 Methodology article
 Open Access
Analysis of energybased algorithms for RNA secondary structure prediction
 Monir Hajiaghayi^{1}Email author,
 Anne Condon^{1} and
 Holger H Hoos^{1}
https://doi.org/10.1186/147121051322
© Hajiaghayi et al; licensee BioMed Central Ltd. 2012
 Received: 21 June 2011
 Accepted: 1 February 2012
 Published: 1 February 2012
Abstract
Background
RNA molecules play critical roles in the cells of organisms, including roles in gene regulation, catalysis, and synthesis of proteins. Since RNA function depends in large part on its folded structures, much effort has been invested in developing accurate methods for prediction of RNA secondary structure from the base sequence. Minimum free energy (MFE) predictions are widely used, based on nearest neighbor thermodynamic parameters of Mathews, Turner et al. or those of Andronescu et al. Some recently proposed alternatives that leverage partition function calculations find the structure with maximum expected accuracy (MEA) or pseudoexpected accuracy (pseudoMEA) methods. Advances in prediction methods are typically benchmarked using sensitivity, positive predictive value and their harmonic mean, namely Fmeasure, on datasets of known reference structures. Since such benchmarks document progress in improving accuracy of computational prediction methods, it is important to understand how measures of accuracy vary as a function of the reference datasets and whether advances in algorithms or thermodynamic parameters yield statistically significant improvements. Our work advances such understanding for the MFE and (pseudo)MEAbased methods, with respect to the latest datasets and energy parameters.
Results
We present three main findings. First, using the bootstrap percentile method, we show that the average Fmeasure accuracy of the MFE and (pseudo)MEAbased algorithms, as measured on our largest datasets with over 2000 RNAs from diverse families, is a reliable estimate (within a 2% range with high confidence) of the accuracy of a population of RNA molecules represented by this set. However, average accuracy on smaller classes of RNAs such as a class of 89 Group I introns used previously in benchmarking algorithm accuracy is not reliable enough to draw meaningful conclusions about the relative merits of the MFE and MEAbased algorithms. Second, on our large datasets, the algorithm with best overall accuracy is a pseudo MEAbased algorithm of Hamada et al. that uses a generalized centroid estimator of base pairs. However, between MFE and other MEAbased methods, there is no clear winner in the sense that the relative accuracy of the MFE versus MEAbased algorithms changes depending on the underlying energy parameters. Third, of the four parameter sets we considered, the best accuracy for the MFE, MEAbased, and pseudoMEAbased methods is 0.686, 0.680, and 0.711, respectively (on a scale from 0 to 1 with 1 meaning perfect structure predictions) and is obtained with a thermodynamic parameter set obtained by Andronescu et al. called BL* (named after the Boltzmann likelihood method by which the parameters were derived).
Conclusions
Large datasets should be used to obtain reliable measures of the accuracy of RNA structure prediction algorithms, and average accuracies on specific classes (such as Group I introns and Transfer RNAs) should be interpreted with caution, considering the relatively small size of currently available datasets for such classes. The accuracy of the MEAbased methods is significantly higher when using the BL* parameter set of Andronescu et al. than when using the parameters of Mathews and Turner, and there is no significant difference between the accuracy of MEAbased methods and MFE when using the BL* parameters. The pseudoMEAbased method of Hamada et al. with the BL* parameter set significantly outperforms all other MFE and MEAbased algorithms on our large data sets.
Keywords
 Minimum Free Energy
 Base Pair Probability
 Maximum Expected Accuracy
 Bootstrap Percentile Method
 Minimum Free Energy Method
Background
RNA molecules are essential to many functions in the cells of all organisms. For example, these molecules are involved in gene translation and also act as catalysts and as regulators of gene expression [1]. Because function is determined by molecular structure, there is significant investment in computational methods for predicting RNA secondary structure, which in turn is useful for inferring tertiary structure [2].
