Bioinformatics Advance Access originally published online on June 14, 2007
Bioinformatics 2007 23(16):2054-2062; doi:10.1093/bioinformatics/btm314
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Boltzmann probability of RNA structural neighbors and riboswitch detection
1Linnaeus Centre for Bioinformatics, Uppsala University, 75124 Uppsala, Sweden, 2School of Computing Sciences, University of East Anglia, Norwich, NR4 7TJ, UK and 3Department of Biology, Boston College, Chestnut Hill, MA 02467, USA
*To whom correspondence should be addressed.
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ABSTRACT |
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 TOP ABSTRACT 1 INTRODUCTION2 MATERIALS AND METHODS3 RESULTS4 DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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Motivation: We describe algorithms implemented in a new software package, RNAbor, to investigate structures in a neighborhood of an input secondary structure 
of an RNA sequence s. The input structure could be the minimum free energy structure, the secondary structure obtained by analysis of the X-ray structure or by comparative sequence analysis, or an arbitrary intermediate structure.
Results: A secondary structure 
of s is called a 
-neighbor of if and differ by exactly base pairs. RNAbor computes the number (N), the Boltzmann partition function (Z) and the minimum free energy (MFE) and corresponding structure over the collection of all -neighbors of . This computation is done simultaneously for all 
m, in run time O (mn3) and memory O(mn2), where n is the sequence length. We apply RNAbor for the detection of possible RNA conformational switches, and compare RNAbor with the switch detection method paRNAss. We also provide examples of how RNAbor can at times improve the accuracy of secondary structure prediction.
Availability: http://bioinformatics.bc.edu/clotelab/RNAbor/
Contact: clote{at}bc.edu
Supplementary information: Supplementary data are available at Bioinformatics online.
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1 INTRODUCTION |
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TOPABSTRACT1 INTRODUCTION2 MATERIALS AND METHODS3 RESULTS4 DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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In the last few years, there has been intense interest in RNA due to the surprising, previously unsuspected roles played by ribonucleic acid in what until now has been a predominantly protein-centric view of molecular biology. Apart from its roles as messenger RNA and transfer RNA, ribonucleic acid molecules play a catalytic role in the peptidyltransferase reaction in peptide bond formation (Nissen et al., 2000; Weinger et al., 2004) and in intron splicing (Vicens and Cech, 2006), both examples of enzymatic RNAs now termed ribonucleic enzymes or ribozymes (Doudna and Cech, 2002). RNA plays a role in post-transcriptional gene regulation due to the hybridization of mRNA by small interfering RNAs (siRNA) (Harborth et al., 2003; Tuschl, 2003) and microRNAs (miRNA) (Lim et al., 2003). By completely different means, RNA performs transcriptional and translational gene regulation by allostery, where a portion of the 5' untranslated region (5' UTR) of mRNA known as a riboswitch (Penchovsky and Breaker, 2005; Winkler et al., 2002) can undergo a conformational change upon binding a specific ligand, such as adenine, guanine or lysine. RNA is known to play critical roles in various other cellular mechanisms, such as dosage compensation (Brown et al., 1992), protein shuttling (Walter and Blobel, 1982), expansion of the genetic code such as selenocysteine insertion (Commans and Böck, 1999), and ribosomal frameshift (Bekaert et al., 2003; Moon et al., 2004). Illustrative of the growing recognition for the importance of RNA, the 2006 Nobel Prize in Physiology or Medicine was awarded to A.Z. Fire and C.C. Mello for their discovery of RNA interference and gene silencing by double-stranded RNA.
In this article, we develop novel and efficient algorithms to investigate structures in a neighborhood of a given secondary structure 
of an RNA sequence s. We call another secondary structure 
of S a -neighbor of 
, if 
and 
differ by exactly base pairs (see Methods section for more details). We develop algorithms to compute the number, 
, the partition function, 
and the minimum free energy 
(s, 
structure over the collection of all -neighbors 
of 
, where 
denotes the energy of 
with respect to the Turner nearest neighbor energy model (Xia et al., 1999), R is the universal gas constant and T is temperature in Kelvin. Our software, called RNAbor (RNA neighbor), additionally computes graphs of the probability density function p = Z/Z as a function of .
