Free energy estimation of short DNA duplex hybridizations
 Dan Tulpan^{1}Email author,
 Mirela Andronescu^{2} and
 Serge Leger^{1}
https://doi.org/10.1186/1471210511105
© Tulpan et al; licensee BioMed Central Ltd. 2010
Received: 6 October 2009
Accepted: 24 February 2010
Published: 24 February 2010
Abstract
Background
Estimation of DNA duplex hybridization free energy is widely used for predicting crosshybridizations in DNA computing and microarray experiments. A number of software programs based on different methods and parametrizations are available for the theoretical estimation of duplex free energies. However, significant differences in free energy values are sometimes observed among estimations obtained with various methods, thus being difficult to decide what value is the accurate one.
Results
We present in this study a quantitative comparison of the similarities and differences among four published DNA/DNA duplex free energy calculation methods and an extended NearestNeighbour Model for perfect matches based on triplet interactions. The comparison was performed on a benchmark data set with 695 pairs of short oligos that we collected and manually curated from 29 publications. Sequence lengths range from 4 to 30 nucleotides and span a large GCcontent percentage range. For perfect matches, we propose an extension of the NearestNeighbour Model that matches or exceeds the performance of the existing ones, both in terms of correlations and root mean squared errors. The proposed model was trained on experimental data with temperature, sodium and sequence concentration characteristics that span a wide range of values, thus conferring the model a higher power of generalization when used for free energy estimations of DNA duplexes under nonstandard experimental conditions.
Conclusions
Based on our preliminary results, we conclude that no statistically significant differences exist among free energy approximations obtained with 4 publicly available and widely used programs, when benchmarked against a collection of 695 pairs of short oligos collected and curated by the authors of this work based on 29 publications. The extended NearestNeighbour Model based on triplet interactions presented in this work is capable of performing accurate estimations of free energies for perfect match duplexes under both standard and nonstandard experimental conditions and may serve as a baseline for further developments in this area of research.
Keywords
Background
Predicting the stability of a DNA duplex from base sequences is a well studied problem nowadays. Nevertheless, the accuracy of DNA duplex stability predictions largely varies with sequence length, base composition and experimental conditions. The Thermodynamic NearestNeighbour (TNN) Model [1] is a stateoftheart approach that is used to estimate the stability of a single or a pair of DNA (or RNA) molecules based on pairwise base interactions and structural conformations. A large collection of thermodynamic nearestneighbour parameters were acquired by interpolation of results obtained from various experimental processes like NMR [2] and optical melting studies [1, 3]. The accuracy of computing free energies for DNA duplexes is an important aspect for all prediction methods, considering their direct application for selecting, for example, microarray probes that perfectly hybridize with their complements within a prespecified hybridization interval, while avoiding selfhybridization for each probe [4]. Here we select four widely used, publicly available computer programs that implement the TNN Model using large numbers of experimentally derived thermodynamic parameters, namely: the MultiRNAFold v2.0 package [5, 6] with two sets of thermodynamic parameters, the Vienna Package v1.8.1 [7] and the UNAFold v3.5 package [8].
The MultiRNAFold package (including the PairFold program for duplexes) predicts the minimum free energy, suboptimal secondary structures and free energy changes of one, two, or several interacting nucleic acid sequences. The thermodynamic model for the thermodynamic stability of a joint secondary structure for two DNA or RNA molecules at a given temperature is performed similarly to that of a single molecule [9], except that an intermolecular initiation penalty is added. The PairFold algorithm uses dynamic programming to calculate minimum free energy secondary structures and runs in time cubic in the lengths of the input sequences (Θ(n^{3})). PairFold uses RNA thermodynamic parameters from the Turner Laboratory [10] and DNA thermodynamic parameters from the Mathews and SantaLucia laboratories [11, 12].
The Vienna Package consists of a suite of computer programs and libraries for prediction of RNA and DNA secondary structures. Nucleic acid secondary structure prediction is done via free energy minimization using three dynamic programming algorithms for structure prediction: the minimum free energy algorithm of [13], which produces a single optimal structure, the partition function algorithm of [14], which calculates base pair probabilities in a thermodynamic ensemble, and the suboptimal folding algorithm of [15], which generates all suboptimal structures within a given energy range of the optimal energy.
UNAFold, the acronym for "Unified Nucleic Acid Folding", is a software package for RNA and DNA folding and hybridization prediction. UNAFold folds singlestranded RNA or DNA, or two single DNA or RNA strands, by computing partition functions for various states of hybridization. The partition functions will then help to derive base pair probabilities and stochastic samples of foldings or hybridizations. The package provides various energy minimization methods, which compute minimum free energy hybridizations and suboptimal foldings.
All three packages use similar dynamic programming algorithms for prediction of minimum free energy (MFE) and suboptimal structures and for partition function calculations. For the purposes of our work (i.e., DNA duplex MFE secondary structure prediction and free energy of hybridization), the main differences lie in the thermodynamic parameters used (SantaLucia or Mathews), and in the features considered (for example, the Vienna Package does not consider special types of polyC hairpin loops in their model, whereas the other two packages do). Thus our first goal is to quantify the impact of these differences on the accuracy of DNA duplex free energy approximations. Throughout the paper, we use a set of measures that reflect the degree of similarity of calculated and experimental secondary structures and free energies. Based on these measures we quantify the accuracy of the predictions of the aforementioned programs using a collection of 695 experimental DNA duplex data that we collected from 29 publications.
We also introduce in this work an extended NearestNeighbour Model for perfect matches based on triplet interactions, that can approximate free energies for DNA duplexes under a wide range of temperatures, sodium and sequence concentrations. The model is similar to the one introduced in 1999 by Owczarzy et al. [16], the main difference residing in the inclusion of only triplet interactions for our model, rather than a mixture of singlets, doublets and triplets for the other. Thus, our second goal is to show that such a model outperforms simpler models based on doublet interactions and produces more accurate free energy approximations for DNA duplex hybridizations occurring in nonstandard experimental environments (for example for different sodium concentrations or at different temperatures).
Results and Discussion
In this work, we compare similarities and correlations of free energy values calculated using three publicly available packages, namely MultiRNAFold, UNAFold and Vienna Package and a NearestNeighbour (NN) Model for perfect matches based on triplet interactions. For this purpose, we collected and used a data set with 695 pairs of short DNA sequences and we investigated what method produces the closest value to the experimental free energy and under what circumstances. We acknowledge the fact that not all sequence lengths are equally represented in the benchmark data set simply due to their availability and thus our analysis may apply better to shorter sequences. The majority (91.37%) of experimental free energy calculations were obtained for perfect (0 mismatches) and almost perfect matches (1 mismatch), thus the current DNA parameter sets tend to have higher accuracy for closeto perfect match DNA duplexes. Another bias in the analysis may come from the fact that some authors have already tried to reconcile the existing differences in free energy model parameters [17, 18] by optimizing sets of DNA parameters using the same sequences already present in the benchmark data set.
Comparison of absolute differences between experimental and estimated free energies (MFE_AD)
Summary of features for the data sets used in this study
Set  Num. duplexes  Seq. len.  T[C]  [Na]^{+} [M]  Seq. conc. [M] 

