Efficient Algorithms for Co-folding of Multiple RNAs | SpringerLink
Skip to main content
Springer Nature Link
Log in

Efficient Algorithms for Co-folding of Multiple RNAs

  • Conference paper
  • First Online:
Biomedical Engineering Systems and Technologies (BIOSTEC 2020)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1400))

Abstract

The simplest class of structures formed by \(N\ge 2\) interacting RNAs consists of all crossing-free base pairs formed over linear arrangements of the constituent RNA sequences. For each permutation of the N strands the structure prediction problem is algorithmically very similar – but not identical – to folding of a single, contiguous RNA. The differences arise from two sources: First, “nicks”, i.e., the transitions from one to the next piece of RNA, need to be treated with special care. Second, the connectedness of the structures needs to guaranteed. For the forward recursions, i.e., the computation of folding energies or partition functions, these modifications are rather straightforward and retain the cubic time complexity of the well-known folding algorithms. This is not the case for a straightforward implementation of the corresponding outside recursion, which becomes quartic. Cubic running times, however, can be restored by introducing linear-size auxiliary arrays. Asymptotically, the extra effort over the corresponding algorithms for a single RNA sequence of the same length is negligible in both time and space. An implementation within the framework of the ViennaRNA package conforms to the theoretical performance bounds and provides access to several algorithmic variants, include the handling of user-defined hard and soft constraints.

An earlier version of this contribution appeared in the Proceedings of the 13th International Joint Conference on Biomedical Engineering Systems and Technologies – Volume 3: Bioinformatics [26]. This work was supported in part by the German Federal Ministry of Education and Research (BMBF, project no. 031A538A, de.NBI-RBC, to PFS and project no. 031L0164C, RNAProNet, to PFS), and the Austrian science fund FWF (project no. I 2874 “Prediction of RNA-RNA interactions”, project no. F 80 “RNAdeco” to ILH).

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

Access this chapter

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

Chapter
JPY 3498
Price includes VAT (Japan)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
JPY 11439
Price includes VAT (Japan)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
JPY 14299
Price includes VAT (Japan)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Alkan, C., Karakoç, E., Nadeau, J.H., Sahinalp, S.C., Zhang, K.Z.: RNA-RNA interaction prediction and antisense RNA target search. J. Comput. Biol. 13, 267–282 (2006). https://doi.org/10.1089/cmb.2006.13.267

    Article  MathSciNet  MATH  Google Scholar 

  2. Andronescu, M., Zhang, Z.C., Condon, A.: Secondary structure prediction of interacting RNA molecules. J. Mol. Biol. 345, 987–1001 (2005). https://doi.org/10.1016/j.jmb.2004.10.082

    Article  Google Scholar 

  3. Badelt, S., Grun, C., Sarma, K.V., Wolfe, B., Shin, S.W., Winfree, E.: A domain-level DNA strand displacement reaction enumerator allowing arbitrary non-pseudoknotted secondary structures. J. R. Soc. Interface 17, 20190866 (2020). https://doi.org/10.1098/rsif.2019.0866

    Article  Google Scholar 

  4. Bernhart, S.H., Mückstein, U., Hofacker, I.L.: RNA accessibility in cubic time. Algorithms Mol. Biol. 6, 3 (2011). https://doi.org/10.1186/1748-7188-6-3

    Article  Google Scholar 

  5. Bernhart, S.H., Tafer, H., Mückstein, U., Flamm, C., Stadler, P.F., Hofacker, I.L.: Partition function and base pairing probabilities of RNA heterodimers. Algorithms Mol. Biol. 1, 3 (2006). https://doi.org/10.1186/1748-7188-1-3

    Article  Google Scholar 

  6. Bindewald, E., Afonin, K., Jaeger, L., Shapiro, B.A.: Multistrand RNA secondary structure prediction and nanostructure design including pseudoknots. ACS Nano 5, 9542–9551 (2011). https://doi.org/10.1021/nn202666w

    Article  Google Scholar 

  7. Busch, A., Richter, A., Backofen, R.: IntaRNA: efficient prediction of bacterial sRNA targets incorporating target site accessibility and seed regions. Bioinformatics 24, 2849–2856 (2008). https://doi.org/10.1093/bioinformatics/btn544

