Principles of Helix-Helix Packing in Proteins: The Helical Lattice Superposition Model

Keywords

Helix-Helix Packing, Helical Lattice Superposition Model, Protein Structure, Alpha Helix, Packing Geometry, Structural Biology

Reference

DOI: https://doi.org/10.1006/jmbi.1996.0044

Abstract

This paper comprehensively analyzes the geometry of helix-helix packing in globular proteins using a Helical Lattice Superposition Model, which involves unrolling the helical cylinders onto a plane and representing each residue as a point.
By exploring the relationship between helix parameters — radius (R), twist angle (ω), and rise per residue (Δ) — the model provides a mathematical framework for evaluating perfect lattice matches for various helix types (α, π, 3₁₀).
Three well-separated solutions for interhelical packing angles emerge from the equations, allowing optimal packing. Surprisingly, even static geometric constraints give rise to diverse and sensitive packing modes, notably showing high flexibility when alanine forms the core.
Beyond ideal packings, suboptimal “knobs into holes” hydrophobic patterns and alternative “knobs onto knobs” models are explored, offering a richer understanding of helix association.
Comparison with experimental data confirms many theoretical predictions, but highlights deviations explained by side-chain Cα–Cβ vector orientation, emphasizing the interplay of geometry and side-chain chemistry.

Notes

1. General Summary

• Helical Lattice Superposition Model unrolls helices onto a plane to analyze packing.
• Study investigates the mathematical relationship among helix radius (R), twist angle (ω), and rise per residue (Δ) for optimal packing.
• Three distinct interhelical packing angles are mathematically derived, corresponding to different helix types and radii.
• Observed α-helix parameters (ω = 99.1°, Δ = 1.45 Å) yield three packing angles: −37.1°, −97.4°, +22.0°.
• Near a twist angle ωₜᵣᵢₚₗₑ = 96.9°, all three packing angles converge to a common radius Rₜᵣᵢₚₗₑ = 3.46 Å, close to poly(Ala) helix radius, indicating alanine’s unique role in flexible packing.
• Suboptimal packing types, such as “knobs into holes,” “knobs onto knobs,” and mixed modes, explain associations stabilized by hydrogen bonds, salt bridges, disulfide bonds, and hydrophobic contacts.
• Experimental observations validate many predictions but also reveal deviations from model periodicity (180°) due to side-chain vector orientations.

2. Preknowledge and Background

• Dihedral angle: geometric angle describing helix-helix orientation.
• Crick (1953):
  • Proposed “knobs into holes” model, suggesting +20° dihedral packing angle for coiled-coils based on optimal fitting of side chains.
• Richmond & Richards (1978):
  • Suggested inverse relationship between packing angle and helix radius.
  • Identified three packing classes based on amino acid preferences.
• Chothia et al. (1977):
  • “Ridges into grooves” model, matching sequential residues (spacing i, j) between helices.
  • Identified packing angles, notably −52° (i=4, j=4).
• Efimov (1979):
  • Related packing angles to rotamer preferences of side-chains, distinguishing polar/apolar types.
• Reddy & Blundell (1993):
  • Correlated interhelical distances with amino acid interface volume.
• Other studies explored energetic aspects, e.g., hydrophobic burial and atomic energy minimization.

3. Core Findings and Insights

• Mathematical framework yields three optimal packing angles per helix set, dependent on helix geometry.
• Flexibility near alanine-rich helices: At ωₜᵣᵢₚₗₑ = 96.9°, radius convergence (R = 3.46 Å) highlights alanine’s core-packing propensity.
• Experimental packings agree with model but show deviations explained by Cα–Cβ side-chain vector orientation, affecting real protein structures.
• Three main packing angles for α-helix with typical geometry: −37.1°, −97.4°, +22.0°, with 180° periodicity.
• “Knobs into holes” scheme favored over “ridges into grooves”, although both contribute to diverse packing types.
• Suboptimal and alternative packing patterns (hydrogen bonds, salt bridges, mixed types) enable diverse helix associations.
• Analysis provides insight into why i=4 (−52°) is optimal in Chothia’s model: relates to easier and more stable packing of side chains, possibly the most compatible angular offset for hydrophobic core formation.

4. RD’s Thoughts and Learnings

• RD finds this paper mathematically challenging — a reread (and maybe revisiting Crick’s 1953 paper first) is needed to fully digest the lattice model math.
• The concept of superposing helical lattices to derive packing angles is conceptually beautiful and rigorous.
• Fascinated by the convergence of radii at ωₜᵣᵢₚₗₑ = 96.9° and its alignment with poly(Ala) properties — explains alanine’s dominance in core packing.
• Great synergy between mathematical prediction and experimental confirmation — even discrepancies lead to deeper insights (Cα–Cβ vectors).
• The comparison of “knobs into holes” and “ridges into grooves” provides a framework for thinking about side-chain interactions, applicable beyond helices (e.g., β-sheet interfaces, coiled-coils).
• Potential application to residue interaction networks (RINs): could similar geometric packing principles explain network connectivity?
• RD wonders if these packing angles correlate with phosphorylation-dependent structural rearrangements (e.g., helix reorientation upon PTMs).

Take-home Messages

• Helix-helix packing in proteins can be mathematically modeled using helical lattice superposition, revealing three key packing angles and corresponding radii.
• Alanine plays a crucial role in flexible core formation due to ideal radius (≈3.46 Å) at converging twist angles.
• “Knobs into holes” remains a dominant model, but mixed and alternative packing stabilize diverse associations.
• Deviations from theory explained by side-chain geometry (Cα–Cβ orientation), emphasizing chemistry + geometry synergy.
• Provides a powerful design principle for protein engineering, predicting optimal helix-helix associations.
• RD recommends reading Crick (1953) first for foundational understanding, but finds this paper rich in insights and applicable to broader structural studies.

RD needs to reread this — but this is mind-blowingly deep work! ✨📚