SHORT COMMUNICATION

Helix-Helix Packing Angle Preferencesfor Finite Helix AxesDirk Walther,1* Clayton Springer,1 and Fred E. Cohen1,2,3

1Department of Cellular and Molecular Pharmacology, University of California at San Francisco,San Francisco, California2Department of Medicine, University of California at San Francisco, San Francisco, California3Department of Biochemistry and Biophysics, University of California at San Francisco, San Francisco, California

ABSTRACT Recently, James Bowie ad-dressed the question of how to normalize cor-rectly the distribution of observed helix-helixpacking angles in proteins (Bowie, NatureStruct. Biol. 4:915–917, 1997). A hitherto unreal-ized yet significant bias toward crossed pack-ing angles was revealed. However, the derivedrandom reference distribution of packingangles requires that helices have to be as-sumed as infinite in length. Here, we comple-ment Bowie’s analysis by consideration of themore realistic case where helices are of finitelength. As a result, the statistical bias towardnear perpendicular packings appears to beeven stronger. Proteins, 33:457–459, 1998.r 1998 Wiley-Liss, Inc.

Protein structures determined at atomic resolu-tion reveal an exceptionally high internal packingefficiency.1 It was therefore assumed that assembliesof a-helices in proteins would be subject to strongpacking constraints dictating their mutual orienta-tions. Widely known models for investigating theseconstraints, such as ‘‘knobs-into-holes,’’2,3 or ‘‘ridges-into-grooves,’’4 examine the appropriate interdigita-tion of amino acid sidechains in inter-helical inter-faces by means of unrolling two idealized helixsurfaces onto a plane normal to the global line ofclosest approach between the helical axes. Helixpacking is thereby viewed as an interaction along avector strictly perpendicular to both helix axes inthese models (Fig. 1A). Contacts across the terminiof helices fall outside this geometric definition andshould thus be excluded in studies on the validity ofsteric packing models (Fig. 1A, right panel).

Bowie derives the random probability of any par-ticular inter-axial (dihedral) packing angle V inspace as proportional to sin(V).5 This result correctlydescribes the distribution of angles between any two

random vectors. With the exception of the sign, thisthree-point angle coincides with the dihedral (orpacking) angle defined by four points in space (twoaxes vectors and a connecting vector around whichthe dihedral angle is measured) only at the globalline of closest approach between any two lines inspace (see Fig. 1A for an illustration). Consequently,the axes have to be assumed as infinite to guaranteethe existence of such a global line of closest approachperpendicular to both axes. For finite axes, therelevant case for helices in protein structures, theglobal line of closest approach does not necessarilyhave to fall within their finite segments but willoften lie beyond the termini of their axes (Fig. 1A).However, the helices do not physically exist at thisvirtual contact site at which steric models wouldapply, and their relative orientation cannot be deter-mined by such virtual interactions. In these cases,the packing requirement, that an interaction vectorperpendicular to both axes exist, is not fulfilled, andthus they should not be considered as helix-helixpacking (Fig. 1A). Therefore, in studying the validityof steric packing models, the random reference distri-bution of packing angles should be derived under theconditions of 1) finite helix axes and 2) the line ofclosest approach perpendicular to both finite helicalaxes, with the latter defining the packing condition.

As illustrated in Figure 1B, the finite-length-condition introduces another and equally significantfactor to the probability calculation derived by Bowie.5

Instead of the packing angle being proportional tosin(V), it now is proportional to sin2(V). Thus, theprobability that a packing angle V will fall into an

Grant sponsor: Deutsche Forschungsgemeinschaft; Grantsponsor: National Institutes of Health.

*Correspondence to: Dirk Walther, Department of Cellularand Molecular Pharmacology, University of California at SanFrancisco, San Francisco, California 94143–0450. E-mail:walther@cmpharm.ucsf.edu

Received 8 May 1998; Accepted 13 August 1998

PROTEINS: Structure, Function, and Genetics 33:457–459 (1998)

r 1998 WILEY-LISS, INC.

interval of packing angles P(V1,V2) can be calculatedfrom Equation 1:

P(V1,V2) 5e

V1

V2sin2 (V) dV

e0

2psin2 (V) dV

51

2p 3V2 2 V1 11

2(sin (2V1) 2 sin (2V2))4 . (1)

It should be noted that this result is independentof the length of the helix axes (as long as they arefinite and longer than zero in length) and that thereis no excluded volume effect. Wedges of disallowedangles due to a second cylindrical object do notinfluence the weighting for different angles. In pro-teins, only a narrow range of helix-helix contactdistances is observed. This restriction, however, doesnot change the calculation of random interhelicalangular preferences.

The random packing angle distribution functionfor finite axes (Eq. 1) is compared with the case ofinfinite axes5 in Figure 2A. The statistical biasmakes crossed packings even more likely when finitehelix axes are considered. By contrast, aligned pack-ings (V < 0°, 180°) are highly unlikely to occur bychance.

