Freezing transition of the random bond RNA model: Statistical properties of the pairing weights

To characterize the pairing specificity of RNA secondary structures as a function of temperature, we analyze the statistics of the pairing weights as follows: for each base $\left(i\right)$ of the sequence of length $N$, we consider the $(N-1)$ pairing weights $w_i\left(j\right)$ with the other bases $(j\ne i)$ of the sequence. We numerically compute the probability distributions $P_1\left(w\right)$ of the maximal weight $w_i^{\max }={\max }_j\left[w_i\left(j\right)\right]$, the probability distribution $\Pi \left(Y_2\right)$ of the parameter $Y_2\left(i\right)={\sum }_j{w}_i^2\left(j\right)$, as well as the average values of the moments $Y_k\left(i\right)={\sum }_j{w}_i^k\left(j\right)$. We find that there are two important temperatures $T_c<T_{\text{gap}}$. For $T>T_{\text{gap}}$, the distribution $P_1\left(w\right)$ vanishes at some value $w_0\left(T\right)<1$, and accordingly the moments $\overline{Y_k\left(i\right)}$ decay exponentially as ${\left[w_0\left(T\right)\right]}^k$ in $k$. For $T<T_{\text{gap}}$, the distributions $P_1\left(w\right)$ and $\Pi \left(Y_2\right)$ present the characteristic Derrida-Flyvbjerg singularities at $w=1∕n$ and $Y_2=1∕n$ for $n=1,2,\dots$. In particular, there exists a temperature-dependent exponent $\mu \left(T\right)$ that governs the singularities $P_1\left(w\right)\sim {(1-w)}^{\mu \left(T\right)-1}$ and $\Pi \left(Y_2\right)\sim {(1-Y_2)}^{\mu \left(T\right)-1}$ as well as the power-law decay of the moments $\overline{Y_k\left(i\right)}\sim 1∕k^{\mu \left(T\right)}$. The exponent $\mu \left(T\right)$ grows from the value $\mu (T=0)=0$ up to $\mu \left(T_{\text{gap}}\right)\sim 2$. The study of spatial properties indicates that the critical temperature $T_c$ where the large-scale roughness exponent changes from the low temperature value $\zeta \sim 0.67$ to the high temperature value $\zeta \sim 0.5$ corresponds to the exponent $\mu \left(T_c\right)=1$. For $T<T_c$, there exists frozen pairs of all sizes, whereas for $T_c<T<T_{\text{gap}}$, there exists frozen pairs, but only up to some characteristic length diverging as $\xi \left(T\right)\sim 1∕{(T_c-T)}^{\nu }$ with $\nu \simeq 2$. The similarities and differences with the weight statistics in Lévy sums and in Derrida’s random energy model are discussed.

Figure 1

Pairing weight landscape $u\left(\frac{N}{2}\right)=[-\ln w_{N∕2}\left(j\right)]$ seen by the middle monomer [see Eq. (21)], for $N=200$ at low and high temperatures (a) at low temperature $(T=0.02)$, only a few weights dominate [note the scale of $\ln w_i\left(j\right)$], at some random positions (b) at high temperature $(T=0.5)$, the disorder is a small perturbation with respect to the entropy of the pure case that favors the neighbors.

Figure 2

(Color online) Probability distribution $P_1\left(w\right)$ of the largest weight seen by a given monomer [see Eq. (24)] (a) at $T=0.05$ (low-temperature phase) for $N=50,100,200,400$: the characteristic Derrida-Flyvbjerg singularities at $w=1$ and $w=1∕2$ are clearly visible. (b) at $T=0.4$ (high-temperature phase) for $N=50,100,200$: the distribution $P_1\left(w\right)$ does not reach $w=1$ anymore, but vanishes at some maximal value $w_0\left(T\right)<1$.

Figure 3

(Color online) Probability distribution $P_1\left(w\right)$ of the largest weight seen by a given monomer [see Eq. (24)] (a) at $T_1\sim 0.095$ where $\mu \left(T_1\right)=1$ for $N=50,100,200,400$: $P_1\left(w\right)$ does not diverge anymore as $w\to 1$ but remains finite. (b) At $T_2=0.15$, where $\mu \left(T_2\right)\sim 2$ for $N=50,100,200,400$: the distribution $P_1\left(w\right)$ vanishes linearly as $w\to 1$.

Figure 4

(Color online) Probability distributions $P_1\left(w\right)$ and $P_2\left(w\right)$ of the two largest weights seen by a given monomer [see Eq. (24)] (a) for $T=0.02$ (here $N=200$) there exists an infinite slope for $P_2\left(w\right)$ and $P_1\left(w\right)$ as $w\to {(1∕2)}^{-}$ (b) for $T_1\sim 0.095$ (here $N=400$), there exists a cusp for $P_2\left(w\right)$ and $P_1\left(w\right)$ as $w\to {(1∕2)}^{-}$.

Figure 5

(Color online) Probability distribution $\Pi \left(Y_2\right)$ of the parameter $Y_2$ [see Eq. (26)] (a) at $T=0.05$ (low-temperature phase) for $N=50,100,200,400$: the characteristic Derrida-Flyvbjerg singularities at $Y_2=1$, $Y_2=1∕2$, and at $Y_2=1∕3$ are clearly visible. (b) At $T=0.4$ (high-temperature phase) for $N=50,100,200$: the distribution $\Pi \left(Y_2\right)$ does not reach $Y_2=1$ anymore, but presents a gap.

