 109.40 183.2 82.5 -94.8 2.4

N-Terminus 1.3497 1.1987 1.0883 124.61 113.27

C-Terminus 1.3514 0.9973 0.9954 117.04 120.15

1.3527 1.1980 1.0878 123.78 113.91

* * * 118.80 *

1.3439

1.3459

* *

* *

123.02

123.78

*

1.3438 0.9955 0.9957 123.44 117.66

*

* * 119.82 *

“Results in all tables obtained from 6311++G**//631+G* optimized calculations. The prime denotes the H trans to keto oxygen of amide group. The prime denotes the H cis to keto oxygen of amide group.

TABLE 2: Total Energies of Completely Optimized and Planar Optimized Geometries and Deviation from Planarity o CS c 7 CS-7’ helii E(optimized)

*

-375.86275 -375.86034 -789.61011 -789.60861 -0.94

E(p1anar) -375.86393 -375.86261 -375.86004 AE(kcal/mol) 0 0.09 0.19 w(O=C-N-Ht) (deg) N-terminus 180 184.2 193.72 C-terminus 180 179.9 160.65 * * Glyi-Glyz * GlytGlys

*

*

172.2 183.8 182.1 178.0

tetrapeptides, inferring the presence of intramolecular hydrogenbonded interaction^.^^ Using geometrical and energetic criteria, they argued for the presence of a hydrogen bond in C, but not in Cs. The ability to characterize the presence and properties of a hydrogen bond in terms of the topology of the electron density has been established in many systems, both strongly and weakly interacting, using both c a l ~ u l a t e d ~and ~ ~experimenta120 ~~~~”~~ .density distributions. No bond path linking the H of N-H with the keto oxygen was found within the glycyl group for the planar geometry II, and no five-membered ring structure is present. However, a slight distortion of the equilibrium geometry, increasing the energy by only 0.9 kcal mol-’, resulted in the formation of an atomic interaction line to yield the Cs structure shown in Figure 2. The distance between 0 and H was reduced by 0.12 A, and the C,CO and C,NH valence angles decreased by 2 O and 3O, respectively, to yield an N-H-O angle of 108.S0. The forces created by moving 0 and H closer to each other on these two internal coordinates are quite small, equaling 0.036 and 0.016 hartree/rad, respectively. The H-N-O angle is far from linear, the five-membered ring is strained, and the ring structure is unstable. The small elongation of the N-O separation required to take the geometry back to that

The Journal of Physical Chemistry, Vol. 98. No. 16, I994

4476

Popelier and Bader

Figure 2 Portrayal of the gradient v e ~ t o rfield of the electron density. in terms of the trajenorics of Vp, for the plane of the heavy nuclei of the glycylgroupintheCrstructureofN-formylglycineamide.Abondcritical point, denoted by a solid dot, is the point common to each bond path and the intersectionof its associated interatamicsurfacewiththe plane. Note the proximity of the ring critical paint. denoted by a solid triangle, with thebondcritical paintofthe0-H hydrogen bond, theseparationequaling 0.19 A. The flat nature of the OIH interatomic surface is characteristic of hydrogen-bonded interactions.

TABLE 3: Geometry and Bond Critical Point Data for Hydrogen-Bonded Ring Systems CJ(distorted)' C7 Geometry (A and deg)

H-N

o=o 0-H &N OHN

0.9973 1.2014 2.1018 2.5967

108.5l

0.9954 1.2020 2.2380 3.0520 137.87

3.613 0.9972 1.2160 1.9208 2.8949 165.07

Bond (b) and Ring (r) Critical Point Data (atomic units) Pb VZPb PI

0.022 0.111 0.021 1.084

Ib(H)

0.858

1.274

0.013 0.052 0.007 0.053 0.853 1.394 0.778

0.022 0.106 0.002 0.092 0.691 1.230 1.739

dist[bcprcp] 0.193 Optimized Cs structure is 0.9 kcal/mol lower in energy.

of Il will result in the coalesence and mutual annihilation of the O-H bond and ring critical points which are in close proximity (Figure 2) and whose p values, p b and pr. are nearly equal, (Table 3). The potential instability of the ring structure isalso reflected in the high ellipticity for the 0-H interaction. Avignon and La~combe'~ give spectroscopic evidence in terms of changes in the values of the N-H stretching frequency for the presence of a strained five-membered hydrogen bond and a more stable C7 structure for glycyl blocked with acetyl- and diethylamide in both CCla solutions and an argon matrix. A stable, seven-membered hydrogen bond appears in the C, conformation, III, corresponding to the presence of a bond path linking the H of the terminal amino group with the keto oxygen of the terminal formyl group. A larger N-H-O angle is possible in a seven-memhered ring and the bond and ring critical points are well separated (Table 3). The value of pr is therefore significantly less than that of pb. the ellipticity of the bond is

