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 analyze 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 not equal i) of the sequence. We numerically compute the probability distributions P1(w) of the maximal weight w[{i}{max}=max{j}[w{i}(j)] , the probability distribution Pi(Y(2)) of the parameter Y2(i)= summation operator{j}w{i}{2}(j) , as well as the average values of the moments Y{k}(i)= summation operator_{j}w_{i}{k}(j) . We find that there are two important temperatures T_{c}<T_{gap} . For T>T_{gap} , the distribution P1(w) vanishes at some value w{0}(T)<1 , and accordingly the moments Y{k}(i) decay exponentially as [w{0}(T)]{k} in k . For T<T{gap} , the distributions P1(w) and Pi(Y2) present the characteristic Derrida-Flyvbjerg singularities at w=1n and Y{2}=1/n for n=1,2,... . In particular, there exists a temperature-dependent exponent mu(T) that governs the singularities P1(w) approximately (1-w){mu(T)-1} and Pi(Y2) approximately (1-Y{2}){mu(T)-1} as well as the power-law decay of the moments Y{k}(i) approximately 1k{mu(T)} . The exponent mu(T) grows from the value mu(T=0)=0 up to mu(T{gap}) approximately 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 approximately 0.67 to the high temperature value zeta approximately 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) approximately 1(T{c}-T){nu} with nu approximately 2 . The similarities and differences with the weight statistics in Lévy sums and in Derrida's random energy model are discussed.