A continuous analog for RNA folding

A linear segment in which a number of pairs of intervals of equal length are identified as potential stems is the subject of a folding problem analogous to inference of RNA secondary structure. A quantity of free energy (or equivalently, energy per unit length) is associated with each stem, and the various types of loops are assigned energy costs as a function of their lengths. Inference of stable structures can then be carried out in the same way as in RNA folding. More important, perturbation of stem lengths and energy densities (modelling various mutational processes affecting nucleotide sequences) allows the delineation of domains of stability of various foldings, through the explicit calculation of their boundaries, in a low-dimensional parameter space.     

Statistics of RNA secondary structures

A statistical reference for RNA secondary structures with minimum free energies is computed by folding large ensembles of random RNA sequences. Four nucleotide alphabets are used: two binary alphabets, AU and GC, the biophysical AUGC and the synthetic GCXK alphabet. RNA secondary structures are made of structural elements, such as stacks, loops, joints, and free ends. Statistical properties of these elements are computed for small RNA molecules of chain lengths up to 100. The results of RNA structure statistics depend strongly on the particular alphabet chosen. The statistical reference is compared with the data derived from natural RNA molecules with similar base frequencies. Secondary structures are represented as trees. Tree editing provides a quantitative measure for the distance d_t, between two structures. We compute a structure density surface as the conditional probability of two structures having distance t given that their sequences have distance h. This surface indicates that the vast majority of possible minimum free energy secondary structures occur within a fairly small neighborhood of any typical (random) sequence. Correlation lengths for secondary structures in their tree representations are computed from probability densities. They are appropriate measures for the complexity of the sequence-structure relation. The correlation length also provides a quantitative estimate for the mean sensitivity of structures to point mutations. © 1993 John Wiley & Sons, Inc.     

The Robustness of Naturally and Artificially Selected Nucleic Acid Secondary Structures

Thermodynamic stability and mutational robustness of secondary structure are critical to the function and evolutionary longevity of RNA molecules. We hypothesize that natural and artificial selection for functional molecules favors the formation of structures that are stable to both thermal and mutational perturbation. There is little direct evidence, however, that functional RNA molecules have been selected for their stability. Here we use thermodynamic secondary structure prediction algorithms to compare the thermal and mutational robustness of over 1000 naturally and artificially evolved molecules. Although we find evidence for the evolution of both types of stability in both sets of molecules, the naturally evolved functional RNA molecules were significantly more stable than those selected in vitro, and artificially evolved catalysts (ribozymes) were more stable than artificially evolved binding species (aptamers). The thermostability of RNA molecules bred in the laboratory is probably not constrained by a lack of suitable variation in the sequence pool but, rather, by intrinsic biases in the selection process.     

Stem kernels for RNA sequence analyses

Several computational methods based on stochastic context-free grammars have been developed for modeling and analyzing functional RNA sequences. These grammatical methods have succeeded in modeling typical secondary structures of RNA, and are used for structural alignment of RNA sequences. However, such stochastic models cannot sufficiently discriminate member sequences of an RNA family from nonmembers and hence detect noncoding RNA regions from genome sequences. A novel kernel function, stem kernel, for the discrimination and detection of functional RNA sequences using support vector machines (SVMs) is proposed. The stem kernel is a natural extension of the string kernel, specifically the all-subsequences kernel, and is tailored to measure the similarity of two RNA sequences from the viewpoint of secondary structures. The stem kernel examines all possible common base pairs and stem structures of arbitrary lengths, including pseudoknots between two RNA sequences, and calculates the inner product of common stem structure counts. An efficient algorithm is developed to calculate the stem kernels based on dynamic programming. The stem kernels are then applied to discriminate members of an RNA family from nonmembers using SVMs. The study indicates that the discrimination ability of the stem kernel is strong compared with conventional methods. Furthermore, the potential application of the stem kernel is demonstrated by the detection of remotely homologous RNA families in terms of secondary structures. This is because the string kernel is proven to work for the remote homology detection of protein sequences. These experimental results have convinced us to apply the stem kernel in order to find novel RNA families from genome sequences.     

