Molecular Geometry

  • Carlile Lavor
  • Sebastià Xambó-Descamps
  • Isiah Zaplana
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


The 3D structure of a molecule is fundamental for understanding its function, especially in the case of proteins [28]. The calculation of protein structures can be tackled experimentally, through Nuclear Magnetic Resonance (NMR) spectroscopy and X-ray crystallography [9], or theoretically, via molecular potential energy minimization [56, 57].


  1. 1.
    A. Agra, R. Figueiredo, C. Lavor, N. Maculan, A. Pereira, C. Requejo, Feasibility check for the distance geometry problem: an application to molecular conformations. Int. Trans. Oper. 24(5), 1023–1040 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    R. Alves, C. Lavor, Geometric algebra to model uncertainties in the discretizable molecular distance geometry problem. Adv. Appl. Clifford Algebr. 27(1), 439–452 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    R. Alves, C. Lavor, C. Souza, M. Souza, Clifford algebra and discretizable distance geometry. Math. Methods Appl. Sci. 41, 3999–4346 (2018)CrossRefGoogle Scholar
  4. 8.
    S. Billinge, P. Duxbury, D. Gonçalves, C. Lavor, A. Mucherino, Assigned and unassigned distance geometry: applications to biological molecules and nanostructures. 4OR 14(4), 337–376 (2016)MathSciNetCrossRefGoogle Scholar
  5. 9.
    A. Brünger, M. Nilges, Computational challenges for macromolecular structure determination by X-ray crystallography and solution NMR-spectroscopy. Q. Rev. Biophys. 26(1), 49–125 (1993)CrossRefGoogle Scholar
  6. 14.
    A. Cassioli, B. Bordeaux, G. Bouvier, M. Mucherino, R. Alves, L. Liberti, M. Nilges, C. Lavor, T. Malliavin, An algorithm to enumerate all possible protein conformations verifying a set of distance constraints. BMC Bioinf. 16, 16–23 (2015)CrossRefGoogle Scholar
  7. 15.
    A. Cassioli, O. Gunluk, C. Lavor, L. Liberti, Discretization vertex orders in distance geometry. Discrete Appl. Math. 197, 27–41 (2015)MathSciNetCrossRefGoogle Scholar
  8. 17.
    P. Chys, Application of geometric algebra for the description of polymer conformations. J. Chem. Phys. 128(10), 104107(1)–104107(12) (2008)CrossRefGoogle Scholar
  9. 18.
    P. Chys, P. Chacón, Spinor product computations for protein conformations. J. Comput. Chem. 33(21), 1717–1729 (2012)CrossRefGoogle Scholar
  10. 21.
    T. Costa, H. Bouwmeester, W. Lodwick, C. Lavor, Calculating the possible conformations arising from uncertainty in the molecular distance geometry problem using constraint interval analysis. Inf. Sci. 415–416, 41–52 (2017)MathSciNetCrossRefGoogle Scholar
  11. 23.
    G. Crippen, T. Havel, Distance Geometry and Molecular Conformation (Wiley, New York, 1988)zbMATHGoogle Scholar
  12. 28.
    B. Donald, Algorithms in Structural Molecular Biology (MIT Press, Cambridge, 2011)Google Scholar
  13. 31.
    A. Dress, T. Havel, Distance geometry and geometric algebra. Found. Phys. 23(10), 1357–1374 (1993)MathSciNetCrossRefGoogle Scholar
  14. 35.
    D. Gonçalves, A. Mucherino, Discretization orders and efficient computation of Cartesian coordinates for distance geometry. Optim. Lett. 8(7), 2111–2125 (2014)MathSciNetCrossRefGoogle Scholar
  15. 56.
    C. Lavor, Analytic evaluation of the gradient and Hessian of molecular potential energy functions. Phys. D Nonlinear Phenom. 227(2), 135–141 (2007)MathSciNetCrossRefGoogle Scholar
  16. 57.
    C. Lavor, N. Maculan, A function to test methods applied to global minimization of potential energy of molecules. Numer. Algorithms 35(2–4), 287–300 (2004)MathSciNetCrossRefGoogle Scholar
  17. 58.
