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Nucleic Acids Res. 2005; 33(2): 519–524.
Published online Jan 26, 2005. doi:  10.1093/nar/gkh983
PMCID: PMC548333

A statistical analysis of RNA folding algorithms through thermodynamic parameter perturbation

Abstract

Computational RNA secondary structure prediction is rather well established. However, such prediction algorithms always depend on a large number of experimentally measured parameters. Here, we study how sensitive structure prediction algorithms are to changes in these parameters. We found already that for changes corresponding to the actual experimental error to which these parameters have been determined, 30% of the structure are falsely predicted whereas the ground state structure is preserved under parameter perturbation in only 5% of all the cases. We establish that base-pairing probabilities calculated in a thermal ensemble are viable although not a perfect measure for the reliability of the prediction of individual structure elements. Here, a new measure of stability using parameter perturbation is proposed, and its limitations are discussed.

INTRODUCTION

In an endeavor to understand the functions of an organism, one cannot ignore the importance of RNA (1). RNA molecules transmit genetic information through the cell. They are also intimately involved in many important biological processes, such as translation, regulation and splicing. In addition to its importance for organisms of the present time, RNA is also an interesting molecule to study owing to its probable role as a major player during the origin of life (2).

The function of a given RNA is determined by its physical structure. This structure is encoded in the sequence of four nucleotides (or bases), A, U, G and C, from which each RNA molecule is composed. Determining the structure of RNA in the laboratory is a laborious, and often unsuccessful, task. Thus, it has become an interdisciplinary task to determine these structures from the sequences alone.

The encoding of a structure in the sequence is realized by specific interactions between the bases. To date, most important of these interactions is the formation of A–U and G–C base pairs, also known as Watson–Crick pairs. With the formation of each base pair, the Gibbs free energy of the structure is lowered, and thus, the stability of the structure is increased. Since the sequence of bases that defines the RNA is finite, the number of possible structures into which a given RNA can fold is also finite. The most thermodynamically probable structure to be formed is the structure with the lowest free energy known as the minimum-free-energy (mfe) structure.

Although the number of possible structures for a given sequence is enormous, computer algorithms, such as the Vienna Package (3) or MFOLD (4), can find the mfe structure or the full partition function of the ensemble of all structures given a sequence in a time that is proportional to the third power of the sequence length due to a recursive relationship (57). The problem with these algorithms lies in the calculation of the free energies of the structures. The contribution to the free energy attributed to a base pairing or the formation of various substructures, such as various kinds of loops, is measured experimentally and used as parameters in the RNA folding algorithm. For various reasons, these parameters contain errors. Several effects, such as steric interactions between different regions of the structure, pseudo-knots, base triplets or even interactions with proteins or other RNA molecules, are not reflected at all in the underlying free-energy model. All these effects result in systematic errors in the free-energy parameters. On the other hand, there are ordinary, non-systematic errors of measurement associated with these parameters as well. Thus, while the algorithm guarantees to find the minimum energy structure within the energy model provided by the experimentally determined parameters, this structure need not be the true (and thus biologically realized) mfe structure.

The goal of this paper is not to discuss the causes of these errors, but to investigate the consequences these errors have regarding structure prediction. Our approach is to randomly modify the measured free-energy parameters within a range comparable with the experimental errors and to record how much the predicted mfe structures change. We found that ~30% of the structures are changed when the free-energy parameters are varied within the experimental error. Although this is a rather sobering result, we at least were able to found that base-pairing probabilities evaluated in a thermal ensemble are very good priors to estimate which parts of the structure prediction are reliable and which are not.

MATERIALS AND METHODS

In this section, we will discuss how RNA structure prediction algorithms work and how we model experimental error in the free-energy parameters. This will provide the necessary background for our study.

RNA secondary structure

The strongest interaction between the bases of an RNA molecule is the formation of Watson–Crick base pairs. Therefore, one distinguishes two levels of structure, namely secondary and tertiary structure (with the primary structure just being another name for the sequence of the molecule). A secondary structure is defined as the collection of all base pairs that have been formed without regard to any spatial organization of the backbone. Subsequently, the tertiary structure includes the actual spatial organization and elements formed by less stable interactions than base pairing, such as base triplets and backbone contacts mediated by divalent ions. Since base pairing is energetically more important than the other interactions, it is meaningful to talk about the secondary structure of an RNA molecule without considering the tertiary structure (8). Here, this point of view of the algorithms has been studied; therefore, we will only discuss the secondary structure in the remainder of this paper.

