Geometric Restrictions on Producible Polygonal Protein Chains

Fixed-angle polygonal chains in three dimensions serve as an interesting model of protein backbones. Here we consider such chains produced inside a machine modeled crudely as a cone, and examine the constraints this model places on the producible chains. We call this notion producible, and prove as our main result that a chain whose maximum turn angle is α is producible in a cone of half-angle ≥ α if and only if the chain is flattenable, that is, the chain can be reconfigured without self-intersection to lie flat in a plane. This result establishes that two seemingly disparate classes of chains are in fact identical. Along the way, we discover that all producible configurations of a chain can be moved to a canonical configuration resembling a helix. One consequence is an algorithm that reconfigures between any two flat states of a nonacute chain in O(n) moves, improving the O(n2)-move algorithm in [ADD+]. Finally, we prove that the producible chains are rare in the following technical sense. A random chain of n links is defined by drawing the lengths and angles from any regular (e.g., uniform) distribution on any subset of the possible values. A random configuration of a chain embeds into ℝ3 by in addition drawing the dihedral angles from any regular distribution. If a class of chains has a locked configuration (and no nontrivial class is known to avoid locked configurations), then the probability that a random configuration of a random chain is producible approaches zero geometrically as n → ∞