1471-2105-9-122 1471-2105 Research article <p>Strategies for measuring evolutionary conservation of RNA secondary structures</p> Gruber R Andreas agruber@tbi.univie.ac.at Bernhart H Stephan berni@tbi.univie.ac.at Hofacker L Ivo ivo@tbi.univie.ac.at Washietl Stefan wash@tbi.univie.ac.at

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

Bioinformatics Group, Department of Computer Science, University of Leipzig, Härtelstrasse 16-18, D-04109 Leipzig, Germany

EMBL-European Bioinformatics Institute, Wellcome Trust Genome Campus, Hinxton, Cambridge CB10 1SD, UK

 BMC Bioinformatics 1471-2105 2008 9 1 122 http://www.biomedcentral.com/1471-2105/9/122 18302738 10.1186/1471-2105-9-122 26 11 2007 26 2 2008 26 2 2008 2008 Gruber et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Background

Evolutionary conservation of RNA secondary structure is a typical feature of many functional non-coding RNAs. Since almost all of the available methods used for prediction and annotation of non-coding RNA genes rely on this evolutionary signature, accurate measures for structural conservation are essential.

Results

We systematically assessed the ability of various measures to detect conserved RNA structures in multiple sequence alignments. We tested three existing and eight novel strategies that are based on metrics of folding energies, metrics of single optimal structure predictions, and metrics of structure ensembles. We find that the folding energy based SCI score used in the RNAz program and a simple base-pair distance metric are by far the most accurate. The use of more complex metrics like for example tree editing does not improve performance. A variant of the SCI performed particularly well on highly conserved alignments and is thus a viable alternative when only little evolutionary information is available. Surprisingly, ensemble based methods that, in principle, could benefit from the additional information contained in sub-optimal structures, perform particularly poorly. As a general trend, we observed that methods that include a consensus structure prediction outperformed equivalent methods that only consider pairwise comparisons.

Conclusion

Structural conservation can be measured accurately with relatively simple and intuitive metrics. They have the potential to form the basis of future RNA gene finders, that face new challenges like finding lineage specific structures or detecting mis-aligned sequences.

Background

RNA secondary structures serve important functions in many non-coding RNAs and cis-acting regulatory elements of mRNAs 12. They mediate RNA-protein/RNA-RNA interactions in many different biological pathways and some even show enzymatic activity themselves. Functional constraints lead to evolutionary conservation of the RNA structure that in many cases can exceed the level of sequence conservation. Therefore, conserved structures are characteristic evolutionarily signatures of functional RNAs. Most programs developed for the detection of novel functional RNAs rely on these signatures.

QRNA 3 was the first program that detects conserved RNAs. It models RNA structure in a pair of sequences using a stochastic context free grammar. Similarly, EvoFold 4 models the structure of a multiple alignment taking into account a phylogenetic tree (phylo-SCGF). AlifoldZ 5 also analyzes multiple alignments. It uses, however, a thermodynamic folding model based on the RNAalifold algorithm 6. All three programs fold and evaluate the conservation of the potential RNA at the same time. As a consequence, their scores combine contributions of RNA stability and conservation.

RNAz 7 disentangles both contributions by calculating two separate scores for stability and conservation. The latter, dubbed structure conservation index (SCI), is thus a measure for structural conservation only. Two other programs, MSARi 8 and ddbRNA 9, are available that also calculate a pure conservation score.

In this paper, we revisit the problem and propose a series of other possible strategies to measure structural conservation and compare their performance on a large data set of structural RNA families. The main motivation is to explore alternatives and possible improvements to currently applied measures, especially the SCI used in RNAz. This study seems worthwhile, since comparative approaches like RNAz and others are starting to get extensively used to annotate RNA structures on a genome wide scale 4101112131415161718192021. At the same time, however, the increasing availability of additional sequence data makes it necessary to already reconsider and adapt these strategies. For example, while for the first prototype-screens in the human genome 415 only 7 vertebrate genomes were available, we now face the challenge of analyzing alignments of up to 28 species 22. While the signal from RNA stability is important when only few sequences are available, more emphasis has to be put on the evolutionary signature in future screens. This might improve the specificity of the predictions, a major limitation of current algorithms 23.

