Fast algorithm for predicting the secondary structure of single-stranded RNA.

Abstract

A computer method is presented for-finding the most stable secondary structures in long single-stranded RNAs. It is 1-2 orders of magnitude faster than existing codes.The time required for its application increases as N3 for a chain N nucleotides long. As many as 1000 nucleotides can be searched in a single run. The approach is systematic and builds an optimal structure in a straightforward inductive procedure based on an exact mathematical algorithm. Two simple halfmatrices are constructed and the best folded form is read directly from the second matrix by a simple back-tracking procedure. The program utilizes published values for base-pairing energies to compute one structure with the lowest free energy.Due to the rapid increase in our knowledge of the nucleotide sequence of many long single-stranded RNAs, it is of interest to attempt to predict the secondary and tertiary structure of these molecules.A simple method for estimating the free energy of loops found in single-stranded RNA based on their sequence was developed several years ago (1-6). By utilizing this method, the most probable loop structure for a given sequence is obtained from comparison of the relative stability of all of the possible structures that can form. Although this approach alone works easily for short nucleotide sequences, longer sequences require that many alternate structures be assessed and computer assistance becomes essential.A number of algorithms have been developed to apply free energy rules to polynucleotide chains (2, 7-9). The basic method in all of these approaches has been similar. Perfectly matched helices in the sequence are identified. Consistent sets of these helices are then assembled, and the overall free energy of each assembled structure is calculated individually. For long chains, the combinatorial aspects of this approach are very large (10, 11) and the time required for the calculations is extremely long.We have developed an approach to computer folding of large polynucleotide chains in which the algorithm is about 100 times faster than existing approaches. Two simple half-matrices are constructed by an inductive procedure which considers the energy contributions of individual base pairs. The loop structure with the lowest free energy is read directly from the second matrix in a simple fashion. The basic algorithm and its mathematical proof have been presented (12). It was developed initially simply to maximize base pairing along a polynucleotide chain. More recently, we realized that the rules for calculating loop stability based on free energy can be incorporated into the algorithm as well.This presentation provides a simplified explanation of the original algorithm for maximal matching as well as a description of the procedure developed for incorporating energy rules. METHODS Basic Formulation of the Method. The algorithm is designed to evaluate the contribution of individual base pairs to the secondary structure of a polynucleotide chain. The basic principle on which it rests is best understood by c...

The folding of large RNAs studied by hybridization to arrays of complementary oligonucleotides

“…The large differences seen in the mfold-predicted structures would be expected to have profound effects on the accessibility to duplex formation by oligonucleotides and lead to qualitatively different patterns rather than the variation in duplex yield observed+ Several thermodynamic-based methods were devised in the 1980s to predict the secondary structure of RNAs (e+g+, Nussinov & Jacobson, 1980;Dumas & Ninio, 1982) and were improved subsequently+ These methods were mostly based on the known structure of tRNAs and thermodynamic studies of small RNA fragments in solution (e+g+, Tinoco et al+, 1973)+ tRNAs are comparatively small and probably exist in the minimal global free energy state and so their secondary structure can be accurately predicted by computational methods+ The algorithms devised by Zuker and colleagues (e+g+, Zuker, 1989) became widely used (e+g+, mfold) because of their ability to calculate the free energy of folds of longer sequences+ For long RNA molecules, energy calculations often return a number of different structures with similar free energies: there is difficulty in choosing the correct one+ We analyzed the full length B5 transcript to see if the free energy of the most stable fold differed significantly from that of the fold in which the 59 end was constrained to its most stable fold+ The minimum free energy of the full transcript was Ϫ250+9 kcal/mol and the sum of the free energies of the two parts folded separately was Ϫ248+2 kcal/mol, showing that there was little difference between them+ This result reemphasizes the problem of using global free energy calculations+ Our results further indicate that the predicted structures have limited usage in biological application such as designing antisense oligonucleotides+ Ho et al+ (1998) have also shown that accessible sites cannot be mapped on predicted structures and that there is no obvious structural difference between accessible and inaccessible regions on the computer folded structures+ Gaspin & Westhof (1995) have devised an approach according to the RNA hierarchical folding view that allows for the dynamic incorporation of folding constraints, enabling the user to participate in the computational folding of RNA+ They predicted the secondary structure of Group I intron Td and RNAseP of Escherichia coli with this method, and the results were similar to those predicted by phylogenetic comparison+ Despite its demonstrated usefulness, such an approach requires substantial experimental data for input that may not be available for some biological applications, in which case empirical approaches become more useful+ Our results suggest ways in which hybridization to arrays may be used in predicting the secondary interactions in RNA molecules+ First, the hybridization data can help in testing short range interactions+ Studies of tRNA (K+U+ Mir & E+M+ Southern, submitted) suggest that strong interactions of oligonucleotides with the target, which are easily identified on the arrays, indicate certain stem-loop structures, and may point to stack interfaces+ The 59 regions of B5 mRNA that hybridize strongly to oligonucleotides, shown against the predicted secondary structure of lowest free energy (Fig+ 6), conform with the partial rules derived from the analysis of tRNA+ Second, long-range interactions are indicated by the loss of hybridization that results from extending the transcript+ The sequences of the oligonucleotides whose hybridization is blocked by the secondary interactions can be read directly from the array to locate the interacting regions p...…”

Structure of viroid replicative intermediates: physico-chemical studies on SP6 transcripts of cloned oligomeric potato spindle tuber viroid

“…Calculation of the most stable secondary structures of the different RNAs and of their possible alternative states (which were defined by fixing certain base pairs) was performed using a slightly modified Zuker-Nussinovprogram (20)(21)(22) on a VAX 11/750 and a Siemens 7.580. The calculation of thermal denaturation curves, from a knowledge of the secondary structures existing at different temperatures, was performed as described earlier (20).…”

Exact calculation of loop formation probability identifies folding motifs in RNA secondary structures

“…The most popular structure prediction algorithms, implemented in Mfold/UNAFold (Zuker 2003), the ViennaRNA Package (Lorenz et al 2011), and RNAstructure (Reuter and Mathews 2010;Bellaousov et al 2013), use a nearest neighbor model that can estimate the Gibbs free energy of folding for an RNA molecule. A search using dynamic programming can find the structure with the lowest (i.e., most negative) free energy, which is the most probable structure at equilibrium (Nussinov and Jacobson 1980;Zuker and Stiegler 1981). In one benchmark, 61.2% of pairs in predicted structures are present in accepted structures, and 68.9% of accepted pairs are in the predicted structure (Bellaousov and Mathews 2010), which is sufficient accuracy to develop testable hypotheses about the structure.…”