Our interest is in assessing the merits of some recent advances in secondary structure prediction in statistically robust ways. We focus on thermodynamically informed approaches for predicting pseudoknot free secondary structures from the base sequence. A widely used method finds the minimum free energy (MFE) structure with respect to the nearest neighbour thermodynamic model of Mathews, Turner and colleagues [3]. Some recent advances in secondary structure prediction are the new maximum expected accuracy (MEAbased) and maximum pseudoexpected accuracy (pseudoMEAbased) methods of Lu et al. [4] and Hamada et al. [5, 6]. These approaches generally maximize (pseudo) expected base pair accuracy as a function of base pair probabilities calculated using a partition function method and have higher average accuracy than the MFE algorithm on the Turner and Andronescu et al. energy parameters.
Knudsen et al. [7] and Do et al. [8] also presented some RNA secondary structure prediction methods based on probabilistic models of structures. But since their probabilistic approaches are not determined using a thermodynamic model, we don't include their methods in our later comparisons. Another advance is estimation of new energy parameters from both thermodynamic and structural data using stateoftheart estimation techniques. Andronescu et al. [9] derived two parameter sets by inference from energies that were derived from optical melting experiments as well as from structural data. The two energy parameter sets are called BL* and CG*, named after the Boltzmann likelihood and constraint generation methods used to infer them. These parameter sets have yielded significant improvements in prediction accuracy of the MFE method, compared with the Turner parameters, with the BL* parameters being slightly better than the CG* parameters. Here and throughout, the accuracy of a prediction refers to its Fmeasure, which is the harmonic mean of sensitivity and positive predictive value (see Methods section for definitions of these measures). All of this work assesses algorithm accuracy on specific classes of RNAs, such as introns or transfer RNAs, as well as overall average accuracy on RNAs taken over all such classes.
This recent work motivates the following questions. Are comparisons of the (pseudo)MEAbased and MFE approaches on specific RNA classes reliable when the size of available datasets is small? Do the MEA or pseudoMEAbased approaches produce significantly more accurate predictions than MFE on the latest energy parameter sets? What is the best combination of algorithm and thermodynamic model? To answer these questions, we report on the accuracy of both (pseudo)MEAbased and MFE methods with respect to two versions of the Turner parameters as well as the recent BL* and CG* parameters of Andronescu et al., on datasets for specific RNA classes as well as large datasets that combine multiple RNA classes.
We present three main findings. First, we show that Fmeasure accuracies on our large datasets are likely to be reliable estimates of accuracy of a population represented by such sets, in the sense that highconfidence interval widths for Fmeasure obtained using the bootstrap percentile method are within a small, 2% range. Average accuracy on smaller classes is less reliable. For example, confidence intervals for both MEA and MFE have an 8% range on a class of 89 Group I introns that has been used previously in benchmarking algorithms. Second, there is a clear "winner" in terms of overall prediction accuracy, namely the pseudoMEAbased method of Hamada et al. [6]. However, the relative accuracy of the MFE and MEAbased approaches depends on the underlying energy parameters: using a permutation test we find that, at a statistically significant level, the accuracy of MFEbased prediction on our large datasets is better on two of the four energy parameter sets that we consider, while MEAbased prediction is better than MFEbased prediction on a third parameter set. Finally, both MEAbased and MFE methods achieve the highest accuracy when using the fourth parameter set we consider, namely the BL* energy parameters of Andronescu et al. [9].
Methods
In this section we first describe the datasets, thermodynamic models, and algorithms considered in this paper. We then describe the accuracy measures and statistical methods used in our analyses.
Datasets
We use three datasets, as follows

SFull is a comprehensive set of 3,245 RNA sequences and their secondary structures that has been assembled from numerous reliable databases [10]. Sequences in this and our other datasets have length at most 700 nucleotides; in some cases these were derived by partitioning larger sequences such as 16S Ribosomal RNA sequences. The average length of sequences in SFull is 270nt.

MT was used by Lu et al. [4] in their study of the MEA algorithm and contains RNAs from the following classes: 16S Ribosomal RNA, 23S Ribosomal RNA, 5S Ribosomal RNA, Group I intron, Group II intron, Ribonuclease P RNA, Signal Recognition RNA and Transfer RNA.