Aboulela et al. [32]  34  16  25, 50  1  [11e6,440e6] 
Allawi et al.1 [37]  24  9  12  37  1  1e4 
Allawi et al.2 [20]  28  9  14  37  1  1e4 
Allawi et al.3 [21]  22  9  14  37  1  1e4 
Bommarito et al. [43]  37  8  9  37  1  n.r. 
Breslauer et al. [26]  12  6  16  25  1  n.r. 
Clark et al. [44]  1  24  37  0.15  2.5e6 
Doktycz et al. [19]  140  8  25  1  2e6 
Gelfand et al. [45]  4  13  25  1  5e5 
LeBlanc et al. [46]  7  10  11  25  1  5e5 
Leonard et al. [22]  5  12  25  1  4e4 
Lesnik et al. [39]  14  8  21  37  0.1  4e6 
Li et al. [23]  12  8  10  25  1  6.1e6 
Nakano et al. [40]  21  6  14  37  0.1  8e6 
Petruska et al.1 [47]  4  9  37  n.r.  n.r. 
Petruska et al.2 [36]  2  30  37  0.17  1e4 
Peyret et al. [48]  52  9  12  37  1  1e4 
Pirrung et al. [49]  2  25  25  0.1  1e6 
Plum et al. [50]  2  13  25  1  6e6 
Ratmeyer et al. [51]  2  12  37  1  6e6 
SantaLucia et al.1 [29]  23  4  16  37  1  4e4 
SantaLucia et al.2 [29]  10  12  24.85  1  5e6 
Sugimoto et al.1 [30]  50  5  14  37  1  5e6 
Sugimoto et al.2 [38]  1  8  37  n.r.  1e4 
Sugimoto et al.3 [52]  8  6  8  37  1  n.r. 
Tanaka et al. [34]  126  12  25  37  1  5e5 
Tibanyenda et al. [33]  3  16  24.85  1  17.5e6 
Wilson et al. [35]  3  11  25  0.4  n.r. 
Wu et al. [53]  48  5  11  25, 37  1  1e4 
TOTAL:  695 
Summary of results for free energy measurements obtained with EVALSS methods.
Method  Stats  MFE_AD [kcal/mol]  Pearson coeff. (r)  SSSI  Sens.  PPV  Fmeasure  

