Skip to main content
Log in

On the optimality of finding DMDGP symmetries

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

The Discretizable Molecular Distance Geometry Problem (DMDGP) is a subclass of the Distance Geometry Problem, which aims to embed a weighted simple undirected graph in a Euclidean space, such that the distances between the points correspond to the values given by the weighted edges in the graph. The search space of the DMDGP is combinatorial, based on a total vertex order that implies symmetry properties related to partial reflections around planes defined by the Cartesian coordinates of three immediate and consecutive vertices that precede the so-called symmetry vertices. Since these symmetries allow us to know a priori the cardinality of the solution set and to calculate all the DMDGP solutions, given one of them, it would be relevant to identify these symmetries efficiently. Exploiting mathematical properties of the vertices associated with these symmetries, we present an optimal algorithm that finds such vertices.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  • Billinge S, Duxbury P, Gonçalves D, Lavor C, Mucherino A (2016) Assigned and unassigned distance geometry: applications to biological molecules and nanostructures. 4OR 14:337–376

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Carvalho R, Lavor C, Protti F (2008) Extending the geometric build-up algorithm for the molecular distance geometry problem. Inf Process Lett 108:234–237

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Cassioli A, Bordiaux B, Bouvier G, Mucherino A, Alves R, Liberti L, Nilges M, Lavor C, Malliavin T (2015) An algorithm to enumerate all possible protein conformations verifying a set of distance constraints. BMC Bioinform 16:16–23

    Article  Google Scholar 

  • Crippen G, Havel T (1988) Distance geometry and molecular conformation. Wiley, Oxford

    MATH  Google Scholar 

  • Fidalgo F, Gonçalves D, Lavor C, Liberti L, Mucherino A (2018) A symmetry-based splitting strategy for discretizable distance geometry problems. J Global Optim 71:717–733

    Article  MathSciNet  Google Scholar 

  • Lavor C, Liberti L, Maculan N (2005) Grover’s algorithm applied to the molecular distance geometry problem. In: Proceedings of the 7th Brazilian congress of neural networks

  • Lavor C, Liberti L, Maculan N (2006) Computational experience with the molecular distance geometry problem. In: Pintér J (ed) Global optimization: scientific and engineering case studies. Springer, Berlin, pp 213–225

    Chapter  Google Scholar 

  • Lavor C, Lee J, Lee-St JA, Liberti L, Mucherino A, Sviridenko M (2012) Discretization orders for distance geometry problems. Optim Lett 6:783–796

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Lavor C, Liberti L, Maculan N, Mucherino A (2012) Recent advances on the discretizable molecular distance geometry problem. Eur J Oper Res 219:698–706

    Article  MathSciNet  Google Scholar 

  • Lavor C, Liberti L, Lodwick W, da Costa TM (2017) An introduction to distance geometry applied to molecular geometry. Springer, Berlin

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Liberti L, Lavor C (2017) Euclidean distance geometry: an introduction. Springer, Berlin

    Book  Google Scholar 

  • Liberti L, Lavor C (2018) Open research areas in distance geometry. In: Migalas A, Pardalos P (eds) Open problems in optimization and data analysis. Springer, Berlin, pp 183–223

    Chapter  Google Scholar 

  • Liberti L, Lavor C, Maculan N (2008) A branch-and-prune algorithm for the molecular distance geometry problem. Int Trans Oper Res 15:1–17

    Article  MathSciNet  Google Scholar 

  • Liberti L, Lavor C, Mucherino A, Maculan N (2010) Molecular distance geometry methods: from continuous to discrete. Int Trans Oper Res 18:33–51

    Article  MathSciNet  Google Scholar 

  • Liberti L, Lavor C, Alencar J, Resende G (2013) Counting the number of solutions of $^{K}$DMDGP instances. Lect Notes Comput Sci 8085:224–230

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Liberti L, Masson B, Lee J, Lavor C, Mucherino A (2014) On the number of realizations of certain Henneberg graphs arising in protein conformation. Discret Appl Math 165:213–232

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • Marquezino F, Portugal R, Lavor C (2019) A primer on quantum computing. Springer, Berlin

    Book  Google Scholar 

  • Mucherino A, Lavor C, Liberti L (2012) Exploiting symmetry properties of the discretizable molecular distance geometry problem. J Bioinform Comput Biol 10:1242009

    Article  Google Scholar 

  • Mucherino A, Lavor C, Liberti L, Maculan N (eds) (2013) Distance geometry: theory, methods, and applications. Springer, Berlin

    MATH  Google Scholar 

  • Nucci P, Nogueira L, Lavor C (2013) Solving the discretizable molecular distance geometry problem by multiple realization trees. In: Mucherino A, Lavor C, Liberti L, Maculan N (eds) Distance geometry: theory, methods, and applications. Springer, New York, pp 161–176

    Chapter  Google Scholar 

  • Wüthrich K (1989) Protein structure determination in solution by nuclear magnetic resonance spectroscopy. Science 243:45–50

    Article  Google Scholar 

Download references

Acknowledgements

We would like to thank the Brazilian research agencies, CNPq and FAPESP, and the careful reading and important comments made by the reviewers.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Wagner Rocha.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by Gabriel Haeser.

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lavor, C., Oliveira, A., Rocha, W. et al. On the optimality of finding DMDGP symmetries. Comp. Appl. Math. 40, 98 (2021). https://doi.org/10.1007/s40314-021-01479-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-021-01479-6

Keywords

Mathematics Subject Classification

Navigation