Copyright:  2018 Raden et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: This work was supported by the German Research Foundation (DFG, ) with grant [BA 2168/12], the E-learning promotion award 2016 of Albert-Ludwigs-University Freiburg (ALU, -freiburg.de), and the German Federal Ministry of Education and Research (BMBF, ) with grant [031 A538A] RBC (de.NBI). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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Bioinformatics analyses have become indispensable to biological research. While platforms like Galaxy enable the setup of tool pipelines without expert knowledge [1, 2], one requires a general understanding of underlying concepts and algorithms to be able to successfully apply and adapt these pipelines to biological data [3, 4]. Thus, bioinformatics is taught n both computer science and biology studies.

It has been established that, when teaching mathematics, a combination of reflective example study and problem solving by hand fosters learning. This learning effect is heightened when done iteratively with increasing difficulty [5]. Thus, diverse examples covering different aspects of the topic have to be provided to guide the learning process. This is even more important in an e-learning or self-study context, in which the study of examples that show different aspects of a problem might compensate for the missing interaction with a teacher [6, 7].

RNA structure prediction topics covered within this manuscript are the formalization of RNA secondary structures and simplified energy models, computation of the number of structures with regards to the given model [19, 20], identification of the minimum free energy structure [21, 22], computation of partition functions [23], probability calculation for single base pairs and unpaired regions [23, 24], and identification of the maximum expected accuracy structure [25, 26].

In the following, we will briefly introduce the available algorithms and their respective applications to life science. Most algorithms are dynamic programming approaches. Thus, we also provide the corresponding recursions for the simplified RNA structure model, which we introduce first.

The figure illustrates for a given subsequence Si..j a unique nested secondary structure decomposition based on the distinction of all possible pairing states of the last nucleotide Sj. Note, this scheme applies to all RNA structure-related algorithms presented here.

For the identification of functional structures or the study of structural alternatives, the enumeration of suboptimal structures is of interest. A generic approach was introduced by Stefan Wuchty and coworkers [43] that enables the enumeration of all structures that are in a certain range of the minimal energy. An implementation is also available in our web interface.

Our interactive user interface enables the computation of both optimal and suboptimal structures. For a user defined sequence as well as recursion and traceback parameters, the dynamic programming table is provided along with a list of (sub)optimal structures. On selection, the according traceback is highlighted within the matrix. This is complemented with a graphical representation of the structure using Forna [44].

Different recursions can be chosen to examine the effects of ambiguous recursions versus the original one. In the following, such an ambiguous variant from [17] is presented.(3)While this recursion also computes the same entries of N and thus maximal number of possible base pairs (N1,n), it is not using a unique decomposition of the structure, i.e., the same structural variant is considered by different recursion cases.

This causes duplicated enumeration of (sub)optimal structures when using Wuchty's traceback algorithm, which can be studied in our web server for different recursions. Furthermore, it is not possible to use variants of ambiguous recursions like Eq 3 to count structures (consider relation of Eqs 2 and 1) or to compute the partition function of the structural ensemble (as discussed next), since both requires a unique consideration of each structure.

In 1981, Michael Zuker and Patrick Stiegler introduced a dynamic programming approach that efficiently computes minimum free energy structures using a Nearest Neighbor energy model [22]. Using further restriction, the same time and space complexity compared to Nussinov's algorithm is kept. The approach with according decomposition depictions and how it relates to Nussinov's algorithm is introduced in detail, e.g., in [45]. Implementations like UNAFold [46] (formerly mfold [47]) or RNAfold [31, 37] are the current state-of-the-art tools for RNA secondary structure prediction.

The nominator of Eq 4 is called Boltzmann weight (of structure P). The denominator is called canonical partition function Z, which is the sum of the Boltzmann weights of all structures in . Since grows exponentially, its exhaustive enumeration to compute Z is impracticable.