In order to make secondary structure prediction computationally feasible, it is necessary to exclude the so-called pseudoknots from the allowed secondary structures. Such a pseudoknot exists if bases i and j form a base pair and bases k and l form a base pair and these two base pairs are nested as i < k < j < l or k < i < l < j. Although these pseudoknots do appear in the biological structures, they are found to be short due to kinetic constraints. Thus, they can be omitted in the secondary structure prediction (8) and be considered as a part of the tertiary structure of a molecule.

Energy model

A secondary structure as defined above can be drawn as shown in Figure 1. It can be decomposed into a large number of loops, such as stacking loops, bulges, interior loops, hairpins and multiloops, as shown in the figure. The main assumption of the generally accepted free-energy model is that the total free energy of a secondary structure is the sum of independent contributions from all of its loops. These loop contributions depend on the identity of the bases in a loop and on the length of the loop. For example, the free energy of a stacking loop depends on the identity of all four bases that form the 2 bp. A stacking loop is formed by leading to (after taking into account symmetry) 21 different free-energy parameters for stacking loops (if in addition to G–C, and A–U also the wobble base pair G–U is allowed). For short bulges and interior loops, the number of parameters increases by a factor of four for every unpaired base in the loop. Longer loops are typically only characterized by their length and by the identity of the unpaired bases immediately next to the base pairs defining the loop in order to avoid an explosion of parameters. Nevertheless, a complete free-energy model is described in the order of thousand parameters that are determined experimentally (9). Since all these parameters are true free energies, i.e. differences of energetic contributions, such as chemical binding energy and bending energy, and entropic contributions from the integrated out spatial degrees of freedom of the backbone and the surrounding water, where each parameter depends on the temperature. In our study, we keep the temperature constant at the physiological 37°C.

Figure 1
Schematic representation of an RNA secondary structure. The solid line represents the backbone of the molecule while the dashed lines symbolize base pairs. Each such structure can be decomposed into stacking loops (s), bulges (b), interior loops (i), ...

Perturbations of the energy model

In order to study the sensitivity of structure prediction to thermodynamic parameters, one must perturb the parameters. For simplicity, we assume that the error in the parameters is roughly Gaussian distributed. We model these errors by the addition of a Gaussian random variable to every single free-energy parameter with mean zero and standard deviation, ε, i.e. with a probability density function

ρ(x)=12πεex2/2ε2.1

In doing so, we take great care to preserve the inherent symmetry in the parameters (e.g. the energy of a stacking loop obtained from a GC-pair and an AU-pair (–GA– paired with –UC–) being equal to the energy of a stacking loop obtained from an UA-pair and a CG-pair, i.e. –UC– paired with –GA–). The parameter, ε, serves as a measure of the magnitude of the experimental error inherent in the parameters.

We will explore a whole range of different values for the parameter, ε, to understand the sensitivity of predicted structures to perturbations of the free-energy parameters. To get an idea what the experimental errors on the free-energy parameters are in reality, it is most illustrative to look at the stacking energies as stacking energies have been measured in many different laboratories. In the case of most-studied DNA stacking energies, seven independent measurements have been systematically compared (10). In addition, this study reports the free-energy parameters, which details the stacking energies averaged over the different types of stacking for each of the seven independent experiments. If we consider the average of these averages, we find it to be −1.4 ± 0.3 kcal/mol. Since the experimental procedure for the determination of RNA free-energy parameters is the same as for DNA free-energy parameters, we conclude that a good estimate for the experimental error is 0.3 kcal/mol. Another indication that this is the order of magnitude for the experimental error of the stacking free energies is that the additive free-energy model itself is experimentally known to break down at this level of precision (11). This implies that these uncertainties are not due to the lack of experimental techniques of higher precision (which could in principle be overcome by new experimental developments), but these uncertainties are unavoidable on principle grounds. As we do not possess very good estimates of the experimental error of the other free-energy parameters, we uniformly apply the same error estimate to all free-energy parameters. Since we expect the experimental error to be larger for the other free-energy parameters, our results are thus a conservative estimate of the error in structure prediction resulting from the uncertainty in the free-energy parameters.