F i w 3. (a) Representation of the 3.61, helical hydrogen-bonded structure of the blocked triglycyl peptide HC(-o)p"l(G",l~',~H~ obtained by optimizing the starting geometry with fixed torsion angles and planarpcptidcbonds. Thestructureindicatedinthefigunism e e d in terms of the bond paths defined by the topology of p and summarized in Tables 3 and 5. (b) Representation of the completely optimized stmctureof blocked triglycyl peptide. ThedistanceseparatingO(1) and H( 13) is just beyond that enabling the formation of a hydrogen bond to yield a 310 helical structnrc.

small, and the ring structure is stable. While topologically more stable in Itl than in the distorted CS, the ring possesses a longer N-O separation and the H-O link is weaker, as evidenced by its smaller value of pb. The C7 system is only 0.1 kcal/mol more stablethan thedistortedCs,illustratingthesoftnessof thedihedral angles 4 and $which are -85.3O and 6 6 . Y . respectively. T h e values place the structure in the @-sheetregion of the Ramachandran map. While the Csand C7structures are the stable forms of isolated dipeptides30 and in solutions of dipeptides,19 they occur only infrequently in polypeptides. Schifer et al.)" find the repeated C7 structure to be the most stable form of the blocked dialanyl peptide (with three peptide bonds) but find the bend and helical structures to bc the most stable forms in bexapeptides. No hydrogen bond is found in the transition state

(W.

Geometries and Stnfhues of NFormyltriglycincAmide. The most important of the Structures proposed by Pauling and Corey for aprotein is thea-helix.w,'' In thisstructureeach turncontains

Geometry of N-Formyltriglycine Amide

TABLE 4

The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 4477

Atomic Properties of Hvdrwen-Bonded H ~~~~

molecule property, atomic units

c 5

c 7

N(H)

0.518 0.482 -0.4222 22.46 0.150 -0.06 -0.05 0.11

0.519 0.481 -0.4216 21.87 0.142 -0.07 -0.06 0.13

q(H) E(H)

u(H)

IM(W QiW) Q2W) Q3(W

3.613 0.496 0.504 -0.4144 17.80 0.125 -0.08 -0.09 0.16

3.6 residues. The distance between adjacent turns (referred to as the pitch) is 5.4 A, and each N-H group is hydrogen bonded with the C=O group of the fourth preceding amino residue. The number of atoms in the ring formed by each hydrogen bond is 13, and the helix is designated as 3.613 or as the CYRhelix in its right-handed form. We begin by modeling the CYR helix with values for the torsion angles, 4 = -67Oand = -45O,characteristic of the CYRzone in the Ramachandran map with w = 0,42and by choosing standard values for the bond lengths and bond angles. The terminal N-H bond and C=O groups were configured to form a hydrogen bond, with the 0-H length equal to 1.55 A and the 0-H-N angle equal to 167’ (Figure 3a). However, optimization results in a partial unfoldingof the starting geometry, and no hydrogen bond is formed. The final 4 angles are close to the original with the exception of that for G3 which becomes more negative, while all of the values assume considerably more positivevalues, giving the structure illustrated in Figure 3b. The result of these changes is to increase the distance between the oxygen O( 1) of the terminal keto group and the hydrogens of the terminal amino group to over 4 A, and no 3.613 hydrogen bond is present in the optimized structure. This result suggests that the stability of the CYR-heliX is the result of forming multiple 3.613 hydrogen bonds in a plypeptide, as opposed to the single such bond possible in the tetrapeptide consideredhere. The effect of thechangeingeometryis todecreasethedistanceofthe terminal oxygen O( 1) to H( 13) of the amide nitrogen linking residues 2 and 3 (Figure 3). The resulting O( 1)-H( 13) and O( 1)-N( 12) separations of 2.30 and 3.29 A, respectively, are just beyond those required for the formation of a 10-membered hydrogenbonded ring structure. The optimized conformation is similar to that associated with the known 310 helix which possesses values for in the neighborhood of -5O. Schilfer et al.30 find the minimum-energy helical structures obtained at the 4-2 1G level for glycyl and alanyl hexapeptides to be closer to 310 than they are to CYR. In fact, the (4,+) values obtained here for the three glycyl residues in the tetrapeptide exhibit the same trends and are in reasonable absolute agreement with the values obtained for residues 3, 4, and 5 in f~rmylpentaglycylamide.~~ Neither the present calculation nor those of Schgfer et al.’O have demonstrated that these findings are independent of the starting geometry, which would require a determination of the energy over the complete Ramachandran space. However, the similarity in the findings obtained by both groups regarding the relative stability of the 310over the (YR structure in different systems for different starting geometries would argue against this result being an artifact. Abandoning the constraint of planarity of the peptide groups is not found to lead to a significantly different overall optimized geometry. The largest deviations are found for the N-terminus of the helix with a C-C, difference of 0.0025 A. Table 2 summarizes the deviations from planarity for the peptide bonds as measured by the torsion angle w. The transition state is the most sensitive to this constraint, although in all cases the energy difference between constrained and unconstrained geometry is vanishingly small. The N-terminal peptide bond in the helical structure is most affected by the allowed nonplanarity, resulting in a deviation of almost 8 O .