RNAbor was motivated by Moulton et al. (2000), who suggested that the stability of a secondary structure might depend on the number of structural neighbors at varying distances from the given structure—for instance from the minimum free energy (MFE) structure. It turns out that the number of structural neighbors at varying distances is not sufficient to distinguish between structural RNA and random RNA having the same MFE structure; see Figure 1. However, we do see a distinction when computing a weighted count of structural neighbors, where low energy structures are more heavily weighted. Formally, this is the Boltzmann partition function with respect to all structural neighbors at a given base pair distance . Figure 1 displays a density plot produced by RNAbor which clearly suggests that precursor miRNA dme-mir-1 from D.melanogaster (AE003667) has a single, well-defined native structure, whereas a random sequence with the same MFE structure has several alternate low energy secondary structures with different topologies.
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We show that the probability density plot can be used to detect RNA conformational switches and compare RNAbor with the conformational switch detection program paRNAss (Giegerich et al., 1999). In paRNAss, a structural RNA switch is predicted by means of studying properties of the energy landscape of the RNA. Secondary structures are sampled from the structure space using RNAsubopt (Wuchty et al., 1999) or mfold (Zuker, 1994). Pairwise distances are calculated between the sampled structures using two different distance measures (e.g. pairs energy barrier, morphological, tree alignment or string edit distance). Using a standard clustering method the structures are clustered into two clusters based on the distance measures. If the RNA is a conformational switch, then it has two stable structures and hence two clusters are expected (in a multi-switch, more than two stable structures are expected). As an additional test, the consensus structure of the clusters are computed and for each sample structure the distances to the two consensus structures are plotted against each other. If the RNA is really a conformational switch, then paRNAss output should display two clouds of points—one near the x-axis and one near the y-axis.
Note that since paRNAss calls RNAsubopt from the Vienna RNA Package program (Hofacker, 2003), it requires a user-defined energy bound, E, in order to generate all secondary structures within E kcal/mol of the MFE.
The plan for the rest of this article is as follows. In the Methods section, we describe the algorithms for computing the number N, the Boltzmann partition function Z and the minimum free energy MFE structure over the collection of all -neighbors. In the Results section, we present graphs of the number N and the Boltzmann probability density p = Z/Z of structural neighbors, which differ by base pairs from a given secondary structure. In addition, we compare the output of RNAbor with the conformational switch detection program paRNAss (Giegerich et al., 1999). In the Discussion section, we conclude by presenting some possible future applications of our algorithms. Pseudocode for our implementation is presented in the Supplementary Material. Additional data obtained by running RNAbor on all SAM riboswitches from Rfam (Griffiths-Jones et al., 2003) is available in the Supplementary Material at http://bioinformatics.bc.edu/clotelab/RNAbor/webSupplement.
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2 MATERIALS AND METHODS |
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TOPABSTRACT1 INTRODUCTION2 MATERIALS AND METHODS3 RESULTS4 DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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Given an RNA nucleotide sequence s, consider a fixed secondary structure
of s. In this section, we describe how to efficiently compute the number N of -neighbors of 
, the partition function Z for -neighbors, and the minimum free energy MFE together with the corresponding structure over all -neighbors. N, Z and MFE are all computed for a fixed temperature. The temperature is set to 37°C by default, but can be changed by the user.
2.1 The number of -neighbors of a fixed secondary structure
Let s = s1, ..., sn denote an RNA sequence, i.e. a sequence of letters in the alphabet of nucleotides {A,C,G,U}. A secondary structure 
on s is a set of base pairs (i, j), where 1 i i + 
< j n and 
0 is an integer (corresponding to minimum hairpin loop size, which we usually set to 3), such that if (k,l) is a base pair, then k=i 
l=j (a nucleotide is involved in at most one base pair) and i < k < j i < l < j (no pseudoknots). We say that 
is compatible with s if for every base pair (i, j) in 
the pair si sj is contained in the set = {AU, UA, GC, CG, GU, UG} (i.e. the set of Watson–Crick base pairs together with wobbles). Given two secondary structures 
, 
on s, we define the base pair distance dBP between 
and 
to be the number of base pairs that they have that are not in common, i.e.
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| (1) |
For the rest of this section, we consider both s as well as the secondary structure 
on s to be fixed. We now provide recursions for determining the number of secondary structures 
compatible with s that are at precisely base pair distance to 
.
