principles of helix-helix packing in proteins: the helical

J. Mol. Biol. (1996) 255, 536–553

Principles of Helix-Helix Packing in Proteins: TheHelical Lattice Superposition Model

Dirk Walther 1*, Frank Eisenhaber 1,2 and Patrick Argos 1

The geometry of helix-helix packing in globular proteins is comprehen-1European Molecular BiologyLaboratory, Meyerhofstraße 1 sively analysed within the model of the superposition of two helix latticesPostfach 10.2209, 69012 which result from unrolling the helix cylinders onto a plane containing

points representing each residue. The requirements for the helix geometryHeidelberg, Germany(the radius R, the twist angle v and the rise per residue D) under perfect2Biochemisches Institut der match of the lattices are studied through a consistent mathematical model

Charite, der Humboldt- that allows consideration of all possible associations of all helix types (a-,Universitat zu Berlin, p- and 310). The corresponding equations have three well-separatedHessische Straße 3–4 10115 solutions for the interhelical packing angle, V, as a function of the helixBerlin, Germany geometric parameters allowing optimal packing. The resulting functional

relations also show unexpected behaviour. For a typically observed a-helix(v = 99.1°, D = 1.45 Å), the three optimal packing angles are Va,b,c = −37.1°,−97.4° and +22.0° with a periodicity of 180° and respective helix radiiRa,b,c = 3.0 Å, 3.5 Å and 4.3 Å. However, the resulting radii are very sensitiveto variations in the twist angle v. At vtriple = 96.9°, all three solutions yieldidentical radii at D = 1.45 Å where Rtriple = 3.46 Å. This radius is close to thatof a poly(Ala) helix, indicating a great packing flexibility when alanine isinvolved in the packing core, and vtriple is close to the mean observed twistangle. In contrast, the variety of possible theoretical solutions is limited forthe other two helix types. Besides the perfect matches, novel suboptimal‘‘knobs into holes’’ hydrophobic packing patterns as a function of the helixradius are described. Alternative ‘‘knobs onto knobs’’ and mixed modelscan be applied in cases where salt bridges, hydrogen bonds, disulphidebonds and tight hydrophobic head-to-head contacts are involved inhelix-helix associations.

An analysis of the experimentally observed packings in proteins con-firmed the conclusions of the theoretical model. Nonetheless, the observeda-helix packings showed deviations from the 180° periodicity expected fromthe model. An investigation of the actual three-dimensional geometry ofhelix-helix packing revealed an explanation for the observed discrep-ancies where a decisive role was assigned to the defined orientation of theCa-Cb vectors of the side-chains. As predicted from the model, helices withdifferent radii (differently sized side-chains in the packing core) wereobserved to utilize different packing cells (packing patterns). In agreementwith the coincidence between Rtriple and the radius of a poly(Ala) helix, Alawas observed to show greatest propensity to build the packing core. The ap-plication of the helix lattice superposition model suggests that the packingof amino acid residues is best described by a ‘‘knobs into holes’’ schemerather than ‘‘ridges into grooves’’. The various specific packing modes madesalient by the model should be useful in protein engineering and design.

7 1996 Academic Press Limited

Keywords: protein; helix; protein folding; helix packing; protein*Corresponding author secondary structure

Introduction

The topic of helix-helix pairwise packing inproteins was addressed soon after helical structures

had been suggested. Several models were devel-oped and were mostly devoted to surface comple-mentarities upon packing. Crick’s model (Crick,1953), later referred to as ‘‘knobs into holes’’,

0022–2836/96/030536–18 $12.00/0 7 1996 Academic Press Limited

Helix-helix Packing 537

introduced the unrolling of regular helices onto aplane and then finding the best fit of the resultinglattices (one point per residue). This was achievedby superposition in a face-to-face mannerthrough rotation followed by translation suchthat residues of one helix (knobs) fit into cellsformed by neighbouring residues in the otherhelix (holes). Assuming a helix radius R of 5.0 Aand a twist angle v of 100.0° between residuesalong the helix path, he found optimal packingat a dihedral packing angle V between the helixaxes at +20° (coiled-coil structures) and a subopti-mal packing at V = −70°. Richmond & Richards(1978) also pursued the knobs into holes modeland concluded further that the packing angle isinversely correlated to the helix radius. Theysuggested three possible classes of helix-helixpacking and, for each class, listed possible aminoacids central to the contact. These preferences wereutilized to predict spatial helical arrangements fromprimary structural information (Richmond &Richards, 1978; Cohen et al., 1979; Cohen & Kuntz,1987).

Chothia et al. (1977, 1981) introduced another andnow widely accepted interpretation of the superim-posed ‘‘helical’’ lattices. Instead of ‘‘knobs intoholes’’ packing, they coined ‘‘ridges into grooves’’.Here, the ridges formed by residues with sequentialspacing i in the first helix fit into grooves formed byresidues in the second helix with spacing j. Byassuming mean observed helix geometries, theyfound three basic packing types by varying i and j;namely, Vi=1,j = 4 = −105°, Vi=4,j = 4 = −52° andVi=3,j = 4 = +23°. In principle, yet other combinationsof i and j were possible (e.g. Vi=3,j = 3 = −109°);however, as they noted, these classes werebarely distinguishable from the former because oftheir similar packing angle and pattern of aminoacid contacts. They introduced yet another packingclass (‘‘crossed ridge’’ packing), where the ridges oftwo helices cross with expected packing angles at+55°, −15° and −105°. Chothia and his co-workersalso argued that the observed preference forpacking angles around V = −52° can be understoodin that ridges, formed by contact residues spaced byi = 4, dominate the shape and surface of the helicalface since they make the smallest angle to the helixaxis.

Efimof (1979) attempted to relate the packingangle with preferred rotational states of theside-chains along the helix. He distinguishedtwo types of packing; polar and apolar, eachgiving rise to different combinations of rotationalisomeric states of the contacting amino acidresidues. For a best fit, he proposed three discretepacking angles for the apolar case (V1+30°,1−30°, 190°) and a range of possible dockingangles in the polar case (−30°EVE30°). Reddy &Blundell (1993) correlated the distance of closestapproach between two packed helices to thevolume of the interface-forming amino acidresidues and used the resulting linear dependencyto predict the interhelical distance of structurally

equivalent helices in homologous proteins. Otherefforts have focused on the energetic aspects ofhelix-helix packing where different interactionpotentials ranging from burial of hydrophobicresidues (Ptitsyn & Rashin, 1975) and othersimplified interaction potentials (Solovyov &Kolchanov, 1984) to atomic energy minimizationand Monte-Carlo sampling (Chou et al., 1983,1984; Tuffery & Lavery, 1993) have been applied.Murzin & Finkelstein (1988) attempted to predictthe topology and orientation of certain helicalassemblies by arranging them in polyhedralshells. Harris et al. (1994) have performed a care-ful study of the diversity in four-helix bundleproteins.

The work presented here was stimulated by theobservation that observed helix-helix packingangles demonstrate a pronounced preference forV1−50°/130°. It is difficult to imagine whythis preference should be a result of the relativelength of one ridge along one helical side, asargued by Chothia et al. (1981), or due to the lesssplayed character of residues in the i = 4 ridge(Chothia et al., 1981; Hutchinson et al., 1994). Thecontact-forming residues in helix associationneed not belong to one and the same ridge.Maximizing the burial of hydrophobic surface uponcontact (presumably favouring smaller packingangles) or an easier and fitter packing of amino acidside-chains at a certain packing angle wouldseem to provide more natural explanations. Thus,the model of unrolled helix lattices was furtherinvestigated and treated mathematically in arigorous fashion. Which set of helix parameters (theradius of the helix R, twist angle v and the rise perresidue D) guarantees an optimal match andassociation of two identical and ideal helicallattices in a face-to-face manner after translating oneof them (homogeneous packing)? Can theambiguities in the ridges-into-groove model beresolved by considering optimization of thepacking density? To approach these questions,the conditions for optimal packing were mathemat-ically formulated to allow careful consideration ofall solutions. To check the theoretical model, astatistical analysis of experimentally determinedhelix-helix packings was effected. The lattershowed that a 180° periodic selection in Vwas not uniform. An explanation for this isprovided here based on the tertiary structuralconfiguration of helices, especially the Ca–Cb

bond direction. To the authors’ knowledge, thetreatment here is mathematically rigorous incontrast to all the previous works wheremore visual approaches were adopted and varioushelical geometric parameters were held fixed.The non-uniform V distribution has not beenpreviously addressed. The various optimal andsuboptimal packing modes made salient by themodel should aid in protein engineering anddesign, especially in selection of residue types toachieve specific helical contact sites or axialorientations.

