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Microscopic Pathways to Helix Formation: Packing Stabilization and Sticky Interactions in Chiral Polymer Condensates

Biman Bagchi

Abstract

Helices are not generic outcomes of polymer collapse. Collapsed conformations of semiflexible polymers with isotropic attractions typically form globules, toroids, or rod-like structures, as seen in simulations and described by coarse-grained necklace and surface-tension models. Helical conformations, in contrast, are generally absent in minimal theories based solely on bending elasticity and isotropic cohesion, since such descriptions lack any mechanism to select torsion, pitch, or periodic packing. Here we identify two minimal and physically distinct routes by which helices can become stable without invoking biochemical specificity. Route (A) is geometric and steric: combining a tube-like packing (thickness) constraint with generic attractions selects an ideal helical packing with finite radius and pitch. Left- and right-handed helices remain exactly degenerate in free energy, so chirality emerges spontaneously even without explicit chiral interactions. Route (B) is energetic and commensurate: periodic "sticker" attractions between monomers separated by a fixed contour distance $m$ enforce a registry between interaction spacing and chain geometry. This commensurability stabilizes helical states by enabling repeated contacts along the backbone, naturally connecting to classical Gibbs-DiMarzio and Zimm-Bragg mechanisms. For both routes, we derive analytical relations for helix radius and pitch, curvature and bending energy, contact-distance constraints, and crossover conditions to toroidal and rod-like morphologies, expressed in terms of persistence length $L_p$, interaction strength, and chain length $N$. This framework explains why helices are non-generic in polymer collapse, identifies the physical ingredients required for their stabilization, and provides testable predictions for when helical and chiral condensates should emerge.

Microscopic Pathways to Helix Formation: Packing Stabilization and Sticky Interactions in Chiral Polymer Condensates

Abstract

Helices are not generic outcomes of polymer collapse. Collapsed conformations of semiflexible polymers with isotropic attractions typically form globules, toroids, or rod-like structures, as seen in simulations and described by coarse-grained necklace and surface-tension models. Helical conformations, in contrast, are generally absent in minimal theories based solely on bending elasticity and isotropic cohesion, since such descriptions lack any mechanism to select torsion, pitch, or periodic packing. Here we identify two minimal and physically distinct routes by which helices can become stable without invoking biochemical specificity. Route (A) is geometric and steric: combining a tube-like packing (thickness) constraint with generic attractions selects an ideal helical packing with finite radius and pitch. Left- and right-handed helices remain exactly degenerate in free energy, so chirality emerges spontaneously even without explicit chiral interactions. Route (B) is energetic and commensurate: periodic "sticker" attractions between monomers separated by a fixed contour distance enforce a registry between interaction spacing and chain geometry. This commensurability stabilizes helical states by enabling repeated contacts along the backbone, naturally connecting to classical Gibbs-DiMarzio and Zimm-Bragg mechanisms. For both routes, we derive analytical relations for helix radius and pitch, curvature and bending energy, contact-distance constraints, and crossover conditions to toroidal and rod-like morphologies, expressed in terms of persistence length , interaction strength, and chain length . This framework explains why helices are non-generic in polymer collapse, identifies the physical ingredients required for their stabilization, and provides testable predictions for when helical and chiral condensates should emerge.

Paper Structure

This paper contains 47 sections, 71 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction and motivation
  2. Helical geometry and energetic ingredients: Equation of helix
  3. Curvature, torsion, and bending energy
  4. Distance between two points along the helix
  5. Generic cohesive contributions
  6. Cooperative stabilization and relation to classical helix--coil theories
  7. Periodicity-assisted helix stabilization: commensurability and stickers (Route B)
  8. Periodic sticker contact condition
  9. Commensurability and axial separation
  10. Dimensionless formulation
  11. Sticker contribution and stability
  12. Cooperativity and relation to classical helix--coil theories
  13. Free-energy consolidation and morphological selection
  14. Helices in coarse-grained theories
  15. Helix free energies from Route (A) and Route (B)
  16. ...and 32 more sections

Figures (1)

  • Figure 1: Two minimal routes to helix formation in collapsed polymers. (a) Packing/thickness route (Route A): a polymer treated as a tube of finite thickness $t$ forms a helix characterized by radius $R$ and pitch $P$ when packing and curvature constraints are satisfied. (b) Hydrogen-bond / sticker route (Route B): periodic attractive interactions between monomers separated by a contour distance $m$ stabilize a helical registry through repeated contacts of strength $\varepsilon_s$.