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The ends of a large RNA molecule are necessarily close

We show on general theoretical grounds that the two ends of single-stranded (ss) RNA molecules (consisting of roughly equal proportions of A, C, G and U) are necessarily close together, largely independent of their length and sequence. This is demonstrated to be a direct consequence of two generic properties of the equilibrium secondary structures, namely that the average proportion of bases in pairs is ∼60% and that the average duplex length is ∼4. Based on mfold and Vienna computations on large numbers of ssRNAs of various lengths (1000–10 000 nt) and sequences (both random and biological), we find that the 5′–3′ distance—defined as the sum of H-bond and covalent (ss) links separating the ends of the RNA chain—is small, averaging 15–20 for each set of viral sequences tested. For random sequences this distance is ∼12, consistent with the theory. We discuss the relevance of these results to evolved sequence complementarity and specific protein binding effects that are known to be important for keeping the two ends of viral and messenger RNAs in close proximity. Finally we speculate on how our conclusions imply indistinguishability in size and shape of equilibrated forms of linear and covalently circularized ssRNA molecules.


Supplementary Data General Comparison of Linear and Circular ssRNAs
We have analyzed 10,000 randomly-permuted single-stranded (ss) RNA sequences of length 400 nucleotides (nt), 5,000 of length 1,000 nt, 2,000 of length 4,000 nt, and 1,000 of length 8,000 nt. All sequences were of uniform nucleotide composition. These differ from the randomly permuted sequences depicted in Fig. 3 only in that the latter are of "viral-like" (rather than uniform) composition. As explained in the main text, the external loop size, , is insensitive to moderate changes in composition. Each sequence was folded with RNAsubopt, a program in the vienna RNA folding package, version 1.7 (1,2), as both a linear ssRNA and a circular ssRNA. Thus, the sets of sequences used in determining the statistics for the circular ssRNAs were the same as those used for the linear ssRNAs. Table S1 provides a comparison of the predicted secondary structures of the linear and circular variants, for several characteristics. These include: • Minimum free energy (MFE).
• Ensemble free energy (EFE). The EFE is given by where the sum is over all structures in the ensemble; note that EFE < MFE.
• Frequency of occurrence of the MFE structure in the ensemble. This is given , where the sum is over all structures in the ensemble. • Ensemble diversity, which is a measure of structural diversity within the ensemble (3). The ensemble diversity is the Boltzmann-weighted mean pairwise "structural distance" between all structures in the ensemble. The "structural distance" between two structures is the number of base pairs present in the first structure but not the second, plus the number present in the second but not the first. • Maximum and average ladder distance (MLD and LD), which are measures of the "extendedness" of a secondary structure (4,5). • Fraction of nucleotides paired ( ).

Ladder Distance Distribution
The effective circularization of a linear ssRNA resulting from secondary structure formation is not confined to the 5´ and 3´ ends, but rather extends to the molecule globally. Here we analyzed 2,000 randomly-permuted sequences of length 4,000, all of uniform composition. Each sequence was folded with RNAsubopt as both a linear and circular ssRNA. Fig. S1 compares the secondary structures of the linear and circular variants by plotting the distribution of ladder distances associated with nucleotides whose positions on the primary sequence differ by . is the minimum number of base pairs ("rungs" on a "ladder") that must be crossed to traverse the secondary structure from base to base . It serves as a measure of the effective distance separating bases within an RNA secondary structure (4,5).
The configurational statistics of a denatured linear ssRNA (one without secondary structure) are the same as those for any linear, flexible, homopolymer. In particular, the average (root-mean-square) distance between bases and increases monotonically with , e.g., as for ideal chains. By contrast, in a denatured circular ssRNA, the average distance initially increases with , reaches a maximum, and then decreases-as expected, since bases 1 and are as close as bases 1 and 2. When the circular ssRNA is allowed to form a secondary structure, the relationship between and the distance separating bases and remains qualitatively the same. However, when the linear ssRNAs are allowed to form secondary structures, their distance relationships become, on average, almost identical to those for the circular ssRNAs. Fig. S1 shows as a function of for both linear and circular ssRNAs; the curves are essentially identical. By construction, for circular ssRNAs, ; but, in addition, for the linear ssRNAs, .