RESULTS

If the predicted mfe structure, also referred to as the ground state, has a far lower free energy than any alternative structure, the ground state is said to be thermodynamically stable. For our study, we are interested in another type of stability. We will call a structure unstable with respect to parameter perturbation if the predicted mfe structure requires a strict adherence to one or more thermodynamic parameters in order to remain as the predicted mfe structure. We will quantify this stability in two different ways.

Distance of structures

In the first way, we study this instability by looking at what fraction of a structure is still predicted correctly once the parameters are perturbed. To this end, we need some quantitative method of comparison for the structures. In this study, we will use the normalized tree distance (3) to quantify the amount by which mfe structures at different free-energy parameter choices differ. We convinced ourselves that other solely structure-based measures, such as the string distance (3), led to results similar to the ones presented here. The tree distance is based on a metric that views a secondary structure as being defined by a tree diagram where the leaves of the tree are the bases and the topology of the tree represents the structure in an intuitive way. The tree distance is then defined as the number of elementary operations on the tree such as cutting a branch and attaching it at a different place in the tree [for a more detailed description of this difference measure see (12)]. Since the tree distance is a number between zero (for identical structures) and the length of the sequence, we rescale the tree distance by the length of the sequence. This scaling allows us to compare the stability of sequences of different lengths and permits a more intuitive interpretation of the data. For example, a scaled distance of 0.2 stands for a 20% difference in the structure.

For our study, we chose a series of natural sequences (namely group I introns) with length varying between 227 and 685, that is, RNA sequences that have been observed in biological systems. For each of these sequences, the mfe structure is predicted by the computer algorithm from the Vienna package (3) using the accepted experimentally measured parameters (9). In addition, we determine the mfe structures of the same sequences for 100 sets of randomly perturbed energy parameters for each uncertainty ε. We calculate the scaled tree distance between each mfe structure calculated with perturbed parameters and the corresponding mfe structure obtained with unperturbed parameters (i.e. the experimentally measured values without alteration) and average these distances over 100 realizations for each sequence and each ε.

Figure 2 shows these averaged distances as a function of the perturbation parameter, ε. One should note the overall instability of the structure prediction of these natural sequences. As shown in Figure 2, there is already a significant deviation of ~30% from the ground state structure at ε ≈ 0.3 kcal/mol that roughly corresponds to the actual experimental error of the parameters (10). For ε ≈ 1 kcal/mol, already half of the structure can no longer be predicted.

Figure 2
The average scaled tree distances of various natural sequences for different perturbations of the parameter ε. The error bars denote the statistical error after averaging over 100 different realizations of perturbed free-energy parameter sets. ...

Ground state probability

An alternative, and as it turns out even more sensitive, measure of stability is the probability that the ground state structure will be the predicted structure at a given perturbation ε. To estimate this quantity, we determine the mfe structure of our sequences for 1000 sets of randomly perturbed parameters for each perturbation strength, ε, and classify the parameter sets according to the mfe structures. Since we catalog the parameter sets that produce each structure, the frequency at which the ‘correct’ (i.e. calculated with the accepted free-energy parameters) structure occurs can be determined, as well as the number of alternative structures possible at a given ε.

As can be seen in Table 1, the ground state structure is most probably not the true structure for one sequence of ε as small as 0.02, and for all sequences of ε = 0.22. At the experimental error rate, i.e. ε ≈ 0.32, only 5% of the parameter perturbations reproduce the same ground state. For this part of the study, we also include the data on randomly generated sequences of varying length in addition to the data on the group I introns used. Table 1 shows that the ground state stability for these sequences is even worse than the group I introns. From these data, one can see that the current error in the thermodynamic parameters casts serious doubt upon the structural predictions made by folding algorithms. Since this error is at least to a good part owing to fundamental limitations in the energy model and thus cannot be significantly reduced just by better measurements, predictions from folding algorithms should never be taken at face value but always be subjected to critical crosschecking.