    C. Lavor, L. Liberti, N. Maculan, Computational experience with the molecular distance geometry problem, in Global Optimization, ed. by J. Pintér, Nonconvex Optimization and Its Applications, vol. 85 (Springer, New York, 2006), pp. 213–225Google Scholar
  18. 59.
    C. Lavor, L. Liberti, N. Maculan, A. Mucherino, The discretizable molecular distance geometry problem. Comput. Optim. Appl. 52(1), 115–146 (2012)MathSciNetCrossRefGoogle Scholar
  19. 60.
    C. Lavor, L. Liberti, N. Maculan, A. Mucherino, Recent advances on the discretizable molecular distance geometry problem. Eur. J. Oper. Res. 219(3), 698–706 (2012)MathSciNetCrossRefGoogle Scholar
  20. 61.
    C. Lavor, L. Liberti, A. Mucherino, The interval BP algorithm for the discretizable molecular distance geometry problem with interval data. J. Glob. Optim. 56(3), 855–871 (2013)CrossRefGoogle Scholar
  21. 62.
    C. Lavor, R. Alves, W. Figueiredo, A. Petraglia, N. Maculan, Clifford algebra and the discretizable molecular distance geometry problem. Adv. Appl. Clifford Algebr. 25(4), 925–942 (2015)MathSciNetCrossRefGoogle Scholar
  22. 65.
    L. Liberti, C. Lavor, Six mathematical gems from the history of distance geometry. Int. Trans. Oper. Res. 23(5), 897–920 (2016)MathSciNetCrossRefGoogle Scholar
  23. 66.
    L. Liberti, C. Lavor, N. Maculan, A branch-and-prune algorithm for the molecular distance geometry problem. Int. Trans. Oper. Res. 15(1), 1–17 (2008)MathSciNetCrossRefGoogle Scholar
  24. 67.
    L. Liberti, C. Lavor, A. Mucherino, N. Maculan, Molecular distance geometry methods: from continuous to discrete. Int. Trans. Oper. Res. 18(1), 33–51 (2011)MathSciNetCrossRefGoogle Scholar
  25. 68.
    L. Liberti, C. Lavor, N. Maculan, A. Mucherino, Euclidean distance geometry and applications. SIAM Rev. 56(1), 3–69 (2014)MathSciNetCrossRefGoogle Scholar
  26. 69.
    L. Liberti, B. Masson, J. Lee, C. Lavor, A. Mucherino, On the number of realizations of certain Henneberg graphs arising in protein conformation. Discrete Appl. Math. 165, 213–232 (2014)MathSciNetCrossRefGoogle Scholar
  27. 75.
    A. Mucherino, C. Lavor, L. Liberti, The discretizable distance geometry problem. Optim. Lett. 6(8), 1671–1686 (2012)MathSciNetCrossRefGoogle Scholar
  28. 76.
    A. Mucherino, C. Lavor, L. Liberti, N. Maculan (eds.), Distance Geometry: Theory, Methods, and Applications (Springer, New York, 2013)zbMATHGoogle Scholar
  29. 80.
    J. Pesonen, O. Henriksson, Polymer conformations in internal (polyspherical) coordinates. J. Comput. Chem. 31(9), 1873–1881 (2009)Google Scholar
  30. 84.
    J. Saxe, Embeddability of weighted graphs in k-space is strongly NP-hard, in Proceedings of 17th Allerton Conference in Communications, Control and Computing (1979), pp. 480–489Google Scholar
  31. 87.
    C. Seok, E. Coutsias, Efficiency of rotational operators for geometric manipulation of chain molecules. Bull. Kor. Chem. Soc. 28(10), 1705–1708 (2007)CrossRefGoogle Scholar
  32. 89.
    M. Souza, C. Lavor, A. Muritiba, M. Maculan, Solving the molecular distance geometry problem with inaccurate distance data. BMC Bioinf. 14(9), S7(1)–S7(6) (2013)Google Scholar
  33. 94.
    K. Wütrich, Protein structure determination in solution by nuclear magnetic resonance spectroscopy. Science 243, 45–50 (1989)CrossRefGoogle Scholar

Copyright information

© The Author(s), under exclusive licence to Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Carlile Lavor
    • 1
  • Sebastià Xambó-Descamps
    • 2
  • Isiah Zaplana
    • 3
  1. 1.Department of Applied Maths (IMECC-UNICAMP)University of CampinasCampinasBrazil
  2. 2.Departament de MatemàtiquesUniversitat Politècnica de CatalunyaBarcelonaSpain
  3. 3.Institut d’Org. i Control de Sist. Ind.Universitat Politècnica de CatalunyaBarcelonaSpain

Personalised recommendations