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Principles of helix-helix packing in proteins: the helical lattice superposition model.

The geometry of helix-helix packing in globular proteins is comprehensively analysed within the model of the superposition of two helix lattices which result from unrolling the helix cylinders onto a plane containing points representing each residue. The requirements for the helix geometry (the radius R, the twist angle omega and the rise per residue delta) under perfect match of the lattices are studied through a consistent mathematical model that allows consideration of all possible associations of all helix types (alpha-, pi- and 3(10)). The corresponding equations have three well-separated solutions for the interhelical packing angle, omega, as a function of the helix geometric parameters allowing optimal packing. The resulting functional relations also show unexpected behaviour. For a typically observed alpha-helix (omega = 99.1 degrees, delta = 1.45 A), the three optimal packing angles are omega a,b,c = -37.1 degrees, -97.4 degrees and +22.0 degrees with a periodicity of 180 degrees and respective helix radii Ra,b,c = 3.0 A, 3.5 A and 4.3 A. However, the resulting radii are very sensitive to variations in the twist angle omega. At omega triple = 96.9 degrees, all three solutions yield identical radii at delta = 1.45 A where Rtriple = 3.46 A. 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 omega triple 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 confirmed the conclusions of the theoretical model. Nonetheless, the observed alpha-helix packings showed deviations from the 180 degrees periodicity expected from the model. An investigation of the actual three-dimensional geometry of helix-helix packing revealed an explanation for the observed discrepancies where a decisive role was assigned to the defined orientation of the C alpha-C beta vectors of the side-chains. As predicted form 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 application 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.