+

+

+

TABLE 5: Comparhn of Bond Critical Poiat Data (&I atomic units) for CIycyl Residues and Terminal Croup C’I Cs-77 G“l c 5 G”3 Central Group 0.426 -0.007 0.039 0.758 1.513 0.271 -0.795 0.085 1.381 1.489 0.275 -0.675 0.060 0.921 1.792 0.353 -2.037 0.059 1.429 0.456 0.291 -1.073 0.036 1.311 0.740 0.291 -1.073 0.036 1.311 0.740

0.426 -0.003 0.051 0.758 1.514 0.269 -0.784 0.036 1.403 1.484 0.259 -0.518 0.041 0.914 1.829 0.356 -1.960 0.062 1.408 0.473 0.294 -1.096 0.026 1.314 0.731 0.297 -1.120 0.029 1.309 0.730

0.426 0.026 0.045 0.757 1.511 0.297 -1.146 0.023 1.326 0.732 0.333 -0.755 0.019 0.853 1.682

0.424 0.007 0.035 0.758 1.513 0.299 -1.154 0.022 1.329 0.121 0.333 -0.783 0.021 0.856 1.682

0.330 -0.813 0.001 0.861 1.686 0.352 -1.983 0.058 1.417 0.465 0.355 -1.951 0.061 1.406 0.413

0.332 -0.826 0.017 0.861 1.682 0.353 -1.963 0.056 1.412 0.470 0.352 -2.053 0.055 1.431 0.454

0.425 -0.031 0.041 0.757 1.512 0.269 -0.783 0.052 1.388 1.496 0.268 -0.634 0.015 0.924 1.803 0.356 -1.970 0.060 1.410 0.472 0.297 -1.118 0.029 1.307 0.732 0.293 -1.089 0.025 1.306 0.740

0.424 -0.036 0.037 0.759 1.516 0.269 -0.782 0.092 1.399 1.479 0.260 -0.535 0.038 0.914 1.828 0.356 -1.944 0.062 1.408 0.475 0.296 -1.108 0.040 1.322 0.722 0.292 -1.083 0.036 1.308 0.737

N-Terminus 0.426 0.058 0.040 0.756 1.510 0.298 -1.150 0.021 1.327 0.729 0.330 -0.829 0.015 0.864 1.687

C-Terminus 0.330 -0.859 0.012 0.866 1.688 0.353 -1.960 0.053 1.414 0.471 0.355 -1.963 0.057 1.410 0.471

0.426 0.425 -0.032 -0.054 0.040 0.036 0.759 0.759 1.514 1.516 0.267 0.267 -0.767 -0.773 0.098 0.104 1.410 1.393 1.477 1.491 0.261 0.263 -0.487 -0.513 0.029 0.027 0.903 0.905 1.829 1.822 0.358 0.355 -1.988 -2.036 0.062 0.057 1.408 1.425 0.470 0.459 0.297 0.296 -1.113 -1.112 0.041 0.037 1.322 1.325 0.719 0.715 0.292 0.293 -1.084 -1.088 0.041 0.045 1.305 1.308 0.739 0.737

*

0,427 0.058 0.044 0.755 1.509 0.299 -1.155 0.022 1.332 0.724 0.329 -0.830 0.011 0.865 1.691

*

*

0.335 -0.840 0.025 0.862 1.678

0.333 -0.843 0.032 0.865 1.679

*

*

*

*

0.333 -0.809 0.005 0.858 1.681 0.353 -2.048 0.055 1.426 0.455 0.353 -1.962 0.055 1.411 0.470

To obtain an rur-helix and study the properties of its hydrogen bond, the starting geometry described above for the helix was optimized at the 3-21G level with fixed torsion angles and planar peptide bonds. We find, as do Head-Gordon et a1.,22that 3-21G geometries providegenerally good agreement with thoseobtained using a higher level basis. After optimization of the bond lengths and valence angles, the pitch, the 0-H and N-O distances, and the 0-H-N angle equaled 5.67 A, 1.92 A, 2.90 A, and 165.1°, respectively. A single-point calculation was performed for this partially constrained geometry using the 6-31 l++G** basis set.

4478 The Journal of Physical Chemistry, Vol. 98, No. 16, 1994

Popelier and Bader

TABLE 6: C o ” r h a of Atomic Pmmtia in Atomic Udb for Glycyl R~&~KYJ .ad Tewtarl Groupr C3

C

0

N

0

CS-7’

0.09 0.982 0.018 -0.6233 47.23 0.127 -0.24 -0.15 0.39 0.982 0.018 -0.6233 47.23 0.127 -0.24 -0.15 0.39 29.999 -206.8650

