Oriented Conformal Geometric Algebra and the Molecular Distance Geometry Problem

  • Carlile LavorEmail author
  • Rafael Alves
Part of the following topical collections:
  1. Homage to Prof. W.A. Rodrigues Jr


The problem of 3D protein structure determination using distance information from nuclear magnetic resonance (NMR) experiments is a classical problem in distance geometry. NMR data and the chemistry of proteins provide a way to define a protein backbone order such that the distances related to the pairs of atoms \(\{i-3,i\},\{i-2,i\},\{i-1,i\}\) are available, implying a combinatorial method to solve the problem, called branch-and-prune (BP). There are two main steps in BP algorithm: the first one (the branching phase) is to intersect three spheres centered at the positions for atoms \( i-3,i-2,i\), with radius given by the atomic distances \( d_{i-3,i},d_{i-2,i},d_{i-1,i}\), respectively, to obtain two possible positions for atom i; and the second one (the pruning phase) is to check if additional spheres (related to distances \(d_{j,i}\), \(j<i-3\)) can be used to select one of the two possibilities for atom i. Differently from distances \(d_{i-2,i},d_{i-1,i}\) (associated to bond lenghts and bond angles), distances \(d_{j,i}\), \(j\le i-3\), may not be precise. BP algorithm has difficulties to deal with uncertainties, and this paper proposes the oriented conformal geometric algebra to take care of intersection of spheres when their centers and radius are not precise.


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



We would like to thank the Brazilian research agencies CNPq, CAPES, and FAPESP, for their financial support, and also Leo Dorst, for his comments and suggestions.