Let 
denote the restriction of 
to interval [i, j] of s, i.e. the set of base pairs 
. A secondary structure 
on s is a -neighbor of 
if 
. For all 0 m, and all 1 i j n, let 
denote the number of secondary structures 
compatible with s such that 
. In the following, we may omit the sequence S and secondary structure 
in our notation since these are fixed. In particular, we put 
(s, 
.
is computed recursively. The initial conditions for computing 
are given by
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| (2) |
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| (3) |
+ 2 nucleotides, and so there are no -neighbors for > 0. The recursion used to compute 
for > 0 and j > i + is
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| (4) |
and 0 otherwise, and 
. This holds since in a secondary structure 
on [i, j] that is a -neighbor of 
, either nucleotide j is unpaired in [i, j] or it is paired to a nucleotide k such that i k < j. In this latter case it is enough to study the smaller sequence segments [i,k–1] and [k+1, j–1] noting that, except for (k, j), base pairs outside of these regions are not allowed. In addition, for 
to be fulfilled it is necessary for w + w' = – b to hold, where 
and 
, since b is the number of base pairs that differ between 
and a structure 
, due to the introduction of the base pair (k, j).
Pseudocode for computing 
for values of between 0 and m is given in the Supplementary Material. The algorithm runs in time O(mn3) and space O(mn2) where, as defined above, n is the length of s and m is the maximum value of .
2.2 Probability analog
In this section, we explain how to extend our approach of computing 
to compute the partition function contribution of the set of structures compatible with a given RNA sequence s at a fixed base pair distance from an RNA structure 
compatible with s. This allows us to compute the probability of the set of structures compatible with s at distance from 
.
It is straight–forward to extend the previous approach to compute partition functions for the Nussinov–Jacobson energy model (Nussinov and Jacobson, 1980). In particular, by simply replacing recursion (4) with
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| (5) |
. The base pair energy E(k,j) takes the value –1 if sk sj 
and 0 otherwise. Note that the energy contribution can be altered for different base pairs e.g. –3 for GC, –2 for AU and –1 for GU are weights used in Nussinov and Jacobson, (1980). Employing a substantially more complicated algorithm, similar to the dynamic programming calculation of the partition function described in McCaskill (1990), the partition function contributions can also be computed according to the Turner energy model. In the Turner energy model a secondary structure is decomposed into loops, as described in Zuker and Sankoff (1984), and the energy is computed as a sum of the energy contributions of the loops. A k-loop consists of k–1 base pairs (excluding the closing base pair) and u unpaired bases. The energies of 1-loops (hairpins) and 2-loops (stacks if u=0, bulges or interior loops if u > 0) are based on experimental data (Mathews and Turner, 2002; Mathews et al., 1999) and are dependent on k and u as well as the RNA sequence. In the Turner model, the energies for multi-loops (k > 2) are generally determined by the approximate linear model EM = a + b(k – 1) + cu, where a, b and c are constants.
As before, from now on we regard s and 
to be a fixed RNA sequence with compatible secondary structure 
. The partition function for S is then defined as 
, where the sum is taken over all structures 
compatible with s, and 
is the energy of the structure 
. We aim to compute the restriction 
(s, 
, i.e. the sum of 
taken over all structures 
that are compatible with s and at base pair distance from 
. The probability for finding a structure at a distance from 
is then given by p = Z/Z.
As with the usual McCaskill partition function calculations (McCaskill, 1990), in the dynamic programming we use three matrices Z, ZB and ZM for recursively computing Z instead of the single matrix N used for computing N in the previous section. In particular, for the sequence segment [i, j] of s, define 
, where the sum is over all structures 
compatible with s and such that 
. Also, define the restricted partition function 
as the sum of 
taken over all structures 
such that 
, and 
, which is the partition function contribution if the sequence segment [i, j] is part of a multi-loop. The matrices Z, ZB and ZM are filled using the following three recursions.
To compute Z we use the recursions
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| (6) |
. Note that the first term of this recursion corresponds to the case where j is unpaired (and hence has no energy contribution) in [i, j]. The second term includes all other structures on [i, j]. The sum is taken over all possible base pairs (k, j) with i k < j. If (k, j) is a base pair the partition function for [k, j] is given by 
, the partition function for [i, k–1] is given by 
.
