Analysis of the secondary and tertiary structures of Rous sarcoma virus RNA

The secondary and tertiary structures of the 35S RNA of Rous Sarcoma virus have been investigated. T1 RNase partial digests have been first resolved into their components by gel electrophoresis under non denaturing conditions and then each component analyzed further by various techniques. More than one hundred structured fragments have thus been isolated and shown to consist of several individual nucleotide sequences located far apart on the genome. On the basis of the results, a cloverleaf model for the structure of RSV 35S RNA is proposed that has implications for the various biological functions of this RNA.     

The evolution of 5S RNA secondary structures

We have applied the Pipas-McMahon algorithm based on free energy calculations to the search for a 5S RNA base-pair structure common to all known sequences. We find that a 'Y' shaped model is consistently among the structures having the lowest free energy using 5S RNA sequences from either eukaryotic or prokaryotic sources. Compaison of this 'Y' structure with models which have recently been proposed show these models to be remarkably similar, and the minor differences are explicable based on the technique used to obtain the model. That prokaryotic and eukaryotic 5S RNA can adopt a similar secondary structure is strong support for its resistance to change during evolution.     

Quaternion representation of RNA sequences and tertiary structures

A quaternion representation of nucleotides is proposed, with representation of RNA sequences by vectors whose elements are quaternions. Structure and transition matrices in quaternion representation are defined. Correspondence between diagrammatic technique in complex-number and quaternion representation of nucleotides is delineated.     

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.     

RNA secondary structures and their prediction

This is a review of past and present attempts to predict the secondary structure of ribonucleic acids (RNAs) through mathematical and computer methods. Related areas covering classification, enumeration and graphical representations of structures are also covered. Various general prediction techniques are discussed, especially the use of thermodynamic criteria to construct an optimal structure. The emphasis in this approach is on the use of dynamic programming algorithms to minimize free energy. One such algorithm is introduced which comprises existing ones as special cases. Issued as NRCC No. 23684.     

Assembly of protein tertiary structures from secondary structures using optimized potentials

We present a simulated annealing-based method for the prediction of the tertiary structures of proteins given knowledge of the secondary structure associated with each amino acid in the sequence. The backbone is represented in a detailed fashion whereas the sidechains and pairwise interactions are modeled in a simplified way, following the LINUS model of Srinivasan and Rose. A perceptron-based technique is used to optimize the interaction potentials for a training set of three proteins. For these proteins, the procedure is able to reproduce the tertiary structures to below 3 A in root mean square deviation (rmsd) from the PDB targets. We present the results of tests on twelve other proteins. For half of these, the lowest energy decoy has a rmsd from the native state below 6 A and, in 9 out of 12 cases, we obtain decoys whose rmsd from the native states are also well below 5 A.     

Automated motif extraction and classification in RNA tertiary structures

We used a novel graph-based approach to extract RNA tertiary motifs. We cataloged them all and clustered them using an innovative graph similarity measure. We applied our method to three widely studied structures: Haloarcula marismortui 50S (H.m 50S), Escherichia coli 50S (E. coli 50S), and Thermus thermophilus 16S (T.th 16S) RNAs. We identified 10 known motifs without any prior knowledge of their shapes or positions. We additionally identified four putative new motifs.     

Human telomerase RNA pseudoknot and hairpin thermal stability with glycine betaine and urea: preferential interactions with RNA secondary and tertiary structures

