Advertisement

Advances in Applied Clifford Algebras

, Volume 27, Issue 1, pp 439–452 | Cite as

Geometric Algebra to Model Uncertainties in the Discretizable Molecular Distance Geometry Problem

  • Rafael Alves
  • Carlile Lavor
Article

Abstract

The discretizable molecular distance geometry problem (DMDGP) is related to the determination of 3D protein structure using distance information detected by nuclear magnetic resonance (NMR) experiments. The chemistry of proteins and the NMR distance information allow us to define an atomic order \({v_{1},\ldots,v_{n}}\) such that the distances related to the pairs \({\{v_{i-3},v_{i}\},\{v_{i-2},v_{i}\},\{v_{i-1},v_{i}\}}\), for \({i > 3}\), are available, which implies that the search space can be represented by a tree. A DMDGP solution can be represented by a path from the root to a leaf node of this tree, found by an exact method, called branch-and-prune (BP). Because of uncertainty in NMR data, some of the distances related to the pairs \({\{v_{i-3},v_{i}\}}\) may not be precise values, being represented by intervals of real numbers \({[\underline{d}_{i-3,i},\overline{d}_{i-3,i}]}\). In order to apply BP algorithm in this context, sample values from those intervals should be taken. The main problem of this approach is that if we sample many values, the search space increases drastically, and for small samples, no solution can be found. We explain how geometric algebra can be used to model uncertainties in the DMDGP, avoiding sample values from intervals \({[\underline{d}_{i-3,i},\overline{d}_{i-3,i}]}\) and eliminating the heuristic characteristics of BP when dealing with interval distances.

Keywords

Conformal geometric algebra Distance geometry Branch and prune algorithm 3D protein structure 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Berger B., Kleinberg J., Leighton T.: Reconstructing a three-dimensional model with arbitrary errors. J. ACM 46, 212–235 (1999)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Cassioli A., Bordeaux B., Bouvier G., Mucherino A., Alves R., Liberti L., Nilges M., Lavor C., Malliavin T.: An algorithm to enumerate all possible protein conformations verifying a set of distance constraints. BMC Bioinform. 16, 16–23 (2015)CrossRefGoogle Scholar
  3. 3.
    Cassioli A., Gunluk O., Lavor C., Liberti L.: Discretization vertex orders in distance geometry. Discrete Appl. Math. 197, 27–41 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Crippen G., Havel T.: Distance Geometry and Molecular Conformation. Wiley, New York (1988)MATHGoogle Scholar
  5. 5.
    Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry. The Morgan Kaufmann Series in Computer Graphics (2007)Google Scholar
  6. 6.
    Franchini, S., Vassalo, G., Sorbello, F.: A brief introduction to Clifford algebra. Technical report no. 2/2010. Dipartimento di Ingegneria Informatica, Università degli Studi di Palermo (2010)Google Scholar
  7. 7.
    Havel, T.: Distance geometry. In: Grant, D., Harris, R. (eds.) Encyclopedia of Nuclear Magnetic Resonance, pp. 1701–1710. Wiley, New York (1995)Google Scholar
  8. 8.
    Hestenes, D.: Old wine in new bottles: a new algebraic framework for computational geometry. In: Advances in Geometric Algebra with Applications in Science and Engineering, pp. 1–14 (2001)Google Scholar
  9. 9.
    Hildenbrand D.: Foundations of Geometric Algebra Computing. Springer, Berlin (2012)CrossRefMATHGoogle Scholar
  10. 10.
    Hildenbrand, D.: Home page of Gaalop. http://www.gaalop.de
  11. 11.
    Hildenbrand, D., Fontijne, D., Perwass, C., Dorst, L.: Geometric algebra and its application to computer graphics. Tutorial 3. In: Proceedings of Eurographics (2004)Google Scholar
  12. 12.
    Lavor C., Liberti L., Maculan N., Mucherino A.: Recent advances on the discretizable molecular distance geometry problem. Eur. J. Oper. Res. 219, 698–706 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lavor C., Liberti L., Maculan N., Mucherino A.: The discretizable molecular distance geometry problem. Comput. Optim. Appl. 52, 115–146 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lavor C., Liberti L., Mucherino A.: The interval BP algorithm for the discretizable molecular distance geometry problem with interval data. J. Global Optim. 56, 855–871 (2013)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lavor C., Alves R., Figueiredo W., Petraglia A., Maculan N.: Clifford algebra and the discretizable molecular distance geometry problem. Adv. Appl. Clifford Algebra 25, 925–942 (2015)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Liberti L., Lavor C., Maculan N.: A branch-and-prune algorithm for the molecular distance geometry problem. Int. Trans. Oper. Res. 15, 1–17 (2008)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Liberti L., Lavor C., Mucherino A., Maculan N.: Molecular distance geometry methods: from continuous to discrete. Int. Trans. Oper. Res. 18, 33–51 (2010)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Liberti L., Lavor C., Maculan N., Mucherino A.: Euclidean distance geometry and applications. SIAM Rev. 56, 3–69 (2014)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Mucherino, A., Lavor, C., Liberti, L., Maculan, N. (eds.): Distance Geometry: Theory, Methods, and Applications. Springer, New York (2013)Google Scholar
  20. 20.
    Perwass, C.: Geometric Algebra with Applications in Engineering. Springer (2009)Google Scholar
  21. 21.
    Pesonen J., Henriksson O.: Polymer conformations in internal (polyspherical) coordinates. J. Comput. Chem. 31, 1874–1881 (2009)Google Scholar
  22. 22.
    Souza M., Lavor C., Muritiba A., Maculan N.: Solving the molecular distance geometry problem with inaccurate distance data. BMC Bioinform. 14, S71–S76 (2013)CrossRefGoogle Scholar
  23. 23.
    Wütrich K.: Protein structure determination in solution by nuclear magnetic resonance spectroscopy. Science 243, 45–50 (1989)ADSCrossRefGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.University of Campinas (IMECC-UNICAMP)CampinasBrazil

Personalised recommendations