### Abstract

We examine a few computational geometric problems concerning the structures of polymers. We use a standard model of a polymer, a polygonal chain (path of line segments) in three dimensions. The chain can be reconfigured in any manner as long as the edge lengths and the angles between consecutive edges remain fixed, and no two edges cross during the motion. We discuss preliminary results on the following problems. Given a chain, select some interior edge uv, defining two subchains which are adjacent to uv. We keep the two subchains individually rigid and rotate one around uv while leaving the other fixed in space, while maintaining the vertex-angles at uv. We call this motion an edge spin at uv. An O(n^{2}) algorithm for this problem is given as well s an Ω (n log n) lower bound on the time complexity. In determining whether a chain can be reconfigured from one conformation to another, it is useful to consider reconfiguring through some canonical conformation. In our three-dimensional case, the most obvious choice is to flatten a chain into the plane. However, we demonstrate that determining if a given chain can be reconfigured into the plane without self-intersecting is NP-hard, even if the restriction that it must lie monotonically is added. We then provide an O(n) algorithm to decide if a chain has a non-crossing convex coil conformation (where all angles turn in the same direction), although we cannot yet decide if a sequence of motions to reconfigure a chain into a convex coil conformation exists.

Original language | English (US) |
---|---|

Pages (from-to) | 303-318 |

Number of pages | 16 |

Journal | Journal of Mathematical Chemistry |

Volume | 27 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2000 |

### Fingerprint

### Keywords

- Edge spin
- Flattening
- Macromolecule conformations
- Polymer conformations
- Polymer motions

### ASJC Scopus subject areas

- Chemistry(all)
- Applied Mathematics

### Cite this

*Journal of Mathematical Chemistry*,

*27*(4), 303-318. https://doi.org/10.1023/A:1018823806289

**Geometric and computational aspects of polymer reconfiguration.** / Soss, Michael; Toussaint, Godfried.

Research output: Contribution to journal › Article

*Journal of Mathematical Chemistry*, vol. 27, no. 4, pp. 303-318. https://doi.org/10.1023/A:1018823806289

}

TY - JOUR

T1 - Geometric and computational aspects of polymer reconfiguration

AU - Soss, Michael

AU - Toussaint, Godfried

PY - 2000/1/1

Y1 - 2000/1/1

N2 - We examine a few computational geometric problems concerning the structures of polymers. We use a standard model of a polymer, a polygonal chain (path of line segments) in three dimensions. The chain can be reconfigured in any manner as long as the edge lengths and the angles between consecutive edges remain fixed, and no two edges cross during the motion. We discuss preliminary results on the following problems. Given a chain, select some interior edge uv, defining two subchains which are adjacent to uv. We keep the two subchains individually rigid and rotate one around uv while leaving the other fixed in space, while maintaining the vertex-angles at uv. We call this motion an edge spin at uv. An O(n2) algorithm for this problem is given as well s an Ω (n log n) lower bound on the time complexity. In determining whether a chain can be reconfigured from one conformation to another, it is useful to consider reconfiguring through some canonical conformation. In our three-dimensional case, the most obvious choice is to flatten a chain into the plane. However, we demonstrate that determining if a given chain can be reconfigured into the plane without self-intersecting is NP-hard, even if the restriction that it must lie monotonically is added. We then provide an O(n) algorithm to decide if a chain has a non-crossing convex coil conformation (where all angles turn in the same direction), although we cannot yet decide if a sequence of motions to reconfigure a chain into a convex coil conformation exists.

AB - We examine a few computational geometric problems concerning the structures of polymers. We use a standard model of a polymer, a polygonal chain (path of line segments) in three dimensions. The chain can be reconfigured in any manner as long as the edge lengths and the angles between consecutive edges remain fixed, and no two edges cross during the motion. We discuss preliminary results on the following problems. Given a chain, select some interior edge uv, defining two subchains which are adjacent to uv. We keep the two subchains individually rigid and rotate one around uv while leaving the other fixed in space, while maintaining the vertex-angles at uv. We call this motion an edge spin at uv. An O(n2) algorithm for this problem is given as well s an Ω (n log n) lower bound on the time complexity. In determining whether a chain can be reconfigured from one conformation to another, it is useful to consider reconfiguring through some canonical conformation. In our three-dimensional case, the most obvious choice is to flatten a chain into the plane. However, we demonstrate that determining if a given chain can be reconfigured into the plane without self-intersecting is NP-hard, even if the restriction that it must lie monotonically is added. We then provide an O(n) algorithm to decide if a chain has a non-crossing convex coil conformation (where all angles turn in the same direction), although we cannot yet decide if a sequence of motions to reconfigure a chain into a convex coil conformation exists.

KW - Edge spin

KW - Flattening

KW - Macromolecule conformations

KW - Polymer conformations

KW - Polymer motions

UR - http://www.scopus.com/inward/record.url?scp=0034338019&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034338019&partnerID=8YFLogxK

U2 - 10.1023/A:1018823806289

DO - 10.1023/A:1018823806289

M3 - Article

VL - 27

SP - 303

EP - 318

JO - Journal of Mathematical Chemistry

JF - Journal of Mathematical Chemistry

SN - 0259-9791

IS - 4

ER -