Figure 6

(Color online) Probability distribution $\Pi \left(Y_2\right)$ of the parameter $Y_2$ [see Eq. (26)] (a) at $T_1\sim 0.095$ where $\mu \left(T_1\right)=1$ for $N=50,100,200,400$: $\Pi \left(Y_2\right)$ does not diverge anymore as $Y_2\to 1$ but remains finite and (b) at $T_2=0.15$, where $\mu \left(T_2\right)\sim 2$ for $N=50,100,200,400$: the distribution $\Pi \left(Y_2\right)$ vanishes linearly as $Y_2\to 1$.

Figure 7

(Color online) Density $f\left(w\right)$ of weights seen by a given monomer [see Eq. (27)] (a) at $T=0.05$ (low-temperature phase) for $N=50,100,200,400$. Near $w\to 1$, $f\left(w\right)$ presents the same singularity as $P_1\left(w\right)$. (b) At $T=0.4$ (high-temperature phase) for $N=50,100,200$: $f\left(w\right)$ vanishes at some value $w_0\left(T\right)<1$.

Figure 8

(Color online) Density $f\left(w\right)$ of weights seen by a given monomer [see Eq. (27)] as compared to $f_{\text{Levy}}\left(w\right)$ [Eq. (A7)] for the corresponding value of $\mu$ (thick curve) (a) at $T=0.02$, where $\mu \left(0.02\right)\sim 0.13$ (for $N=100,200,400$): $f\left(w\right)$ is rather close to the density $f_{\text{Levy}}\left(w\right)$. (b) At $T=0.08$ where $\mu \left(0.08\right)\sim 0.77$ (for $N=50,100,200$): there is now a big difference between the density $f\left(w\right)$ measured for RNA and the density $f_{\text{Levy}}\left(w\right)$.

Figure 9

(Color online) (a) Decay of the moments $\overline{Y_k\left(i\right)}$ of Eq. (36) as a function of $k⩽100$ for $N=800$ and $T=0.01,0.02,0.05,0.095$. (b) Exponent $\mu \left(T\right)$ as measured from the slope of the log-log decay in the asymptotic region.

Figure 10

(Color online) Density $f_l\left(w\right)$ of weights for pair lengths $l=3,7,15,31,63$ for size $N=200$ (a) at $T=0.05$, all curves diverge as $w\to 1$. (b) at $T=0.4$, all curves display an $l$-dependent gap.

Figure 11

(Color online) Densities $f_l\left(w\right)$ of weights for pair lengths $l=3,7,15,31,63$ for size $N=200$ (a) at $T_1\sim 0.095$, all densities $f_l\left(w\right)$ still reach the value $w=1$ (b) at $T_{\text{gap}}=0.15$, the densities $f_l\left(w\right)$ of small sizes $l$ still reach the value $w=1$, whereas the densities $f_l\left(w\right)$ of large sizes $l$ do not.

Figure 12

(Color online) Ratio $B_N\left(T\right)$ defined in Eq. (54): (a) as a function of temperature $T$ for sizes $N=100$ (엯), $N=200$ (◻), $N=400$ (◇), $N=600$ (▵), $N=800$ (▷). (b) Finite size scaling of the same data in terms of the variable $x=(T-T_c)N^{1∕\nu }$ [see Eq. (56)] with $T_c=0.095$ and $\nu =2$.

Figure 13

(Color online) Scaling form of the probability distribution $P_N^{\mathrm{pref}}\left(l\right)$: log-log plot of $N^{\eta }{P}_N^{\mathrm{pref}}\left(l\right)$ in terms of $x=l∕N$ (a) for $T=0.05$ (low-$T$ phase) the rescaling is done with $\eta =1.33$ [see Eq. (59)] and (b) for $T_c\sim 0.095$ the rescaling is done with ${\eta }_c=1.5$ [see Eq. (61)].

Figure 14

(Color online) (a) Plot of $v_N\left(T\right)=\overline{l^{\mathrm{pref}}\left(N\right)}∕N^{0.5}$ as a function of $T$ for the sizes $N=100$ (엯), $N=200$ (엯), $N=400$ (◇), $N=600$ (▵), $N=800$ (▷). (b) Finite size scaling of the same data in terms of the variable $x=(T-T_c)N^{1∕\nu }$ [see Eq. (62)] with $T_c=0.095$ and $\nu =2$.

Figure 15

(Color online) Height scaling: (a) the curves $z_N\left(T\right)=\overline{h_{\mathrm{med}}}∕N^{0.67}$ of various sizes $N=50$ (*), $N=100$ (엯), $N=200$ (◻), $N=400$ (◇), $N=600$ (▵) present crossings shifting towards $T_c$. (b) Finite size scaling of the same data in terms of the variable $x=(T-T_c)N^{1∕\nu }$ [see Eq. (66)] with $T_c=0.095$ and $\nu =2$.

Figure 16

(Color online) Critical behavior of the ratio $R_N\left(T\right)$ defined in Eq. (71): (a) curves $K_N\left(T\right)=N^{0.17}{R}_N\left(T\right)$ for the sizes $N=50$ (*) $N=100$ (엯), $N=200$ (◻), $N=400$ (◇), $N=600$ (▵), $N=800$ (▷). (b) Finite size scaling of the same data in terms of the variable $x=(T-T_c)N^{1∕\nu }$ [see Eq. (72)] with $T_c=0.095$ and $\nu =2$.