New structural variation in evolutionary searches of RNA neutral networks

RNA secondary structure is an important computational model to understand how genetic variation maps into phenotypic (structural) variation. Evolutionary innovation in RNA structures is facilitated by neutral networks, large connected sets of RNA sequences that fold into the same structure. Our work extends and deepens previous studies on neutral networks. First, we show that even the 1-mutant neighborhood of a given sequence (genotype) G0 with structure (phenotype) P contains many structural variants that are not close to P. This holds for biological and generic RNA sequences alike. Second, we analyze the relation between new structures in the 1-neighborhoods of genotypes Gk that are only a moderate Hamming distance k away from G0, and the structure of G0 itself, both for biological and for generic RNA structures. Third, we analyze the relation between mutational robustness of a sequence and the distances of structural variants near this sequence. Our findings underscore the role of neutral networks in evolutionary innovation, and the role that high robustness can play in diminishing the potential for such innovation.     

Single nucleotide RNA choreography

New structural analysis methods, and a tree formalism re-define and expand the RNA motif concept, unifying what previously appeared to be disparate groups of structures. We find RNA tetraloops at high frequencies, in new contexts, with unexpected lengths, and in novel topologies. The results, with broad implications for RNA structure in general, show that even at this most elementary level of organization, RNA tolerates astounding variation in conformation, length, sequence and context. However the variation is not random; it is well-described by four distinct modes, which are 3-2 switches (backbone topology variations), insertions, deletions and strand clips.     

Visualizing the global secondary structure of a viral RNA genome with cryo-electron microscopy

The lifecycle, and therefore the virulence, of single-stranded (ss)-RNA viruses is regulated not only by their particular protein gene products, but also by the secondary and tertiary structure of their genomes. The secondary structure of the entire genomic RNA of satellite tobacco mosaic virus (STMV) was recently determined by selective 2′-hydroxyl acylation analyzed by primer extension (SHAPE). The SHAPE analysis suggested a single highly extended secondary structure with much less branching than occurs in the ensemble of structures predicted by purely thermodynamic algorithms. Here we examine the solution-equilibrated STMV genome by direct visualization with cryo-electron microscopy (cryo-EM), using an RNA of similar length transcribed from the yeast genome as a control. The cryo-EM data reveal an ensemble of branching patterns that are collectively consistent with the SHAPE-derived secondary structure model. Thus, our results both elucidate the statistical nature of the secondary structure of large ss-RNAs and give visual support for modern RNA structure determination methods. Additionally, this work introduces cryo-EM as a means to distinguish between competing secondary structure models if the models differ significantly in terms of the number and/or length of branches. Furthermore, with the latest advances in cryo-EM technology, we suggest the possibility of developing methods that incorporate restraints from cryo-EM into the next generation of algorithms for the determination of RNA secondary and tertiary structures.     

Tracing the evolution of RNA structure in ribosomes

The elucidation of ribosomal structure has shown that the function of ribosomes is fundamentally confined to dynamic interactions established between the RNA components of the ribosomal ensemble. These findings now enable a detailed analysis of the evolution of ribosomal RNA (rRNA) structure. The origin and diversification of rRNA was studied here using phylogenetic tools directly at the structural level. A rooted universal tree was reconstructed from the combined secondary structures of large (LSU) and small (SSU) subunit rRNA using cladistic methods and considerations in statistical mechanics. The evolution of the complete repertoire of structural ribosomal characters was formally traced lineage-by-lineage in the tree, showing a tendency towards molecular simplification and a homogeneous reduction of ribosomal structural change with time. Character tracing revealed patterns of evolution in inter-subunit bridge contacts and tRNA-binding sites that were consistent with the proposed coupling of tRNA translocation and subunit movement. These patterns support the concerted evolution of tRNA-binding sites in the two subunits and the ancestral nature and common origin of certain structural ribosomal features, such as the peptidyl (P) site, the functional relay of the penultimate stem helix of SSU rRNA, and other structures participating in ribosomal dynamics. Overall results provide a rare insight into the evolution of ribosomal structure.     