We compute ZB using the recursion
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| (7) |
, and 
. Here, 
(x,y) is the Kronecker function, which equals 1 if x=y, and else 0. Note that since the above equation computes 
, it follows that (i, j) forms a base pair in the neighboring structures 
(if this is not possible, then 
). The first term in the recursion takes care of the case where (i, j) is the only base pair in [i, j], i.e. (i, j) closes a hairpin loop. The second term handles the case where there is an interior loop (or a bulge or a stack) closed by (i, j) and (k, l). The third term takes care of all the structures where (i, j) closes a multi-loop. To reduce complexity of the algorithm, the interior and bulge loop size can be limited to a maximum size of L, by requiring that l > j–L in the above recursion.
The final recursion, for computing ZM, is
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| (8) |
and 
. Note that since 
computes the partition function contribution under the assumption that [i, j] is part of a multi-loop, there will be exactly one stem-loop structure in this region (the ZB term) or more than one (the ZB-ZM term).
Note that the recursions for computing the number of -neighbors and the partition function analogues are non-redundant in that each structure is counted once and only once.
Pseudocode for computing Z is given in the Supplementary Material. The complexity is the same as for computing the number of -neighbors, O(mn2) in space and O(mn3) in time, if the size of internal loops and bulges are limited to a fixed length such as 30, following the convention of Vienna RNA Package.
2.3 Minimum free energy -neighbors
Given an RNA nucleotide sequence s and secondary structure 
, the minimum free energy -neighbor is that secondary structure 
of s, which has base pair distance with 
, and which has least free energy MFE among all structures having base pair distance with 
. Free energy is measured according to the Turner energy model (Mathews et al., 1999; Xia et al., 1999), where our treatment of dangles follows that of Vienna RNA package with –d2 option.
In this section, we describe a novel algorithm capable of computing the MFE structures, for all . As in our partition function computation, the run time [resp. space requirement] to compute all MFE structures for m is O(mn3) [resp. O(mn2)]. This algorithm is obtained from the algorithm in Section 2.2 essentially by replacing Boltzmann factor 
by free energy 
and by replacing the operations of addition [resp. multiplication] by minimization [resp. addition]. In future work, we plan to analyze the structure morphological changes in proceeding from 
to MFE0, MFE1, MFE2, etc. As indicated in the Results section, such an analysis could prove useful in conformational switch detection and other applications.
Fix RNA nucleotide sequence s = s1, ..., sn and secondary structure 
of s. To compute MFE we use the following recursions:
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| (9) |
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| (10) |
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| (11) |
Once the minimum free energy of -neighbors (MFE) is computed the corresponding MFE structures can be computed by a simple traceback for each MFE.
For reasons of space, the pseudocode for computing MFE is not presented; given our previous description of MFE and the pseudocode for computing the partition function Z, appearing in the Supplementary Material, the reader will have no difficulty to reconstruct the pseudocode for MFE.
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3 RESULTS |
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TOPABSTRACT1 INTRODUCTION2 MATERIALS AND METHODS3 RESULTS4 DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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In this section, we present probability density graphs for a variety of conformational switches and for some non-switches. Additional data is provided for all SAM riboswitches in our Supplementary Material. We also compare the output of RNAbor with distance plots generated by the web server paRNAss (Giegerich et al., 1999), that uses a heuristic to determine whether there appear to be two or more clusters of distinct secondary structures for a given RNA sequence. Some example are also presented that indicate that RNAbor can be used to provide improved secondary structure predictions as compared with the MFE structure.
3.1 Detecting conformational switches
In this section, we define a conformational switch to be an RNA sequence which has exactly two distinct low energy secondary structures. By multi-switch we mean an RNA sequence which can adopt two or more distinct low energy secondary structures. For a given RNA sequence s and secondary structure 
of s, we use RNAbor to compute p = Z/Z. Taking 
to be the MFE structure, or alternatively the structure determined by comparative sequence alignment (Cannone et al., 2002), our intuition is that a conformational switch should display a bi-modal probability density graph.
