Recent results on assigned and unassigned distance geometry with applications to protein molecules and nanostructures

Abstract

In the 2 years since our last 4OR review of distance geometry methods with applications to proteins and nanostructures, there has been rapid progress in treating uncertainties in the discretizable distance geometry problem; and a new class of geometry problems started to be explored, namely vector geometry problems. In this work we review this progress in the context of the earlier literature.

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Fig. 1

(Image from Gujarathi 2014)

Fig. 2

(Reproduced with permission from Juhás et al. 2006)

Fig. 3
Fig. 4

(Reproduced with permission from Duxbury et al. 2016)

Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

(Reproduced with permission from Billinge and Levin 2007)

Fig. 11

(Reroduced with permission from Juhás et al. 2006, 2008)

Notes

  1. 1.

    \(\langle (X,y), (\hat{X},\hat{y}) \rangle := \text {tr} (X \hat{X}^T) + y^T \hat{y}\).

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Acknowledgements

The authors are thankful to the editors for their invitation to submit an updated version of our survey that was previously published in 2016 in the Quarterly Journal of Operations Research. We wish to thank FAPESP and CNPq for financial support. Support for work at Michigan State University by the MSU foundation is gratefully acknowledged. Collaborations with Pavol Juhas, Luke Granlund, Saurabh Gujarathi, Chris Farrow and Connor Glosser are much appreciated. PMD, CL and AM would like to thank Leo Liberti for interesting and motivating discussions. Work in the Billinge group was supported by the U.S. National Science Foundation through grant DMREF-1534910.

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Correspondence to Douglas S. Gonçalves.

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This is an updated version of the paper “Assigned and unassigned distance geometry: applications to biological molecules and nanostructures” that appeared in 4OR—Q J Oper Res (2016) 14: 337–376.

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Billinge, S.J.L., Duxbury, P.M., Gonçalves, D.S. et al. Recent results on assigned and unassigned distance geometry with applications to protein molecules and nanostructures. Ann Oper Res 271, 161–203 (2018). https://doi.org/10.1007/s10479-018-2989-6

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Keywords

  • Distance geometry
  • Graph rigidity
  • Molecular conformations
  • Nanostructures