Helix-helix Packing538

Table 1. Definition of symbolsSymbol Definition

Radius of a helixRD Rise per residue along the helix axis

Angular twist per residue along the helix pathvV Dihedral packing angle (sign conventions as in Chothia et al., 1981)

Identifiers for the three optimal solutions of the functions describing the lattice superpositiona, b, cIdentifiers for the three optimal solutions of the model equations where the mean values of D and v are taken fromam , bm , cm

helices observed in protein tertiary structuresVN Dihedral packing angle using the 180° rotation symmetry of ideal helix-helix packing; i.e. VN = V + 180° if V < 0°;

otherwise VN = Vt Angle measuring the deviation of the helix axis from the contact plane of the helix pair; i.e. the plane normal

to the line of closest approach. The angle is non-zero when, for straight or curved helix axes, the line of closestapproach is not perpendicular to at least one of the respective helix axes and crosses the axes at the helicalterminiDistance of closest approach between two fitted helix axesd

di Distance of closest approach between a local helix axis assigned to the residue i of the first helix and the secondfitted helix axis; i.e. the shortest distance between the position obtained by drawing a perpendicular fromthe geometric centre of the side-chain atoms of residue i (Ca for Gly) to its fitted helix axis to the second fitted helixaxis

a Skew angle (Harris et al., 1994) between a vector, obtained by drawing a perpendicular from the geometric centreof the side-chain (Ca for Gly) to its fitted helix axis and the local line of closest approach between the interactinghelicesSite of contact between a residue of one helix and a second helix; i.e. the geometric centre of the positions of twoPc

side-chain atoms of the residue of the first helix that are closest to the second helix axis (Ca only for Gly and Cb onlyfor Ala)Position of the side-chain atom of a contacting helical residue that is furthest from the fitted helix axis (Ca for Gly)Ptip

Ra Apparent helix radius defined as the distance of the Ptip-atom to the fitted helix axis; atomic radii were not consideredand were assumed to be compensated by side-chain–side-chain interdigitation upon packing

Several of these parameters are illustrated by Figure 1.

Mathematical Description

Homogeneous (hydrophobic) packing

Optimal (perfect) packing

The model used here assumes regular andstraight helices of radius R, twist angle v betweensuccessive residues along the helix path and a rise

per residue D along the helix axis. Various symbolsutilized throughout the text are listed in Table 1 andtheir definitions are illustrated in Figure 1.Unrolling an ideal helix onto a plane towards theobserver results in a regular lattice, as shownin Figure 2, where each point represents aresidue. In associating a-helices, each of thesame geometry, one lattice must be rotated relativeto the other about a lattice position such that thepoints of the two lattices overlap. Then anappropriately chosen translation of one lattice mustbe effected so that the knobs (points in onelattice) fall into the centre of parallelograms(holes) in the other helix (Figure 2). The parallelo-grams are formed by connecting four neighbouringpoints in one of the lattices. This packingoptimization, where infinite lattices of unrolledhelices overlap, is justified by the assumption thatthe global two-dimensional optimum coincideswith the best possible local packing optimum inthree dimensions.

This phenomenon can be mathematicallyformulated. Each lattice can be respectively de-scribed with two base vectors (v1 and v2; v"1 andv"2 ). In face-to-face packing of the helices, thebase vectors are related through mirror sym-metry: v'x;1,2 = −vx;1,2 and v'z;1,2 = vz;1,2, where x and zrepresent respective vector components and themirror plane contains the z-axis (Figure 2).Superposition requires rotation of one latticesuch that v"1,2 = RVv'1,2, where RV is a rotationmatrix corresponding to two helices with axialpacking angle V. The lattice point Pi is the centre ofrotation (Figure 2). Under the condition of perfect

Figure 1. Schematic drawing of the parameters used forthe description of helix-helix packing geometries. A'1 andA'2 correspond to the helix axes projected onto the contactplane, which is normal to the line of closest approach.Definitions of the parameters and associated symbols aregiven in Table 1.


Figure 2. Detail of a helical lattice created byunrolling an infinite regular helix onto a plane towardsthe observer and specified by an xz-coordinate system.The origin is set onto one lattice point Pi representingresidue i. The vectors correspond to possible base vectorsfor the lattice. The indices of the points refer to respectiveamino acid sequence positions along the helix relativeto residue i, where k = [2p/v] + 1, where 2p/v ismade integral through truncation and v is in radians.The sequence separations for average a-helices are givenin parentheses. Two of any of the three base vectorsshown can be chosen to specify the lattice (threepossibilities). The grey coloured parallelograms corre-spond to the three possible topologically possible packingcells (holes) into which the lattice points (knobs) can befit, resulting in helix-helix packing. The identifier of agiven cell is calculated from indices of lattice pointsassociated with the cell, which are summed afterbecoming powers to the base 2, and k is assumed as 4; forexample, a cell bonded by i, i + 3, i + 4 and i + 7 isidentified by 153 = 20 + 23 + 24 + 27; similarly for cell 27(20 + 21 + 23 + 24); cell 51 and so forth.

Without loss of generality, two base vectors withrespective components can be selected (Figure 2)such that:

v1 = Pi+1 − Pi = 0ARD 1 (3)

v2 = Pi+k − Pi = 0BRkD1 (4)

where A = v, B = kv − 2p and k = (2p/v) + 1, where2p/v is truncated to an integral value and v is inradians. Since the helices are considered ascylinders, the x-co-ordinate corresponds to arcs onthe cylinder. As shown in the Appendix, this systemof equations yields three distinct solutions (Table 2)for the packing angle V, corresponding to packingclasses designated here as a, b and c. The solutionswere found to possess a 180° periodicity asindicated by their signs. The functions for eachclass, corresponding to particular values of n1..4, aresubsequently given where the (b) relationships forR/D are derived by squaring and summing the (a)relationships:

class a:

cos(V) = 2(1 − k)B + kAkA − B ,

hG

G

J

jsin(V) = 2RD

2AB − B2

B − kA (5a)

0RD1

2

= kA(2k − 4) + kB(2 − k)B3 − 4AB2 + 4BA2 (5b)

class b:

cos(V) = 2A − kBkA − B , sin(V) = 2R

DA2 − B2

B − kA (6a)

0RD1

2

= k2 − 1A2 − B2 (6b)

overlap, the mirrored and rotated lattice can bedescribed through a linear combination of theoriginal base vectors (v1,2) with four integer factors(n1, n2, n3 and n4) such that:

v"1 = n1v1 + n2v2 (1)

v"2 = n3v1 + n4v2 (2)

where n1..4 = 0, +1 or −1.

Table 2. The six solutions for optimal superposition of ideal helical latticesCorrespondinga Vb Rc Classd

n1 n2 n3 n4 solution equation: (deg.) (A) ident.

