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Principles of Helix-Helix Packing in Proteins: the Helical Lattice Superposition Model Dirk Walther1*, Frank Eisenhaber1,2 and Patrick Argos1

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Principles of Helix-Helix Packing in Proteins: the Helical Lattice Superposition Model Dirk Walther1*, Frank Eisenhaber1,2 and Patrick Argos1 J. Mol. Biol. (1996) 255, 536–553 Principles of Helix-Helix Packing in Proteins: The Helical Lattice Superposition Model Dirk Walther1*, Frank Eisenhaber1,2 and Patrick Argos1 1European Molecular Biology The geometry of helix-helix packing in globular proteins is comprehen- Laboratory, Meyerhofstraße 1 sively analysed within the model of the superposition of two helix lattices Postfach 10.2209, 69012 which result from unrolling the helix cylinders onto a plane containing Heidelberg, Germany points representing each residue. The requirements for the helix geometry (the radius R, the twist angle v and the rise per residue D) under perfect 2Biochemisches 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-, Universita¨t zu Berlin, p- and 310). The corresponding equations have three well-separated Hessische Straße 3–4 10115 solutions for the interhelical packing angle, V, as a function of the helix Berlin, 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 radii Ra,b,c = 3.0 Å, 3.5 Å and 4.3 Å. However, the resulting radii are very sensitive ° to variations in the twist angle v. At vtriple = 96.9 , all three solutions yield identical radii at D = 1.45 Å where Rtriple = 3.46 Å. This radius is close to that of a poly(Ala) helix, indicating a great packing flexibility when alanine is involved in the packing core, and vtriple is close to the mean observed twist angle. In contrast, the variety of possible theoretical solutions is limited for the other two helix types. Besides the perfect matches, novel suboptimal ‘‘knobs into holes’’ hydrophobic packing patterns as a function of the helix radius are described. Alternative ‘‘knobs onto knobs’’ and mixed models can be applied in cases where salt bridges, hydrogen bonds, disulphide bonds and tight hydrophobic head-to-head contacts are involved in helix-helix associations. An analysis of the experimentally observed packings in proteins con- firmed the conclusions of the theoretical model. Nonetheless, the observed a-helix packings showed deviations from the 180° periodicity expected from the model. An investigation of the actual three-dimensional geometry of helix-helix packing revealed an explanation for the observed discrep- ancies where a decisive role was assigned to the defined orientation of the Ca-Cb vectors of the side-chains. As predicted from the model, helices with different radii (differently sized side-chains in the packing core) were observed to utilize different packing cells (packing patterns). In agreement with the coincidence between Rtriple and the radius of a poly(Ala) helix, Ala was observed to show greatest propensity to build the packing core. The ap- plication of the helix lattice superposition model suggests that the packing of amino acid residues is best described by a ‘‘knobs into holes’’ scheme rather than ‘‘ridges into grooves’’. The various specific packing modes made salient 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 had been suggested. Several models were devel- oped and were mostly devoted to surface comple- The topic of helix-helix pairwise packing in mentarities upon packing. Crick’s model (Crick, proteins was addressed soon after helical structures 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 a equivalent helices in homologous proteins. Other plane and then finding the best fit of the resulting efforts have focused on the energetic aspects of lattices (one point per residue). This was achieved helix-helix packing where different interaction by superposition in a face-to-face manner potentials ranging from burial of hydrophobic through rotation followed by translation such residues (Ptitsyn & Rashin, 1975) and other that residues of one helix (knobs) fit into cells simplified interaction potentials (Solovyov & formed by neighbouring residues in the other Kolchanov, 1984) to atomic energy minimization helix (holes). Assuming a helix radius R of 5.0 A˚ and Monte-Carlo sampling (Chou et al., 1983, and a twist angle v of 100.0° between residues 1984; Tuffe´ry & Lavery, 1993) have been applied. along the helix path, he found optimal packing Murzin & Finkelstein (1988) attempted to predict at a dihedral packing angle V between the helix the topology and orientation of certain helical axes at +20° (coiled-coil structures) and a