Multiple coronaviruses including MERS-CoV causing Middle East Respiratory Syndrome, SARS-CoV causing SARS, and SARS-CoV-2 causing COVID-19, use a mechanism known as -1 programmed ribosomal frameshifting (-1 PRF) to replicate. SARS-CoV-2 possesses a unique RNA pseudoknotted structure that stimulates -1 PRF. Targeting -1 PRF in SARS-CoV-2 to impair viral replication can improve patients' prognoses. Crucial to developing these therapies is understanding the structure of the SARS-CoV-2 -1 PRF pseudoknot. Our goal is to expand knowledge of -1 PRF structural conformations. Following a structural alignment approach, we identify similarities in -1 PRF pseudoknots of SARS-CoV-2, SARS-CoV, and MERS-CoV. We provide in-depth analysis of the SARS-CoV-2 and MERS-CoV -1 PRF pseudoknots, including reference and noteworthy mutated sequences. To better understand the impact of mutations, we provide insight on -1 PRF pseudoknot sequence mutations and their effect on resulting structures. We introduce Shapify, a novel algorithm that given an RNA sequence incorporates structural reactivity (SHAPE) data and partial structure information to output an RNA secondary structure prediction within a biologically sound hierarchical folding approach. Shapify enhances our understanding of SARS-CoV-2 -1 PRF pseudoknot conformations by providing energetically favourable predictions that are relevant to structure-function and may correlate with -1 PRF efficiency. Applied to the SARS-CoV-2 -1 PRF pseudoknot, Shapify unveils previously unknown paths from initial stems to pseudoknotted structures. By contextualizing our work with available experimental data, our structure predictions motivate future RNA structure-function research and can aid 3-D modeling of pseudoknots.

Motivation: The computational prediction of RNA secondary structure by free energy minimization has become an important tool in RNA research. However in practice, energy minimization is mostly limited to pseudoknot-free structures or rather simple pseudoknots, not covering many biologically important structures such as kissing hairpins. Algorithms capable of predicting sufficiently complex pseudoknots (for sequences of length n) used to have extreme complexities, e.g. Pknots has O(n6) time and O(n4) space complexity. The algorithm CCJ dramatically improves the asymptotic run time for predicting complex pseudoknots (handling almost all relevant pseudoknots, while being slightly less general than Pknots), but this came at the cost of large constant factors in space and time, which strongly limited its practical application (∼200 bases already require 256 GB space). Results: We present a CCJ-type algorithm, Knotty, that handles the same comprehensive pseudoknot class of structures as CCJ with improved space complexity of Θ(n3+Z)-due to the applied technique of sparsification, the number of 'candidates', Z, appears to grow significantly slower than n4 on our benchmark set (which include pseudoknotted RNAs up to 400 nt). In terms of run time over this benchmark, Knotty clearly outperforms Pknots and the original CCJ implementation, CCJ 1.0; Knotty's space consumption fundamentally improves over CCJ 1.0, being on a par with the space-economic Pknots. By comparing to CCJ 2.0, our unsparsified Knotty variant, we demonstrate the isolated effect of sparsification. Moreover, Knotty employs the state-of-the-art energy model of 'HotKnots DP09', which results in superior prediction accuracy over Pknots. Availability and implementation: Our software is available at https://github.com/HosnaJabbari/Knotty. Supplementary information: Supplementary data are available at Bioinformatics online.

While computational RNA secondary structure prediction is an important tool in RNA research, it is still fundamentally limited to pseudoknot-free structures (or at best very simple pseudoknots) in practice. Here, we make the prediction of complex pseudoknots - including kissing hairpin structures - practically applicable by reducing the originally high space consumption. For this aim, we apply the technique of sparsification and other space-saving modifications to the recurrences of the pseudoknot prediction algorithm by Chen, Condon and Jabbari (CCJ algorithm). Thus, the theoretical space complexity of free energy minimization is reduced to Theta(n^3+Z), in the sequence length n and the number of non-optimally decomposable fragments ("candidates") Z. The sparsified CCJ algorithm, sparseCCJ, is presented in detail. Moreover, we provide and compare three generations of CCJ implementations, which continuously improve the space requirements: the original CCJ implementation, our first modified implementation, and our final sparsified implementation. The two latest implementations implement the established HotKnots DP09 energy model. In our experiments, using 244GB of RAM, the original CCJ implementation failed to handle sequences longer than 195 bases; sparseCCJ handles our pseudoknot data set (up to about length 400 bases) in this space limit. All three CCJ implementations are available at https://github.com/HosnaJabbari/CCJ.