MA is a subset of the SFull set, containing exactly those sequence of SFull that are in the RNA classes included in MT. We formed the MA dataset in order to compare our algorithms on the same classes as did Lu et al., while using sets of RNAs from these classes with as large a size as possible.
Overview of the different RNA classes in MT and MA data sets
RNA class  No. in MT  mean ± std of length  Avg. similarity  No. in MA  mean ± std of length  Avg. similarity 

16S Ribosomal RNA  89  377.88 ± 167.18  0.60  675  485.66 ± 113.02  0.62 
23S Ribosomal RNA  27  460.6 ± 151.3  0.53  159  453.44 ± 117.85  0.57 
5S Ribosomal RNA  309  119.5 ± 2.69  0.88  128  120.98 ±3.21  0.88 
Group I intron  16  344.88 ± 66.42  0.63  89  368.49 ±103.58  0.63 
Group II intron  3  668.7 ± 70.92  0.70  2  578 ± 47  0.72 
Ribonuclease P RNA  6  382.5 ± 41.66  0.74  399  332.78 ±52.34  0.72 
Signal Recognition RNA  91  267.95 ± 61.72  0.71  364  227.04 ± 109.53  0.65 
Transfer RNA  484  77.48 ± 4.8  0.96  489  77.19 ± 5.13  0.95 
Total  1024  2305 
Lu et al. reported their results on a restricted format of the MT set in which certain bases of some sequences are in lower case, indicating that the base is unpaired in the reference structure. They have used this structural information in their predictions. However, we have not employed this information, and therefore our accuracy measures on the MT set are different from those of Lu et al. [4].
Thermodynamic Models
A thermodynamic model consists of featuressmall structural motifs such as stacked pairsplus free energy change parameters, one per feature. The first model we use, the Turner model [3] called "Turner99", is the most widely used energy model for prediction of RNA secondary structures. The model has over 7600 features, which are based on Turner's nearest neighbor rules and reflect the assumption that the stability of a base pair or loop depends on its sequence and on the adjacent base pair or unpaired bases. The model is additive in that the overall free energy change of a secondary structure for a given sequence is the sum of the free energy changes for features of the structure. The parameters of the model were derived from optical melting experiments, the most commonly used experimental approach to determine the free energy change of RNA structures.
We also consider variants of the Turner model, used by Andronescu et al. [9]. The T99MultiRNAFold (T99MRF) model is derived from the Turner99 model but includes only 363 features. Parameters for features in the Turner99 model can be obtained by extrapolation from the parameters of the T99MRF model. Using maximum likelihood and constraint optimization methods, Andronescu et al. [9] derived new free energy change parameters for these 363 features; the resulting models are called BL* (for Boltzmann likelihood) and CG* (for constraint generation) respectively. We used all three models, namely T99MRF, BL* and CG*, in this work, in order to assess the dependence of algorithm accuracy on model parameters. We note that in Lu et al.'s work [4], the parameters of a newer version of Turner model, called Turner2004, were used for one of the structural motifs, coaxial stacking. However, for the rest of structural motifs the parameters of Turner99 model were engaged. So, we also call the parameter set used for the Lu et al.'s benchmark Turner99 since most of its parameters are Turner99 ones and also the coaxial stacking motif is not employed in the other model that we study. Zakov et al. [11] also obtained parameters that improve RNA structure prediction. But we don't consider their parameters in this work, since those are not applicable for the partition function calculation and therefore for the probability calculation required for the MEA method.
Algorithms
We analyze four RNA secondary structure prediction algorithms. The first predicts secondary structures that have minimum free energy (MFE) with respect to a given thermodynamic model. The second is the maximum expected accuracy (MEA) algorithm as introduced by Lu et al. [4], which maximizes expected base pair accuracy as a function of base pair probabilities calculated using a partition function method. We implemented the MEA algorithm for use with the MultiRNAFold models. As a result, we worked not only with the algorithms of Lu et al., which we refer to as rsMFE and rsMEA, but also with the MFE algorithm of Andronescu et al., referred to as ubcMFE, and a new implementation of MEA which we developed, called ubcMEA. Lu et al.'s benchmark [4] showed that rsMEA gives the best prediction accuracy when its γ parameter (which controls the relative sensitivity and positive predictive value) is equal to 1. Accordingly, we also set γ to be 1 in ubcMEA.
