**Reference**:- Erik D. Demaine, Stefan Langerman, and Joseph O'Rourke, “Geometric Restrictions on Producible Polygonal Protein Chains”,
*Algorithmica*, volume 44, number 2, February 2006, pages 167–181. Special issue of selected papers from the 14th Annual International Symposium on Algorithms and Computation, 2003. **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
*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*(*n*^{2})-move algorithm in [ADD^{+}02].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**R**^{3}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*→ ∞. **Comments**:- This paper is also available from SpringerLink.
**Copyright**:- Copyright held by the authors.
**Length**:- The paper is 17 pages.
**Availability**:- The paper is available in PostScript (5391k), gzipped PostScript (3069k), and PDF (464k).
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**Related papers**:- ProteinMachine_ISAAC2003 (Geometric Restrictions on Producible Polygonal Protein Chains)

See also other papers by Erik Demaine.

Last updated January 15, 2021 by Erik Demaine.