Why Helices Rarely Emerge in Simulations of Polymer Collapse: Geometric and Cooperative Routes to Chiral Condensates

Collapsed conformations of semiflexible polymers with isotropic attractions generally form globules, toroids, or rod-like structures, as observed in simulations and described by coarse-grained necklace and surface-tension models. Helical conformations, by contrast, are not generic outcomes of polymer collapse. Minimal theories based solely on bending elasticity and isotropic cohesion provide no mechanism that selects 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 in origin: combining a tube-like packing (thickness) constraint with generic attractions selects an ideal helical packing with a finite radius and pitch. In this mechanism, left- and right-handed helices are exactly degenerate in free energy, so that chirality can arise through spontaneous symmetry breaking or be selected by a weak external bias. 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 selectively stabilizes helical states by allowing the same set of monomers to form repeated contacts along the backbone, naturally connecting the theory to classical Gibbs-DiMarzio and Zimm-Bragg mechanisms relevant for biopolymers. Within this framework, even a small intrinsic chiral bias─such as that introduced by chiral monomers─can be amplified by cooperative helix formation to select a dominant handedness over long segments. For both routes, we derive analytical relations for the helix radius and pitch, the associated curvature and bending energy, contact-distance constraints, and crossover conditions to toroidal and rod-like morphologies, expressed in terms of the persistence length L_p, interaction strength, and chain length N. The theory thus explains why helices are nongeneric in polymer collapse, identifies the minimal physical ingredients required to stabilize them, and yields testable predictions for when helical and chiral condensates should appear.