Impact of Loop Statistics on the Thermodynamics of RNA Folding

Loops are abundant in native RNA structures and proliferate close to the unfolding transition. By including a statistical weight $\sim {\ell }^{-c}$ for loops of length $\ell$ in the recursion relation for the partition function, we show that the heat capacity depends sensitively on the presence and value of the exponent $c$, even for a short explicit tRNA sequence. For long homo-RNA, we analytically calculate the critical temperature and critical exponents which exhibit a nonuniversal dependence on $c$.

Figure 1

Schematic representation of a secondary RNA structure. Solid lines denote the RNA backbone, broken lines base pairs, and gray lines non-nested backbone bonds that are counted by the variable $M$; here $M=11$.

Figure 2

Experimental heat capacity of the tRNA-phe of yeast for NaCl concentrations $20\mathrm{m}M$ (triangles) and $150\mathrm{m}M$ (squares) [14]. Solid lines show results using Eq. (1) with loop exponents $c=3.0$, 2.16, 1.76, 0 (from left to right), compared with the results from the program RNAheat in the Vienna package [20] which uses a linearized multiloop entropy for large loops (dashed curve). The dotted curve is obtained with $c=3$ and the same energy parameter set as for the solid curves, except for the loop initiation penalty which was omitted by setting $g^i=g^t$. The inset sketches the low-temperature secondary RNA structure obtained from Eq. (1), which perfectly matches experimental crystal structures.

Figure 3

(a) Phase diagram for three different values of the loop exponent $c$ as a function of the base-pairing weight $w$ and force fugacity $s$ featuring an unfolded phase (bottom) and a folded compact phase (top), following Eq. (8). The phase boundaries diverge at the vertical dotted lines. For $c=c^{*}\simeq 2.479$, the phase boundary approaches $s=1$ and therefore only the unfolded phase exists. For $c\le 2$, there is only the folded phase. The dots denote the unfolding transition in the absence of external force, i.e., $s=1$, which is considered in (b) and (c): Temperature dependence of the (b) specific heat $C$ and (c) fraction of bound bases $\theta$ for $c=2.3$. The insets show the third derivatives $C^{\prime \prime \prime }={\mathrm{d}}^{3}C/\mathrm{d}{T}^3$ and ${\theta }^{\prime \prime \prime }={\mathrm{d}}^3\theta /\mathrm{d}{T}^3$ which clearly exhibit singular behavior. Squares denote numerical solutions of Eqs. (5, 6) ($T<T_{\mathrm{cr}}$) or (7) ($T>T_{\mathrm{cr}}$). The solid lines show the leading order expansion around $T_{\mathrm{cr}}$ (denoted by vertical dotted lines), according to which $C^{\prime \prime \prime }$ diverges with the exponent $\chi =2/3$ for $c=2.3$, see Eq. (10), and ${\theta }^{\prime \prime \prime }$ is characterized by the exponent $\lambda =1/3$, see Eq. (13). In the inset of (c), the analytical result including the next-leading order is also shown (broken line). The horizontal broken line in (c) denotes the residual fraction of bound pairs at infinite temperature.