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Geometric Restrictions on Producible Polygonal Protein Chains

  • Erik D. Demaine
  • Stefan Langerman
  • Joseph O’Rourke
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2906)

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 → ∞.

Keywords

Dihedral Angle Edge Length Turn Angle Regular Distribution Geometric Restriction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [ADD+02]
    Aloupis, G., Demaine, E., Dujmović, V., Erickson, J., Langerman, S., Meijer, H., Streinu, I., O’Rourke, J., Overmars, M., Soss, M., Toussaint, G.: Flat-state connectivity of linkages under dihedral motions. In: Bose, P., Morin, P. (eds.) ISAAC 2002. LNCS, vol. 2518, pp. 369–380. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. [ADM+02]
    Aloupis, G., Demaine, E.D., Meijer, H., O’Rourke, J., Streinu, I., Toussaint, G.: Flat-state connectedness of fixed-angle chains: Special acute chains. In: Proc. 14th Canad. Conf. Comp. Geom., pp. 27–30 (2002)Google Scholar
  3. [BDD+02]
    Biedl, T., Demaine, E., Demaine, M., Lubiw, A., O’Rourke, J., Overmars, M., Robbins, S., Streinu, I., Toussaint, G.T., Whitesides, S.: On reconfiguring tree linkages: Trees can lock. Discrete Mathematics 117, 293–297 (2002)zbMATHCrossRefGoogle Scholar
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    Soss, M., Toussaint, G.T.: Geometric and computational aspects of polymer reconfiguration. J. Math. Chemistry 27(4), 303–318 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Erik D. Demaine
    • 1
  • Stefan Langerman
    • 2
  • Joseph O’Rourke
    • 3
  1. 1.MIT Laboratory for Computer ScienceCambridgeUSA
  2. 2.Département d’informatiqueUniversité Libre de BruxellesBruxellesBelgium
  3. 3.Department of Computer ScienceSmith CollegeNorthamptonUSA

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