(1) −1 1 0 1 (5a, b): + sign −37.1 3.0 a(2) 1 −1 0 −1 (5a, b): − sign 142.9 3.0 a(3) 0 1 1 0 (6a, b): + sign −97.4 3.5 b(4) 0 −1 −1 0 (6a, b): − sign 82.6 3.5 b(5) −1 0 −1 1 (7a, b): + sign 22.0 4.3 c(6) 1 0 1 −1 (7a, b): − sign −158.0 4.3 c

a Solution corresponding to the given sign conditions in the specified equations.b Values given for the packing angle assume a twist angle v = 99.1°, which is the mean observed

value in known protein structures.c The helical radius given assumes v = 99.1° and D = 1.45 A, values most often observed in actual

helices.d Class identification (see the text).


Figure 3. Graphical representation for the three solutions (packing classes a, b and c) of optimally overlapped helicallattices. Calculated optimal packing angle VN and calculated ratio R/D are shown as functions of the helix twist anglev for the given interval. The vertical lines correspond to the mean twist angles for the three helix types(a − (�v� = 99.1215.0°), 310 and p-helices). The filled circles in B correspond respectively to the ratios R/D for differentamino acids obtained by dividing the mean observed distances of the ‘‘tip’’ atoms (Ptip) of the residues Gly, Ala, Valand Leu to the axis of a-helices (i.e. the apparent helix radius) with the corresponding mean rise per residue of therespective helix type (�D� = 1.45 (21.1) A) for the a-helix, 2.0 A (310-helix) and 1.1 A (p-helix)). The radii for the 310 andthe p-helix were corrected for their different Ca based radii (1.9 and 2.8 A, respectively). �v�a-helix was taken from themean observed twist angle between the geometric centres of side-chain atoms for two consecutive helix residues. Thesemean values (�v� and �D�) did not include observations based on the four residues at either helix terminus wherethe helix axis is less accurately determined (see the text). Values for the 310 and p-helices were taken from Schulz &Schirmer (1979).

class c:

cos(V) = 2B + (k − 1)AkA − B ,

hG

G

J

jsin(V) = 2RD

2AB − A2

B − kA (7a)

0RD1

2

= A(2k − 1) + B(2 − 4k)A3 − 4A2B + 4AB2 (7b)

It is clear that, for each class, the packing angleV is a function of the twist angle v, as is the ratioR/D. VN and R/D can be plotted against v (Fig-ure 3) for each class. As evident from the equations,each solution has a restricted definition space andthere are singularities at different twist angles. Fordifferent k dependent upon v, the solutions repeatbut have different periods. The solutions for allthree classes recurrently cross at points (Figure 3) inthe radius dependency (for the triple point closestto the mean observed twist angle for a-helices �v�a,vtriple = 96.9°). The corresponding helix lattices areregular hexagons such that the three solutions for

the packing angle can accommodate the same helixradius. At �v�a, the three packing classes displaydifferent radii. The smallest accommodated radiusis found in packing class a (R = 3.0 A) and thelargest in class c (R = 4.3 A) assuming the meanobserved rise per residue �D� = 1.45 A. Only forhelices of the a-type is the simultaneous occurrenceof all three packing classes allowed and the meanobserved value �v�a is found closer to a triple pointthan the mean twist angle of the other helix types(p and 310). Thus, the helical lattice obtained fromunrolling an average a-helix more closely resemblesa regular hexagonal lattice, which allows threepacking angles simultaneously. The p-helix does notcorrespond to solution c and the mean twist anglefor 310 helices (�v�310 = 120.0°) coincides with anambiguity where the solution switches from k = 4 tok = 3 with increasing v. Furthermore, the requiredhelical radii corresponding to v intervals about theobserved mean of a-helices fall in a biologicallyreasonable range (Figure 3B), whereas for the otherhelix types, the required radius for one solution hasto be either infinite (solution c for p-helices) or isambiguously defined (solution a and b for 310


helices). Solution c for 310 helices is found just at thetransition to an infinite radius.

Suboptimal packing

Apart from analysing perfect lattice overlap,helix-helix axial association angles may be observedcorresponding to local packing optima where, forexample, base vectors are superimposed such thatregularity is achieved in only one lattice direction.Since helices often pack over a few turns, thecondition of infinite lattice superposition may betoo strong and not fulfilled. Thus, only the nearestsix neighbours around a central lattice point areconsidered to determine suboptimal packing. Asbefore, the central points of two sublattices arebrought into coincidence. Subsequently, one sub-lat-tice is rotated at the angle V around the centralpoint. A packing parameter SP was constructed tomeasure the degree of overlap between the twosublattices:

SP(VN, R) = sm

min 0=Pm (R) − Qn=1..6(VN, R)=R 1 (8)

where m = 1..6 corresponds to the six neighbouringlattice points of Pi (Figure 2), the lattice vectors topoints Pm are held fixed and Qn are the vector

positions in the mirrored and rotated lattice. Theangle VN covers the range 0 to 180° and implies asecond possible packing angle at V = VN − 180° dueto the rotational symmetry of ideal helices. Since inthe model an increased helix radius implies thesame for the helix-forming side-chains, the dis-tances between the lattice points were normalizedto the helical radius (R).

The behaviour of the function SP(VN, R) is shownin Figure 4, where its value is plotted against VN

with D and v taken as the mean observed values fora-helices. The three minima of the optimal packings(SP(VN, R) = 0) can be readily identified. For a givenradius more than one packing angle meets therequirements of little steric clash, increasing thepossible number of helix-helix packing angles. Forexample, at larger radii, in addition to the steepminimum at VN120°, a broad but shallowminimum is found near VN = 120°.

Determination of the translation vector for thesuperimposed lattice

Only optimal superposition of lattice pointsthrough rotation has been thus far considered. Foractual helix-helix packing, the translation vector bywhich one of the superimposed lattices is shifted tobring the knobs (lattice points) into holes (lattice

Figure 4. Three-dimensional plot of the function SP = SP(VN, R). The colour spectrum corresponds to differentisosurfaces where violet colors belong to minimal values of SP. The twist angle and the rise per residue were respectivelytaken as the mean observed values, v = 99.1° and D = 1.45 A. The grey shadows are shown merely for dimensionalperspective and reflect the direction of the light source.


Figure 5. Size of the packing cells 27, 51 and 153 as afunction of the helix radius. The size is defined by thelength of the smaller diagonal associated with therespective packing cells. The twist angle was set to 99.1°and the rise per residue D to 1.45 A, the respective averageobserved values. Arrows indicate observed radii ofhelices composed only of Gly (G), Ala (A), Val (V) andLeu (L).

bigger radii, side-chains should prefer to pack intocell 153. According to the size criterion adoptedhere, cell 51 is never the largest packing cell. For thisreason, and since in cylindrically shaped helices cell51 is mostly oriented away from the helix-helixinterface, it will not be considered further.

Figure 6 shows the three possible perfectlysuperimposed helix lattices where the meanobserved twist angle is taken. The selected packingcells are cell 27 for solutions am and bm and cell 153for cm .