The third algorithm that we analyze is the generalized centroid estimator method of Hamada et al. [5]. This is similar to the MEA method, but uses a somewhat different objective function, namely a gammacentroid estimator, to infer a structure from base pair probabilities. This method, which we refer to as gCg1, employs a parameter γ to balance sensitivity and positive predictive value; based on the results of Hamada et al., we set γ = 1. The fourth method is another algorithm of Hamada et al. [6] that generalizes the centroid estimator by using a pseudoexpected accuracy maximization technique that automatically selects γ to balance sensitivity and positive predictive value for a given input. We refer to this algorithm as gCpMFmeas (for generalized centroid maximized pseudoexpected accuracy).
The rsMEA and rsMFE algorithms always use the parameters of Turner99 model as their free energy parameter set, while ubcMEA and ubcMFE use parameter sets in the MultiRNAFold model format, namely the BL*, CG* and T99MRF sets. The gCg1 and gCpMFmeas algorithms also employ BL* parameters using a Turner99 format.
Accuracy Measures
We use three measures for determining the structural prediction accuracy, namely sensitivity (also called precision or precision rate), positive predictive value or PPV (also called recall), and Fmeasure, which combines the sensitivity and PPV into a single measure.
Sensitivity is the ratio of correctly predicted base pairs to the total base pairs in the reference structures. PPV is the fraction of correctly predicted base pairs, out of all predicted base pairs. Fmeasure is the harmonic mean of the sensitivity and PPV. This value is equal to the arithmetic mean when sensitivity and PPV are equal. However, Fmeasure becomes smaller than the arithmetic mean as one of the numbers approaches 0 (while the other is fixed). The possible values for these three measures are between 0 and 1; the closer to 1, the better prediction.
Throughout this paper, we calculate three types of averages for a given measure "M" (which can be any of PPV, sensitivity, or Fmeasure), namely unweighted averages, weighted averages and Sweighted averages, defined below. The weighted average counts each sequence equally, regardless of which class it belongs to. The unweighted average, on the other hand, counts each class equally and was used by Lu et al [4]. A potential problem with the unweighted average is that an RNA class with many highly similar sequences can have disproportionate influence on the overall accuracy, relative to its sequence diversity. Therefore, we introduce the Sweighted average, which takes into account the similarities between RNA sequences in each RNA class and gives a weight to each one according to its average normalized similarity, in such a way that classes with highly similar sequences have lower weight.
where s_{ i }is the mean of the normalized similarities measured between the reference structures of any two of RNA sequences in the corresponding RNA class C_{ i }. Our normalized similarities were computed using the SimTree procedure by Eden et al. [12], which takes into account secondary structure similarities in addition to sequence similarities. The average normalized similarities are reported in Table 1 and always fall between zero and one, where an average normalized similarity of one means that all sequences in the set are identical. The Sweighted average equals the weighted average when s_{ i }is zero for all RNA classes, and it equals the unweighted average when s_{ i }is one for all classes. For the remaining cases, the value of the Sweighted average would fall between that of the unweighted and weighted averages.
Bootstrap Percentile Confidence Intervals
We calculated the confidence intervals of average Fmeasure for various RNA classes, by using the bootstrap percentile method [13, 14] following the recent work by Aghaeepour and Hoos [15]. We first took 10^{4} resamples with replacement from the original Fmeasure values obtained by the prediction method under investigation on the given reference dataset, where all resamples had the same size as the original sample. We then computed the average Fmeasures of the resamples and employed these into the bootstrap distribution. Finally, we determined the lower and upper limits of the 95% bootstrap percentile confidence interval as the 2.5th and 97.5th percentile of the bootstrap distribution, respectively. These calculations were all performed using the "boot" package of the R statistics software environment [16].
