Unassigned distance geometry and molecular conformation problems | Journal of Global Optimization Skip to main content
Springer Nature Link
Log in

Unassigned distance geometry and molecular conformation problems

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

3D protein structures and nanostructures can be obtained by exploiting distance information provided by experimental techniques, such as nuclear magnetic resonance and the pair distribution function method. These are examples of instances of the unassigned distance geometry problem (uDGP), where the aim is to calculate the position of some points using a list of associated distance values not previoulsy assigned to the pair of points. We propose new mathematical programming formulations and a new heuristic to solve the uDGP related to molecular structure calculations. In addition to theoretical results, computational experiments are also provided.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Alves, R., Lavor, C.: Geometric algebra to model uncertainties in the discretizable molecular distance geometry problem. Adv. Appl. Clifford Algebr. 27, 439–452 (2017)

    Article  MathSciNet  Google Scholar 

  2. Alves, R., Lavor, C., Souza, C., Souza, M.: Clifford algebra and discretizable distance geometry. Math. Methods Appl. Sci. 41, 3999–4346 (2018)

    Article  MathSciNet  Google Scholar 

  3. Baez-Sanchez, A., Lavor, C.: On the estimation of unknown distances for a class of Euclidean distance matrix completion problems with interval data. Linear Algebr. Appl. 592, 287–305 (2020)

    Article  MathSciNet  Google Scholar 

  4. Bartmeyer, P., Lyra, C.: A new quadratic relaxation for binary variables applied to the distance geometry problem. Struct. Multidiscip. Optim. 62, 2197–2201 (2020)

    Article  MathSciNet  Google Scholar 

  5. Bendsoe, M., Sigmund, O.: Topol. Optim. Theory. Methods and Applications, Springer, New York (2003)

    Google Scholar 

  6. 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)

  7. Billinge, S., Duxbury, P., Gonçalves, D., Lavor, C., Mucherino, A.: Recent results on assigned and unassigned distance geometry with applications to proteinmolecules and nanostructures. Ann. Oper. Res. 271, 161–203 (2018)

    Article  MathSciNet  Google Scholar 

  8. Cassioli, A., Gunluk, O., Lavor, C., Liberti, L.: Discretization vertex orders in distance geometry. Discret. Appl. Math. 197, 27–41 (2015)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  10. Dambrosio, C., Ky, V., Lavor, C., Liberti, L., Maculan, N.: New error measures and methods for realizing protein graphs from distance data. Discret. Comput. Geom. 57, 371–418 (2017)

    Article  MathSciNet  Google Scholar 

  11. Duxbury, P., Granlund, L., Gujarathi, S., Juhas, P., Billinge, S.: The unassigned distance geometry problem. Discret. Appl. Math. 204, 117–132 (2016)

    Article  MathSciNet  Google Scholar 

  12. Fontoura, L., Martinelli, R., Poggi, M., Vidal, T.: The minimum distance superset problem: formulations and algorithms. J. Glob. Optim. 72, 27–53 (2018)

    Article  MathSciNet  Google Scholar 

  13. Gonçalves, D., Mucherino, A., Lavor, C., Liberti, L.: Recent advances on the interval distance geometry problem. J. Glob. Optim. 69, 525–545 (2017)

    Article  MathSciNet  Google Scholar 

  14. Gujarathi, S., Farrow, C., Glosser, C., Granlund, L., Duxbury, P.: Ab-initio reconstruction of complex Euclidean networks in two dimensions. Phys. Rev. E 89, 053311 (2014)

    Article  Google Scholar 

  15. Juhás, P., Cherba, D., Duxbury, P., Punch, W., Billinge, S.: Ab initio determination of solid-state nanostructure. Nature 440, 655–658 (2006)

    Article  Google Scholar 

  16. Lavor, C.: On generating instances for the molecular distance geometry problem. In: Liberti, L., Maculan, N. (eds.) Global Optimization: From Theory to Implementation, pp. 405–414. Springer, Berlin (2006)

    Chapter  Google Scholar 

  17. 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)

    Article  MathSciNet  Google Scholar 

  18. Lavor, C., Liberti, L., Maculan, N., Mucherino, A.: The discretizable molecular distance geometry problem. Comput. Optim. Appl. 52, 115–146 (2012)

    Article  MathSciNet  Google Scholar 

  19. 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)

    Article  Google Scholar 

  20. Lavor, C., Alves, R., Figueiredo, W., Petraglia, A., Maculan, N.: Clifford algebra and the discretizable molecular distance geometry problem. Adv. Appl. Clifford Algebr. 25, 925–942 (2015)

    Article  MathSciNet  Google Scholar 

  21. Lavor, C., Liberti, L., Lodwick, W., Mendonça da Costa, T.: An Introduction to Distance Geometry applied to Molecular Geometry. SpringerBriefs, New York (2017)

    Book  Google Scholar 

  22. Lavor, C., Xambó-Descamps, S., Zaplana, I.: A Geometric Algebra Invitation to Space-Time Physics. Robotics and Molecular Geometry. SpringerBriefs, New York (2018)

    Book  Google Scholar 

  23. Lavor, C., Liberti, L., Donald, B., Worley, B., Bardiaux, B., Malliavin, T., Nilges, M.: Minimal NMR distance information for rigidity of protein graphs. Discret. Appl. Math. 256, 91–104 (2019)

