Analytical description of finite size effects for RNA secondary structures

The ensemble of RNA secondary structures of uniform sequences is studied analytically. We calculate the partition function for very long sequences and discuss how the crossover length, beyond which asymptotic scaling laws apply, depends on thermodynamic parameters. For realistic choices of parameters this length can be much longer than natural RNA molecules. This has to be taken into account when applying asymptotic theory to interpret experiments or numerical results.

Figure 1

An RNA secondary structure. The thick line stands for the backbone of the molecule and thin lines stand for base pairings. The solid dots represent monomers. $5^{\prime }$ and $3^{\prime }$ show the head and tail of this RNA of length 58. Many different loops formed when RNA folds are also defined in the figure.

Figure 2

Pseudoknots in RNA structures: The base pairings indicated by the arrow in (a) create a pseudoknot. (b) The short pseudoknots (called “kissing hairpins”). (c) The long pseudoknots in three dimensions.

Figure 3

Recursive relation for a simple model of an RNA molecule. The wavy lines stand for undetermined structure and the corresponding partition function. The arch represents a binding pair between bases $k$ and $N$. The assumption of excluding pseudoknots separates the last term into two independent parts since two pairs cannot go across each other.

Figure 4

Separation of stems from the bubble structure.

Figure 5

Recursion relation for stem structure. The dashed lines stand for the undetermined structures. Thick lines represent the backbone and thin lines stand for pair bindings.

Figure 6

Average number of unbound bases per interior loop vs $\tilde{i}$ for different stacking energies. Following the measured free energies [12], we chose the typical values $b=0.01$, $\tilde{b}=0.85$, and $i=0.05$ for the other Boltzmann factors.

Figure 7

Recursion relation for bubble structures. Dashed lines stand for undetermined structures. Thick lines and thin lines stand for the backbone and pair binding, respectively. In (a), the left-hand term represents an undetermined bubble structure with the two end bases paired. The double-dashed line stands for an undetermined multiloop structure and it can be decomposed into the components in (b), as explained in the main text.

Figure 8

Replacing the position of $t$ by stem structures. The left-hand term represents the closed structure with both stem and bubble structures in it.

Figure 9

Ratio between the large-$s$ crossover length and the numerical value of the homogeneous one for many combinations of the parameters: $b,i=\{0,1\}$ and $h,m=\{0.1,0.01,0.001\}$. These choices cover the region of realistic values.

Figure 10

Equivalence between the ${\left(GCA\right)}_n$ sequence and the molten phase: Three consequent bases $GCA$ are mapped to one single base so the $GC∕CG$ stack is equivalent to a binding pair in the molten phase. Then, the smallest interior loop on the left is viewed as a stacking loop and the following interior loop becomes a bulge in the molten phase. The hairpin loop on the right with two units of $GCA$ is considered to have two unbound bases in the molten phase.

Figure 11

Average size of the secondary structure of a ${\left(GCA\right)}_n$ sequence at $37°\mathrm{C}$. The data are fitted by the expected law (20).

Figure 12

Numerical values of homogeneous crossover length for the ${\left(AU\right)}_n$ sequence with respect to temperatures. This quantity is obtained by numerically calculating the branch point and then expanding $\widehat{Q}\left(z\right)$ around this point (see Appendix ).

Figure 13

A ${\left(GCC\right)}_n$ stem in the ground state. The dashed lines stand for the base pairing of ground states. The dotted lines show the possibilities for base $G$ to a pair with neighboring $C$.