Our web implementation enables the computation of both base pair probabilities as well as unpaired probabilities. To provide insights into how the temperature and energy model influence structure and base pair probabilities, the user can alter the used temperature as well as Ebp. Besides a visualization of the partition function tables Q and Qbp, the user is provided with a visualization of the base pair and unpaired probabilities using the established dot plot format (e.g., used also by UNAfold/mfold [46, 47] or RNAfold [37, 50]). Within this matrix-like illustration, each base pair probability is represented by a dot of proportional size, i.e., the higher the probability, the larger the dot and small probabilities are not visible. With a bit of visual practice, dot plots enable an easy identification of highly probable substructures and the study of structural alternatives.

So far, individual structures were evaluated based on their number of base pairs or energy. This focus on single structures might hide that some substructures (base pairs or unpaired positions) are very common among highly probable structures but not found, e.g., in the most probable structure and thus are lost from the prediction. To face this problem, the expected accuracy can be used for structure evaluation [25, 26, 51].

Here, we follow Chuong B. Do and coworkers [25] and define the expected accuracy of a structure P by(13)It is basically the weighted sum of all base pair probabilities of the respective structure, together with unpaired probability estimates for all its positions k not involved in any base pair, i.e., features of the whole structural ensemble are mapped to individual structures. The position-wise unpaired probability is computed by(14)from base pair probabilities, which is equivalent to Prss(k,k) from Eq 11. Base pair probabilities in Eq 13 are weighted by a factor of two to reflect that two sequence positions are covered. Furthermore, a weighting factor  is introduced, which scales the importance of unpaired versus base pair probabilities.

Given this measure, we can compute the maximum expected accuracy (MEA) structure, i.e., a structure formed by the most accurate/likely base pairs rather than simply maximizing their number (or minimizing the overall energy). To calculate the MEA and an according structure, a variant of the Nussinov algorithm (Eq 2) can be applied, i.e.,(15)where unpaired positions are weighted by Pru (case 1) and base pairs with (case 2). M is initialized with 0. The MEA is found in M1,n while a corresponding structure can be identified via traceback. A recursion variant adapting Eq 3 can be found in [25].

Our MEA web interface computes base pair and unpaired probabilities using the recursions introduced above for the simplified energy model. Thus, the effects of temperature or base pair energy Ebp on MEA computations can be directly studied. As for the Nussinov algorithm, structure and traceback visualization is enabled as well as suboptimal MEA enumeration using our generic implementation of Wuchty's algorithm [43]. An alteration of the  weighting factor for base pair probabilities provides insights into its importance for accurate structure prediction.

Given two RNA sequences S1 and S2 of lengths n and m, respectively, we denote with the reversely indexed S2 to simplify the index notation, since RNA molecules interact in antiparallel orientation. The latter applies to both intra- and intermolecular base pairing. When considering S1 and , we can design a dynamic programming approach for the simplified energy model using a two-dimensional matrix H. An entry Hi,j will provide the maximal number of intermolecular base pairs for the prefixes and .

As already mentioned, Eq 16 is a variant of the global sequence alignment approach introduced by Saul B. Needleman and Christian D. Wunsch [52] using an adapted scoring scheme (base pair instead of match/mismatch scoring for and no gap cost). Thus, initializing all Hi,0/H0,j with 0, the entry Hn,m provides the maximal number of intermolecular base pairs that can be formed, and a traceback starting at Hn,m yields the respective interaction details. This approach enables very low runtimes (O(nm)), as observed by Brian Tjaden and coworkers, who presented in [30] a variant of Eq 16. When computing hybridization-only interactions via minimizing a more sophisticated energy model, the strategy has to be altered to follow a scheme similar to local sequence alignment as defined by Temple Smith and Michael S. Waterman [53], which is detailed in [30].

The web interface of our implementation identifies and reports all optimal interaction sites. For each, an American Standard Code for Information Interchange (ASCII) visualization of the intermolecular base pairs is provided. Note, to reduce code redundancy, we do not use an implementation of Eq 16 but use a base pair-maximization variant of Eq 19, which is discussed in the next section. 152ee80cbc

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