Geometric Restrictions on Producible Polygonal Protein Chains

Abstract

Fixed-angle polygonal chains in 3D 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 α-producible, and prove as our main result that a chain is α-producible if and only if it is flattenable, that is, it 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 α-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(n ²)-move algorithm in [ADD⁺02].

Finally, we prove that the α-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 we know of no nontrivial class that avoids locked configurations), then the probability that a random configuration of a random chain is α-producible approaches zero geometrically as n → ∞.