Thermal denaturation of the human telomerase RNA (hTR) DeltaU177 pseudoknot and hTR p2b hairpin was investigated by dual UV-wavelength absorbance spectroscopy in aqueous glycine betaine and urea solutions. The hTR DeltaU177 pseudoknot contains two helix-loop interactions that comprise the tertiary structure, as well as a GC-rich 6 bp stem (stem 1) and an AU-rich 9 bp stem (stem 2). The p2b hairpin also contains GC-rich stem 1 and a unique uridine-rich helix with a pentaloop. Glycine betaine stabilizes the pseudoknot tertiary structure in 135 mm NaCl and facilitates only a minor destabilization of tertiary structure in 40 mm NaCl. As with double-helical DNA, glycine betaine interacts more strongly with the surface area exposed upon unfolding of GC-rich stem 1 than either AU-rich stem 2 or the hairpin uridine-rich helix. Urea was shown to destabilize all RNA pseudoknot and hairpin secondary and tertiary structures but exhibits a stronger preferential interaction with AU-rich stem 2. Correlating these interactions with water-accessible surface area calculations indicates that the extent of interaction of glycine betaine with the surface area exposed upon RNA unfolding decreases as the nonpolar character of the unfolded RNA surface increases. As expected, the extent of interaction of urea with the surface area exposed for unfolding RNA increases as the fraction of amide functional groups increases. However, interaction of urea with amide functional groups alone cannot explain the stronger preferential interaction of urea with AU-rich stem 2. Interaction of urea with adenine relative to guanine and cytosine bases or sequence-dependent hydration is proposed for the stronger preferential interaction of urea with AU-rich duplexes.     

Combinatorics of locally optimal RNA secondary structures

It is a classical result of Stein and Waterman that the asymptotic number of RNA secondary structures is $1.104366 \cdot n^{-3/2} \cdot 2.618034^n$ . Motivated by the kinetics of RNA secondary structure formation, we are interested in determining the asymptotic number of secondary structures that are locally optimal, with respect to a particular energy model. In the Nussinov energy model, where each base pair contributes $-1$ towards the energy of the structure, locally optimal structures are exactly the saturated structures, for which we have previously shown that asymptotically, there are $1.07427\cdot n^{-3/2} \cdot 2.35467^n$ many saturated structures for a sequence of length $n$ . In this paper, we consider the base stacking energy model, a mild variant of the Nussinov model, where each stacked base pair contributes $-1$ toward the energy of the structure. Locally optimal structures with respect to the base stacking energy model are exactly those secondary structures, whose stems cannot be extended. Such structures were first considered by Evers and Giegerich, who described a dynamic programming algorithm to enumerate all locally optimal structures. In this paper, we apply methods from enumerative combinatorics to compute the asymptotic number of such structures. Additionally, we consider analogous combinatorial problems for secondary structures with annotated single-stranded, stacking nucleotides (dangles).     

Folding of a universal ribozyme: the ribonuclease P RNA

Ribonuclease P is among the first ribozymes discovered, and is the only ubiquitously occurring ribozyme besides the ribosome. The bacterial RNase P RNA is catalytically active without its protein subunit and has been studied for over two decades as a model system for RNA catalysis, structure and folding. This review focuses on the thermodynamic, kinetic and structural frameworks derived from the folding studies of bacterial RNase P RNA.     

Base-pair probability profiles of RNA secondary structures

Dynamic programming algorithms are able to predict optimal andsuboptimal secondary structures of RNA. These suboptimal oralternative secondary structures are important for the biologicalfunction of RNA. The distribution of secondary structures presentin solution is governed by the thermodynamic equilibrium betweenthe different structures. An algorithm is presented which approximatesthe total partition function by a Boltzmann–weighted summationof optimal and suboptimal secondary structures at several temperatures.A clear representation of the equilibrium distribution of secondarystructures is derived from a two-dimensional bonding matrixwith base–pairing probability as the third dimension.The temperature dependence of the equilibrium distribution givesthe denaturation behavior of the nucleic acid, which may becompared to experimental optical denaturation curves after correctionfor the hypochromicities of the different base-pairs. Similarly,temperature-induced mobility changes detected in temperature-gradientgel electrophoresis of nucleic acids may be interpreted on thebasis of the temperature dependence of the equilibrium distribution.Results are illustrated for natural circular and synthetic linearpotato spindle tuber viroid RNA respectively, and are comparedto experimental data.     