MultiRNAFold  min  0.000  0.7565  4.35  40.00  0.1667  1  0.2857 
(Mathews)  q1  0.340  100.00  1.0000  1  1.0000  
median  0.860  100.00  1.0000  1  1.0000  
mean  2.681  95.83  0.9547  1  0.9711  
q3  3.590  100.00  1.0000  1  1.0000  
max  18.400  100.00  1.0000  1  1.0000  
stddev  3.429  10.56  0.1224  0  0.09236  
MultiRNAFold  min  0.000  0.7663  4.131  40.00  0.1667  1  0.2857 
(SantaLucia)  q1  0.330  100.00  1.0000  1  1.0000  
median  0.720  100.00  1.0000  1  1.0000  
mean  2.528  96.44  0.9608  1  0.9747  
q3  3.510  100.00  1.0000  1  1.0000  
max  17.200  100.00  1.0000  1  1.0000  
stddev  3.269  10.23  0.1189  0  0.08966  
min  0.000  0.7660  3.992  40.00  0.1667  1  0.2857  
q1  0.256  100.00  1.0000  1  1.0000  
median  0.630  100.00  1.0000  1  1.0000  
UNAFold  mean  2.374  96.08  0.9571  1  0.9724  
q3  3.016  100.00  1.0000  1  1.0000  
max  11.880  100.00  1.0000  1  1.0000  
stddev  3.212  10.66  0.1231  0  0.09234  
Vienna  min  0.010  0.7630  3.667  5.882  0.0000  0.0000  0.0000 
Package  q1  1.700  100.000  1.0000  1.0000  1.0000  
median  2.330  100.000  1.0000  1.0000  1.0000  
mean  3.025  95.210  0.9467  0.9856  0.9616  
q3  3.935  100.000  1.0000  1.0000  1.0000  
max  15.400  100.000  1.0000  1.0000  1.0000  
stddev  2.075  13.74  0.1581  0.1192  0.1387 
Comparison of root mean squared errors (RMSE)
Summary of results for free energy measurements obtained with EVALFE methods.
Method  Statistics  MFE_AD [kcal/mol]  Pearson coeff. (r)  RMSE 

MultiRNAFold  min  0.0000  0.7352  4.418 
(Mathews)  q1  0.300  
median  0.800  
mean  2.672  
q3  3.395  
max  18.400  
stddev  3.521  
MultiRNAFold  min  0.0000  0.7456  4.223 
(SantaLucia)  q1  0.330  
median  0.680  
mean  2.553  
q3  3.390  
max  17.200  
stddev  3.367  
min  0.0000  0.7434  4.101  
q1  0.2528  
median  0.6128  
UNAFold  mean  2.4110  
q3  2.9970  
max  13.0000  
stddev  3.319  
Vienna  min  0.0000  0.7413  3.876 
Package  q1  1.820  
median  2.440  
mean  3.167  
q3  3.965  
max  15.400  
stddev  2.236 
Estimated free energy parameters
ID  Doublet  Counts  ID  Doublet  Counts  