subopti- assemblies by arranging them in polyhedral mal packing at V = −70°. Richmond & Richards shells. Harris et al. (1994) have performed a care- (1978) also pursued the knobs into holes model ful study of the diversity in four-helix bundle and concluded further that the packing angle is proteins. inversely correlated to the helix radius. They The work presented here was stimulated by the suggested three possible classes of helix-helix observation that observed helix-helix packing packing and, for each class, listed possible amino angles demonstrate a pronounced preference for acids central to the contact. These preferences were V1−50°/130°. It is difficult to imagine why utilized to predict spatial helical arrangements from this preference should be a result of the relative primary structural information (Richmond & length of one ridge along one helical side, as Richards, 1978; Cohen et al., 1979; Cohen & Kuntz, argued by Chothia et al. (1981), or due to the less 1987). splayed character of residues in the i = 4 ridge Chothia et al. (1977, 1981) introduced another and (Chothia et al., 1981; Hutchinson et al., 1994). The now widely accepted interpretation of the superim- contact-forming residues in helix association posed ‘‘helical’’ lattices. Instead of ‘‘knobs into need not belong to one and the same ridge. holes’’ packing, they coined ‘‘ridges into grooves’’. Maximizing the burial of hydrophobic surface upon Here, the ridges formed by residues with sequential contact (presumably favouring smaller packing spacing i in the first helix fit into grooves formed by angles) or an easier and fitter packing of amino acid residues in the second helix with spacing j. By side-chains at a certain packing angle would assuming mean observed helix geometries, they seem to provide more natural explanations. Thus, found three basic packing types by varying i and j; the model of unrolled helix lattices was further − ° − ° namely, Vi=1,j = 4 = 105 , Vi=4,j = 4 = 52 and investigated and treated mathematically in a ° Vi=3,j = 4 = +23 . In principle, yet other combinations rigorous fashion. Which set of helix parameters (the − ° of i and j were possible (e.g. Vi=3,j = 3 = 109 ); radius of the helix R, twist angle v and the rise per however, as they noted, these classes were residue D) guarantees an optimal match and barely distinguishable from the former because of association of two identical and ideal helical their similar packing angle and pattern of amino lattices in a face-to-face manner after translating one acid contacts. They introduced yet another packing of them (homogeneous packing)? Can the class (‘‘crossed ridge’’ packing), where the ridges of ambiguities in the ridges-into-groove model be two helices cross with expected packing angles at resolved by considering optimization of the +55°, −15° and −105°. Chothia and his co-workers packing density? To approach these questions, also argued that the observed preference for the conditions for optimal packing were mathemat- packing angles around V = −52° can be understood ically formulated to allow careful consideration of in that ridges, formed by contact residues spaced by all solutions. To check the theoretical model, a i = 4, dominate the shape and surface of the helical statistical analysis of experimentally determined face since they make the smallest angle to the helix helix-helix packings was effected. The latter axis. showed that a 180° periodic selection in V Efimof (1979) attempted to relate the packing was not uniform. An explanation for this is angle with preferred rotational states of the provided here based on the tertiary structural side-chains along the helix. He distinguished configuration of helices, especially the Ca–Cb two types of packing; polar and apolar, each bond direction. To the authors’ knowledge, the giving rise to different combinations of rotational treatment here is mathematically rigorous in isomeric states of the contacting amino acid contrast to all the previous works where residues. For a best fit, he proposed three discrete more visual approaches were adopted and various packing angles for the apolar case (V1+30°, helical geometric parameters were held fixed. 1−30°, 190°) and a range of possible docking The non-uniform V distribution has not been angles in the polar case (−30°EVE30°). Reddy & previously addressed. The various optimal and Blundell (1993) correlated the distance of closest suboptimal packing modes made salient by the approach between two packed helices to the model should aid in protein engineering and volume of the interface-forming amino acid design, especially
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