Non-homogeneous packing

The knobs-into-holes model assumes that theamino acid side-chains pack isotropically into a holeformed by four side-chains of the second helix andwith parallelogram cross-section. This might not benecessary if some other interaction joined the twohelices, such as bonds formed between theassociating amino acids, including disulphidebonds, salt bridges, hydrogen bonds or tighthydrophobic head-to-head contact (knobs ontoknobs packing). If the packing site merely consistedof this type of contact alone, the preferred packingangles would remain the same as those derivedfrom superposition of the helical lattices but notranslation would be necessary. This situation isunlikely. Nonetheless, a mixture of knobs-into-holesand bonded contacts are yet possible. Chothia et al.(1981) have coined the term ‘‘crossed ridge helixpacking’’ for these cases. The possible packingangles for this association type can be obtained bya consideration of suboptimal packing in the modeldeveloped here. The superposition of the twocentral lattice points can now be interpreted as aresidue-residue bond of any type. Since theneighbouring residues should still obey the normalknobs-into-holes scheme but without shifting, thefunction SP(VN, R) for the sublattice has now to bemaximal instead of minimal. In agreement with theangles predicted by Chothia for the crossed ridgecase, Figure 4 reveals three isolated maxima at

cells) must be applied. In accordance with the threetopologically possible lattice cells (referred to as153, 27 and 51), three translation vectors arepossible where lattice points are shifted to theircentres (Figure 2). The cellular designations areexplained in the legend to Figure 2. To achieve themost homogeneous and dense packing, the largestpossible cell must be selected for association with aside-chain of an interacting helix. The length of thesmaller diagonal of each cell was chosen as a simpleestimate of cellular size. (The area of an inscribedcircle, an alternative, does not exist for parallelo-grams.) The plots of Figure 5 demonstrate that acell’s capacity depends on the helix radius and thusdifferent cells are preferentially occupied atdifferent helix radii. For helices with smaller radii,cell 27 should be favoured, while for helices with

Figure 6. Helical lattices according to the theoretical solutions for packing classes am , bm and cm at the mean observedparameters (v = 99.1, D = 1.45 A) and the corresponding packing angles V(am , bm , cm ) = −37.1°, −97.4° and +22.0°,respectively. Starting from perfect superposition achieved by rotation of the mirrored lattice (open circles), one latticewas shifted to centre the lattice points in the appropriately chosen packing cells (see the text). The continuous (broken)line denotes the helix axis of the lattice with the filled (open) circles.


VN155°, 115° and 175°. In addition, helices withlarge radii should also pack at VN175°.

Analysis of Observed Helix-HelixAssociations

The theoretical model used here assumes thatideal helices pack. In the following section, theexperimental verification of the conclusions drawnfrom the mathematical lattice superposition modelis discussed.

Data

A total of 220 protein tertiary structures,determined at 2.0 A resolution or better and withmutual sequence similarity less than 35% asselected by the program OBSTRUCT (Heringaet al., 1992, available via World-Wide-Web; URL:

http://www.embl-heidelberg.de/obstruct/

obstruct info.html)

were used for a statistical analysis of helix-helixassociation (the set is available upon request bye-mail to [email protected]). The assign-ments of the a-helical stretches were taken from theprogram DSSP (Kabsch & Sander, 1983). The anglebetween two consecutive carbonyl bonds was notallowed to exceed 65°; otherwise, the helix wasdivided into two at this residue. Two helices weredefined to be in close contact if at least threeresidues of each helix had at least one interhelicalatom-atom contact with maximal threshold distanceof 4.5 A between atom centres. The resulting datasetof proteins used in this study contained 687 closelypacked pairs of helices. Membrane proteins werenot included and only heavy atoms were con-sidered.

Definition of the helix axis

The definition of the helix axis from which manycontact characteristics are measured bears criticallyon the results. Since helices can be bent, a procedureto fit a local helix axis, Ai , to every residue i alongthe helix was adopted. It takes advantage of astraightforward algorithm for the overall axialdefinition given by Chothia et al. (1981). The vectorcoincident with the local helix axis of residue i, ui ,can be determined from the cross product of thevectors Bi and Bi+1 such that:

ui = Bi × Bi+1 (9)

where:

Bi = ri + ri+2 − 2ri+1 (10)

and r is the position vector of the Ca atom in residuei. At the C terminus of the helix, where the residueindices would go beyond those in the helix, the localline vector is taken from the closest helicalconstituent residues. A point on the local axis Ai isassigned by calculating the geometric centre of the

closest four consecutive Ca positions around theresidue i (i.e. Ca,

j + i − 1 . . . Ca,j + i + 2 where j = 0 for the

inner helical residues and appropriately chosen atthe helix termini and the points are correspondinglyshifted along ui ). The length of the local axis is firstset to 1.5 A. The direction of the local axis Ai

associated with residue i is then smoothed by takingthe average direction of three consecutive localvectors centred at i (two at the helix ends). Toachieve a continuous axial curve over the entirehelix, the new starting and ending points ofconsecutive local lines are joined by calculating themiddle point between the end point of the first localstretch and the starting point of the next localstretch. The new lengths and directions of the localaxes are then recalculated. This smoothing pro-cedure is repeated three times. Despite thesimplicity of this algorithm, the improvement forthe fit of the local axis is considerable. The standarddeviation of the distances of each Ca atom to thehelix axis decreased from s = 0.34 A for a globallydefined axis (obtained by averaging the vectors ui

over the whole helix and taking the geometric centreof every Ca position) to s = 0.14 A when using thelocal axes. When only the inner helical residueswere considered (four residues subtracted at eitherhelix termini), the accuracy was improved. Thestandard deviation decreased from s = 0.37 A forthe global axes to s = 0.07 A for local axes.

The packing angles are positive if the backgroundhelix is rotated clockwise with respect to the frontalhelix when facing them. The helices are parallelwith respect to their sequence direction at V = 0°.The packing angles are sometimes normalized to theinterval 0° < VN<180° because of the 2-fold rotationaxis of ideal helices; i.e. VN = V + 180° if V < 0 ;otherwise VN = V. The line of closest approachbetween two helices was calculated by determiningthe line of closest approach with minimum lengthamongst all local axes pairs. To ensure a face-to-facepacking, distorted helix-helix associations wherethe line of closest approach intersected at the helixtermini and t > 5° (Table 1 and Figure 1) wereomitted, leaving 449 helix pairs for analysis.

Packing cell determination

The packing cell of a second helix utilized by acontacting residue in the first helix was determinedby the sequence separation of four residuescontaining, respectively, one of the four closestatoms (one closest atom per residue) to thegeometric centre of the two atoms in the contactingresidue (Ca for Gly and Cb for Ala) in the first helixthat are closest to the axis of the second helix(position of contact, Pc; see Figure 1 for illustration).To ensure a real packing conformation, the thirdclosest residue of the second helix to the contactingresidue in the first helix was required to be withina distance of 6.0 A. Despite the imprecision of thecell-determining procedure, the observed ranking ofcell usage is as predicted. The three topologically


possible cells (153, 27 and 51) were detected mostoften with respective counts 767, 647 and 228. Otherdetermined cells such as 23 and 275 had frequenciesof 48 and 42, and were followed by others.

Algorithm for interhelical ‘‘bond’’determination

Interhelical bonds were determined on the basisof geometric pattern recognition. A bond wasidentified between two side-chains in differenthelices if their corresponding Pc sites were mutuallythe closest to each other. The Pc sites must be nomore than 4 A apart and the closest Pc site for otherresidues in the same helix must be 5 A or greater.Furthermore, the angle between the local line ofclosest approach and the vector joining the twomutually closest positions Pc was required to besmaller than 45°. These conditions assured knobs-onto-knobs packing, and that identified residuesliterally faced each other and did not pack into a cellformed by the oppositely facing helix. For 95helix-helix pairs, this definition was fulfilled; 80such pairs had only one interhelical bond while 15displayed two.

Results

The distribution of the observed global (perhelix-helix pair) and local (per amino acid residuealong the two packed helices) dihedral packingangles in the selected set of proteins is shown inFigure 7. To a certain extent, the histogram of thelocal packing angles biases the observations to moreparallel or antiparallel associations because oflonger possible contact regions. Yet, it allowsconsiderations at which angle packings are possibleover a longer stretch where the lattice model iscertainly more critical. To account for possiblerestrictions due to short loops connecting twosuccessive helices along the chain, which disallowsparallel packing, the condition of more than 20intervening residues was applied in a secondhistogram. In a third histogram, all helix-helix pairswere used except those displaying interhelicalbonds.

The two largest peaks occur in the intervals−70°EVE−20° and +110°EVE+140°. The mediumpeaks are found at −170°EVE−150° and−110°EVE−90°. Fewer helical pairs pack at+10°EVE+60° and +160°EVE+180° . As indi-cated by arrows in Figure 7, in the negative angularrange the optimal solutions (am , bm and cm ) of thehelical lattice superposition model match theobserved peaks well. In the positive range, the classa peak at V1+142° misses the observed peak by20°, which rather corresponds to the angle of thepredicted suboptimal solution for larger helix radii.The positive class b peak falls at a peak shoulder.The expected peak at V1+22° is little observed inthe histogram of the global packing angles.