Central Group 5.445 5.424 0.576 0.555 -37.4993 -37.5036 49.02 49.38 0.589 0.582 -0.63 -0.55 -0.30 -0.5 1 0.93 1.06 4.230 4.213 1.770 1.787 -36.7329 -36.7182 3 1.65 3 1.26 0.828 0.806 -1 .00 -1.06 -0.42 -0.38 1.43 1.42 9.346 9.352 -1.352 -1.346 -75.6339 -75.6368 137.15 138.34 0.527 0.522 -0.53 -0.59 -0.06 -0.04 0.65 0.57 8.485 8.468 -1.485 -1.468 -55.3020 -55.3000 102.39 102.35 0.283 0.276 -2.27 -2.14 0.73 0.83 1.55 1.31 0.565 0.569 0.435 0.431 -0.4496 -0.4474 27.76 27.70 0.165 0.165 -0.04 -0.04 -0.04 -0.03 0.07 0.08 0.978 0.967 0.022 0.033 0.6266 -0.6179 43.85 44.82 0.121 0.124 -0.22 -0.22 -0.17 -0.17 0.39 0.39 0.983 0.971 0.017 0.029 -0.6240 -0.6221 47.03 45.68 0.123 0.124 -0.24 -0.25 -0.18 -0.15 0.42 0.40 30.000 29.996 -206.8641 -206.8502

4.174 1.820 -36.6890 38.79 0.784 -1.20 -0.29 1.49 9.339 -1.339 -75.6253 138.39 0.523 -0.57 -0.05 0.62

4.179 1.821 -36.6896 38.51 0.787 -1.22 -0.26 1.48 9.346 -1.346 -75.6244 132.13 0.520 -0.55 0.05 0.50

5.421 0.579 -37.4847 48.97 0.600 -0.73 -0.37 1.10 4.227 1.772 -36.7108 32.30 0.836 -1.02 -0.37 1.39 9.348 -1.348 -75.6510 133.59 0.517 -0.63 0.10 0.53 8.512 -1.512 -55.3464 101.83 0.275 -2.37 0.92 1.47 0.527 0.473 -0.4265 23.74 0.154 -0.06

-0.04

C

c7

N-Terminus 4.183 1.817 -36.9605 38.86 0.782 -1.22 -0.26 1.48 9.343 -1.343 -7 5.6244 137.14 0.531 -0.57 0.02 0.54

W’l

G’i

5.450 0.550 -37.501 0 49.71 0.561 -0.77 -0.59 1.35 4.220 1.780 -36.7267 30.09 0.805 -1.05 -0.36 1.40 9.354 -1.354 -75.6503 131.75 0.518 -0.52 0.00 0.52 8.487 -1.487 -55.2948 99.69 0.246 -2.15 0.63 1.52 0.572 0.428 -0.4510 27.69 0.166 -0.05 -0.03 0.08 0.948 0.052 -0.6107 44.46 0.126 -0.22 -0.15 0.37 0.974 0.026 -0.6187 46.52 0.122 -0.24 -0.18 0.42 30.005 -206.8532

5.445 0.555 -37.4961 49.50 0.568 -0.83 -0.58 1.40 4.221 1.779 -36.7205 3 1.24 0.795 -1.10 -0.34 1.44 9.351 -1.351 -75.6468 135.10 0.520 -0.65 0.08 0.57 8.503 -1.503 -55.3367 84.90 0.244 -2.02 0.54 1.49 0.566 0.434 0.4510 25.60 0.162 -0.07 -0.02 0.09 0.942 0.058 -0.6083 43.58 0.125 -0.21 -0.15 0.37 0.98 1 0.019 -0.6221 46.69 0.124 -0.25 -0.17 0.42 30.008 -206.88 15

4.195 1.805 -36.7025 39.02 0.784 -1.21 -0.26 1.47 9.337 -1.337 -75.6444 131.31 0.546 -0.54 -0.01 0.55

G”3 5.442 0.558 -37.4971 50.40 0.572 -0.86 -0.54 1.40 4.216 1.784 -36.7153 3 1.54 0.820 -1.05 -0.39 1.44 9.359 -1.359 -75.6359 138.50 0.509 -0.57 -0.10 0.67 8.521 -1.521 -55.3410 94.44 0.241 -2.00 0.64 1.36 0.534 0.466 -0.4349 20.96 0.141 -0.09

-0.04 0.13 0.930 0.070 -0.6022 41.91 0.121 -0.24 -0.14 0.38 0.977 0.023 -0.6213 46.63 0.125 -0.25 -0.16 0.41 29.979 -206.8477

*

(I

*

The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 4479

Geometry of N-Formyltriglycine Amide

TABLE 6 (Continued) c 5

C%7+

c 7

G" 1

N-Terminus 0.983 0.017 -0.6239 48.22 0.123 -0.32 -0.13 0.45

0.987 0.01 3 -0.6254 48.42 0.123 -0.32 -0.12 0.45

C-Terminus 8.395 -1.395 -55.1799 122.72 0.267 -2.81 0.84 1.98 0.546 0.454 -0.4355 26.86 0.160 -0.04 -0.03 0.07 0.565 0.435 -0.445 1 27.76 0.164 -0.04 4.04 0.08 54.011

8.418 -1.418 -55.1997 120.40 0.256 -2.75 0.79 1.96 0.558 0.442 -0.4391 27.46 0.164 -0.04 -0.04 0.07 0.519 0.481 -0.4216 21.87 0.142 -0.07 -0.06 0.13 53.999