However, the results presented here are not only of relevance for comparative de novo ncRNA prediction. The SCI, for example, has also been used to measure structural similarity in a clustering approach to find new ncRNA families within one species 1324. In principle, conservation measures of that kind could also be useful for general RNA homology search algorithms that combine sequence and structure conservation 25.

Moreover, using a structure conservation measure on an alignment of sequences that are known to have a conserved RNA structure can help to assess the quality of the alignment. This idea has been used to benchmark the performance of multiple alignment programs on structural RNAs 2627, and more recently to detect mis-aligned sequences and assist in the semi-automatic improvement of RNA alignments 28.

Finally it must be noted that assessing structural conservation, at the same time, means measuring change of RNA structures throughout evolution. Exploring different ways to quantify such structural changes can help inferring structure based phylogenies 2930 and might improve our understanding of RNA structure evolution 3031.

Methods for measuring structural conservation

Structural conservation can be measured on different levels. In the following sections we describe 11 different methods that are based on (i) comparison of predicted minimum free energies (i.e. not on their minimum free energy structures), (ii) comparison of single structures, (iii) comparison of ensembles of structures representing the whole folding space, and (iv) the two specialized methods used by ddbRNA and MSARi. A short summary of all methods is given in Table 1.

<p>Table 1</p>

Overview of methods

Category

Methods

Description

References

Methods based on folding energies

SCI

RNAalifold consensus energy normalized by dividing by the average energy of the single sequences folded independently.

[7]

SCIRN Aeval

Evaluation of energies of sequences under the constraint of being forced to fold into the structures of the other sequences in comparison to the unconstrained energies.

this work

Methods based on single structures

Base-pair distance

Number of base-pairs not shared by two structures.

[64]

Mountain metric

Distance as the difference of two mountain functions, which give the number of base-pairs enclosing a position.

[40]

Tree editing

Based on the representation of RNA secondary structures as trees. A distance is deined as the cost of transforming one tree into the other.

[41-43]

Methods considering the entire folding space

Ensemble distance

Base-pair distance extended to compare ensembles of structures.

this work

Ensemble mountain metric

Distance as the difference of two mountain functions, which give the number of base-pairs that are, on average, expected to enclose a position.

[47]

RNApdist like distance

Distance measure based on the comparison of vectors of probabilities of being paired upstream, paired downstream, and unpaired.

[33,48]

RNAshapes

Similarity measure based on probabilities of abstract shapes.

[49]

Other Methods

ddbRNA

Evaluates compensatory mutations in all possible stem loops in all sequences of an alignment.

[9]

MSARi

Evaluation of the statistical significance of short, contiguous potential base-pair regions under different distribution models.

[8]

Methods based on folding energies

The idea to evaluate structure similarity indirectly through the minimum free energy (MFE) rather than by direct comparison of the structure itself seems to be counter-intuitive at the first glance. The principle, however, becomes clear when considering the RNAalifold algorithm. RNAalifold implements a consensus folding algorithm for a set of aligned RNA sequences. It extends standard dynamic programming algorithms for RNA secondary prediction 32 by averaging the energy contributions over all sequences and incorporating covariation terms into the energy model to reward compensatory mutations and to penalize non-compatible base-pairs. This procedure results in a "consensus MFE" for the alignment. The absolute value of the consensus MFE is of little value to assess the conservation of structures since it mainly reflects the folding energy that is heavily dependent on the nucleotide composition and the length of the alignment. Therefore, the consensus MFE Econs is normalized by the average MFE E¯single MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGafmyrauKbaebadaWgaaWcbaGaem4CamNaemyAaKMaemOBa4Maem4zaCMaemiBaWMaemyzaugabeaaaaa@3564@ of the single sequences as computed by RNAfold giving the structure conservation index

S C I = E c o n s / E ¯ s i n g l e MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaem4uamLaem4qamKaemysaKKaeyypa0Jaemyrau0aaSbaaSqaaiabdogaJjabd+gaVjabd6gaUjabdohaZbqabaGccqGGVaWlcuWGfbqrgaqeamaaBaaaleaacqWGZbWCcqWGPbqAcqWGUbGBcqWGNbWzcqWGSbaBcqWGLbqzaeqaaaaa@41CA@

If the sequences show equally stable folding energies if forced to fold into a common structure compared to being folded independently, this indicates a conserved structure and the SCI is high. The lower bound of the SCI is zero, indicating that RNAalifold is not able to find a consensus structure, while a SCI close to one corresponds to perfect structure conservation. Compensatory mutations adding additional bonus energies to the consensus MFE can even give rise to a SCI higher than one.