1.  AA/TT  0.838948  84  6.  CC/GG  1.698997  74 
2.  AC/TG  1.394988  102  7.  CG/GC  0.967002  106 
3.  AG/TC  1.323547  102  8.  GA/CT  0.938327  101 
4.  AT/TA  0.375235  130  9.  GC/CG  0.711466  126 
5.  CA/GT  1.406794  95  10.  TA/AT  0.144092  136 
ID  Triplet  Counts  ID  Triplet  Counts  
1.  AAA/TTT  0.844597  10  17.  CAG/GTC  1.625284  23 
2.  AAC/TTG  1.841904  19  18.  CCA/GGT  1.568813  18 
3.  AAG/TTC  1.201194  17  19.  CCC/GGG  2.396507  17 
4.  AAT/TTA  0.991596  19  20.  CCG/GGC  1.888906  22 
5.  ACA/TGT  1.121939  20  21.  CGA/GCT  1.668273  19 
6.  ACC/TGG  1.793995  23  22.  CGC/GCG  2.195726  23 
7.  ACG/TGC  1.615048  30  23.  CTA/GAT  0.871636  40 
8.  ACT/TGA  0.781693  23  24.  CTC/GAG  1.198450  16 
9.  AGA/TCT  1.103536  15  25.  GAA/CTT  1.317278  18 
10.  AGC/TCG  1.528461  36  26.  GAC/CTG  1.498999  29 
11.  AGG/TCC  1.323278  18  27.  GCA/CGT  1.454430  21 
12.  ATA/TAT  0.562379  46  28.  GCC/CGG  1.973081  24 
13.  ATC/TAG  1.157521  29  29.  GGA/CCT  1.696158  20 
14.  ATG/TAC  1.263601  26  30.  GTA/CAT  1.158422  32 
15.  CAA/GTT  0.988509  16  31.  TAA/ATT  0.519499  27 
16.  CAC/GTG  2.088824  17  32.  TCA/AGT  1.042342  19 
Comparison of Pearson correlation coefficients (r)
Model training
Require: A thermodynamic model T, an input set S with perfect match DNA duplexes. 

Ensure: An optimal set of thermodynamic DNA parameters X for the input model 
1: Initialize counts matrix F with zeros for all unique doublets/triplets 
2: Initialize results matrix R with experimentally approximated free energies for each duplex 
3: for i = 0 to S do 
4: Count unique doublets/triplets in duplex S[i] and update F 
5: end for 
6: Solve the equation X = arg min_{ X }(F × X  R)^{2} 
7: return X 
Model evaluation
Comparison of secondary structure similarity indexes of experimental and predicted secondary structures (SSSI)
The accuracy of secondary structure prediction for various methods can be evaluated by using the newly introduced measure described in equation 5. The SSSI measure simply calculates the percentage of correctly predicted secondary structure bonds corresponding to the positions in each secondary structure (corresponding to each sequence in the duplex) that match the position in the experimental secondary structure, normalized by the sum of sequence lengths. Comparable mean SSSI values were produced by all methods with a maximal value of 96.44% attained by PairFoldSantaLucia. The lowest value (95.21%) was obtained with Vienna Package (see column 6 in Table 2). All methods have large standard deviation for SSSI values, thus suggesting a wide sample distribution.
Comparison of SENS, PPV and F for predicted secondary structures
Comparison of free energy parameters for DNA doublets measured at 37°C and 1 M sodium concentration
Conclusions
In this work we showed that no major differences exist among free energy estimations of short DNA duplex hybridization when comparing four publicly available programs that employ various sets of thermodynamic parameters.
Here we introduce a simplified TNN Model based on triplets interactions for perfect match hybridizations of DNA duplexes. The model is able to approximate free energies for DNA duplexes under various experimental conditions with higher accuracy and lower RMSEs compared to the four publicly available programs considered in this work. The improvement is more noticeable for DNA duplexes at nonstandard experimental temperature conditions (for example at 25°C). This improvement obtained with the TNN Model based on triplets could be explained by the presence of a larger set of parameters consisting of 32 unique triplets (compared to only 10 unique doublets in the classical TNN Model) that better capture the impact of sequence components on the overall free energy of a DNA duplex. An alternative and potential complementary explanation of these improvements is the use of a wider variety of experimental data points in the thermodynamic parameter extrapolation process (the model training stage) compared to the smaller and less diverse data sets used in the other four programs. Nevertheless, we notice that additional experimental data employing longer and more diverse sequences is required in order to obtain a better approximation of free energies for DNA duplexes at other nonstandard experimental conditions.
Three extensions of the TNNTripletsPM Model might improve its performance, given that additional experimental data that covers a higher percentage of the parameters and experimental condition combinations is obtained experimentally: (i) the model can incorporate weighted additive terms that account for hybridization initialization, temperature, pH, sodium concentration or sequence concentrations; (ii) the model can incorporate symmetrical and asymmetrical internal loops, multibranch loops, dangling ends and hairpin rules similar to those already existent in the classical TNN Model; (iii) the model can also incorporate positional dependencies of triplets with respect to the 5' and 3' ends of the sequences.
Methods
The present study is divided into two major sections:

Evaluation of free energy estimates (EVALFE): a comparative assessment of free energies calculated for DNA duplexes using different methods when both the duplex sequence and the duplex experimental secondary structure are given.

Evaluation of secondary structure predictions (EVALSS): an accuracy assessment of secondary structure predictions when only the duplex sequence is given and the secondary structure is predicted.
Data
Sequence concentrations range from 17.5 × 10^{6} M in [33] to 10^{4} M in [20, 21, 31, 37, 38]. The sodium concentration varies from 0.1 M in [39] and [40] to 1 M in 20 out of 29 sources. The reported free energies were measured at reaction temperatures ranging between 24.85°C [33, 41] and 50°C [32].
Free energy calculations
In this study, three publicly available packages were used to calculate and compare the free energies for pairs of short DNA sequences: MultiRNAFold (with Mathews and SantaLucia parameters), UNAFold and the Vienna Package. All packages implement the TNN Model based on base doublet parameters.
where G° is the free energy measured, H° is the enthalpy, T is the absolute temperature measured in degrees Kelvin and S° is the entropy.
where R is the gas constant (1.98717 cal/(mol K)), T is the absolute temperature, and k is the equilibrium constant.
Computational procedures
The two main sections of this study, namely the evaluation of free energy estimates and the evaluation of secondary structure predictions, employ computational procedures made available in the corresponding software packages. The evaluation of free energy estimates (EVALFE) includes the following procedures:

The function free energy pairfold (sequence1, sequence2, known structure) is provided by the MultiRNAFold package to compute the free energy for two sequences when the known secondary structure is given. The pairfold wrapper has been slightly modified to accept as parameters: two sequences, the temperature, the set of parameters (Mathews or SantaLucia), the nucleic acid (DNA or RNA) and the type of hybridization (with or without intramolecular interactions between nucleotides).

The function RNAeval is provided by the Vienna Package to compute the free energy for two sequences when the known secondary structure is provided. We wrote a Python wrapper that calls this function with the following parameters: T temperature, P dna.par. The wrapper also preprocesses the sequence and structure input so to satisfy the interactivity requirements of the RNAeval function.

The function ctEnergy is provided by the UNAFold Package to compute the free energy for two sequences when the known secondary structure is given. We wrote a Python wrapper that preprocesses the sequences and structures into a CTformatted input file and calls the function with the following parameters: n DNA, t temperature, N sodium concentration.
The evaluation of secondary structure predictions (EVALSS) includes the following procedures:

The function pairfold mfe (sequence1, sequence2, output structure) is provided by the MultiRNAFold package to compute the minimum free energy secondary structure for two DNA sequences that fold into 'output structure'. The pairfold wrapper has been slightly modified as described above.

The function RNAcofold is provided by the Vienna Package to predict the free energy secondary structure for two sequences. A wrapper has been created for this function to accommodate the input and the parameters for the interactive interface as described above.

