Visualizing RNA base-pairing probabilities with RNAbow diagrams

The authors introduce a visualization tool for RNA base pair probabilities yielding intuitive diagrams. The visualization aids in disentangling incompatible structure folds contained in the partition function.

Abstract

There are many effective ways to represent a minimum free energy RNA secondary structure that make it easy to locate its helices and loops. It is a greater challenge to visualize the thermal average probabilities of all folds in a partition function sum; dot plot representations are often puzzling. Therefore, we introduce the RNAbows visualization tool for RNA base pair probabilities. RNAbows represent base pair probabilities with line thickness and shading, yielding intuitive diagrams. RNAbows aid in disentangling incompatible structures, allow comparisons between clusters of folds, highlight differences between wild-type and mutant folds, and are also rather beautiful.

INTRODUCTION

Graphical representations can profoundly influence our conception of physical reality or interpretation of data. For example, in conventional representations of RNA secondary structure the stems (regions of stacked base pairs) and loops (gaps between) are easily identified; however, showing only one set of base pairs makes invisible the prevalence of thermal fluctuations. In fact, the likelihood of being in even the most probable structure is exceedingly small, and thermal fluctuations allow the molecule to explore many states. To characterize RNA structures in thermal equilibrium, better visualization methods are needed.

Much work has gone into developing computational methods to predict the secondary structure from the sequence, including minimizing free energy (Zuker 1989; Mathews et al. 2004; Markham and Zuker 2008), computing the partition function (McCaskill 1990; Hofacker et al. 1994; Hofacker 2003; Mathews et al. 2004; Markham and Zuker 2008), stochastically sampling the partition function (Ding and Lawrence 2003), enumerating states (Wuchty et al. 1999), kinetic approaches (Isambert and Siggia 2000; Hofacker et al. 2010), maximum-expected accuracy approaches (Do et al. 2006; Lu et al. 2009), comparative analysis (Cannone et al. 2002; Wiebe and Meyer 2010), and statistical methods (Andronescu et al. 2010; Gardner et al. 2011; Rivas et al. 2012). The accuracy of predictions has received scrutiny (Doshi et al. 2004; Dowell and Eddy 2004; Layton and Bundschuh 2005; Hajiaghayi et al. 2012). Our particular interest is to visualize ensembles of structural states in thermal equilibrium as predicted by partition-function-based methods.

A number of tools have also been developed to visualize RNA secondary structures. The minimum free energy (MFE) or other secondary structures can be depicted in two-dimensional “airport terminal” diagrams, in which the backbone defines the perimeter and lines or dots between bases denote the pairs, as in Figure 1A. A classic “rainbow” diagram, see Figure 1B, encodes the same information, but instead of the backbone sequence forming the perimeter, it is stretched horizontally with the base pairs making long arcs. In circle diagrams (Nussinov et al. 1978) the backbone is arranged in a circle with arcs again marking the pairs. Most compact is bracket notation (Hofacker et al. 1994), see Figure 1C, in which unpaired bases are periods and matching parentheses indicate paired bases. To represent non-nested pseudoknot structures, bracket notation requires additional delimiters, like [ ] or { }.

FIGURE 1.

Depictions of secondary structures for the L. collosoma Spliced Leader sequence: (A) two-dimensional “airport terminal” diagram of the Minimum Free Energy (MFE) state; (B) classic “rainbow” diagram (MFE); (C) bracket notation with periods representing unpaired bases and parentheses indicating paired bases (MFE). (D) A dot plot with partition function probabilities P_ij with base i vertical and base j horizontal. Color is assigned on the basis of the logs of probabilities. (Graphics adapted from RNAstructure.) (E) ViennaRNA’s prediction (using slightly different free energy rules) with bases color-coded according to their partition function probabilities. (Graphics adapted from ViennaRNA.) (F) A Dot Plot available from ViennaRNA uses box-size proportional to probability (top triangle), but the grid obscures low-probability pairs. (G) An AllPairs RNAbow diagram with the line width and darkness proportional to the probability of the base pairs. (H) A Clusters RNAbow diagram after resolving into the two dominant clusters, with probability 0.57 (red) and 0.43 (blue); note that the MFE state (B) belongs to the less-probable blue cluster.

Partition function-based computational methods predict the thermal average probabilities P_ij of RNA base pairs rather than one single structure. The P_ij information is often represented in dot plots—a grid is made and the size or color of the dot at (i, j) indicates the probability of pairing base i with base j, as in Figure 1D,F. Dots along diagonals indicate stems.

Because the eye naturally groups similar objects together (Metzger 2006), the dot plot representation in Figure 1D subliminally suggests that each color represents a unique structure. But closer examination reveals, for example, that base 41 along the horizontal axis forms red pairs with bases 4, 9, 13, and 35 along the vertical axis. So, if there is not a single red structure, can one figure out which dots are consistent?

The Figure 1E hybrid approach adds to the MFE structure a color-coding of the bases according to their probability of pairing (De Rijk and De Wachter 1997). This approach may leave the impression of a single static structure in which the predictions vary in certainty, rather than of a fluctuating molecule exploring many states and many local minima.

RESULTS

We introduce RNAbow diagrams as a more intuitive way to visualize RNA structures in thermal equilibrium. RNAbows are the partition function analog of rainbow diagrams. In RNAbow diagrams, we use the line thickness and shade of the arcs to represent the probability of a base pair. The single AllPairs RNAbow displays the entire partition function. In Figure 1G it is simple to see the two local minima structures because the eye naturally groups parallel lines. With RNAbows, our perceptual inclinations help us rather than hinder us.