    Article  MathSciNet  Google Scholar 

  24. Lavor, C., Alves, R.: Oriented conformal geometric algebra and the molecular distance geometry problem. Adv. Appl. Clifford Algebr. 29, 1–19 (2019)

    Article  MathSciNet  Google Scholar 

  25. Lavor, C., Souza, M., Mariano, L., Liberti, L.: On the polinomiality of finding \(^{K}\)DMDGP re-orders. Discret. Appl. Math. 267, 190–194 (2019)

    Article  Google Scholar 

  26. Lavor, C., Souza, M., Mariano, L., Gonçalves, D., Mucherino, A.: Improving the sampling process in the interval Branch-and-Prune algorithm for the discretizable molecular distance geometry problem. Appl. Math. Comput. 389, 125586 (2021)

    MathSciNet  MATH  Google Scholar 

  27. 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)

    Article  MathSciNet  Google Scholar 

  28. Liberti, L., Lavor, C., Maculan, N., Mucherino, A.: Euclidean distance geometry and applications. SIAM Rev. 56, 3–69 (2014)

    Article  MathSciNet  Google Scholar 

  29. Liberti, L., Lavor, C.: Six mathematical gems from the history of distance geometry. Int. Trans. Oper. Res. 23, 897–920 (2016)

    Article  MathSciNet  Google Scholar 

  30. Liberti, L., Lavor, C.: Euclidean Distance Geometry: An Introduction. Springer, New York (2017)

    Book  Google Scholar 

  31. Liberti, L., Lavor, L.: Open research areas in distance geometry. In: Pardalos, P., Migdalas, A. (eds.) Open Problems in Optimization and Data Analysis, pp. 183–223. Springer, New York (2018)

    Chapter  Google Scholar 

  32. Maioli, D., Lavor, C., Gonçalves, D.: A note on computing the intersection of spheres in \({\mathbb{R}}^{n}\). ANZIAM J. 59, 271–279 (2017)

    Article  MathSciNet  Google Scholar 

  33. Malliavin, T., Mucherino, A., Lavor, C., Liberti, L.: Systematic exploration of protein conformational space using a distance geometry approach. J. Chem. Inform. Model. 59, 4486–4503 (2019)

    Article  Google Scholar 

  34. Martínez, J.M.: A note on the theoretical convergence properties of the SIMP method. Struct. Multidiscipl. Optim. 29, 319–323 (2005)

    Article  MathSciNet  Google Scholar 

  35. Menger, K.: Untersuchungen uber allgemeine Metrik. Mathematische Annalen 100, 75–163 (1928)

    Article  MathSciNet  Google Scholar 

  36. Moreira, N., Duarte, L., Lavor, C., Torezzan, C.: A novel low-rank matrix completion approach to estimate missing entries in Euclidean distance matrix. Comput. Appl. Math. 37, 4989–4999 (2018)

    Article  MathSciNet  Google Scholar 

  37. Mucherino, A., Lavor, C., Liberti, L., Maculan, N. (eds.): Distance Geometry: Theory, Methods, and Applications. Springer, New York (2013)

    MATH  Google Scholar 

  38. Neto, L.S., Lavor, C., Lodwick, W.: A constrained interval approach to the generalized distance geometry problem. Optim. Lett. 14, 483–492 (2020)

    Article  MathSciNet  Google Scholar 

  39. Santiago, C., Lavor, C., Monteiro, S., Kroner-Martins, A.: A new algorithm for the small-field astrometric point-pattern matching problem. J. Glob. Optim. 72, 55–70 (2018)

    Article  MathSciNet  Google Scholar 

  40. Saxe, J.: Embeddability of weighted graphs in k-space is strongly np-hard, Proceeding of the 17th Allerton Conference in Communications, Control and Computing, 480–489 (1979)

  41. Skiena, S., Smith, W., Lemke, P.: Reconstructing sets from interpoint distances, Proceedings of the Sixth ACM Symposium on Computational Geometry, 332–339 (1990)

  42. 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)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank the Brazilian research agencies CNPq and FAPESP, for their financial support, and the reviewers for their valuable comments.

Author information

Authors and Affiliations

  1. Michigan State University, Natural Science Building, 288 Farm Lane, East Lansing, MI, USA

    Phil Duxbury

  2. IMECC-Unicamp, Cidade Universitaria Zeferino Vaz, Campinas, Brazil

    Carlile Lavor

  3. CNRS LIX, Ecole Polytechnique, 91128, Palaiseau, France

    Leo Liberti

  4. Science and Technology Institute, Federal University of Sao Paulo, Sao Jose dos Campos, Brazil

    Luiz Leduino de Salles-Neto

Authors
  1. Phil Duxbury
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carlile Lavor
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Leo Liberti
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Luiz Leduino de Salles-Neto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luiz Leduino de Salles-Neto.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Fapesp and CNPq-Brazil.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Duxbury, P., Lavor, C., Liberti, L. et al. Unassigned distance geometry and molecular conformation problems. J Glob Optim 83, 73–82 (2022). https://doi.org/10.1007/s10898-021-01023-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-021-01023-0

Keywords

Search

Navigation