The Cartesian forces on the nuclei for the geometry optimized at the 3-21G level computed using the density obtained from the 6-3 11G** basis remain small, the root-mean-square value being only 0.014 hartree/bohr. This electron density distribution predicts a bond path linking the terminal keto oxygen with a hydrogen of the terminal amide group to yield the anticipated 13-membered ring structure. Reference to Table 3 shows this to be a stable system with the bond critical point well separated from the ring critical point in value and in position and exhibiting only a small ellipticity. Although this hydrogen bond has a noticeably longer N-O separation than is found in the C5 system, the bonds are of equal order as measured by the values of Pb. The reason for this is the much more favorable value for the N-H-O angle in the 13membered ring, a value deviating by only from 15" from the preferred linear structure. The values of Pb and VZpbfor all three hydrogen bonds are typical and agree withvalues reported earlier. The N-H-O hydrogen bonds in formamide dimer38and in H20NH3,37for example, both have bond energies of 6 kcal mol-' and Pb values equal to 0.021 au. In forming a hydrogen bond, a hydrogen atom loses electronic charge, increases in energy, and undergoes a decrease in ~ o l u m e . A ~ ~comparison ,~~ of the values for the hydrogen-bonded H given in Table 4. compared to the average values for a free H bonded to an amide nitrogen, namely N(H) = 0.565, E(H) = -0.45 au, and u(H) = 27 au, indicates that these characteristic changes are found for the molecules discussed here, thelargest being found for the a-helix. Thecharge reorganization on the hydrogen is such that it facilitates the approach of two closed-shell systems, the H of the acid and the keto oxygen, by removing charge density from along the axis of approach to a toruslike distribution about the axis.36 This is reflected in a near doubling of the positive component of the quadrupole moment tensor (Q3(H) from its normal value of 0.08 au for an amide H when it is hydrogen bonded.

8.371 -1.371 -55.1697 121.18 0.265 -2.70 0.80 1.90 0.560 0.440 -0.4411 27.55 0.164 -0.04 -0.03 0.08 0.562 0.438 -0,4441 27.52 0.163 -0.05 -0.03 0.08 54.006

0.973 0.027 -0.6192 47.92 0.122 -0.31 -0.13 0.44

Gff3

Gf'2

*

*

*

*

*

*

*

*

8.415 -1.415 -55.1965 121.31 0.266 -2.79 0.82 1.97 0.525 0.475 -0.4255 22.49 0.144 -0.07 -0.04 0.12 0.559 0.441 -0.4409 27.70 0.164 -0.04 -0.04 0.07

114.015

Conformational Dependence of the Electronic Charge Distribution

The dependence of the charge distributions of the amino acid residues on the twisting of the peptide chain and on the nonplanarity of the peptide bond is summarized in terms of the properties of p(r) at the bond critical points (Table 5 ) and in terms of the atomic properties (Table 6). We shall be primarily concerned with the changes in the properties of the three glycyl groups G"i of the tetrapeptide which accompany changes in IP and $ away from the planar case found in C5. These three groups have (+,$) values characteristic of the helical structure. The same general observations apply to the C7 conformation as the reader may verify. The reader is referred to the paper by Chang and Bader2for a discussion of the bonding in a peptide chain, as summarized in terms of the critical point data listed in Table 5 and the atomic properties in Table 6. The data in Table 1 indicate that the primary changes in bond lengths and bond angles of the glycyl groups G", are those associated with C,. The lengths C,-N and C,-C increase by a maximum of 0.015 and 0.008 A, respectively, in the three glycyl fragments of the tetrapeptide. The angle of NC,C opens by a maximum of 7", while C,CO decreases by approximately 4" and C,CN increases by 3". The angle HNC also decreases by approximately 3", but the remaining bond lengths and bond angles undergo smaller changes, the next largest being associated with the C,-H bonds. The twisting of the peptide chain results in a differentiation of the hydrogens bonded to C,. The least affected set, hydrogens 23,25, and 27 of Figure 3b, with a bond shortening of -0.003 A, approach the plane of the C,NH nuclei with dihedral angles in the range 10-40" and move out of the plane of the C , C 4 nuclei, with dihedral angles in the range 80-55". This is labeled the trans set. The second set, hydrogens 22, 24, and 26, with a bond shortening of -0.005 A, exhibit the reverse behavior, forming larger dihedral angles with the C,NH group,