  1. 1.
    Agra, A., Figueiredo, R., Lavor, C., Maculan, N., Pereira, A., Requejo, C.: Feasibility check for the distance geometry problem: an application to molecular conformations. Int. Trans. Oper. Res. 24, 1023–1040 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alves, R., Lavor, C.: Geometric algebra to model uncertainties in the discretizable molecular distance geometry problem. Adv. Appl. Clifford Algebra 27, 439–452 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alves, R., Lavor, C., Souza, C., Souza, M.: Clifford algebra and discretizable distance geometry. Math. Methods Appl. Sci. 41, 3999–4346 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Billinge, S., Duxbury, P., Gonçalves, D., Lavor, C., Mucherino, A.: Assigned and unassigned distance geometry: applications to biological molecules and nanostructures. 4OR 14, 337–376 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Billinge, S., Duxbury, P., Gonçalves, D., Lavor, C., Mucherino, A.: Recent results on assigned and unassigned distance geometry with applications to protein molecules and nanostructures. Ann. Oper. Res. 271, 161–203 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cameron, J., Lasenby, J.: Oriented conformal geometric algebra. Adv. Appl. Clifford Algebra 18, 523–538 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cassioli, A., Gunluk, O., Lavor, C., Liberti, L.: Discretization vertex orders in distance geometry. Discrete Appl. Math. 197, 27–41 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    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
  9. 9.
    Costa, T., Bouwmeester, H., Lodwick, W., Lavor, C.: Calculating the possible conformations arising from uncertainty in the molecular distance geometry problem using constraint interval analysis. Inform. Sci. 415–416, 41–52 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Crippen, G., Havel, T.: Distance Geometry and Molecular Conformation. Wiley, New York (1988)zbMATHGoogle Scholar
  11. 11.
    Donald, B.: Algorithms in Structural Molecular Biology. MIT Press, Cambridge (2011)Google Scholar
  12. 12.
    Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry. Morgan Kaufman, San Mateo (2007)Google Scholar
  13. 13.
    Dress, A., Havel, T.: Distance geometry and geometric algebra. Found. Phys. 23, 1357–1374 (1993)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Fidalgo, F., Gonalves, D., Lavor, C., Liberti, L., Mucherino, A.: A symmetry-based splitting strategy for discretizable distance geometry problems. J. Glob. Optim. 71, 717–733 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gonçalves, D., Mucherino, A.: Discretization orders and efficient computation of Cartesian coordinates for distance geometry. Optim. Lett. 8, 2111–2125 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gonçalves, D., Mucherino, A., Lavor, C., Liberti, L.: Recent advances on the interval distance geometry problem. J. Glob. Optim. 69, 525–545 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hestenes, D.: Old wine in new bottles: a new algebraic framework for computational geometry. In: Corrochano E. B., Sobczyk G. (eds.) Geometric Algebra with Applications in Science and Engineering. Birkhäuser, Boston (2001)Google Scholar
  18. 18.
    Hildenbrand, D.: Foundations of Geometric Algebra Computing. Springer, Berlin Heidelberg (2012)zbMATHGoogle Scholar
  19. 19.
    Lavor, C., Xambó-Descamps, S., Zaplana, I.: A Geometric Algebra Invitation to Space-Time Physics Robotics and Molecular Geometry. SpringerBriefs in Mathematics. Springer, Berlin Heidelberg (2018)CrossRefGoogle Scholar
  20. 20.
    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)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lavor, C., Liberti, L., Maculan, N., Mucherino, A.: The discretizable molecular distance geometry problem. Comput. Optim. Appl. 52, 115–146 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Lavor, C., Liberti, L., Mucherino, A.: The interval BP algorithm for the discretizable molecular distance geometry problem with interval data. J. Glob. Optim. 56, 855–871 (2013)CrossRefGoogle Scholar
  23. 23.
    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)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lavor, C., Liberti, L., Lodwick, W., Mendonça da Costa, T.: An Introduction to Distance Geometry applied to Molecular Geometry. SpringerBriefs in Computer Science. Springer, Berlin Heidelberg (2017)CrossRefGoogle Scholar
  25. 25.
    Lavor, C., Liberti, L., Donald, B., Worley, B., Bardiaux, B., Malliavin, T., Nilges, M.: Minimal NMR distance information for rigidity of protein graphs. Discrete Applied Mathematics (2018) (to appear) Google Scholar
  26. 26.
    Li, H., Hestenes, D., Rockwood, A.: Generalized Homogeneous Coordinates for Computational Geometry. In: Sommer, G. (ed.) Geometric Computing with Clifford Algebra, pp. 25–58. Springer, Berlin Heidelberg (2001)Google Scholar
  27. 27.
    Liberti, L., Lavor, C.: Open Research Areas in Distance Geometry. In: Pardalos, P., Migdalas, A. (eds.) Open Problems in Optimization and Data Analysis. Springer, Berlin Heidelberg (2018). (to appear)Google Scholar
  28. 28.
    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)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Liberti, L., Lavor, C., Mucherino, A., Maculan, N.: Molecular distance geometry methods: from continuous to discrete. Int. Trans. Oper. Res. 18, 33–51 (2010)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Liberti, L., Lavor, C., Maculan, N., Mucherino, A.: Euclidean distance geometry and applications. SIAM Rev. 56, 3–69 (2014)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Liberti, L., Masson, B., Lee, J., Lavor, C., Mucherino, A.: On the number of realizations of certain Henneberg graphs arising in protein conformation. Discrete Appl. Math. 165, 213–232 (2014)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Liberti, L., Lavor, C.: Six mathematical gems from the history of distance geometry. Int. Trans. Oper. Res. 23, 897–920 (2016)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Liberti, L., Lavor, C.: Euclidean Distance Geometry. An Introduction. Springer, Berlin (2017)CrossRefGoogle Scholar
  34. 34.
    Menger, K.: Untersuchungen uber allgemeine Metrik. Math. Ann. 100, 75–163 (1928)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Mucherino, A., Lavor, C., Liberti, L.: The discretizable distance geometry problem. Optim. Lett. 6, 1671–1686 (2012)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Mucherino, A., Lavor, C., Liberti, L., Maculan, N. (eds.): Distance Geometry: Theory, Methods, and Applications. Springer, Berlin (2013)zbMATHGoogle Scholar
  37. 37.
    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
  38. 38.
    Stolfi, J.: Oriented Projective Geometry—A Framework for Geometric Computations. Academic Press, Cambridge (1991)zbMATHGoogle Scholar
  39. 39.
    Worley, B., Delhommel, F., Cordier, F., Malliavin, T., Bardiaux, B., Wolff, N., Nilges, M., Lavor, C., Liberti, L.: Tuning interval branch-and-prune for protein structure determination. J. Glob. Optim. 72, 109–127 (2018)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Wütrich, K.: Protein structure determination in solution by nuclear magnetic resonance spectroscopy. Science 243, 45–50 (1989)ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.University of Campinas (IMECC-UNICAMP)CampinasBrazil
  2. 2.Federal University of ABC (CMCC-UFABC)Santo AndréBrazil

Personalised recommendations