The SCI, as given above, requires the computation of a consensus structure for the whole alignment. Alternatively, one can consider formulating a similar measure based on pairwise comparisons of all sequences. To this end, the folding energy of each sequence is evaluated when forced to fold into the structures of the other sequences. The pairwise SCI for an alignment A MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8haXheaaa@3747@ is given by

S C I R N   A e v a l ( A ) = ∑ x , y ∈ A x ≠ y E ( x | S y ) ( N − 1 ) ∑ x ∈ A E ( x | S x ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7277@

where E(x|Sy) denotes the free energy of sequence x when adopting the minimum free energy structure Sy of sequence y, and N is the number of sequences in the alignment. The free energies for a given sequence in a given structure can be easily evaluated with the program RNAeval from the Vienna RNA package 33. Therefore, we refer to this method as the "RNAeval" method.

Methods based on single structures

A more intuitive way to assess structural similarity is by comparing structures themselves rather than comparing the energies associated with these structures. Conservation measures derived from various structure metrics are described in this section. Unlike the energy based methods from the previous section that are inherently linked to thermodynamic folding, the following methods do not depend on the way of how structures are predicted. There are several different ways, like thermodynamic energy minimization 34, kinetic folding 35 or probabilistic models 363738, but the choice of the method will not influence the underlying concept. However, since the goal of this study is not to compare the accuracy of different folding algorithms, we use here exclusively energy minimization (RNAfold) to ensure comparability between all methods.

Base-pair distance

The most simple distance measure between two sequences is the Hamming distance, i.e. the number of positions with different nucleotides. For RNA structures, one could think of calculating the Hamming distance of two strings in dot bracket notation with the three characters "(", ".", ")". However, this does not account for the correlations between the opening and closing positions that are characteristic for the structure.

An alternative to the Hamming distance more suitable for secondary structures is the so-called base-pair distance. The base-pair distance between to RNA secondary structures Sx and Sy is defined as the number of base-pairs not shared by the two structures. Formally it can be described in terms of set theory, where the base-pair distance corresponds to the cardinality of the symmetric set difference:

d B P ( S x , S y ) = | ( S x \ S y ) ∪ ( S y \ S x ) | = | S x ∪ S y | − | S x ∩ S y | = | S x | + | S y | − 2 | S x ∩ S y | = ∑ i < j ( δ i j x + δ i j y − 2 δ i j x δ i j y ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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r7aKnaaDaaaleaacqWGPbqAcqWGQbGAaeaacqWG5bqEaaGccqGHsislcqaIYaGmcqWF0oazdaqhaaWcbaGaemyAaKMaemOAaOgabaGaemiEaGhaaOGae8hTdq2aa0baaSqaaiabdMgaPjabdQgaQbqaaiabdMha5baakiabcMcaPaWcbaGaemyAaKMaeyipaWJaemOAaOgabeqdcqGHris5aaaaaaa@A357@

with δijx MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaacciGae8hTdq2aa0baaSqaaiabdMgaPjabdQgaQbqaaiabdIha4baaaaa@31DD@ = 1 if (i,j) is a base-pair of structure Sx, and δijx MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaacciGae8hTdq2aa0baaSqaaiabdMgaPjabdQgaQbqaaiabdIha4baaaaa@31DD@ = 0 otherwise. dBP itself is not a suitable measure for comparison as long it is not set in relation to the union of the base-pairs in Sx and Sy. The normalized base-pair distance scaled to the interval [0, 1] between two structures is given by

D B P ( S x , S y ) = | S x ∪ S y | − | S x ∩ S y | | S x ∪ S y | = ∑ i < j ( δ i j x + δ i j y − 2 δ i j x δ i j y ) ∑ i < j ( δ i j x + δ i j y − δ i j x δ i j y ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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r7aKnaaDaaabaGaemyAaKMaemOAaOgabaGaemiEaGhaaiabgUcaRiab=r7aKnaaDaaabaGaemyAaKMaemOAaOgabaGaemyEaKhaaiabgkHiTiab=r7aKnaaDaaabaGaemyAaKMaemOAaOgabaGaemiEaGhaaiab=r7aKnaaDaaabaGaemyAaKMaemOAaOgabaGaemyEaKhaaiabcMcaPaqaaiabdMgaPjabgYda8iabdQgaQbqabiabggHiLdaaaaaa@9DF5@