    Article  Google Scholar 

  8. Chappell, J., Watters, K.E., Takahashi, M.K., Lucks, J.B.: A renaissance in RNA synthetic biology: new mechanisms, applications and tools for the future. Curr. Opin. Chem. Biol. 28, 47–56 (2015). https://doi.org/10.1016/j.cbpa.2015.05.018

    Article  Google Scholar 

  9. Chitsaz, H., Salari, R., Sahinalp, S.C., Backofen, R.: A partition function algorithm for interacting nucleic acid strands. Bioinformatics 25, i365–i373 (2009). https://doi.org/10.1093/bioinformatics/btp212

    Article  Google Scholar 

  10. Dimitrov, R.A., Zuker, M.: Prediction of hybridization and melting for double-stranded nucleic acids. Biophys. J. 87, 215–226 (2004). https://doi.org/10.1529/biophysj.103.020743

    Article  Google Scholar 

  11. Ding, Y., Chan, C.Y., Lawrence, C.E.: Sfold web server for statistical folding and rational design of nucleic acids. Nucleic Acids Res. 32, W135–W141 (2004). https://doi.org/10.1093/nar/gkh449

    Article  Google Scholar 

  12. Dirks, R.M., Bois, J.S., Schaeffer, J.M., Winfree, E., Pierce, N.A.: Thermodynamic analysis of interacting nucleic acid strands. SIAM Rev. 49, 65–88 (2007). https://doi.org/10.1137/060651100

    Article  MathSciNet  MATH  Google Scholar 

  13. Durand, G., et al.: A combinatorial approach to the repertoire of RNA kissing motifs; towards multiplex detection by switching hairpin aptamers. Nucleic Acids Res. 44, 4450–4459 (2016). https://doi.org/10.1093/nar/gkw206

    Article  Google Scholar 

  14. Dutta, T., Srivastava, S.: Small RNA-mediated regulation in bacteria: a growing palette of diverse mechanisms. Gene 656, 60–72 (2018). https://doi.org/10.1016/j.gene.2018.02.068

    Article  Google Scholar 

  15. Forties, R.A., Bundschuh, R.: Modeling the interplay of single stranded binding proteins and nucleic acid secondary structure. Bioinformatics 26, 61–67 (2010). https://doi.org/10.1093/bioinformatics/btp627

    Article  Google Scholar 

  16. Gong, J., Ju, Y., Shao, D., Zhang, Q.C.: Advances and challenges towards the study of RNA-RNA interactions in a transcriptome-wide scale. Quant. Biol. 6(3), 239–252 (2018). https://doi.org/10.1007/s40484-018-0146-5

    Article  Google Scholar 

  17. Guil, S., Esteller, M.: RNA-RNA interactions in gene regulation: the coding and noncoding players. Trends Biochem. Sci. 40, 248–256 (2015). https://doi.org/10.1016/j.tibs.2015.03.001

    Article  Google Scholar 

  18. Hofacker, I.L., Fontana, W., Stadler, P.F., Bonhoeffer, L.S., Tacker, M., Schuster, P.: Fast folding and comparison of RNA secondary structures. Monatshefte für Chemie 125, 167–188 (1994). https://doi.org/10.1007/BF00818163

    Article  Google Scholar 

  19. Hofacker, I.L., Reidys, C.M., Stadler, P.F.: Symmetric circular matchings and RNA folding. Discret. Math. 312, 100–112 (2012). https://doi.org/10.1016/j.disc.2011.06.004

    Article  MathSciNet  MATH  Google Scholar 

  20. Huang, F.W.D., Qin, J., Reidys, C.M., Stadler, P.F.: Partition function and base pairing probabilities for RNA-RNA interaction prediction. Bioinformatics 25, 2646–2654 (2009). https://doi.org/10.1093/bioinformatics/btp481

    Article  Google Scholar 

  21. Isaacs, F.J., Dwyer, D.J., Collins, J.J.: RNA synthetic biology. Nat. Biotechnol. 24, 545–554 (2006). https://doi.org/10.1038/nbt1208

    Article  Google Scholar 

  22. King, D.E.: Dlib-ml: a machine learning toolkit. J. Mach. Learn. Res. 10, 1755–1758 (2009). /http://dlib.net/

    Google Scholar 

  23. Legendre, A., Angel, E., Tahi, F.: RCPred: RNA complex prediction as a constrained maximum weight clique problem. BMC Bioinform. 20, 128 (2019). https://doi.org/10.1186/s12859-019-2648-1

