Freezing of Random RNA

We study secondary structures of random RNA molecules by means of a renormalized field theory based on an expansion in the sequence disorder. We show that there is a continuous phase transition from a molten phase at higher temperatures to a low-temperature glass phase. The primary freezing occurs above the critical temperature, with local islands of stable folds forming within the molten phase. The size of these islands defines the correlation length of the transition. Our results include critical exponents at the transition and in the glass phase.

Figure 1

Secondary structures of a random RNA molecule at distant times. Base pairings can be nested, such as ($s,t$) and ($s^{\prime },t^{\prime }$), or independent, such as ($s,t$) and ($s^{\prime \prime },t^{\prime \prime }$). The pairing overlap is defined by the common base pairings between the left and right configuration (the corresponding bases are shown in black). (a) Above $T_c$, the molecule contains conserved subfolds on scales up to the correlation length $\xi$ (marked by shading) and is molten on larger scales. (b) Below $T_c$, the molecule is locked into its minimum-energy structure on all scales, up to rare fluctuations (unshaded).

Figure 2

Overlap correlations in Eq. (6). (a) Two-replica one-point function $⟨{\Psi }_{\alpha \beta }(s_1,t_1){⟩}_0$. (b) Two-replica two-point function $⟨{\Psi }_{\alpha \beta }(s_1,t_1){\Psi }_{\alpha \beta }(s_2,t_2){⟩}_0$. (c) Three-replica two-point function $⟨{\Psi }_{\alpha \beta }(s_1,t_1){\Psi }_{\alpha \gamma }(s_2,t_2){⟩}_0$.

Figure 3

The critical exponents ${\rho }^{*}$ and ${\theta }^{*}$ as a function of $\epsilon$ for (a) $p=2$, (b) $p=1$, and (c) $p=0$. Exact results (thick solid), renormalization-group results valid to all orders (thick dashed) or to first order (thin dashed), presumably exact values (thin solid) (see text), reference line ${\rho }_0(\epsilon )$ (dotted).