RNA multi-structure landscapes

Statistical properties of RNA folding landscapes obtained by the partition function algorithm (McCaskill 1990) are investigated in detail. The pair correlation of free energies as a function of the Hamming distance is used as a measure for the ruggedness of the landscape. The calculation of the partition function contains information about the entire ensemble of secondary structures as a function of temperature and opens the door to all quantities of thermodynamic interest, in contrast with the conventional minimal free energy approach. A metric distance of structure ensembles is introduced and pair correlations at the level of the structures themselves are computed. Just as with landscapes based on most stable secondary structure prediction, the landscapes defined on the full biophysical GCAU alphabet are much smoother than the landscapes restricted to pure GC sequences and the correlation lengths are almost constant fractions of the chain lengths. Correlation functions for multi-structure landscapes exhibit an increased correlation length, especially near the melting temperature. However, the main effect on evolution is rather an effective increase in sampling for finite populations where each sequence explores multiple structures. Correspondence to: P. Schuster     

RNA Sampler: a new sampling based algorithm for common RNA secondary structure prediction and structural alignment

MOTIVATION: Non-coding RNA genes and RNA structural regulatory motifs play important roles in gene regulation and other cellular functions. They are often characterized by specific secondary structures that are critical to their functions and are often conserved in phylogenetically or functionally related sequences. Predicting common RNA secondary structures in multiple unaligned sequences remains a challenge in bioinformatics research. Methods and RESULTS: We present a new sampling based algorithm to predict common RNA secondary structures in multiple unaligned sequences. Our algorithm finds the common structure between two sequences by probabilistically sampling aligned stems based on stem conservation calculated from intrasequence base pairing probabilities and intersequence base alignment probabilities. It iteratively updates these probabilities based on sampled structures and subsequently recalculates stem conservation using the updated probabilities. The iterative process terminates upon convergence of the sampled structures. We extend the algorithm to multiple sequences by a consistency-based method, which iteratively incorporates and reinforces consistent structure information from pairwise comparisons into consensus structures. The algorithm has no limitation on predicting pseudoknots. In extensive testing on real sequence data, our algorithm outperformed other leading RNA structure prediction methods in both sensitivity and specificity with a reasonably fast speed. It also generated better structural alignments than other programs in sequences of a wide range of identities, which more accurately represent the RNA secondary structure conservations. AVAILABILITY: The algorithm is implemented in a C program, RNA Sampler, which is available at http://ural.wustl.edu/software.html     

The human immunodeficiency virus type 1 encapsidation site is a multipartite RNA element composed of functional hairpin structures.

We analyzed the leader region of human immunodeficiency virus type 1 (HIV-1) RNA to decipher the nature of the cis-acting E/psi element required for encapsidation of viral RNA into virus particles. Our data indicate that, for RNA encapsidation, there are at least two functional subregions in the leader region. One subregion is located at a position immediately proximal to the major splice donor, and the second is located between the splice donor and the beginning of the gag gene. This suggests that at least two discrete cis-acting elements are recognition signals for encapsidation. To determine whether specific putative RNA secondary structures serve as the signal(s) for encapsidation, we constructed primary base substitution mutations that would be expected to destabilize these potential structures and second-site compensatory mutations that would restore secondary structure. Analysis of these mutants allowed the identification of two discrete hairpins that facilitate RNA encapsidation in vivo. Thus, the HIV-1 E/psi region is a multipartite element composed of specific and functional RNA secondary structures. Compensation of the primary mutations by the second-site mutations could not be attained in trans. This indicates that interstrand base pairing between these two stem regions within the hairpins does not appear to be the basis for HIV-1 RNA dimer formation. Comparison of the hypothetical RNA secondary structures from 10 replication-competent HIV-1 strains suggests that a subset of the hydrogen-bonded base pairs within the stems of the hairpins is likely to be required for function in cis.     

Does Hybridization Increase Evolutionary Rate? Data from the 28S-rDNA D8 Domain in Echinoderms