Table 1
Frequency at which the ground state is predicted (right) and the number of alternate structures predicted (left) as a function of the parameter perturbation ε

Reliability estimation

Given that we obviously have to accept the fact that 30% of a secondary structure will be incorrectly predicted just because of the uncertainties in the free-energy parameters, the question comes up whether it is at least possible to find out which parts of the structure are the reliable ones and which are the unreliable ones. If the RNA secondary structure prediction algorithm is used to calculate the full partition function for a given sequence instead of the mfe structure only, it can assign to every pair (i,j) of bases a probability pi,j that these two bases are paired within a thermal ensemble (7). Since these probabilities can be calculated in the same time as the mfe structure, these probabilities are a convenient measure for the reliability of the prediction of an individual base pair. However, it is not a priori clear if a high probability in the thermal ensemble corresponds to stability with respect to uncertainties in the free-energy parameters.

To study if the thermal probabilities have any meaning for the stability of a base pair with respect to parameter changes, we compare the thermal ensemble directly to an ensemble of mfe structures calculated with the perturbed parameters. To this end, we calculate the mfe structure for a given sequence for 1000 different sets of perturbed free-energy parameters. We catalog the resulting base pairs, and determine the frequency with which they occur. Then, we compare this frequency to the thermal ensemble probability pi,j calculated for each individual base pair at the accepted parameters.

The comparison of base pair frequencies versus the thermal ensemble probabilities is shown in Figure 3 for a representative sequence (we convinced ourselves that the results are qualitatively similar for all sequences). We observe that at small ε, since few or no alternative structures are predicted, the plot appears to be very much a step function; base pairs which the thermal ensemble predicts to have a significant probability (≈40% or more) occur while less likely base pairs do not. As ε is increased, more alternative structures begin to appear, and one can see the edges of the step begin to smooth. When ε = 0.32 kcal/mol, a strong correlation is apparent even though there is a clear spread. If ε increased beyond 0.32 kcal/mol, the correlation between the base pair frequency and the thermal ensemble probability differs in no significant qualitative way. From these correlations, we deduce that as the level of parameter perturbations reaches the value of ε = 0.32 kcal/mol, the pool of alternative secondary structures minimizing perturbed energy parameters and the pool of suboptimal structures probed in the thermal ensemble become similar. This is in a way surprising since the thermal energy itself is ~0.6 kcal/mol, and thus much smaller than the perturbation of the total-free energy of a structure obtained by perturbing every free-energy parameter by 0.32 kcal/mol. It might imply that the base pair probability of an individual base pair is only sensitive to perturbations of a few key free-energy parameters that delineate different low-free-energy structures from each other. Whatever the reason for the observed correlation, we can conclude that the easily calculable thermal probabilities are a good estimate for the sensitivity of a given base pair to parameter perturbations.

Figure 3
Correlation between the probability to find a base pair in an mfe structure with perturbed parameters and the probability of the base pair in the thermal ensemble for the sequence with accession number ...

DISCUSSION

We studied the sensitivity of RNA secondary structure prediction to perturbations of the free-energy parameters. The main result is that if the free-energy parameters are perturbed within a range that is supposed to be the experimental uncertainty with which these parameters have been determined, ~30% of the structure turns out to be unreliable and the chance of predicting the same ground state as with the unperturbed parameters is only 5% even for moderately sized sequences with lengths up to 426. Given this imprecision, we found that at least base-pairing probabilities calculated in a thermal ensemble are reasonably well correlated with the probabilities that a base pair will be unaffected by uncertainties in the free-energy parameters. These results support the commonly employed method of using thermal ensemble probabilities to sort out which parts of the structure can be trusted and which cannot. However, although calculation of the thermal ensemble (13,14) is expedient, it offers only knowledge of how base pairings will behave on a individual basis, and not how they will behave in concert. The ground state probability method not only gives a probabilistic measure of the accuracy of the prediction, but also provide all the probable alternative structures and some gauge of their likelihood of being the true structure. If the user has computer then he/she should always check the ground state and individual base pair probabilities using the method outlined in this paper. Doing so is imperative if the thermal ensemble suggests a dubious structure prediction. The amount of time sacrificed for the additional information is dependent upon how accurate one wishes the probabilities to be. With advances in computer chip technology, the extra factor of 100–1000 in computation time involved in using the ground state probability as opposed to using the thermal ensemble may soon become a more practical investment in cases where it is of big importance to know in addition to the predicted structure which parts of the structure are likely to be correctly predicted and which should be discarded as simple artifacts of the imprecisions of the free-energy model.

Acknowledgments

This research was supported by the Research Experience for Undergraduates programs of the National Science Foundation through grant number PHY-0242665.

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