Combinatorial properties of RNA secondary structures.

The secondary structure of an RNA molecule is of great importance and possesses influence, e.g., on the interaction of tRNA molecules with proteins or on the stabilization of mRNA molecules. The classification of secondary structures by means of their order proved useful with respect to numerous applications. In 1978, Waterman, who gave the first precise formal framework for the topic, suggested to determine the number a(n,p) of secondary structures of size n and given order p. Since then, no satisfactory result has been found. Based on an observation due to Viennot et al., we will derive generating functions for the secondary structures of order p from generating functions for binary tree structures with Horton-Strahler number p. These generating functions enable us to compute a precise asymptotic equivalent for a(n,p). Furthermore, we will determine the related number of structures when the number of unpaired bases shows up as an additional parameter. Our approach proves to be general enough to compute the average order of a secondary structure together with all the r-th moments and to enumerate substructures such as hairpins or bulges in dependence on the order of the secondary structures considered.     

Search for conservative secondary structures of RNA

We suggest a new algorithm to search a given set of the RNA sequences for conserved secondary structures. The algorithm is based on alignment of the sequences for potential helical strands. This procedure can be used to search for new structured RNAs and new regulatory elements. It is efficient for the genome-scale analysis. The results of various tests run with this algorithm are shown.     

Computer-aided prediction of RNA secondary structures.

A brief survey of computer algorithms that have been developed to generate predictions of the secondary structures of RNA molecules is presented. Two particular methods are described in some detail. The first utilizes a thermodynamic energy minimization algorithm that takes into account the likelihood that short-range folding tends to be favored over long-range interactions. The second utilizes an interactive computer graphic modelling algorithm that enables the user to consider thermodynamic criteria as well as structural data obtained by nuclease susceptibility, chemical reactivity and phylogenetic studies. Examples of structures for prokaryotic 16S and 23S ribosomal RNAs, several eukaryotic 5S ribosomal RNAs and rabbit beta-globin messenger RNA are presented as case studies in order to describe the two techniques. Anm argument is made for integrating the two approaches presented in this paper, enabling the user to generate proposed structures using thermodynamic criteria, allowing interactive refinement of these structures through the application of experimentally derived data.     

Quantifying the energetic interplay of RNA tertiary and secondary structure interactions

To understand the RNA-folding problem, we must know the extent to which RNA structure formation is hierarchical (tertiary folding of preformed secondary structure). Recently, nuclear magnetic resonance (NMR) spectroscopy was used to show that Mg2+-dependent tertiary interactions force secondary structure rearrangement in the 56-nt tP5abc RNA, a truncated subdomain of the Tetrahymena group I intron. Here we combine mutagenesis with folding computations, nondenaturing gel electrophoresis, high-resolution NMR spectroscopy, and chemical-modification experiments to probe further the energetic interplay of tertiary and secondary interactions in tP5abc. Point mutations predicted to destabilize the secondary structure of folded tP5abc greatly disrupt its Mg2+-dependent folding, as monitored by nondenaturing gels. Imino proton assignments and sequential NOE walks of the two-dimensional NMR spectrum of one of the tP5abc mutants confirm the predicted secondary structure, which does not change in the presence of Mg2+. In contrast to these data on tP5abc, the same point mutations in the context of the P4-P6 domain (of which P5abc is a subdomain) shift the Mg2+ dependence of P4-P6 folding only moderately, and dimethyl sulfate (DMS) modification experiments demonstrate that Mg2+ does cause secondary structure rearrangement of the P4-P6 mutants' P5abc subdomains. Our data provide experimental support for two simple conclusions: (1) Even single point mutations at bases involved only in secondary structure can be enough to tip the balance between RNA tertiary and secondary interactions. (2) Domain context must be considered in evaluating the relative importance of tertiary and secondary contributions. This tertiary/secondary interplay is likely relevant to the folding of many large RNA and to bimolecular snRNA-snRNA and snRNA-intron RNA interactions.