We verified that the bootstrap distributions are close to Gaussian, using the AndersonDarling test, which indicates that the bootstrap percentile intervals can be expected to be reasonably accurate [14].
Permutation Test Method
To assess the statistical significance of the observed performance differences, we used permutation tests, following Aghaeepour and Hoos [15]. Since Lu et al. [4] and Hamada et al. [5] reported that the MEAbased methods outperform MFE, we used a onesided permutation test [14] to determine whether MEAbased methods have significantly better accuracy than MFE on our parameter sets.
The test that we applied proceeds as follows. First, we calculated the difference in means between sets of Fmeasure values obtained by the two given structure prediction procedures, an MEAbased method and MFE. For simplicity, we call these two sets A and B and we denote their sizes as n_{ A }and n_{ B }, respectively. Then we combined the Fmeasure values of sets A and B. Next, we calculated and recorded the difference in sample means for 10^{4} randomly chosen ways of dividing these combined values into two sets of size n_{ A }and n_{ B }. The pvalue was then calculated as the proportion of the sample means thus determined whose difference was less than or equal to that of the means of sets A and B. Then, if the pvalue is less than the 5% significance level, we reject the null hypothesis that MFE and the MEAbased method have equal accuracy and thus accept the alternative hypothesis that the MEAbased method has significantly better accuracy than MFE. Otherwise, we cannot reject the null hypothesis and therefore we cannot accept the alternative hypothesis.
Furthermore, to assess whether the difference in accuracy between the MEAbased methods and the MFE method on a given parameter set is significant, we performed a twosided permutation test. This test works exactly like the onesided permutation test, except that its pvalue is calculated as the proportion of the sampled permutations where the absolute difference was greater than or equal to that of absolute difference of the means of sets A and B. Then, if the pvalue of this test is less than the 5% significance level, we reject the null hypothesis that MFE and MEA have equal accuracy, otherwise, we cannot reject the null hypothesis.
All of these calculations were performed using the "perm" package of the R statistics software environment.
Results and Discussion
In this section, we investigate to which degree the prediction accuracy of the energybased methods studied in this paper is dependent on different datasets and different thermodynamic parameter sets used for prediction.
We start by considering how the size of a dataset will influence the accuracy achieved by RNA secondary structure prediction methods. We then study the dependency of prediction accuracy of the MFE, MEA, gCg1, and gCpMFmeas algorithms on the thermodynamic parameter sets that they use.
Dependency of the Energybased Methods on Data Characteristics
Results of the Energybased Methods on Different Data Sets
Because measures of accuracy on reference datasets are used to assess the quality of prediction achieved by various algorithms, it is important to understand how such measures vary depending on the reference datasets used. Later in this section we also consider how accuracy measures vary depending on the energy model used.