4480

The Journal of Physical Chemistry, Vol. 98, No. 16, 1994

110-160°, and smaller ones with the C,C=O group, 40-60°, with the exception of H(27), for which the latter angle is approximately the same as that for H(26). This is labeled the cis set. It is important to note that, in general, the differences in geometrical parameters are reduced within the set of three glycyl groups, the variation in the bond angles being particularly small. While the G”i have different values for the set (&J.),they derive from the same region of the Ramachandran map. Bond Critical Point Data. The values of the charge density at the bond critical point P b exhibit remarkably small variations throughout all conformations for each type of bond. The largest change from the planar case occurs for the C,-N bond where Pb decreases by a maximum of 0.015 au, the next largest change being a decrease for the C,-C bond of 0.004 au. These decreases reflect the corresponding increases in bond lengths. As noted above, the twisting of the peptide chain differentiates between the two hydrogens bonded to C,. The shortening of the bond to the H trans to the keto oxygen is equally shared between the bonded radii of C, and H, and Pb remains unchanged. The bond to the H cis to the keto groups undergoes the greatest shortening, which has the effect of moving the critical point toward the proton and noticeably increasing the value of Pb. The least affected bond in terms of changes in its critical point data is C=O. This bond and the properties of the keto oxygen atom are essentially unaffected by conformational changes. The small changes that are found in the values of v2pb are a result of the critical point for this bond occurring in a region where the slope of the Laplacian distribution is very steep and close to a change in sign. The P b values for the 10 C-N amide bonds exhibit a range of only 0.006 au, in spite of the deviations from planarity noted in Table 2. The major axes of the ellipticities e of all the bonds in the peptide chain in the planar conformation C5 are perpendicular to the plane containing the nuclei. The largest value of t is for the C,-N bond, and it undergoes a significant decrease with the loss of planarity in forming the helical structure. The ellipticitiesare however, small, indicating that the preferential accumulation of electronic charge in the %-plane” is slight and that the coupling of the bonds by conjugation of charge along the peptide chain is weak.2 Thevariationin values of all the critical point propertieswithin the set of the three helical groups is smaller still, the maximum change in Pb being 0.003 au for the C,-N bond. The bonded radii of the C and N atoms in the links between G”l and G”2 and between Gf‘2 and G”3 differ by less than 0.002 A, leading to correspondingly small changesin amide bond lengths. The bonded radii for the C,-H interactions are constant. Clearly, the bonds in a glycyl group with (+,J.) values characteristic of a helical structure are nearly independent of the exact location in the Ramachandran map, and the data for the group G”2 could be taken as representative of a helical glycyl group. Atomic Properties. The first property to be considered is the requirement of charge neutrality for a peptide group. Reference to Table 6 shows that the sum of the atomic populations N(Q) over each of the five peptide groups equals 30, the sum of the nuclear charges, to within the integration error.43 Thus, each peptide residue behaves as a self-contained unit with respect to charge conservation, changes in the torsion angles, and loss of amide planarity, resulting in a flow of charge within the group but not across its two amidic boundaries. The blocking amino and formyl groups necessarily exhibit a similar constancy in their net charges, of magnitude0.500 f 0.007e. Thus, the electrostatic fields through the -HNI and I C ( 4 ) - surfaces of a glycyl group are equivalent to the presence of -0Se and +0Se charges, respectively,since they are required to match the complementary surfaces of the neighboring groups. A corresponding charge neutrality is found for a similarly blocked serinyl residue (Figure l), its integrated net charges being -0.006e and -0.0002e for

Popelier and Bader torsionangles (4,J.)of (-99.1,8.3) and (165.2,180.8),respectively, as optimized in 6-31 1++G**//6-31+GS calculations. The extent of change in all the atomic properties caused by changes in the torsion angles 4, J., and o away from the planar C5 conformation is mirrored in the changes found for the atomic populations, and these are considered first. The primary changes in atomic populations caused by a twisting of the peptide chain are a transfer of charge to C, in the amount of 0.02-0.03e and a corresponding removal of 0.05-0.03e from its bonded H cis to the keto oxygen, the population of the trans H remaining unchanged. There is a small transfer of charge within the keto group from C to 0, its net charge remaining unchanged. The amidic N loses electronic charge for a rotation about the N-C bond in f ~ r m a m i d e . ~ This ? ~ effect ~ is noted here for the N atom in G”I which at 8 O has the largest departure from planarity. The change in population is less than 0.01e for N in G”2 and G”3 where the degree of nonplanarity is only 2O. The remaining changes in population are associated with the amidic hydrogen. As discussedabove, this hydrogen of the glycyl group in the equilibrium geometry of C5 is involved in incipient hydrogen bond formation, the electron density being on the verge of forming the associated bond critical point. Thus, its properties are close to those given in Table 4 for the same hydrogen in the hydrogen-bonded form of C5. The amidic H of glycyl in CIis not hydrogen bonded, and its properties are typical of those found in a nonassociated amide g r o ~ p . The ~ ~ amidic . ~ ~ H atoms in G”1 and G”2 exhibit properties similar to those for H in C,, but the amidic hydrogen H( 13) in G”3 resembles that in C5. This result is anticipated, as it is this hydrogen that is brought into proximity with the keto oxygen, O( 1) of Figure 3, to yield the conformation typical of the 310 helix. The atomic properties of H(13), like those of the amidic H in the equilibrium geometry of C5, reveal the presence of incipient hydrogen bond formation. The H atom loseselectronicchargeonhydrogen bond formation, and the extent of involvement of the amidic H in such an interaction can be gauged by the magnitude of its net charge, which equals +0.482e and +0.473e in the hydrogen-bonded and equilibrium forms of C5, respectively, and +0.466e in the helix. The amidic H of the terminal amino group, H(18) of Figure 3b, which if hydrogen bonded would yield the 3.613 a-helix, also exhibits properties characteristic of incipient hydrogen bond formation in the helical structure. Its net charge of +0.475e and other properties (Table 6) approach those for the same H in the constrained model of the a-helical structure given in Table 4, whose net charge equals +0.504e. Thus, the atomic properties indicate that the amidic hydrogen on the terminal amino group H( 18), and the corresponding hydrogen H(13) in G”3 are both involved in incipient hydrogen bond formation, indicating that the optimized helical conformation found here for formyltriglycine amide is transitional between the 3.613 and 310 helical structures. The principal changes in atomic populations for the atoms of the glycyl groups in forming a helical structure occur for C, and its bonded hydrogen that is cis to the neighboring keto oxygen. Bringing a hydrogen into the proximity of the negatively charged keto oxygen, q ( 0 ) = -1.35e, as caused by twisting the peptide chain, causes electron density to be transferred from H to C,. This results in a net positive charge on cis H which is 2-3 times that on the H trans to the keto oxygen, the methylenes of all three of the glycyl groups being similarly affected. The monopoles q(Q) and dipoles, as summarized by its magnitude IM(a2)l and components Q,(Q) of the quadrupole moment expressed in its diagonal form, for each type of atom in the three helical glycyl groups exhibit characteristic and similar values. While the small deviation from planarity of the amidic N in G”l decreases the magnitude of its net charge by -0.02e, its remaining moments are less affected. All three of the amidic nitrogen atoms exhibit the quadrupolar polarization characteristic of spZ hybridization.