The overall score for a multiple alignment A MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8haXheaaa@3747@ can either be calculated as the average of all pairwise sequence comparisons

2 ( N − 1 ) N ∑ x , y ∈ A x > y D B P ( S x , S y ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqcfa4aaSaaaeaacqaIYaGmaeaacqGGOaakcqWGobGtcqGHsislcqaIXaqmcqGGPaqkcqWGobGtaaGcdaaeqbqaaiabdseaenaaBaaaleaacqWGcbGqcqWGqbauaeqaaOGaeiikaGIaem4uam1aaSbaaSqaaiabdIha4bqabaGccqGGSaalcqWGtbWudaWgaaWcbaGaemyEaKhabeaakiabcMcaPaWcbaqbaeqabiqaaaqaaiabdIha4jabcYcaSiabdMha5jabgIGioprtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=bq8bbqaaiabdIha4jabg6da+iabdMha5baaaeqaniabggHiLdaaaa@5659@

or as the average of all comparisons of each sequence to a consensus structure

1 N ∑ x ∈ A D B P ( S x , S c o n s e n s u s ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqcfa4aaSaaaeaacqaIXaqmaeaacqWGobGtaaGcdaaeqbqaaiabdseaenaaBaaaleaacqWGcbGqcqWGqbauaeqaaOGaeiikaGIaem4uam1aaSbaaSqaaiabdIha4bqabaGccqGGSaalcqWGtbWudaWgaaWcbaGaem4yamMaem4Ba8MaemOBa4Maem4CamNaemyzauMaemOBa4Maem4CamNaemyDauNaem4CamhabeaakiabcMcaPaWcbaGaemiEaGNaeyicI48enfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8haXheabeqdcqGHris5aaaa@5657@

If not stated otherwise, also all other methods that are based on pairwise comparisons can be calculated either as the average over all (N - 1)N/2 pairwise comparisons, or the average of all N comparisons to the consensus structure.

Mountain metric

The mountain metric is based on the mountain representation of RNA secondary structures 39 and follows the idea that the distance between two structures Sx and Sy can be expressed as the difference of the two mountain graphs. For this purpose, a lp-norm can be defined that induces a metric dMp MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemizaq2aa0baaSqaaiabd2eanbqaaiabdchaWbaaaaa@2FDD@ on two secondary structures Sx and Sy as the difference of the two mountain functions m(Sx) and m(Sy) 40:

d M p ( S x , S y ) : = | | m ( S x ) − m ( S y ) | | : = ( ∑ k = 1 n | m k ( S x ) − m k ( S y ) | p ) 1 p MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6FCF@

The mountain function mk(S) is defined as the number of base-pairs enclosing position k. The effect that base-pairs are weighted differently can be overcome by scaling each base-pair to the range it spans.

m k ( S ) = ∑ i < k ∑ k < j 1 j − i − 1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemyBa02aaSbaaSqaaiabdUgaRbqabaGccqGGOaakcqWGtbWucqGGPaqkcqGH9aqpdaaeqbqaamaaqafajuaGbaWaaSaaaeaacqaIXaqmaeaacqWGQbGAcqGHsislcqWGPbqAcqGHsislcqaIXaqmaaaaleaacqWGRbWAcqGH8aapcqWGQbGAaeqaniabggHiLdaaleaacqWGPbqAcqGH8aapcqWGRbWAaeqaniabggHiLdaaaa@45D4@

As dMp MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemizaq2aa0baaSqaaiabd2eanbqaaiabdchaWbaaaaa@2FDD@ is expected to grow with the length of sequences, we are in the need of defining a normalized distance measure to be able to compare distances for sequence pairs of different length. The maximal distance of a secondary structure Smax on a sequence of length n to the open chain Sopen is obtained if Smax is a stem of maximal height (⌊(n - 3)/2⌋), which is a hairpin loop enclosing three unpaired bases. The normalized mountain metric DMp MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemiraq0aa0baaSqaaiabd2eanbqaaiabdchaWbaaaaa@2F9D@ is then defined as the ratio of the distance dMp MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemizaq2aa0baaSqaaiabd2eanbqaaiabdchaWbaaaaa@2FDD@ (Sx, Sy) of two secondary structures with length n to the maximal distance dMp MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemizaq2aa0baaSqaaiabd2eanbqaaiabdchaWbaaaaa@2FDD@ (Smax, Sopen) at length n:

D M p ( S x , S y ) = d M p ( S x , S y ) d M p ( S m a x , S o p e n ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemiraq0aa0baaSqaaiabd2eanbqaaiabdchaWbaakiabcIcaOiabdofatnaaBaaaleaacqWG4baEaeqaaOGaeiilaWIaem4uam1aaSbaaSqaaiabdMha5bqabaGccqGGPaqkcqGH9aqpjuaGdaWcaaqaaiabdsgaKnaaDaaabaGaemyta0eabaGaemiCaahaaiabcIcaOiabdofatnaaBaaabaGaemiEaGhabeaacqGGSaalcqWGtbWudaWgaaqaaiabdMha5bqabaGaeiykaKcabaGaemizaq2aa0baaeaacqWGnbqtaeaacqWGWbaCaaGaeiikaGIaem4uam1aaSbaaeaacqWGTbqBcqWGHbqycqWG4baEaeqaaiabcYcaSiabdofatnaaBaaabaGaem4Ba8MaemiCaaNaemyzauMaemOBa4gabeaacqGGPaqkaaaaaa@58EE@

Tree editing

RNA secondary structures can be represented as ordered, rooted trees 414243. The tree representation can be deduced from the dot-bracket notation (characters "(" and ")" correspond to the 5' base and the 3' base in the base-pair, respectively, while "." denotes an unpaired base), as the brackets clearly imply parent-child relationships. The ordering among the siblings of a node is imposed by the 5' to 3' nature of the RNA molecule. To avoid formation of an unconnected forest of trees, a virtual root has to be introduced.

The tree representation at full resolution without any loss of information with regard to the dot-bracket notation can be derived by assigning each unpaired base to a leaf node and each base-pair to an internal node. The resulting tree can be rewritten to a homeomorphically irreducible tree (HIT) by collapsing all base-pairs in a stem into a single internal node and adjacent unpaired bases into a single leaf node 43. Each node is then assigned a weight reflecting the number of nodes or leaves that were combined.

Shapiro proposed another encoding that retains only a coarse-grained shape of a secondary structure 41. This is useful in the case of comparison of major structural elements of a RNA molecule but it comes along with a loss of information (cf. section "Abstract shapes"). A secondary structure can be decomposed into stems (S), hairpin loops (H), interior loops (I), multi-loops (M), and external nucleotides (E). While external nucleotides are assigned to a leaf, unpaired bases in a multi-loop are lost. The weighted coarse-grained approach compensates the effect of information reduction at least by assigning to each node or leaf the number of elements that were condensed to it.

Tree editing induces a metric in the space of trees and hence a metric in the space of RNA secondary structures. An edit script, which is a series of edit operations, namely deletion, insertion and relabeling of a node, each assigned a cost can transform any tree Tx into any other tree Ty. The distance between two trees d(Tx, Ty) is then defined as the cost of the edit script with minimal cost. Normalization of the tree editing distance is done by comparing the distance of two trees d(Tx, Ty) to the sum of the costs of deleting either of the two secondary structures, where • denotes a tree consisting solely of a root:

D T ( T x , T y ) = d ( T x , T y ) d ( T x , • ) + d ( • , T y ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemiraq0aaSbaaSqaaiabdsfaubqabaGccqGGOaakcqWGubavdaWgaaWcbaGaemiEaGhabeaakiabcYcaSiabdsfaunaaBaaaleaacqWG5bqEaeqaaOGaeiykaKIaeyypa0tcfa4aaSaaaeaacqWGKbazcqGGOaakcqWGubavdaWgaaqaaiabdIha4bqabaGaeiilaWIaemivaq1aaSbaaeaacqWG5bqEaeqaaiabcMcaPaqaaiabdsgaKjabcIcaOiabdsfaunaaBaaabaGaemiEaGhabeaacqGGSaalcqGHIaYTcqGGPaqkcqGHRaWkcqWGKbazcqGGOaakcqGHIaYTcqGGSaalcqWGubavdaWgaaqaaiabdMha5bqabaGaeiykaKcaaaaa@5356@

Among the methods used here, tree editing is the only one that can act on structures of unequal length. In this work we will focus on two different implementations of tree editing. RNAdistance 33 a tool from the Vienna RNA package implements a tree editing algorithm initially proposed by Shapiro 41 and acts on the full representation, HIT representation 43, coarse-grained and weighted coarse-grained representation 41. Allali & Sagot 44 pointed out some shortcomings of the classic tree editing operations and introduced novel editing operations called node-fusion and edge-fusion, implemented in the program MiGaL. MiGaL uses a new concept of encoding trees at different levels of abstraction called layers 45, which are interconnected to each other via vertex coloring operations.