Figure 7. Frequency histogram for the observeddihedral helix-helix packing angle V (bin width 10°).A, Packing angle about the global line of closest approach.B, Histogram of local packing angles; i.e. the packingangle about the local line of closest approach defined foreach contacting amino acid in the helix-helix pair (Fig-ure 1). The light grey filled histogram corresponds to databased on all observed helix-helix pairs while the darkgrey filled histogram was determined from those withmore than 20 intervening residues between the end of onehelix in a contacting pair and the beginning of the otherhelix. The third histogram (thick line) corresponds to allhelix-helix pairs with no detected interhelical bond.Arrows show the predicted packing angles for the threeoptimal solutions (am , bm and cm ) according to thetheoretical model developed here; the mean observedtwist angle was taken as 99.1°.

The correlation between frequencies of packingangles in the negative range to its periodic angle inthe positive range; i.e. rf(V<0), f(V+180°), was 0.74. Theabsence of a pronounced peak at V1+22° isperhaps explained by the repulsion of the dipolemoments in nearly parallel helices. However, theimportance of this effect is doubted by severalauthors, and theoretical and experimental resultsindicate that the effect may be limited (Chou &Zheng, 1992; Robinson & Sligar, 1993). Further-more, in the histogram of the local packing angles(Figure 7B), the expected peak at V1+22° is betterobserved, indicating that the packings at larger


Figure 8. Observed density of interface atoms atdifferent packing angles. The core region is defined by asphere centred at the middle of the line of closestapproach (length d) between the two helices and withradius r = d/2. Only atoms belonging to contact-formingamino acids have been considered; i.e. at least oneinterhelical atom-atom contact was within 4.5 A. Further-more, it was required that all helix termini (Ca positions)be outside the defined sphere. Arrows correspond to theoptimal packing angles for classes am , bm and cm . Resultswere obtained by a running average over 20 successivepoint clusters of the ordered helix data pairs with respectto VN with a mean standard deviation �s� = 0.013 A−3.

Figure 9. Radius dependency as a function of thedihedral packing angle as expressed through a 30-pointrunning average of the distances of closest approachordered with ascending V values. Arrows correspond tothe optimal packing angles for classes am , bm and cm . Themean standard distribution for the 30-point clusters was1.2 A.

angle of helices with large radii (vide infra).Interestingly, the highest core densities are found atthe most frequently observed packing angles(V1−40°/+130°, Figure 7). The plot shows devi-ations from 180° periodicity.

Radius dependency of helix-helixpacking angles

In the present model, it is assumed that theassociating helices have the same radius. A checkwas made regarding this assumption. The distanceof the side-chain atom that belongs to a contactingresidue in one helix and that is the furthest from itsown helix axis was projected onto the local line ofclosest approach between the two helices and thennormalized by division with the closest approachdistance. Residues were considered only if the anglebetween the perpendicular drawn from thegeometric centre of the side-chain to its helix axisand the local line of closest approach was smallerthan 25° to ensure that the residue points almostdirectly to the second helix along the local line ofclosest approach (‘‘radius’’ at the interaction site). Itwas further required that identified contactingresidues be surrounded by at least three side-chainsof the second helix with closest side-chains atomswithin 6 A to the Pc site of the contacting residue.Good knobs-into-holes packing was thus guaran-teed. The normalized distances for residuesfulfilling these conditions have a mean of 0.44 anda standard deviation s of 0.08 over 491 residues.The relatively low value of s confirms that theassumption of similar radii is reasonable for themany contacting regions.

It is clear from Figure 3B and the behaviour ofthe function SP(VN, R) (Figure 4) that for theoptimal solutions the radii of the helices and thusthe distance of closest approach between packed

positive angles are accommodated by shortercontact regions for which the lattice approach isnaturally less stringent. An examination of thepacking parameter SP (Figure 4) at VN1+60° andV10° indicates steric unacceptability at all helicalradii, which explains the paucity of observationsnear these packing angles where only interhelicalbond packing is possible. Nonetheless, from thetheoretical model described here, the observeddistribution is likely to consist of a mixture ofoptimal and suboptimal solutions that are a functionof the helical radii as well as knobs-onto-knobspacking cases. A detailed structural investigationwill be subsequently described.

Goodness of packing at different packingangles

Helix-helix interfaces should display a high atomdensity at their association site (core). In Figure 8this property is shown as a function of packingangle. The packing core was defined by a spherecentred at the middle position of the line of closestapproach and with a radius of half the length of theline (r = d/2). The running average of the densitieshas pronounced maxima. Except packings at V122°(very few data), high densities are generallyachieved in agreement with the predicted optimalpacking classes am , bm and cm (i.e. highest possiblehomogeneous packing density), which are derivedfrom the theoretical model by using v = 99.1°, themean observed value. The peak at V1140° is broadand extends to V = +120°, a suboptimal packing


helices should increase in the order R(am ) < R(bm )< R(cm ). As shown in Figure 9 (mean closestapproach distances versus the packing angle), thesepredictions are confirmed by the observed data inthe negative angular range. Shortest distances ofclosest approach are found in agreement with theoptimal knobs-into-holes packing classes, whichallow interdigitation of the side-chains. As pre-dicted, the distances increase from am to bm and cm .However, the distribution is, like the histogram ofthe packing angles (Figure 7), not periodic. In thepositive range, very close distances are notobserved. Only a shallow minimum (knobs-into-holes interdigitation) is observed in an intervalwhere a suboptimal packing for larger helix radii ispredicted. This is consistent with the observationthat the main peak in the histogram of packingangles in the positive range (Figure 7) actuallyagrees with the suboptimal packing arrangement ofhelices with larger helix radii (Figure 4). It seems asif the required small helix radii for the optimalsolutions am and bm (short distances of closestapproach), practically corresponding to packings ofGly or Ala in the packing core, are less compatiblewith positive packing angles. This recurrent andconsistent deviation from periodicity can beexplained by the actual spatial packing geometriesof the two periodic solutions that are naturallyexcluded in the two-dimensional lattice model (seeDiscussion).

From Figure 4, it is evident that helices with radiibetween 3.0 A and 4.5 A fall simultaneously intothe three minima of the function SP(VN, R). Thus,residue types that impart such helix radii shouldprovide the largest packing flexibility and thereforebe advantageous over others and correspondinglybe often observed as radius-determining residues inthe packing core; i.e. they should display a smallskew angle a (see Table 1 and Figure 1 for definitionand illustration) to the other helix axis. At larger aangles, the size of the residue becomes significant,such that extended residues can compensate for thecylindrical shape of helices and are observed moreoften because of the closer distance to the secondhelix. It can be shown that the mean skew angleincreases from Ala to Val to Leu (�a�Ala = 41.1°,�a�Val = 43.2° and �a�Leu = 47.7°). The logarithmicrelative frequencies of the 20 amino acids as packingcore residues normalized to the mean relativefrequency of the residue within helices generally isshown in Figure 10. The relative probability ishighest for Ala followed by Val and Ile. The radiusof a poly(Ala) helix is near 3.4 A and 4.5 A forpoly(Val) and 5.2 A for poly(Ile). The preference ofAla and Val is thus explained by their ability toallow flexible packing arrangements to achieve anoptimal protein fold.