To illustrate the behavior of a typical conformational switch, we present examples of RNAbor output for known switches. Consider for instance the 105 nt SAM riboswitch with EMBL accession number AE016750.1/132874-132778 and sequence AACUUAUCAA GAGAAGUGGA GGGACUGGCC CAAAGAAGCU UCGGCAACAU UGUAUCAUGU GCCAAUUCCA GUAACCGAGA AGGUUAGAAG AUAAGGU. Figure 2 displays three secondary structures: the MFE structure, the MFE24 structure and the native structure inferred by comparative sequence analysis of the SAM riboswitch seed multiple sequence alignment from Rfam (Griffiths-Jones et al., 2003). As computed by RNAbor, the MFE structure has free energy of –28.1 kcal/mol and the Boltzmann probability p0 is 0.11, while the MFE24 structure has free energy of –26.7 kcal/mol and p24 is 0.05. Note the similarity of the MFE24 structure with the native structure; in particular, the apical loop regions are correctly computed. There is a second MFE structure–the MFE27 structure, with free energy –28.1 kcal/mol. Although the Boltzmann probability p27 is 0.16, the maximum of p over all , the MFE27 structure is rather different than the native structure (data not shown).
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(s, 
(s, 
, for all secondary structures of RNA sequence s having base pair distance 
. Alternative secondary structures for the riboswitch are shown in the figure. In the upper left is the MFE structure with free energy –28.10 kcal/mol shown, in the middle the MFE24 structure with free energy –26.7 kcal/mol and in the lower right the consensus Rfam structure. Sorted in decreasing order, the most significant Boltzmann probabilities are 0.16, 0.11, 0.090, 0.068, 0.067, 0.057, 0.055, respectively for values of Figure 2 shows the probability density plot, i.e. the probability p
= Z/Z of finding a structure at distance from the input structure, which in this example is the MFE structure. It is not the case that all conformational switches display a bi-modal or multi-modal Boltzmann probability density curve. In particular, the probability density curve is uni-modal for the 101 nt switch (Schlax et al., 2001) with EMBL accession number AE0140031/5850-5961. This mRNA has a pseudoknotted structure, which is responsible for the translational repression of the alpha operon by an entrapment mechanism. Since the algorithm RNAbor, like its predecessors mfold and RNAfold, considers only non-pseudoknotted secondary structures, there is no reason to expect that RNAbor display a multi-modal probability density curve for this conformational switch.
Figure 3a depicts a bi-modal density graph for the artificially engineered bistable switch CUUAUGAGGG UACUCAUAAG AGUAUCC of Flamm et al. (2001). Figure 3b displays the probability density function p for the 76 nt conformational switch (Ke et al., 2004), which controls hepatitis delta virus ribozyme catalysis (PDB ID 1SJ3:R). Both these examples display bi-modal Boltzmann probability curves.
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(s, 
(s, 
, of all 3.2 Comparison with paRNAss
We now compare the ability of RNAbor and paRNAss to predict RNA conformational switches. We have chosen to display the paRNAss distance plot of energy barrier versus morphological distance (Voss et al., 2004), but in all the below examples the distance plots using tree alignment or string edit distance showed similar results.
The Escherichia coli hok (host killing) mRNA folds into two different conformations (Franch et al., 1997). The full length mRNA folds into a stable structure involving a long-range interaction between the 5' and 3'-end. Degradation of the 3'-end leads to a conformational change as the stabilizing long-range interaction is broken. Here, we have investigated the part of the mRNA that undergoes a conformational change (as provided on the paRNAss web server http://bibiserv.techfak.uni-bielefeld.de/parnass).
For this RNA, both RNAbor and paRNAss detect the conformational switch, the RNAbor probability plot shows two distinct peaks suggesting two alternative stable structures and the paRNAss plot shows two clearly separated clusters, both suggesting that all the reasonably stable structures fall into one out of two conformations, see Figure 4.
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Although both RNAbor and paRNAss suggest that the hok gene has two alternative structures, there are some uncertainties in the result. In the RNAbor density plot there are actually three peaks (even though the third peak is significantly smaller than the other two), indicating that there might be more than two alternative structures.
The 5'-untranslated (UTR) region of E.coli thiM mRNA undergoes a change in structure, that is important for regulation (Winkler et al., 2002). Both RNAbor and paRNAss indicate more than one single stable structure for the thiM-leader. As can be seen from Figure 5, there actually seem to be more than two alternative structures. However, the third structure seems to be less important (lower probability), and hence this RNA is predicted as a conformational switch by RNAbor.
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3.3 Improving on the minimum free energy structure
In this section, we discuss several examples where a MFE
structure is closer to the native secondary structure, as extracted from the 3D X-ray structure, than is the MFE structure. This phenomenon generally occurs when the probability density graph indicates a second peak, although sometimes that peak may be modest.