    Article  Google Scholar 

  24. Lorenz, R., et al.: 2D meets 4G: G-quadruplexes in RNA secondary structure prediction. IEEE Trans. Comput. Biol. Bioinf. 10, 832–844 (2013). https://doi.org/10.1109/TCBB.2013.7

    Article  Google Scholar 

  25. Lorenz, R., et al.: ViennaRNA package 2.0. Algorithms Mol. Biol. 6, 26 (2011). https://doi.org/10.1186/1748-7188-6-26

    Article  Google Scholar 

  26. Lorenz, R., Flamm, C., Hofacker, I.L., Stadler, P.F.: Efficient computation of base-pairing probabilities in multi-strand RNA folding. In: de Maria, E., Fred, A., Gamboa, H. (eds.) Proceedings of the 13th International Joint Conference on Biomedical Engineering Systems and Technologies – Volume 3: Bioinformatics. pp. 23–31. Scitepress, Setúbal (2020)

    Google Scholar 

  27. Lorenz, R., Hofacker, I.L., Stadler, P.F.: RNA folding with hard and soft constraints. Algorithms Mol. Biol. 11, 8 (2016). https://doi.org/10.1186/s13015-016-0070-z

    Article  Google Scholar 

  28. McCaskill, J.S.: The equilibrium partition function and base pair binding probabilities for RNA secondary structure. Biopolymers 29, 1105–1119 (1990). https://doi.org/10.1002/bip.360290621

    Article  Google Scholar 

  29. Mneimneh, S., Ahmed, S.A.: Multiple RNA interaction: beyond two. IEEE Trans. Nanobiosci. 14, 210–219 (2015). https://doi.org/10.1109/TNB.2015.2402591

    Article  Google Scholar 

  30. Mückstein, U., et al.: Translational control by RNA-RNA interaction: improved computation of RNA-RNA binding thermodynamics. In: Elloumi, M., Küng, J., Linial, M., Murphy, R.F., Schneider, K., Toma, C. (eds.) BIRD 2008. CCIS, vol. 13, pp. 114–127. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70600-7_9

    Chapter  Google Scholar 

  31. Reidys, C.M.: Combinatorial Computational Biology of RNA. Springer, Heidelberg (2011). https://doi.org/10.1007/978-0-387-76731-4

  32. Schaeffer, J.M., Thachuk, C., Winfree, E.: Stochastic simulation of the kinetics of multiple interacting nucleic acid strands. In: Phillips, A., Yin, P. (eds.) DNA 2015. LNCS, vol. 9211, pp. 194–211. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21999-8_13

    Chapter  Google Scholar 

  33. Serra, M.J., Turner, D.H.: Predicting thermodynamic properties of RNA. Methods Enzymol. 259, 242–261 (1995). https://doi.org/10.1016/0076-6879(95)59047-1

    Article  Google Scholar 

  34. Shear, D.B.: Stability and uniqueness of the equilibrium point in chemical reaction systems. J. Chem. Phys. 48, 4144–4147 (1968). https://doi.org/10.1063/1.1669753

    Article  Google Scholar 

  35. Höner zu Siederdissen, C., Prohaska, S.J., Stadler, P.F.: Algebraic dynamic programming over general data structures. BMC Bioinform. 16(19), S2 (2015). https://doi.org/10.1186/1471-2105-16-S19-S2

  36. Tacker, M., Stadler, P.F., Bornberg-Bauer, E.G., Hofacker, I.L., Schuster, P.: Algorithm independent properties of RNA structure prediction. Eur. Biophys. J. 25, 115–130 (1996). https://doi.org/10.1007/s002490050023

    Article  Google Scholar 

  37. Tafer, H., Kehr, S., Hertel, J., Stadler, P.F.: RNAsnoop: efficient target prediction for box H/ACA snoRNAs. Bioinformatics 26, 610–616 (2010). https://doi.org/10.1093/bioinformatics/btp680

    Article  Google Scholar 

  38. Backofen, R., et al.: RNAs everywhere: genome-wide annotation of structured RNAs. J. Exp. Zool. B: Mol. Dev. Evol. 308(B), 1–25 (2007). https://doi.org/10.1002/jez.b.21130. The Athanasius F. Bompfünewerer RNA Consortium

  39. Turner, D.H., Mathews, D.H.: NNDB: the nearest neighbor parameter database for predicting stability of nucleic acid secondary structure. Nucleic Acids Res. 38, D280–D282 (2010). https://doi.org/10.1093/nar/gkp892

