Melting of Branched RNA Molecules

In this Letter we show that the melting thermodynamics of RNA molecules is very sensitive to the branching geometry. We find that, when pairing interactions are described by a Gō model, unbranched RNA molecules with a linear geometry melt via a conventional continuous phase transition with classical exponents while RNA molecules with the branching geometry of a Cayley tree, with coordination number three, have a free energy that shows no thermodynamic singularity within numerical precision. Nevertheless, we provide an analytical proof that the free energy does have a mathematical singularity at the stability limit of the ordered structure. The correlation length appears to diverge but only on the high-temperature side of this singularity.

Figure 1

Single-stranded RNA molecule having a branched secondary structure that follows the outline of a Cayley tree with coordination number three. Nucleotides are schematically indicated by circles, bonds between nucleotides and complementary pairing by solid lines around the perimeter or across the structure, respectively. (a) Ground state structure with pairing restricted to a complementary “native” pair for each branch of the Cayley tree. (b) In a molten-globule bubble (hatched) all possible pairing interactions are permitted. (c) For a fixed set $S^{\prime }$ of specific base pairs (solid lines) the set $C(S^{\prime })$ is defined as all possible sets of base pairs compatible with the base pairs in set $S^{\prime }$ (dashed lines are one example of such a set). Note that the additional base pairs are constrained to the loops of the structure $S^{\prime }$ so the sum over all possible base-pairing configurations factorizes into a factor from each such loop.

Figure 2

Second derivative of the free energy with respect to the Boltzmann weight $\tilde{q}$ of specifically paired bases plotted as a function of $\tilde{q}$ for different values of the level $k$ of the Cayley tree ground state. The free energy was computed numerically from the recursion relations Eqs. (6, 7) and expressed in units of $N{k}_{B}T$ with $N$ the sequence length of the RNA strand. The arrow denotes the location of the mathematical singularity associated with melting of the root of the Cayley tree. Left-hand inset: Same except that the ground state is a linear hairpin. A thermodynamic singularity develops near ${\tilde{q}}_c=18.4$ with mean-field critical exponents. Right-hand inset: Numerically computed “pinching” free energy $\Delta F(k)$ [see Eq. (8)] versus the level $k$ of the Cayley tree. For $\tilde{q}$ larger then 90, $\Delta F(k)/k_{B}T$ is independent of $k$, consistent with the ordered ground state. For $\tilde{q}$ less than 20, $\Delta F(k)/k_{B}T$ can be fitted by the scaling relation $\Delta F(k)/k_{B}T\simeq \frac{3}{2}(k+2)\ln 2-\ln a^{+}$ associated with the molten-globule state. The crossover point between these two regimes for intermediate values of $\tilde{q}$ marks the size of the ordered, correlated regions in the molten-globule state. For $\tilde{q}$ above 80, the size of these correlated regions exceeds the system size.