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

Abstract

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

20 pages, 32 figures

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