Figure 6 presents the Boltzmann probability density plot and alternate secondary structure models for the S-adenosylmethionine riboswitch mRNA regulatory element with PDB code 2GIS (Montange and Batey, 2006). The figure shows the native secondary structure for 2GIS, determined by extraction from the 3D X-ray structure1, the MFE24 structure, which clearly resembles the native state, and the rather different looking MFE structure. Figure 6 graphs the probability density, where a large second peak is present, centered at = 23.
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Improvement over the MFE structure is not restricted to riboswitches. Indeed, Figure 7 displays a very unrealistic linear MFE structure for Ile-tRNA from E.coli—accession number DI1660 from Sprinzl's tRNA database (Sprinzl et al., 1998), as well as alternative structures predicted by RNAbor. Note the correctly predicted anti-codon GAU.
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4 DISCUSSION |
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TOPABSTRACT1 INTRODUCTION2 MATERIALS AND METHODS3 RESULTS4 DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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In this article we present some novel algorithms for efficiently computing the number (N
), the Boltzmann partition function (Z) and the minimum free energy (MFE) and corresponding structure over the collection of all -neighbors of a secondary structure of a fixed RNA sequence, all of which are implemented in the webserver RNAbor. We find that the output of RNAbor gives useful insights into the landscape of foldings for a single RNA sequence. In addition, we observe that the output of RNAbor compares well with that of the conformational switch detection program paRNAss. In future work, we will make a more extensive comparison of RNAbor with other RNA folding landscape analysis programs, such as RNAshapes (Giegerich et al., 2004; Steffen et al., 2006; Voss et al., 2006) and sfold (Ding and Lawrence, 2003; Ding et al., 2004).
Potential applications of RNAbor which will be pursued in future work include the following.
- Since RNAbor allows one to distinguish whether the given RNA nucleotide sequence s has a single pronounced well of attraction around a given secondary structure
of s, it may be possible to use RNAbor to detect situations where the native secondary structure, as determined by X-ray crystallography, is different than that proposed by mfold and RNAfold (e.g. see Section 3.3). The idea would be to determine if there is no peak around = 0, when 
is taken to be the MFE structure.
- Figure 5a shows an interesting example where the RNA seems to have more than one alternative structure. Does this RNA have more than two alternative structures? Is it the case that the MFE structure is not biologically functional? (In this example, the other two alternative structures seem to be probable.) Using RNAbor, we can determine the minimum free energy structures over all -neighbors, and subsequently focus on MFE structures where the Boltzmann probability p is high. Ultimately, chemical probing experiments might determine whether these MFE structures are the preferred biologically active structure.
- RNAbor is a useful complement to already existing tools for detecting putative conformational switches. Unlike paRNAss, the number of structures to be analyzed and the maximum allowable free energy difference from the MFE structure need not be decided in advance. (These can change the paRNAss result quite dramatically.) Depending on the number of structures to be analyzed and the energy bound, paRNAss can take an exponential amount of time, in contrast to O(mn3) time for RNAbor to compute N, Z and MFE.
- As shown in Figures 6 and 7, RNAbor can sometimes predict a secondary structure, which is closer to the real secondary structure than is the MFE structure, as determined from X-ray structures or comparative sequence analysis.
- As for any bioinformatics software, it will be necessary to perform experimental validation of predictions made by RNAbor. In future work, we intend to include user-defined constraints, which allow the user to require all investigated structures to contain certain specified base pairs and for certain specified nucleotides to remain unpaired. This will allow RNAbor to be used together with chemical probing experiments to determine biologically active conformers.
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ACKNOWLEDGEMENTS |
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TOPABSTRACT1 INTRODUCTION2 MATERIALS AND METHODS3 RESULTS4 DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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Research of P.C. was partially supported by National Science Foundation DBI-0543506, which additionally supported some travel of E.F. All three authors would like to thank Elena Rivas, Eric Westhof and funding agencies for organizing the meeting RNA-2006 in Benasque, Spain, in July 2006, where some of this work was carried out.
Conflict of Interest: none declared.
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FOOTNOTES |
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Associate Editor: Anna Tramontano
1Using the software rnaview (Yang et al., 2003), we first obtained a list of all Watson–Crick cis base pairs. 
Received on April 30, 2007; revised on April 30, 2007; accepted on June 5, 2007
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TOPABSTRACT1 INTRODUCTION2 MATERIALS AND METHODS3 RESULTS4 DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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