The divergent domain D8 of the large ribosomal RNA is very variable and extended in vertebrates compared to other eukaryotes. We provide data from 31 species of echinoderms and present the first comparative analysis of the D8 in nonvertebrate deuterostomes. In addition, we obtained 16S mitochondrial DNA sequences for the sea urchin taxa and analyzed single-strand conformation polymorphism (SSCP) of D8 in several populations within the species complex Echinocardium cordatum. A common secondary structure supported by compensatory substitutions and indels is inferred for echinoderms. Variation mostly arises at the tip of the longest stem (D8a), and the most variable taxa also display the longest and most stable D8. The most stable variants are the only ones displaying bulges in the terminal part of the stem, suggesting that selection, rather than maximizing stability of the D8 secondary structure, maintains it in a given range. Striking variation in D8 evolutionary rates was evidenced among sea urchins, by comparison with both 16S mitochondrial DNA and paleontological data. In Echinocardium cordatum and Strongylocentrotus pallidus and S. droebachiensis, belonging to very distant genera, the increase in D8 evolutionary rate is extreme. Their highly stable D8 secondary structures rule out the possibility of pseudogenes. These taxa are the only ones in which interspecific hybridization was reported. We discuss how evolutionary rates may be affected in nuclear relative to mitochondrial genes after hybridization, by selective or mutational processes such as gene silencing and concerted evolution.     

Transient RNA structure features are evolutionarily conserved and can be computationally predicted

Functional RNA structures tend to be conserved during evolution. This finding is, for example, exploited by comparative methods for RNA secondary structure prediction that currently provide the state-of-art in terms of prediction accuracy. We here provide strong evidence that homologous RNA genes not only fold into similar final RNA structures, but that their folding pathways also share common transient structural features that have been evolutionarily conserved. For this, we compile and investigate a non-redundant data set of 32 sequences with known transient and final RNA secondary structures and devise a dedicated computational analysis pipeline.     

Comparative sequence analysis and patterns of covariation in RNA secondary structures

A novel method of RNA secondary structure prediction based on a comparison of nucleotide sequences is described. This method correctly predicts nearly all evolutionarily conserved secondary structures of five different RNAs: tRNA, 5S rRNA, bacterial ribonuclease P (RNase P) RNA, eukaryotic small subunit rRNA, and the 3' untranslated region (UTR) of the Drosophila bicoid (bcd) mRNA. Furthermore, covariations occurring in the helices of these conserved RNA structures are analyzed. Two physical parameters are found to be important determinants of the evolution of compensatory mutations: the length of a helix and the distance between base-pairing nucleotides. For the helices of bcd 3' UTR mRNA and RNase P RNA, a positive correlation between the rate of compensatory evolution and helix length is found. The analysis of Drosophila bcd 3' UTR mRNA further revealed that the rate of compensatory evolution decreases with the physical distance between base-pairing residues. This result is in qualitative agreement with Kimura's model of compensatory fitness interactions, which assumes that mutations occurring in RNA helices are individually deleterious but become neutral in appropriate combinations.     

Modular evolution and increase of functional complexity in replicating RNA molecules

At early stages of biochemical evolution, the complexity of replicating molecules was limited by unavoidably high mutation rates. In an RNA world, prior to the appearance of cellular life, an increase in molecular length, and thus in functional complexity, could have been mediated by modular evolution. We describe here a scenario in which short, replicating RNA sequences are selected to perform a simple function. Molecular function is represented through the secondary structure corresponding to each sequence, and a given target secondary structure yields the optimal function in the environment where the population evolves. The combination of independently evolved populations may have facilitated the emergence of larger molecules able to perform more complex functions (including RNA replication) that could arise as a combination of simpler ones. We quantitatively show that modular evolution has relevant advantages with respect to the direct evolution of large functional molecules, among them the allowance of higher mutation rates, the shortening of evolutionary times, and the very possibility of finding complex structures that could not be otherwise directly selected.     

RNA secondary structure and compensatory evolution

The classic concept of epistatic fitness interactions between genes has been extended to study interactions within gene regions, especially between nucleotides that are important in maintaining pre-mRNA/mRNA secondary structures. It is shown that the majority of linkage disequilibria found within the Drosophila Adh gene are likely to be caused by epistatic selection operating on RNA secondary structures. A recently proposed method of RNA secondary structure prediction based on DNA sequence comparisons is reviewed and applied to several types of RNAs, including tRNA, rRNA, and mRNA. The patterns of covariation in these RNAs are analyzed based on Kimura's compensatory evolution model. The results suggest that this model describes the substitution process in the pairing regions (helices) of RNA secondary structures well when the helices are evolutionarily conserved and thermodynamically stable, but fails in some other cases. Epistatic selection maintaining pre-mRNA/mRNA secondary structures is compared to weak selective forces that determine features such as base composition and synonymous codon usage. The relationships among these forces and their relative strengths are addressed. Finally, our mutagenesis experiments using the Drosophila Adh locus are reviewed. These experiments analyze long-range compensatory interactions between the 5' and 3' ends of Adh mRNA, the different constraints on secondary structures in introns and exons, and the possible role of secondary structures in RNA splicing.     