Fmeasure prediction accuracy of the MEA, MFE, gCg1, and gCpMFmeas algorithms on the MT dataset
RNA class  Class size  Mean ± std of length  Fmeas (BL*)  Fmeas (Turner99)  

ubcMEA  ubcMFE  gCg1  gCpMFmeas  rsMEA  rsMFE  
16S Ribosomal RNA  88  377.88 ± 167.18  0.649  0.621  0.640  0.659  0.574  0.539 
23S Ribosomal RNA  27  460.6 ± 151.3  0.711  0.683  0.693  0.733  0.681  0.646 
5S Ribosomal RNA  309  119.5 ± 2.69  0.739  0.743  0.725  0.746  0.625  0.642 
Group I intron  16  344.88 ± 66.42  0.705  0.650  0.674  0.708  0.627  0.599 
Group II intron  3  668.7 70.92  0.720  0.739  0.683  0.750  0.744  0.703 
Ribonuclease P RNA  6  382.5 ± 41.66  0.471  0.460  0.519  0.495  0.517  0.522 
Signal Recognition RNA  91  267.95 ± 61.72  0.641  0.621  0.633  0.637  0.518  0.557 
Transfer RNA  484  77.48 ± 4.8  0.718  0.775  0.727  0.782  0.726  0.727 
Unweighted Average  0.669  0.662  0.662  0.689  0.627  0.617  
Weighted Average  0.710  0.732  0.707  0.743  0.660  0.665  
SWeighted Average  0.670  0.652  0.660  0.684  0.612  0.598 
Fmeasure prediction accuracy of the MEA, MFE, gCg1, and gCpMFmeas algorithms on the MA dataset
RNA class  Class size  Mean ± std of length  Fmeas (BL*)  Fmeas (Turner99)  

ubcMEA  ubcMFE  gCg1  gCpMFmeas  rsMEA  rsMFE  
16S Ribosomal RNA  675  485.66 ± 113.02  0.625  0.645  0.630  0.665  0.561  0.521 
23S Ribosomal RNA  159  453.44 ± 117.85  0.645  0.643  0.626  0.664  0.588  0.562 
5S Ribosomal RNA  128  120.98 ± 3.21  0.782  0.780  0.763  0.782  0.616  0.630 
Group I intron  89  368.49 ± 103.58  0.644  0.631  0.642  0.670  0.576  0.550 
Group II intron  2  578 ± 47  0.540  0.609  0.524  0.582  0.472  0.471 
Ribonuclease P RNA  399  332.78 ± 52.34  0.643  0.603  0.656  0.678  0.615  0.575 
Signal Recognition RNA  364  227.04 ± 109.53  0.730  0.721  0.680  0.708  0.609  0.625 
Transfer RNA  489  77.19 ± 5.13  0.706  0.764  0.719  0.773  0.727  0.726 
Unweighted Average  0.668  0.676  0.655  0.690  0.596  0.583  
Weighted Average  0.672  0.682  0.669  0.708  0.618  0.600  
SWeighted Average  0.658  0.660  0.648  0.681  0.588  0.568 
Bootstrap Confidence Intervals for the Prediction Accuracy of the Energybased Methods on different RNA Classes
Following the work of Aghaeepour and Hoos [15], we use bootstrap percentile confidence intervals to assess the accuracy of the average Fmeasures obtained by a secondary structure prediction procedure on a given RNA dataset; these provide a measure of the overall average accuracy on the whole population from which the dataset is drawn (see Methods for a detailed description of bootstrap percentiled confidence intervals). We chose to calculate confidence intervals using weighted average Fmeasure. We note from the data of Table 3 that on the MA dataset, all three averages (i.e., weighted, unweighted, and Sweighted) are qualitatively similar in the sense that if the Fmeasure accuracy of one algorithm is better than another with respect to one average, then the same is true with respect to the other averages. Thus, we would expect the same qualitative conclusions if we had used a different average.
Confidence intervals obtained by the bootstrap percentile method for the MA and SFull sets and different RNA classes in MA
RNA class  Class size  Confidence Interval (CL = 0.95)  

ubcMEA  ubcMFE  rsMEA  rsMFE  gCpMFmeas  
16S Ribosomal RNA  675  (0.611, 0.639)  (0.630, 0.660)  (0.548, 0.574)  (0.507, 0.536)  (0.652, 0.679) 
23S Ribosomal RNA  159  (0.609, 0.677)  (0.607, 0.677)  (0.556, 0.618)  (0.530, 0.593)  (0.633, 0.693) 
5S Ribosomal RNA  128  (0.754, 0.807)  (0.751, 0.806)  (0.573, 0.657)  (0.586, 0.671)  (0.758, 0.804) 
Group I intron  89  (0.605, 0.680)  (0.594, 0.666)  (0.540, 0.611)  (0.513, 0.587)  (0.634, 0.704) 
Ribonuclease P RNA  399  (0.629, 0.659)  (0.588, 0.618)  (0.602, 0.628)  (0.561, 0.588)  (0.665, 0.690) 
Signal Recognition RNA  364  (0.708, 0.750)  (0.698, 0.742)  (0.583, 0.635)  (0.599, 0.651)  (0.685, 0.728) 
Transfer RNA  489  (0.683, 0.723)  (0.742, 0.785)  (0.707, 0.747)  (0.705, 0.748)  (0.754, 0.791) 
MA  2305  (0.664, 0.680)  (0.673, 0.691)  (0.610, 0.627)  (0.591, 0.609)  (0.700, 0.715) 
SFull  3246  (0.673, 0.688)  (0.678, 0.694)  (0.615, 0.631)  (0.598, 0.616)  (0.704, 0.718) 
First, the confidence intervals of all algorithms on the MA and SFull sets have a width of at most 0.018, indicating that the average accuracy measured on these sets is likely to be a good estimate  within 1%  of the accuracy of a population of RNA molecules represented by these sets. However, the interval widths on individual classes can much higher, e.g., 0.075 for ubcMEA on the Group I intron class, suggesting that average accuracy is not a reliable measure of a method's overall accuracy on such classes. We note that, without the use of rigorous statistical methods, a 0.02 difference in accuracy is considered significant in some related work [4].