Radius dependency of packing cell (holes)occupancy

Larger residues of one helix should preferablypack into the largest cell formed by the residues i,

Figure 10. Logarithmic relative probability for theoccurrence of individual amino acids in the packing core.Only side-chains with a corresponding skew angle asmaller than 25° were taken under the condition thatdi < d + 1.0 A (Figure 1, Table 1). The obtained relativefrequencies were normalized by the relative frequenciesof the appropriate amino acid as found in all helices of thedataset. Listed in the plot are the specific helical radiicorresponding to helices composed entirely of a givenamino acid type and the respective observed standarddeviations. Numbers near the bars correspond to thenumber of observations.

i + 3, i + 4 and i + 7 (cell 153) and smaller residueswould prefer the smaller cell formed by residues i,i + 1, i + 3 and i + 4 (cell 27; Figures 2 and 5).Corresponding preferences for these cells areconfirmed qualitatively by the experimentallyobserved data (Figure 11A). The small amino acidsGly and Ala clearly prefer to occupy cell 27. At anapparent helix radius of 4.5 A (distance of the tipatom of the residue occupying the cell to its helixaxis), the occupancy is balanced between cell 27and cell 153. Larger residues (Ile, Leu and Phe)prefer cell 153 as predicted. The assignment of thepacking cell is based on the position of contact (Pc)and not the tip atom (Ptip). For bigger side-chainsthese two positions may differ, likely explaining theweak difference for the two cells in occupancies atR > 7.0 A. It was observed that cell 51 also preferslarger amino acids; since it is somewhat to the sideof the helix-helix interface, extended residues mustreach like arms to fill it. The discrete nature ofresidue sizes used in packing is also evident fromthe clear peaks in the plot of Figure 11A.Furthermore, the more direct approach relating thedistance of closest approach to the predominantlyused packing cell type at a given helix-helixinterface also confirms the theoretical conclusions(Figure 11B). More closely packed helices prepon-derantly utilize cell 27, while cell 153 dominates forhelices further apart at the association site.

Correlation of packing angle and preferredpacking cell

Since the helix radius is related to the packingangle and to the packing cell predominantlyoccupied, a well-defined correlation should exist


Figure 11. Correlation between the helix radius and theoccupied packing cell. A, Normalized histogram ofoccupancies of a specific packing cell (hole) are plotted asa function of the length (size) of the occupying side-chaindefined by the distance of its tip atom to its helix axis(apparent helix radius Ra, Figure 1). Packing cell 27 isindicated by a broken line and cell 153 by a continuousline. For comparison, the mean distances for selectedamino acids as found in all helices of the protein datasetare indicated by the arrows. The bin width was taken as0.25 A. B, Normalized histogram of observed distances ofclosest approach for helices with a predominantly packedcell 27 (broken line, 34 examples) and cell 153 (continuousline, 48 examples). The respective difference in thenumber of occupied packing cells of the two types for agiven helix-helix contact region was larger than 2 toensure cell-type dominance. Conditions that deem a celloccupied are discussed in the text. The bin width wastaken as 1 A.

Figure 12. Relative occupancies of packing cells (holes)as a function of packing angle VN. A helical pair wasassigned to only one cell packing type according to themost frequently occupied cell along the contact. Countsare registered only if the number of occupied cells of type153 (continuous line) is larger by at least +3 than thenumber of occupied cell types 27 for a single pair ofpacked helices (48 examples) and vice versa for cell 27(broken lines, 34 examples).

are clearly distinct for cell 27 at VN1150° where themodel predicts association of helices with smallerradii. Peaks for cell 27 are found also at VN180° andVN140°. The former angle corresponds to anoptimal solution (class b) and the latter can beidentified as suboptimal for helices with smallerradii. The distribution of Sp(VN,R) for helices withlarge radii has two minima, in contrast to that ofhelices with smaller radii, which exhibits threeminima (Figure 4). This is confirmed by the datashown in Figure 12, which reveals that the broadpeak at VN1130° actually comprises two differentpacking modes, optimal packings (cell 27 peak) andsuboptimal packings (cell 153 peak).

Non-homogeneous associations

Besides side-chain interdigitation facilitated byvan der Waals contacts of apolar atoms (knobs intoholes), interhelical salt bridges, disulphide bonds,hydrogen bonds and tight head-to-head van derWaals contacts can constitute interhelical contacts,referred to here as interhelical bond interactions(knobs-onto-knobs). Indeed, cysteine and chargedresidues, and the polar asparagine were found toshow the highest propensity of forming such bonds.Helix-helix associations with only one such interhe-lical bond were found more often at the expectedpacking angles (vide supra), provided that at leastone helix of the pair had less than 12 residues (37examples, data not shown). In longer helices withlarger contact regions, the packing angles behavedaccording to knobs-into-holes where hydrophobiccontacts dominate.

between the packing angle and the preferredpacking cell. At packing angles preferred by larger(smaller) helices, the packing cell 153 (27) should bemainly occupied. By assigning each pair of packedhelices to one cell class determined by theprevailing cell type used, the resulting distributionis in good agreement with the predictions of themodel (Figure 12). Because of the sparseness ofdata, the packings were normalized to the range0°EVNE180°. Cell 153 is preferably occupied overtwo packing angle intervals. It is the dominating cellat VN125° and occurs also at VN1130°, asuboptimal packing angle for larger radii. The peaks


Discussion

Deviation from the 180 ° periodicity, limitationsof the two-dimensional approach

A model for helix-helix packing based onsuperposition of two planar lattices yields 180°periodic solutions in the packing angle V. However,the observed properties show deviations fromperiodicity. In particular, the predicted optimalsolutions am and bm are not convincingly representedby the experimental data in the positive angularrange, neither the packing angles (Figure 7) nor atthe expected smaller radii (Figure 9). What causesthis discrepancy? Why is packing with shortdistances of closest approach (small helix radii)disfavoured in the positive V range? Three mainfeatures of the real spatial structure of a-helicesare not described by a two-dimensional model: (1)the cylindrical shape; (2) the radii along the helixare discrete rather than continuous, as are theside-chain orientations (rotamers); and (3) the non-orthogonal extensions of side-chains; i.e. the Ca–Cb

vectors leave the helix backbone under a definedangle (extension angle) and are not, as assumed bythe model, straight extensions of the perpendiculardrawn from the Ca-positions to the helix axis. Thislatter property has been shown important incausing different oligomerization states of coiledcoils (Harbury et al., 1993).

Principally, for a real three-dimensional butregular helix, the lattice obtained by unrolling sucha helix onto a plane coincides with that used in themodel. Despite an apparently smaller helix radiusfor the same helix-building amino acids caused bythe extension angle, the solutions for the packingangles would still be 180°-periodic but differ only ina translation of one lattice. However, in threedimensions, the extension angle entails differentalignments of the Ca–Cb vectors of side-chainsperforming interhelical contacts (angle g) and thusdifferent mutual orientations for the contactingresidues; i.e. between the knob and correspondinghole residues. The angle g between the Ca–Cb

vectors of two contacting side-chains (at least oneinter-helix atom-atom contact shorter than 4.5 A)correlates at 71% with the angle between thecorresponding Ca geometric-centre-of-side-chainvectors. Obviously, the alignment angle g dependson the packing angle V, as demonstrated byFigure 13. The sinusoidal shape of the observedmean reflects the full-circle rotation in V. Further,not only does gmean vary with the packing angle butalso the observed standard deviations sg. High sg

values reflect side-chain–side-chain contacts ofresidues with alternately nearly parallel (small g)and antiparallel Ca–Cb (large g) vector pairs,whereas smaller deviations point to more regularpacking with the corresponding mean g in the 90 to120° range. In this respect, the optimal solutions am

and bm show more regularity in the negative packingangle range than in the positive range. For solution

Figure 13. Observed angle (g) between the Ca–Cb

vectors of two interhelically contacting side-chains.Shown are the mean value (A) and the standarddeviations (B) as a function of the packing angle of thecorresponding helix-helix pair obtained by a 100-points(black lines) and 50-points (grey lines) running average ofthe V-ordered data points. The black line corresponds toall observed g-angles (4122 events) in A. In B, The rawdata for the black curve were the standard deviations ofg per helix-helix pair. The 100-point clusters of therunning average had a mean standard deviation of 10.3°.The grey lines were obtained for observations where atleast one Ca–Cb vector of the contacting residue-residuepair made an angle to the global line of closest approachoriented to the adjacent helix smaller than 45° (2747events); i.e. residues centrally involved in the helix-helixpacking. Arrows correspond to the three periodic optimalsolutions of the helical lattice superposition model.

