Paper by Erik D. Demaine

Erik D. Demaine, Stefan Langerman, and Joseph O'Rourke, “Geometric Restrictions on Producible Polygonal Protein Chains”, in Proceedings of the 14th Annual International Symposium on Algorithms and Computation (ISAAC 2003), Lecture Notes in Computer Science, volume 2906, Kyoto, Japan, December 15–17, 2003, pages 395–404.

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(n2)-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 → ∞.

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The paper is 10 pages.

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Related papers:
ProteinMachine_Algorithmica (Geometric Restrictions on Producible Polygonal Protein Chains)

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Last updated July 25, 2017 by Erik Demaine.