Second, the confidence interval widths of RNA classes do not strictly decrease as the size of the class increases (i.e., the number of RNAs in the respective part of the reference set). For example, as shown in Table 4, for the ubcMEA (ubcMFE) algorithm the confidence interval width of the Ribonuclease P RNA class is 0.01 (0.013) less than the interval width for the Transfer RNA class, even though the Transfer RNA class contains roughly 1.2 times as many RNAs as are in the Ribonuclease P RNA class. Thus factors specific to the classes, other than class size, must account for confidence interval widths.
One possibility is that classes with greater similarity among structures would have smaller confidence intervals, since in the limit, if all sequences in a class are identical then the confidence interval has zero width. However, data from Tables 1 and 4 did not support such a correlation between average normalized similarity and confidence interval width, even for classes not too different in size. For example, the Ribonuclease P and Transfer RNA classes in the MA set have relatively similar sizes (399 and 489 sequences, respectively), with the Ribonuclease P RNA class having lower normalized similarity (0.72) than the Transfer RNA class (0.95), as one might expect; yet, for ubcMEA, the confidence interval for the Ribonuclease P RNA class is narrower (0.030) than those for the Ribonuclease P RNA class (0.040).
Understanding classspecific factors underlying differences in prediction accuracy remains a goal for further study.
The other important observation from this figure is that, as one could expect, the width of the confidence interval tends to decrease with increasing number of RNAs of a given type in the reference set, but (as previously noted) there are notable exceptions to this trend for sets of size below 500.
The final interesting observation from Figure 3 and Table 4 is that the gCpMFmeas method using the BL* parameter set outperforms all the other methods on most RNA classes and also on our large MA and SFull sets. Hamada et al. [5] also showed that on a small set of 151 RNAs, gCpMFmeas achieves better prediction accuracy than MFE and the other MEAbased methods when using the BL* parameter set. Our results confirm their finding on our large MA and SFull data sets.
Similar graphs for the rsMEA and rsMFE algorithms are provided in the additional files. Additional files 1 and 2, Figures S4 and S5 support findings similar to those reported for ubcMEA, ubcMFE and gCPMFmeas. Since the overall results on the gCg1 algorithm were roughly the same as the results of the ubcMEA method, we report its prediction accuracies only in Tables 2 and 3 and did not study it further.
Dependency of the Energybased Methods on Thermodynamic Parameters
The accuracy of algorithmic methods achieved by energybased secondary structure prediction approaches, such as gCPMFmeas, MEA and MFE, depends on the underlying thermodynamic parameter sets. Lu et al. showed that the accuracy of MEA is better than that of MFE when using the Turner99 parameter set (on the MT reference set of RNAs). But does MEA or gCPMFmeas outperform MFE on other published parameter sets? If so, is the difference in accuracy statistically significant? We address these questions in the following.
