The geometric behavior of polygons under extreme confinement

In nature, macromolecular chains often exist under extreme confinement. Current techniques for modeling these chains by sampling random polygons in confined volumes are limited. In previous work, we introduced a simple model for knots under extreme confinement consisting of random polygons restricted to cylinders, where the edges each connect random points on the top and bottom disks of the given confining cylinder. We presented numerical evidence that polygons generated by this cylindrical model exhibit knotting probabilities that match the predicted knotting probabilities of rooted equilateral unit-length edge random polygons confined within a sphere of radius R ≈ 0.62, a stronger degree of confinement than R = 1. In this paper, we show that certain geometric quantities (total curvature, average crossing number, and writhe) of the cylindrical polygons are consistent with the predictions for rooted equilateral unit-length edge random polygons in spherical confinement with radius below R = 1, varying between 0.62 and 0.72 depending on the quantity. This strengthens the case that this simple model can be used to gain insight into the asymptotic behavior of random polygons under strong spherical confinement. Total torsion is the exception: the cylindrical model does not reproduce the spherical total torsion of phantom polygons at any of the confinement radii in the spherical data, an anomaly we explain geometrically and observe directly in the distribution of individual torsion angles. This anomaly disappears when polygons are grouped by crossing number and total torsion is measured relative to each model's own phantom average. Knot complexity then has a similar relative effect on total torsion in both models.