Methods considering the entire folding space

Distance of structure ensembles

Because the stabilizing energies of base-pair formation are in the same energy range as the thermal energy, RNA molecules in physiological conditions are far away from being caged into one rigid secondary structure. Instead, one usually observes an ensemble of RNA structures, which can be represented by an energy weighted Boltzmann distribution. McCaskill proposed a dynamic programming algorithm 46 that allows to efficiently compute the partition function Q, where ΔG is the conformational Gibb's Free Energy change, R is the gas constant, T is the absolute temperature, and S MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiqaacqWFsc=uaaa@37CF@ is the ensemble of possible secondary structures.

Q = ∑ S ∈ S e − Δ G ( S ) R T MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemyuaeLaeyypa0ZaaabuaeaacqWGLbqzdaahaaWcbeqcfayaamaalaaabaGaeyOeI0IaeuiLdqKaem4raCKaeiikaGIaem4uamLaeiykaKcabaGaemOuaiLaemivaqfaaaaaaSqaaiabdofatjabgIGioprr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbaceaGae8NKWpfabeqdcqGHris5aaaa@49EB@

The probability of a single structure S is then given by

P ( S ) = e − Δ G ( S ) / R T Q MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemiuaaLaeiikaGIaem4uamLaeiykaKIaeyypa0tcfa4aaSaaaeaacqWGLbqzdaahaaqabeaacqGHsislcqqHuoarcqWGhbWrcqGGOaakcqWGtbWucqGGPaqkcqGGVaWlcqWGsbGucqWGubavaaaabaGaemyuaefaaaaa@3DFE@

and hence the probability of a single base-pair (i, j) is

p i j = ∑ S ∈ S P ( S ) δ i j S MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemiCaa3aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGH9aqpdaaeqbqaaiabdcfaqjabcIcaOiabdofatjabcMcaPGGaciab=r7aKnaaDaaaleaacqWGPbqAcqWGQbGAaeaacqWGtbWuaaaabaGaem4uamLaeyicI48efv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiqaacqGFsc=uaeqaniabggHiLdaaaa@4C0E@

where δijS MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaacciGae8hTdq2aa0baaSqaaiabdMgaPjabdQgaQbqaaiabdofatbaaaaa@3193@ is one if (i, j) is a base-pair of structure S, and zero otherwise. Using these assumptions the equation of the base-pair distance can be remodeled to calculate the average base-pair distance 〈dBP(Sy,Sx)〉 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaeyykJeUaemizaq2aaSbaaSqaaiabdkeacjabdcfaqbqabaGccqGGOaaktuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGabaiab=jj8tnaaBaaaleaacqWG5bqEaeqaaOGaeiilaWIae8NKWp1aaSbaaSqaaiabdIha4bqabaGccqGGPaqkcqGHQms8aaa@475B@ between all structures of the two ensembles Sx MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiqaacqWFsc=udaWgaaWcbaGaemiEaGhabeaaaaa@3974@ and Sy MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiqaacqWFsc=udaWgaaWcbaGaemyEaKhabeaaaaa@3976@.

〈 d B P ( S x , S y ) 〉 = ∑ S x ∈ S x ∑ S y ∈ S y [ P ( S x ) P ( S y ) ∑ i < j ( δ i j x + δ i j y − 2 δ i j x δ i j y ) ] = ∑ i < j [ ∑ S x ∈ S x P ( S x ) δ i j x ∑ S y ∈ S y P ( S y ) + ∑ S y ∈ S y P ( S y ) δ i j y ∑ S x ∈ S x P ( S x ) − 2 ∑ S x ∈ S x P ( S x ) δ i j x ∑ S y ∈ S y P ( S y ) δ i j x