cm , the standard deviations are slightly smaller inthe positive range but the orientations of the Ca–Cb

vectors have less impact on the packing becauseof the larger required helix radii. Figure 14 revealsthe consequences of the systematically differentg-angles on the helix-helix packing. In the case ofalternating parallel and antiparallel Ca–Cb vectors(henceforth called alternating packing), where theoptimal solutions am and bm are in the positivepacking angle range (Figure 13), the three-dimen-sional packing differs from the more regular(g-angles) packings (henceforth called regularpacking). Figure 14 illustrates this for solution am .Solutions am and bm require small helix radii(Figure 3) and, consequently, short distances ofclosest approach. This is achieved by small residuesin the packing core (preferentially Gly, Ala or Pro).In three dimensions, the planar lattice approach


may be understood as packings of helices ‘‘un-rolling’’ their side-chains onto the surface of theother. Thus, side-chains outside the packing coremay be larger, thereby fulfilling the planar packingconditions and filling the crevice that would beopened up by the packing of ideal cylinders. This issupported by experimental observations such as theincreasing mean skew angle from Ala to Val to Leu(vide supra). This mutual (‘‘gearwheel’’) unrolling isdifferent for regular and alternating packings. In thealternating packing case, knob-residues repeatedlypack with hole-residues from the other helix withnearly antiparallel Ca–Cb vectors (Figure 14).Obviously, given a corresponding packing angle V,alterations of the side-chain sizes are less tolerablein this case where steric clash of the respectiveside-groups from the two helices can easily resultbecause of the parallel or facing Ca–Cb vectors. Inthe regular case, steric hindrance is less likelybecause the hole-residues point away from theinterface and may even be extended. Consequently,regular packings may have short closest approachdistances and more sequences (greater tolerance todifferent side-chain sizes) fulfil the requirement forsmall helix radii for solutions am and bm . Alternatingpacking generally has larger distances of closestapproach and thus, instead of utilizing the optimal

solution am , they go to the next accessible solutionfor helices with larger radii; i.e. V1120 to 130°(Figure 4). The same principle considerations holdfor the bm periodic solutions. The next accessiblepacking mode for solution bm is also the suboptimalwith V1120 to 130°, resulting in frequent obser-vations for this packing angle range (Figure 7).

Ridges into grooves: a model lackingstructural details

In the work presented here, helix-helix packingwas studied theoretically from the perspective ofthe helical lattice superposition concept, whichallowed all possible associations to be systemati-cally considered from a purely mathematicalperspective and is not found in previous work(Crick, 1953; Efimof, 1979; Chothia et al., 1981).Thus, a more complete understanding of packingoptions, both optimal and suboptimal, has beenachieved.

The lattice superposition model treats the packingproblem on the basis of individual side-chains asthe smallest packing unit, while higher-orderstructures are assumed by the ridges into groove(r/g) model where the dominating shape feature ofhelices are considered smooth, and continuous

Figure 14. Differences in the packing between the two 180°-periodic solutions of class am ; illustration of the regularand alternating packing mode. The pictures show real examples of packed helices with corresponding packing anglesand distances of closest approach illustrating the differences in the Ca–Cb vector alignments: regular packing (PDB entrycodes and sequence numbers) 1dbp, helix 1, 43 to 53, helix 9, 237 to 253; alternating packing (right graph) 1thl, helix2, 137 to 151, helix 3, 159 to 179. The Ca–Cb bonds are drawn in magenta. Interhelical contacts between residues withnearly perpendicular Ca–Cb vectors are denoted by broken red lines. Ca–Cb vectors with antiparallel orientation areindicated by broken blue lines and the ones with nearly parallel orientation are shown with dotted blue lines. The yellowcurved lines are the helix axes, the thin blue continuous lines are the lines of closest approach. The broken dark greylines connecting the Cb positions denote the packing cell (cell 27). The sequences of the helices are given in the one-lettercode.


Figure 15. Expected packing angles VN for the ‘‘ridgesinto groove’’ model as a function of the helix radius. Thenumbers correspond to the combinations of the ridgesand grooves; i.e. in the terminology of the modelpresented here, the oriented angles are given for the sixpossible combinations of the three base vectors (Figure 2)of one helical lattice with the corresponding three basevectors of the other; i.e. mirrored but not rotated lattice(D = 1.45 A, v = 99.1°).

groove is therefore unlikely. There exists a registerallowing only discrete translations where theside-chains of one helix can click into the localdepressions of the other helix (knobs into holes).Only through consideration of these key features ofhelices can successful prediction of the radiusdependencies and occupancies of packing cells(holes) according to packing angle be achieved.Though two continuous ridges can certainly bealigned, others must inevitably cross. This conflictcan be resolved only by assuming a discrete naturefor ridges and grooves. Furthermore, only 27.8% ofthe helical residues make intra-helical side-chain–side-chain contacts (atom-atom distances smallerthan 4.0 A). Thus, smooth ridges hardly predomi-nate.

Through a consistent mathematical treatment,three and only three solutions for the perfectsuperposition of a-helical lattices have beendemonstrated. Not only is suboptimal packingevident in the model but also the relationshipbetween occupancy of packing cells and the helixradius. Further, it is shown that within thepreferred packing angle range 120° < VN < 160°,there are two topologically different packingarrangements (small helix radii/cell 27 occupancyand large helix radii/cell 153 occupancy). Thisresult cannot be inferred from the r/g model wherepacking cells are not considered.

Regularity of helices

The helix superposition model assumes regu-larity and that packing of two helices is strainless.The helix pairs must also display similar radii,possess relatively straight helical axes and constanttwist angles and rises per residue. Significantviolation lessens the applicability of the model.Despite the large variations in the observed twistangles vi,i+1 between consecutive centres of side-chains (standard deviation s of 15°), the side-chainsare covalently bound to the Ca backbone atomswhich are very regular in their vi,i+1 (s = 3.7° forinner helical residues).

Helix radius dependency of packing

By analysing the helix radius dependency of thepacking angle, it was possible to reshape thesuggestions of Richmond & Richards (1978), whoinferred that the radius is inversely correlated withthe packing angles defined as the smaller of the twocomplementary angles with 180°. The model usedhere shows that the dependency is not amonotonous function, as observed also by Reddy &Blundell (1993), albeit without the detailed expla-nations provided here.

The helix geometric parameters that allowoptimized packing were examined. It is noteworthythat the structural characteristics of a-helicesdesigned by nature best and most consistentlysatisfy the requirements in the helix parameters

ridges and grooves are formed by residues atregular sequence separation. These ridges andgrooves correspond to base vectors in the modelpresented here, where helix-helix packing involvestheir alignment in the respective lattices such thatthe condition vi = lRVv'j is fulfilled. The term l is ascalar value, RV is a rotation matrix with thecorresponding packing angle V, v'j and vi are vectorsjoining lattice points with sequence spacing i and j(e.g. i = j = 4 for class 4-4), and the prime denotesthe applied mirror operation corresponding toface-to-face packing. The resulting packing anglesare plotted in Figure 15 as a function of the helixradius. In the helix lattice superposition model, notonly is the direction of a pair of base vectorsconsidered but also their length and the packingproperties of their neighbours. In most cases, thiscoincides with a ‘‘knobs into holes’’ (k/h) packingscheme. The equivalent k/h graph is given inFigure 4. The three k/h optimal solutions (am , bm

and cm ) are found at the intersection points in ther/g model where three different base vectors areinvolved (Figure 15). The k/h treatment deletessome of the possible solutions of the r/g model dueto steric clashes at other lattice points; for example,the 1-4 and 1-3 r/g classes at larger helical radii orthe smaller radial segments of the 3-3 class. In thek/h approach, the optimal solutions delineate thepreferred packing angles. For different classes of ther/g model, packing angles are not as distinguish-able. Nonetheless, both approaches rely on thedirection of base vectors and thus some packingsolutions are commonly predicted.

