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Efficient Algorithms for Co-folding of Multiple RNAs

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Biomedical Engineering Systems and Technologies (BIOSTEC 2020)

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

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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).

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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

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  1. Ronny Lorenz
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  2. Christoph Flamm
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  4. Peter F. Stadler
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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 \(\mathbf {M}_{\alpha ,\kappa }= \mathbf {A}_{\alpha ,\kappa }\sqrt{K_{\kappa }} \exp \left( \frac{1}{2}\sum _{\alpha '} L_{\alpha '} \mathbf {A}_{\alpha ',\kappa }\right) \).

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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

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

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