The script UNAFold.pl is provided by the UNAFold Package to predict the free energy secondary structure for two sequences. We wrote a Python wrapper that preprocesses the sequences and structures into a CTformatted input file and calls the function with the same parameters as for the ctEnergy function.
The TNNTripletsPM Model
These values were obtained by using an over determined system of N equations (3) and solving equation 4 with a leastmean squared optimization function (implemented in the backslash operator for matrices) available in Matlab 7.7. Here N takes the value 228 (67% of 340 perfect match free energies measured at 25°C and 37°C), 132 (67% from 197 perfect match free energies measured at 25°C), or 96 (67% of 145 perfect match free energies measured at 37°C). The system with N equations has been extrapolated by selecting from the initial data set only the free energy measurements for perfect match DNA duplexes and counting the frequency of triplets in each duplex. Thus, for each duplex, the sum of parametric values for each triplet multiplied with its counts equals the experimental free energy. While our model is very simple and currently does not take into consideration mismatches, internal loops, and dangling ends, its strength is given by its ability to estimate perfect match DNA duplex free energies for a wide range of sodium, sequence and target concentrations and temperatures. This strength is given by the presence of a large and mixed training data set that was used to extrapolate the nearestneighbour (NN) parameters for both the doublet and the tripletbased models.
Model training and testing
The training process for the TNNTripletsPM Model is summarized in Table 5. We first process the input set, which contains perfect match DNA sequences and their corresponding experimentally derived free energies. The processing consists of scanning each perfect match sequence from left to right by moving a window of size 3 nucleotides (or 2 for the doublets) and counting the frequency of each of the 32 unique triplets. We record each frequency at corresponding positions (i, j) in matrix F and each experimentally derived free energy is recorded at position i in matrix R. Here i represents the number of the sequence in the set and j represents the number of the triplet (from 1 to 32), whose frequency is recorded. After matrices F and R have been populated, a solution for equation 4 is computed and the value of vector X containing free energy parameters for all the unique triplets is reported.
Comparative measures
We use a large number of measures of similarity between experimental and computed free energies. Some of these measures were previously used by [42] to compare melting temperatures obtained with different methods and by [6] to estimate model parameters for RNA secondary structure prediction. If not stated otherwise, all comparisons in this paper were done on a data set comprising 695 pairs of DNA sequences collected from 29 publications. The measures used in this study are grouped in two categories, namely:
Measures that evaluate accuracy of free energy estimations
The following measures are used for free energy estimations of the known structures, as well as free energy estimations of predicted structures.

the observed absolute difference between experimental and estimated free energies (MFE_AD),

the Pearson correlation coefficient (r),

the root mean squared error (RMSE),
Measures that evaluate accuracy of secondary structure predictions

the secondary structure similarity index of experimental and predicted secondary structures (SSSI)

the prediction sensitivity for secondary structures (SENS)

the positive predictive value for secondary structures (PPV)

the Fmeasure for predicted secondary structures (F)
For MFE_AD, SSSI, SENS, PPV and F we report the minimum, the first quartile, the median, the mean, the third quartile, the maximum and the standard deviation.
where s 1_{ exp }, s 2_{ exp }are two equally long structures obtained experimentally, s 1_{ calc }, s 2_{ calc }are two equally long calculated structures, and SS(a, b) is the total number of identical characters at corresponding positions in both structures. SSSI represents the percentage of positions in which two structures agree.
Unlike similar measures that assign a +1 score for two identical base pairs in two duplex structures, SSSI assigns a +1 score for two base pairs that have either the start or the end positions identical. This mechanism allows the differentiation between duplex secondary structures that have either one (score +1) or both (score +2) bases in a base pair correctly predicted.
Computational infrastructure
The entire analysis of this study was done with R version 2.5.1, Perl 5.8.8 and Python 2.5. All computations were carried out on a Open SuSe 10.2 Linux (kernel version 2.6.18.2) machine equipped with a Pentium 4, 2.8 GHz processor with 1 GB of RAM.
Declarations
Acknowledgements
Funding for this work was provided to DT and SL by the National Research Council of Canada. We gratefully acknowledge the helpful comments and suggestions provided by Dr. Anne Condon and Dr. Miroslava CuperlovicCulf. We thank Dr. Fazel Famili, Georges Corriveau and Natalie Hartford for proofreading our article and the anonymous reviewers of this manuscript for their valuable feedback.
Authors’ Affiliations
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