Geometry of N-Formyltriglycine Amide The presence of the anticipatednonbondedu densityperpendicular to the amidic plane is confirmed by the large negative values for the corresponding component of the quadrupolar moment, Ql(N). The data indicate that one could assign values to the monopole, dipole, and quadrupole moments for the atoms in the glycyl group in a helical conformation,which would be of sufficient accuracy for the prediction of its contributions to the electrostatic field of a helix. One must take into account the degree of nonplanarity of the amidic nitrogen and differentiate between hydrogen-bonded and free amidic hydrogens in the assignment of their monopoles.

Conclusion This paper has investigated the result of twisting a peptide chain, with its accompanyingloss of planarity of the amide bond, on geometrical parameters and on the charge distributions of the atoms forming the backbone of the chain. It has been found that charge neutrality of a peptide group, a necessary requirement for its transferability, is preserved in the helical structures. The study indicates that it should be possible to assign characteristic properties to the atoms of a peptide chain by taking into account their dependence on the torsion angles as determined here. The presence of a bond path linking an amidic H and a keto oxygen enables one to establish the existence of a hydrogen bond in the helical structure. While its presence or absence is a discontinuous process, the atomic properties of the amidic hydrogen display a continuous variation between the limiting values characteristic of the unassociated and bonded limits. Thus, the atomicproperties also enable one to detect which hydrogen atoms are in the process of incipient hydrogen bond formation and todetermine theextent of this interaction.

References and Notes (1) Bader, R. F. W. Atomsin Molecules-A Quantum Theory;University of Oxford Press: Oxford, 1990. (2) Chang, C.; Bader, R. F. W. J. Phys. Chem. 1992, 96, 1654. (3) Bader, R. F.W.; Popelier, P. L. A.; Keith, T. A. Angew. Chem. 1994, 33. 620. (4) Zakrzewska, K.; Pullman, A. J . Comput. Chem. 1985,6,265. ( 5 ) Bellido, M. N.; Rullmann, J. A. C. J. Comput. Chem. 1989,10,479. (6) Rein, R. Adv. Quantum Chem. 1973, 7, 335. (7) Mezei, M.; Campbell, E. S . Theor. Chim. Acta 1977,48,227. (8) Pullman, A.; Perahia, D. Theor. Chim. Acta 1978, 48, 29. (9) Lavery, R.; Etchebest, C.; Pullman, A. Chem. Phys. Lett. 1982.85, 266. (10) Rico, J. F.; Alvirez-Collado, J. R.; Paniagua, M. Mol. Phys. 1985, 56, 1145. (11) Sokalski, W. A.; Sawaryn, A. J . Chem. Phys. 1987.87, 526. (12) Vignb-Maeder, F.; Claviere, P. J. Chem. Phys. 1988,88, 4934. (13) Stone, A. J. Chem. Phys. Lett. 1981,83,233. Stone, A. J.; Alderton, M. Mol. Phys. 1985,56, 1047.