With this new reference distribution, the actualpropensities for certain angles need to be re-evaluated. The histogram of helix-helix packingangles observed in a representative protein set alongwith the predicted optimal packing angles accordingto the steric helical lattice superposition model3 areshown in Figure 2B. Only packing geometries fulfill-ing the condition that the line of closest approach isperpendicular to both (finite) helical axes were con-sidered (Fig. 1A). The normalized frequencies(Fig. 2C) reveal an even more prominent overrepre-sentation of packing angles near zero and 180 de-grees and avoidance of crossed packings than hadbeen observed by Bowie.5 The major frequency peaksin Figure 2B, which generally match with the pre-dicted optimal steric packing angles,3 manifest them-selves only as shoulders in the distribution of propen-sities (Fig. 2C). However, the preference values forangles near zero and 180 degrees are far fromcertain. Given a total of 1,027 examined helix-helixpackings, only 0.53 random events would have beenexpected between 0° # V , 10°. This is to becompared with four observed interactions in thisinterval. Evidently, to obtain reliable statistics forthese intervals, a substantially larger database ofdetermined protein structures is required.

While packing precision may turn out to have lessimpact on actual propensities for certain helix-helixpacking angles, steric models have been shown topredict correctly maximal atomic packing densities

Fig. 1. A: Illustration of the packing condition for finite helices.If the angle designated as u adopts values greater than 90°,possibly 180° as, for instance, in a ‘‘T-shaped’’ assembly, ahelix-helix pair does not fulfill the geometric packing conditionimposed by steric packing models (see main text). However, sucharrangements are not excluded in Bowie’s random referencedistribution. B: Packing angle probabilities for finite helix axes. Theprobability of selecting a particular angle was shown to beproportional to sin(V).5 When finite helix axes are assumed, andhelix-helix packing is defined to occur only when the line of closestapproach between the two finite axes intersects both axes at aperpendicular angle, the selection of a starting point for a secondhelix is subject to further restrictions. First, a reference vectorrepresenting helix 1 and a perpendicular vector to this axisdetermining the direction of the line of closest approach to apossible second helix axis are randomly chosen. In the figurethese two vectors are oriented such that the line of closestapproach is normal to the paper plane. A second finite helix axismay then be placed only within a plane parallel to the paper planeand such that its starting point is inside the parallelogram A given aparticular packing angle V. Otherwise, the line of closest approachbetween the two finite axes would no longer be perpendicular toboth of them. The area of A, i.e., the probability of placing thestarting point of helix 2 inside A, is proportional to sin(V) [A 5l1l2sin(V)]. Consequently, the probability for selecting a particularpacking angle is proportional to the product of both contributions:the spherical-polar distribution described by Bowie and the finitelength condition introduced here, i.e., PV~ sin2(V). In three dimen-sions, with variable distances of closest approach and all possibleorientations relative to a fixed reference frame, the parallelogramintegrates into a volume illustrated by the rotation body. However,this volume in turn is proportional to sin(V).

458 D. WALTHER ET AL.

at helix-helix interfaces as a function of the packingangle as well as associated distances between helicalaxes.3 Arguments based on optimal steric interac-tions may well retain their relevance in this context.

ACKNOWLEDGMENTS

D.W. is grateful to James Bowie for the detaileddiscussion of his publication and the encouragementto publish this communication. D.W. was supportedby a post-doctoral fellowship from the DeutscheForschungsgemeinschaft (DFG).

REFERENCES

1. Richards, F. The interpretation of protein structures: totalvolume, group volume distributions and packing density. J.Mol.Biol. 82:1–14, 1974.

2. Crick, F. The packing of a-helices: simple coiled coils. ActaCrystallogr. 6:689–697, 1953.

3. Walther, D., Eisenhaber, F., Argos, P. Principles of helix-helix packing in proteins: the helical lattice superpositionmodel. J. Mol. Biol. 255:536–553, 1996.

4. Chothia, C., Levitt, M., Richardson, D. Helix to helix pack-ing in proteins. J. Mol. Biol. 145:215–250, 1981.

5. Bowie, J.U. Helix packing angle preferences. Nature Struct.Biol. 4:915–917, 1997.

Fig. 2. A: Probability of random packing angles binned into10-degree intervals for infinite5 and finite helix axes (Eq. 1). B: Observeddistribution of packing angles at a 10-degree bin width from a total of1,776 packed helix pairs contained in a protein set of 757 proteinchains determined at 2.4 A resolution or better and less than 30%sequence identity between any two sequences (obtained fromhttp://www.fccc.edu/research/labs/dunbrack/culledpdb.html). Morethan 20 intervening residues between any two packed helices wererequired to reduce a bias from connectivity constraints. Helix defini-tions, helix axes assignments, and packing conditions were taken asused by Walther et al.3 Packings with their line of closest approachintersecting with the termini of the axes (i.e., the actual global line ofclosest approach lies beyond the termini of the axes) were toleratedwhen the deviation from 90° of intersection angles with both axes didnot sum up to more than 1° (angle t as in ref. 3), all others werediscarded, leaving 1,027 pairs for analysis. The three optimal packingmodes (am, bm, and cm) according to the helical lattice superpositionmodel3 are indicated by arrows. Error bars correspond to onestandard deviation s estimated as s 5 sqrt(N) where N is the numberof observations in a given packing angle interval (bin). C: Packingangle propensities obtained by converting the absolute frequenciesincluding the upper and lower boundaries according to the estimatedstandard deviation from B into relative frequencies and subsequentlydividing them by the random probability for a given bin (Eq. 1). Thelogarithm was applied to equally scale avoidance (values , 0) andpreference (values . 0).

459PACKING ANGLES AND FINITE HELIX AXES