Accuracy comparison of different prediction algorithms with various parameter sets on the SFull set
Algorithm  FMeasure  

T99MRF  BL*  CG*  Turner99  
ubcMEA  0.582 (0.574,0.591)  0.680 (0.673,0.688)  0.636 (0.628,0.644)  n/a 
ubcMFE  0.601 (0.592,0.609)  0.686 (0.678,0.694)  0.671 (0.663,0.679)  n/a 
rsMEA  n/a  n/a  n/a  0.623 (0.615,0.632) 
rsMFE  n/a  n/a  n/a  0.607 (0.598,0.615) 
gCpMFmeas    0.711 (0.704, 0.718)     
Pvalue of onesided permutation test for the MEA versus MFE algorithm on different parameter sets
Parameter set  T99MRF  BL*  CG*  Turner99 

pvalue  0.999  0.848  1  0.003 
Pvalue of twosided permutation test for the MEA and MFE algorithms on different parameter sets
Parameter set  T99MRF  BL*  CG*  Turner99 

pvalue  0.003  0.299  0  0.006 
Furthermore, the results in Table 5 show that of the parameter sets we considered, BL* gives the highest prediction accuracy, regardless of whether the MEA or MFE prediction algorithm is used. This is consistent with earlier results by Andronescu et al. [9] regarding the performance of ubcMFE using the BL* parameter set.
Finally, we observe that for a given class of RNAs, depending on the energy model, MFE sometimes performs better than MEA and vice versa (see Table 4 and Figure 2). For example, when considering our set of 16S Ribosomal RNAs and using the BL* parameter set, MFE outperforms MEA by 0.02, while for the Full Turner 99 parameters, MEA outperforms MFE by 0.04 (both performance differences are statistically significant based on a permutation test with significance threshold 0.05).
Overall, we conclude that the relative performance of MEA versus MFE depends on the thermodynamic parameter set used, and that both ubcMEA and ubcMFE with the BL* parameter set have significantly better accuracy than rsMEA with the Turner99 parameter set.
Conclusions
Improvements both in algorithmic methods and in thermodynamic models lead to important advances in secondary structure prediction. In this work, we showed that gCpMFmeas with the BL* parameter set significantly outperforms the other MFE and MEAbased methods we studied. However, the relative performance of MEAbased and MFE methods vary depending on the thermodynamic parameter set used. For example, MEAbased methods significantly outperform minimum free energy (MFE) methods for the Turner99 model but the opposite is true for other models and in fact the difference in performance between MEAbased and MFE methods is negligible on our best thermodynamic model, BL*. Our findings suggest that, as thermodynamic models continue to evolve, a diverse toolbox of algorithmic methods will continue to be important.
We also showed the importance of using large datasets when assessing accuracy of specific algorithms and thermodynamic models. Specifically, we showed that bootstrap percentile confidence intervals for average prediction accuracy on our two largest datasets, MA and SFull, have narrow widths (at most 0.018), indicating that the average accuracies measured on these sets are likely to be good estimates of the accuracies of the populations of RNA molecules they represent. In contrast, interval widths for several of the specific RNA classes studied in the paper were much larger, with no clear correlation between confidence interval width and either class size or average normalized similarity. It may be the case that confidence interval widths are correlated with the distribution of evolutionary distances among the class members, rather than on the more simplistic average normalized similarity that we consider in this paper; studying this further would be an interesting direction for future research. Regardless, our analysis shows that larger datasets are needed to obtain reliable accuracy estimates on specific classes of RNAs, underscoring the importance of continued expansion of RNA secondary structure data sets.
Our work illustrates the use of statistical techniques to gauge the reliability and significance of measured accuracies of RNA secondary structure prediction algorithms. We hope that this work will provide a useful basis for ongoing assessment of the merits of RNA secondary structure prediction algorithms.
Declarations
Acknowledgements
The authors thank and acknowledge Dr. Mirela Andronescu and Dr. David H. Mathews for providing their data sets and software used in the presented work. They also thank the anonymous reviewers for their very helpful comments on an earlier draft of the paper. This research was funded by a grant from the Natural Sciences and Engineering Research Council of Canada (NSERC).
Authors’ Affiliations
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