    Article  Google Scholar 

  40. Wang, F., Lu, C.H., Willner, I.: From cascaded catalytic nucleic acids to enzyme-DNA nanostructures: controlling reactivity, sensing, logic operations, and assembly of complex structures. Chem. Rev. 114, 2881–2941 (2014). https://doi.org/10.1021/cr400354z

    Article  Google Scholar 

  41. Will, C.L., Lührmann, R.: Spliceosome structure and function. Cold Spring Harb. Perspect. Biol. 3, a003707 (2011). https://doi.org/10.1101/cshperspect.a003707

    Article  Google Scholar 

  42. Wuchty, S., Fontana, W., Hofacker, I.L., Schuster, P.: Complete suboptimal folding of RNA and the stability of secondary structures. Biopolymers 49, 145–165 (1999). https://doi.org/10.1002/(SICI)1097-0282(199902)49:2<145::AID-BIP4>3.0.CO;2-G

    Article  Google Scholar 

  43. Zadeh, J.N., et al.: NUPACK: analysis and design of nucleic acid systems. J. Comput. Chem. 32, 170–173 (2011). https://doi.org/10.1002/jcc.21596

    Article  Google Scholar 

  44. Zuker, M., Stiegler, P.: Optimal computer folding of large RNA sequences using thermodynamics and auxiliary information. Nucleic Acids Res. 9, 133–148 (1981). https://doi.org/10.1093/nar/9.1.133

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Theoretical Chemistry, University of Vienna, Währingerstraße 17, 1090, Wien, Austria

    Ronny Lorenz, Christoph Flamm, Ivo L. Hofacker & Peter F. Stadler

  2. Bioinformatics and Computational Biology, Faculty of Computer Science, University of Vienna, Währingerstraße 29, 1090, Wien, Austria

    Ivo L. Hofacker

  3. Bioinformatics Group, Department of Computer Science, Interdisciplinary Center for Bioinformatics, and Competence Center for Scalable Data Services and Solutions Dresden/Leipzig, Universität Leipzig, Härtelstraße 16-18, 04107, Leipzig, Germany

    Peter F. Stadler

  4. Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103, Leipzig, Germany

    Peter F. Stadler

  5. Facultad de Ciencias, Universidad National de Colombia, Sede Bogotá, Colombia

    Peter F. Stadler

  6. Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM, 87501, USA

    Peter F. Stadler

Authors
  1. Ronny Lorenz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christoph Flamm
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ivo L. Hofacker
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Peter F. Stadler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter F. Stadler .

Editor information

Editors and Affiliations

  1. Zhejiang University, Hangzhou, China

    Xuesong Ye

  2. Fraunhofer Portugal Research Center for Assistive Information and Communication Solutions, Porto, Portugal

    Filipe Soares

  3. Nice Sophia Antipolis University, Nice Cedex 2, France

    Elisabetta De Maria

  4. Universidad Politécnica de Madrid, Madrid, Spain

    Pedro Gómez Vilda

  5. University of Milano-Bicocca, Milan, Italy

    Federico Cabitza

  6. Instituto de Telecomunicações, University of Lisbon, Lisbon, Portugal

    Ana Fred

  7. Universidade Nova de Lisboa, Caparica, Portugal

    Hugo Gamboa

Appendix

Appendix

1.1 Gradient and Hessian of h

Efficient optimization of h, Eq. (21), required the gradient and the Hessian of h, which we give here for convenience:

(24)

We note that the Hessian is negative definite since the sum can be written as \(-\mathbf {M} \mathbf {M}^+\) with \(\mathbf {M}_{\alpha ,\kappa }= \mathbf {A}_{\alpha ,\kappa }\sqrt{K_{\kappa }} \exp \left( \frac{1}{2}\sum _{\alpha '} L_{\alpha '} \mathbf {A}_{\alpha ',\kappa }\right) \).

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Lorenz, R., Flamm, C., Hofacker, I.L., Stadler, P.F. (2021). Efficient Algorithms for Co-folding of Multiple RNAs. In: Ye, X., et al. Biomedical Engineering Systems and Technologies. BIOSTEC 2020. Communications in Computer and Information Science, vol 1400. Springer, Cham. https://doi.org/10.1007/978-3-030-72379-8_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-72379-8_10

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-72378-1

  • Online ISBN: 978-3-030-72379-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation