**Reference**:- 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. **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**R**^{3}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*→ ∞. **Comments**:- This paper is also available from SpringerLink.
**Copyright**:- The paper is \copyright Springer-Verlag.
**Length**:- The paper is 10 pages.
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**Related papers**:- ProteinMachine_Algorithmica (Geometric Restrictions on Producible Polygonal Protein Chains)

See also other papers by Erik Demaine.

Last updated May 5, 2021 by Erik Demaine.