Evolution of mutational robustness in an RNA virus

Mutational (genetic) robustness is phenotypic constancy in the face of mutational changes to the genome. Robustness is critical to the understanding of evolution because phenotypically expressed genetic variation is the fuel of natural selection. Nonetheless, the evidence for adaptive evolution of mutational robustness in biological populations is controversial. Robustness should be selectively favored when mutation rates are high, a common feature of RNA viruses. However, selection for robustness may be relaxed under virus co-infection because complementation between virus genotypes can buffer mutational effects. We therefore hypothesized that selection for genetic robustness in viruses will be weakened with increasing frequency of co-infection. To test this idea, we used populations of RNA phage φ6 that were experimentally evolved at low and high levels of co-infection and subjected lineages of these viruses to mutation accumulation through population bottlenecking. The data demonstrate that viruses evolved under high co-infection show relatively greater mean magnitude and variance in the fitness changes generated by addition of random mutations, confirming our hypothesis that they experience weakened selection for robustness. Our study further suggests that co-infection of host cells may be advantageous to RNA viruses only in the short term. In addition, we observed higher mutation frequencies in the more robust viruses, indicating that evolution of robustness might foster less-accurate genome replication in RNA viruses.     

Secondary structure of the central region of bacteriophage MS2 RNA. Conservation and biological significance

The RNA of the Escherichia coli RNA phages is highly structured with 75% of the nucleotides estimated to take part in base-pairing. We have used enzymatic and chemical sensitivity of nucleotides, phylogenetic sequence comparison and the phenotypes of constructed mutants to develop a secondary structure model for the central region (900 nucleotides) of the group I phage MS2. The RNA folds into a number of, mostly irregular, helices and is further condensed by several long-distance interactions. There is substantial conservation of helices between the related groups I and II, attesting to the relevance of discrete RNA folding. In general, the secondary structure is thought to be needed to prevent annealing of plus and minus strand and to confer protection against RNase. Superimposed, however, are features required to regulate translation and replication. The MS2 RNA section studied here contains three translational start sites, as well as the binding sites for the coat protein and the replicase enzyme. Considering the density of helices along the RNA, it is not unexpected to find that all these sites lie in helical regions. This fact, however, does not mean that these sites are recognized as secondary structure elements by their interaction partners. This holds true only for the coat protein binding site. The other four sites function in the unfolded state and the stability of the helix in which they are contained serves to negatively control their accessibility. Mutations that stabilize helices containing ribosomal binding sites reduce their efficiency and vice versa. Comparison of homologous helices in different phage RNAs indicates that base substitutions have occurred in such a way that the thermodynamic stability of the helix is maintained. The evolution of individual helices shows several distinct size-reduction patterns. We have observed codon deletions from loop areas and shortening of hairpins by base-pair deletions from either the bottom, the middle or the top of stem structures. Evidence for the coaxial stacking of some helical segments is discussed.     

Characteristic chemical probing patterns of loop motifs improve prediction accuracy of RNA secondary structures

RNA structures play a fundamental role in nearly every aspect of cellular physiology and pathology. Gaining insights into the functions of RNA molecules requires accurate predictions of RNA secondary structures. However, the existing thermodynamic folding models remain less accurate than desired, even when chemical probing data, such as selective 2′-hydroxyl acylation analyzed by primer extension (SHAPE) reactivities, are used as restraints. Unlike most SHAPE-directed algorithms that only consider SHAPE restraints for base pairing, we extract two-dimensional structural features encoded in SHAPE data and establish robust relationships between characteristic SHAPE patterns and loop motifs of various types (hairpin, internal, and bulge) and lengths (2–11 nucleotides). Such characteristic SHAPE patterns are closely related to the sugar pucker conformations of loop residues. Based on these patterns, we propose a computational method, SHAPELoop, which refines the predicted results of the existing methods, thereby further improving their prediction accuracy. In addition, SHAPELoop can provide information about local or global structural rearrangements (including pseudoknots) and help researchers to easily test their hypothesized secondary structures.