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 $m$ 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.