Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Unassigned distance geometry and the Buckminsterfullerene

Leo Liberti

Abstract

The Buckminsterfullerene is an inorganic molecule consisting of 60 carbon atoms in the shape of a soccer ball. It was used in [Juhas et al., Nature 2006] to showcase algorithms that find the correct shape of a protein from limited data (length of inter-atomic distances) without any further chemical experiment: in that case, by means of a complicated constructive heuristic based on genetic algorithms. In this paper we show that we can reconstruct the Buckminsterfullerene structure by means of mathematical programming, standard solver software, and little else.

Unassigned distance geometry and the Buckminsterfullerene

Abstract

The Buckminsterfullerene is an inorganic molecule consisting of 60 carbon atoms in the shape of a soccer ball. It was used in [Juhas et al., Nature 2006] to showcase algorithms that find the correct shape of a protein from limited data (length of inter-atomic distances) without any further chemical experiment: in that case, by means of a complicated constructive heuristic based on genetic algorithms. In this paper we show that we can reconstruct the Buckminsterfullerene structure by means of mathematical programming, standard solver software, and little else.

Paper Structure

This paper contains 13 sections, 10 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Two problems in distance geometry
  3. Goal of this paper
  4. MINLP formulations for the UDGP
  5. Exact MINLP formulations
  6. An approximate MILP formulation
  7. A matheuristic for the UDGP
  8. The way of proteins and their scale
  9. The algorithm
  10. Solving the MIDDP
  11. Solving the DGP
  12. Finding the shape of the Buckminsterfullerene
  13. Conclusion

Figures (2)

  • Figure 1: Left: the Buckminsterfullerene molecule with a spiral order on the vertices. Center: wrong realizations obtained for $|\delta|=0.5\,N$. Right: the realization for $|\delta|=N$ yields a perfect reconstruction (rotated).
  • Figure 2: Slightly wrong realizations for $|\delta|\in[0.995N,0.999N]$.