It is obvious that the ridges and grooves are‘‘bumpy’’ and that protruding side-chains and localdepressions are more appropriate helical surfacedescriptions. A smooth sliding of a ridge into a


(Figure 3). This would allow considerable andadvantageous flexibility in achieving the proteinfold. Apart from internal structural strains, the cleardisadvantage of other helical types in packingflexibility (p-helix and 310-helix) in viable foldedproteins is evident.

It has been shown that alanine as a helicalconstituent provides the largest flexibility inpossible packing angles, since the radius of apoly(Ala) a-helix is closest to that associated withthe v triple point. Alanine has accordingly beenobserved to be very often involved in helix-helixcontacts as a central, radius-determining amino acid(Figure 10). This alanine preference in helices isthus explained not only by its compatibility withhelical structure as such but also by the attendantvariety allowed for packing arrangements.

The observed relationship between packing angleand helix radius is likely to be of use in theengineering of protein structure. If, for instance, thedesigning task required helices packed with 20° (or−160°), leucine would be the ideal candidate forhydrophobic associations. If a packing angle ofabout −40° is desired, glycine would be the betterchoice. This is supported by the work of Chou et al.(1984) who, in their energetic analysis of helix-helixpacking, found the lowest interaction energies at−154° (VN = 26°) for packing of poly(Leu) helicesand at 144° (VN = 144°) for poly(Ala) helices.

The model in this work also explains the observedincreased occurrence of leucine and the decreasedfrequency of glycine and proline in four-helixbundle proteins, where helices pack at aboutVN120° (Paliakasis & Kokkinidis, 1992). Alaninewas also often involved, which supports the modelin that alanine was shown to possess greatestpacking flexibility. The significance of packing celltype and helix radius, and the corresponding needfor a good residue fit into a specific cell shouldfurther aid in associating helices. Minimally, thenumber of possible interaction sites can be reducedfor any two specific helices. Attempts in thisprediction direction are in progress.

In conclusion, the observed preference forpacking angles near −40° and +130° may not beexplained by a better packing of side-chains alone.The presented study revealed that there are threeoptimal periodic solutions for the packing angle.Furthermore, the preferred angle in the positiverange is not the calculated optimal solution andtherefore corresponds to a suboptimal solution (videsupra), hence, other determinants like entropiceffects or surface burial differences might beimportant.

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Appendix

Under the condition of perfect overlap of the twohelical lattices, equations (1) and (2) must besatisfied. The base vectors v"1 and v"2 descibing thesecond lattice result from v"1,2 = RVv'1,2 where thevectors v'1,2 are mirrors of the base vectors of the firstlattice v1,2 and RV is a rotation matrix (seeMathematical Description). By substituting theselected vectors of equations (3) and (4) intoequations (1) and (2) the following system ofequations results:

−AR cos(V) − D sin(V) = n1AR + n2BR (A1)

−AR sin(V) + D cos(V) = n1D + n2kD (A2)

−BR cos(V) − kD sin(V) = n3AR + n4BR (A3)

−BR sin(V) + kD cos(V) = n3D + n4kD (A4)

Given specific helix geometric parameters (R, v, D),this system of equations would contain fiveunknowns (V, n1, n2, n3 and n4). However, for theinteger variables n1..4, several boundary conditionsapply:

n1,2,3,4 $ (−1, 0, 1) (A5)

n21 + n2

2$0 and n23 + n2

4$0 (A6)

=n1 + n2= < 2 and =n3 + n4= < 2 (A7)

n2n3$n1n4 (A8)

These conditions reflect restrictions in the lengthand orientation of the base vectors. Obviously, n1

and n2 may not be simultaneously zero; the sameholds for n3 and n4 (equation (A6)). The base vectorsmay not exceed in magnitude the distance of theclosest hexagonal lattice points around the point Pi

(Figure 2; equations (A5) and (A7)) and they maynot be linearly dependent (equation (A8)). Theseboundary conditions reduce the number of possiblecombinations of values for n1 to n4 from 81 to 24.Further restrictions can be elicited by reformulatingequations (A1) to (A4). (1) Multiplying equation(A1) and equation (A2) by B and (A3) and (A4) byA and subsequently subtracting equation (A1) from(A3) and (A2) from (A4) yields:

D(B − kA)sin(V) = (AB(n4 − n1)

+ n3A2 − n2B2)R (A9)

D(kA − B)cos(V)

= ((n3 + kn4)A − (n1 + n2kD)B)D (A10)

Dividing equation (A9) by equation (A10), the 180°periodicity of the solutions become obvious;namely:

tan(V) = RD f(v) (A11)

where f indicates a function.(2) Separating R and D in equations (A1) to (A4)

and equating one side of equation (A1) with (A2)and one side of equation (A3) with (A4) yields:

n2(kA − B)cos(V) = A − (n1 + kn2)(n1A + n2B)

(A12)

n3(kA − B)cos(V) = (n3 + kn4)(n3A + n4B) − kB

(A13)

(3) By multiplying equation (A1) and (A3) by Dand equation (A2) and (A4) by AR and sub-sequently subtracting equation (A1) from (A2) and(A3) from (A4) and multiplying equation (A1) and(A3) by kD and (A2) and (A4) by BR andsubsequently subtracting (A1) from (A2) and (A3)from (A4), it can be shown that:

(kD2 − ABR2)sin(V) + DR(kA + B)cos(V)

= n1DR(B − kA) (A14)

(D2 − A2R2)sin(V) + 2ARDcos(V)

= n2DR(kA − B) (A15)

(k2D2 − B2R2)sin(V) + 2kBR cos(V)

= n3DR(B − kA) (A16)

(kD2 − ABR2)sin(V) + DR(kA + B)cos(V)

= −n4DR(B − kA) (A17)

These latter transformations restrict the possiblecombinations of n1,2,3,4. Comparing equations (A14)and (A17), it directly follows that n1 = −n4, given thatB − kA$0. The remaining cases of possible combi-nations must be investigated separately. If n2 = n3 = 0and n1 = −n4 = 21, then it follows from equation(A10) that:

(kA − B)cos(V) = 2(kA − B) (A18)

Since kA − B$0, then cos(V) = 21 and thussin(V) = 0. Under these conditions equations (A15)and (A16) yield 22ARD = 0 and 2kBR = 0. SinceA = v and for any real helix v cannot be zero, thenthe combinations of n-values above are dismissable.If n1 = n4 = 0 and n2,3 = 21, then it follows fromequations (A12) and (A13) that:

n2(kA − B)cos(V) = A − kB (A19)

n3(kA − B)cos(V) = A − kB (A20)

Since kA − B$0, n2 = n3 providing A − kB$0. IfA − kB = 0, the twist angle v must be 2kp/(k2 − 1)and from equation (A14), D/R = 2p/(k2 − 1). Adetailed examination of equation (A9) taken withthese values for v and D/R shows that n2 = n3.


Consequently, six possible combinations of n1,2,3,4

remain. The solutions for the packing angle V cannow be obtained directly by using these possiblesets in equations (A9) and (A10). The allowed setsare given in Table 2 of the main text and correspondto packing classes designated here as a, b and c (each

with 180° periodicity). Note that the equations aresolved under the conditions of lattice superpositionfor two associating helices. V represents the rotationangle required for one lattice to achieve the overlap.Actual packing is, of course, a result of latticetranslation as well.

Edited by B. Honig

(Received 4 July 1995; accepted 9 October 1995)

principles of helix-helix packing in proteins: the helical

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