The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 4481 (14) Price, S. L.; Faerman, C. H.; Murray, C. W. J . Comput. Chem. 1991, 12, 1187. (15) Faerman, C. H.; Price, S . L. J. Am. Chem. SOC.1990, 112,4915. (16) Price,S.;Stone,A. J. J . Chem.Soc.,Faraday Trans. 1992,88,1755. (17) Cooper, D. L.; Stutchbury, N. C. J. Chem. Phys. Lett. 1985, 120, 167. (18) Breneman, C. M. The Application of Charge Density Research to Chemistry and Drug Design; NATO AS1 Series; Jeffrey, G. A., Piniella, J. F., Eds.;Plenum Press: New York, 1991; Vol. 250, p 357. Breneman, C. M. Sterling Winthrop Methodology Report No. 3:12/24/92. (19) Bader, R. F. W.; Popelier, P. L. A. J . Mol. Srruci. (THEOCHEM) 1992, 255, 145. (20) Destro, R.; Bianchi, P.; Gatti, C.; Merati, F. Chem. Phys. Lett. 1991, 186, 47. (21) Head-Gordon, T.; Head-Gordon, M.; Frisch, M. J.; Brooks 111, C.; Pople, J. A. Int. J. Quantum Chem., Quantum Biol. Symp. 1988, 16, 311. (22) Head-Gordon, T.; Head-Gordon, M.;Frisch, M. J.; Brooks 111, C.; Pople, J. A. J. Am. Chem. SOC.1991,113, 5989. (23) Tobias, D. J.; Brooks 111, C. L. J. Phys. Chem. 1992, 96, 3864. (24) Gould, I. R.; Kollman, P. A. J. Phys. Chem. 1992, 96, 9255. (25) Price, S. L.; Andrew, J. S.; Murray, C. W.; Amos, R. D. J. Am. Chem. Soc. 1992, 114, 8268. (26) Bdhm, H.-J.; Brode, S. J. Am. Chem. SOC.1991,113, 7129. (27) Perczel, A.; Angyin, J. G.; Kajtir, M.; Viviani, W.; Rivail, J.-L.; Marcoccia, J.-F.; Csizmadia, I. G. J. Am. Chem. Soc. 1991, 113, 6256. (28) Wright, L. R.; Borkman, R. F. J . Phys. Chem. 1982,86, 3956. (29) Schiifer, L.; Newton, S. Q.; Momany, F. A.; Klimkowski, V. J. J. Mol. Srrucr. (THEOCHEM) 1991,232, 275. (30) SchHfer, L.; Newton, S.Q.;Cao, M.; Peters, A.; Van Alsenoy, C.; Wolinski, K.; Momany, F. A. J . Am. Chem. SOC.1993,115, 272. (31) Gaussian 90; Frisch, M. J.; Head-Gordon, M.; Trucks, G. W.;

Foresman, J. B.; Schlegel, H. B.; Raghavachari, K.; Robb, M. A.; Binkley, J. S.;Gonzalez, C.; DeFrees, D. J.; Fox, D. J.; Whiteside, R. A.; Seeger, R.; Melius, C. F.; Baker, J.; Martin, R. L.; Kahn, L. R.; Stewart, J. J. P.; Topiol, S.;Pople, J. A. Gaussian, Inc., Pittsburgh, PA, 1990. (32) Frisch, M. J.; Pople, J. A.; Binkley, J. S. J . Chem. Phys. 1984, 80, 3265. (33) Popelier, P. L. A.; Bader, R. F. W. Chem. Phys. Lerr. 1992,189,542. (34) Almlof, J.; Faegri, K.; Korsell, K. J. Comput. Chem. 1982,13,385. (35) Peters, D.; Peters, J. J . Mol. Struct. (THEOCHEM) 1979,53, 103; 1980, 62, 229; 1980, 68, 243; 1980, 69, 249. (36) Carroll, M.T.; Bader, R. F. W. Mol. Phys. 1988, 65, 695. (37) Badqr, R. F. W.; Carroll, M. T.; Cheeseman, J. R.; Chang, C. J. Am. Chem. SOC.1987, 109, 7968. (38) Cheeseman, J. R.; Carroll, M. T.; Bader, R. F. W. Chem. Phys. Lerr. 1988, 143,450. (39) Avignon, M.; Lascombe, J. In Conformation ofBiologica1 Molecules and Polymers; Bergman, E. D., Pullman, B., Eds.; Academic Press: New York, 1973; p 97. (40) Corey, R. B.; Pauling, L. Proc. R. SOC.London, Ser. B 1953, 141, 10. (41) Ghelis, C.; Yon, J. In Protein FoldingintheSeriesMolecularBiology; Horecker, B., Kaplan, N. 0.. Marmur, J., Scheraga, H. A., Eds.; Academic Press: New York, 1982. (42) Ramakrishnan, C.; Ramachandran, G. N. Biophys. J . 1965,5,909. (43) The total population of a glycyl group deviates from 30 by less than 0.008e with the exception of that for G”3 where there is an integration error associated with a single atom. The integration error in a population for a total system is, in general, 0.01e or less (Table 6). (44) Wiherg, K. B.; Laidig, K. E. J . Am. Chem. Soc. 1987, 109, 5933. (45) Laidig, K. E.; Bader, R. F. W. J. Am. Chem. SOC.1991,113, 6312.