To facilitate comparisons at a glance we introduce the difference RNAbow diagram, such as Figures 1H, 2, and 3. Two folds, top and bottom, are juxtaposed. Color highlights the differences between folds. When P_ij^top > P_ij^bot, the top arc’s color is set proportional to the relative probability excess X_ij^top = (P_ij^top − P_ij^bot), otherwise X_ij^top = 0. We then either use the (hue, saturation, value) or RGB color models, with

for pair (i, j) on the top. Formulas for bottom arcs are analogous. Pairs with similar weight are colored black, extra weight drives top pairs toward red and bottom pairs toward blue.

FIGURE 2.

The partition functions of wild-type (red) and the U22G mutant (blue) 5′ UTR of ferritin light-chain mRNA are depicted with Difference RNAbows. The colors are set proportional to the difference between the clusters such that common elements are black, while the distinct elements are either red or blue. The dramatic effect of this single nucleotide polymorphism on the secondary structure is evident. Other base changes within loop regions have less influence.

FIGURE 3.

Chemical mapping experiments indicate which bases are unpaired, and this information can also be visualized with Difference RNAbows. Here, for the 5′-UTR region of HIV-1, we compare the constrained (Schroeder et al. 2011) partition function where lowercase bases are forced to be unpaired (top, red) with the unconstrained partition function (bottom, blue).

In Figure 1G we see two dominant structural classes in the total partition function. To visualize each local minima we first have to partition the partition function; we use our PF method, which is described fully in the Supplemental Information. The idea is to identify the base pair (i, j) that is most incompatible with other base pairs. We then split the partition function into two, one with the (i, j) pair Prohibited and one with the (i, j) pair Forced to exist. The resulting P and F clusters describe two local free-energy minima, including fluctuations. These are visualized with a Clusters RNAbow in Figure 1H.

In the more probable red cluster of Figure 1H, one can see the thermal equilibrium between states in which G31 pairs to either U45 or U47. And, one can also see a possible UAAA/UUUG hairpin duplex early in the sequence that has no topological barrier with the later strong hairpin; it is formed only about one-quarter of the time in this cluster.

In the blue cluster of Figure 1H, one sees gradations in the stability of the hairpin’s stem that are not seen in the MFE structure (Fig. 1B) because the MFE bonds either exist or not. Notice also that the MFE structure is one of the states in the least probable of the clusters.

If desired, the PF procedure could be repeated again on each cluster to further disentangle structures. Notice also in Figure 1H that the maximum probabilities P_ij within each daughter cluster approach 1, while the most probable pair in the parent cluster was 0.57, roughly the weight p_P. It is easy to imagine applications to visualizing riboswitches that exhibit a conformational change between two folds.

In Figure 2, we present a Difference RNAbow comparison of the 5′ UTR of Ferritin Light Chain wild-type to the U22G mutant (Halvorsen et al. 2010) associated with Hyperferritinemia cataract syndrome. This single nucleotide polymorphism dramatically changes the folding pattern. In particular, the loss of the Iron Response Element, the brightest red hairpin in Figure 2, disrupts binding by an iron-response protein.

In Figure 3 we present a Difference RNAbow that shows how information about which bases are unpaired obtained from chemical mapping experiments can be incorporated in partition function calculations.

ACCESS

From http://rna.williams.edu/ users can create their own RNAbows with a choice of ViennaRNA, RNAstructure, or UNAFold (Hofacker 2003; Mathews et al. 2004; Markham and Zuker 2008) to compute the partition functions. Three RNAbows tools are available:

• AllPairs to visualize the entire partition function (Fig. 1G) with base pairs denoted by arcs whose width and shading is proportional to the probability of the pair,

• Clusters to split that partition function into two clusters (Fig. 1H) using the PF method described in the Supplemental Information, and

• Difference RNAbows to highlight the differences between the partition functions of two sequences (Figs. 1H, 2, 3).

The RNAbow graphics are rendered in EPS, PDF, or SVG formats for easy import into other applications. All graphics are vector based, so there is no image degradation at any scale.

Advanced users can also import P_ij data they have precomputed using other algorithms. Source code for RNAbows is also available by request for incorporation into other applications.

CONCLUSION

Visualization tools can offer insights into problems beyond mere representation of data. With RNAbows, a partition function’s base pair probability information becomes easier to use and more powerful. Our instinctive pattern-matching ability allows us to quickly compare clusters of structures. Incompatible clusters can be disentangled by eye or by the PF procedure described.

The basic RNAbows tools are extensible to represent any data set with varying coupling strengths and then to highlight differences between conditions. The difference RNAbow juxtaposes arcs and highlights differences with color. The recent R-chie package (Lai et al. 2012) also uses double arcs to visualize helices predicted from multiple alignments with the Transat program (Wiebe and Meyer 2010). Making side-by-side comparisons of multiple folds is less straightforward with dot plots.

Furthermore, the mental misconceptions that come from looking at a static MFE, PDB, or consensus structure—forgetting that thermal fluctuations open and close pairs, or forgetting that not all pairs are equally stable—are challenged subliminally by the gradations of the shadings in RNAbow pairs.

SUPPLEMENTAL MATERIAL

Supplemental material is available for this article.