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RNA secondary structure folding kinetics is known to be important for the biological function of certain processes, such as the hok/sok system in E. coli. Although linear algebra provides an exact computational solution of secondary structure folding kinetics with respect to the Turner energy model for tiny (≈ 20 nt) RNA sequences, the folding kinetics for larger sequences can only be approximated by binning structures into macrostates in a coarse-grained model, or by repeatedly simulating secondary structure folding with either the Monte Carlo algorithm or the Gillespie algorithm.Here we investigate the relation between the Monte Carlo algorithm and the Gillespie algorithm. We prove that asymptotically, the expected time for a K-step trajectory of the Monte Carlo algorithm is equal to N times that of the Gillespie algorithm, where N denotes the Boltzmann expected network degree. If the network is regular (i.e. every node has the same degree), then the mean first passage time (MFPT) computed by the Monte Carlo algorithm is equal to MFPT computed by the Gillespie algorithm multiplied by N ; however, this is not true for non-regular networks. In particular, RNA secondary structure folding kinetics, as computed by the Monte Carlo algorithm, is not equal to the folding kinetics, as computed by the Gillespie algorithm, although the mean first passage times are roughly correlated.Simulation software for RNA secondary structure folding according to the Monte Carlo and Gillespie algorithms is publicly available, as is our software to compute the expected degree of the network of secondary structures of a given RNA sequence -see http://bioinformatics.bc.edu/clote/ RNAexpNumNbors.
RNA secondary structure folding kinetics is known to be important for the biological function of certain processes, such as the hok/sok system in E. coli. Although linear algebra provides an exact computational solution of secondary structure folding kinetics with respect to the Turner energy model for tiny (≈ 20 nt) RNA sequences, the folding kinetics for larger sequences can only be approximated by binning structures into macrostates in a coarse-grained model, or by repeatedly simulating secondary structure folding with either the Monte Carlo algorithm or the Gillespie algorithm.Here we investigate the relation between the Monte Carlo algorithm and the Gillespie algorithm. We prove that asymptotically, the expected time for a K-step trajectory of the Monte Carlo algorithm is equal to N times that of the Gillespie algorithm, where N denotes the Boltzmann expected network degree. If the network is regular (i.e. every node has the same degree), then the mean first passage time (MFPT) computed by the Monte Carlo algorithm is equal to MFPT computed by the Gillespie algorithm multiplied by N ; however, this is not true for non-regular networks. In particular, RNA secondary structure folding kinetics, as computed by the Monte Carlo algorithm, is not equal to the folding kinetics, as computed by the Gillespie algorithm, although the mean first passage times are roughly correlated.Simulation software for RNA secondary structure folding according to the Monte Carlo and Gillespie algorithms is publicly available, as is our software to compute the expected degree of the network of secondary structures of a given RNA sequence -see http://bioinformatics.bc.edu/clote/ RNAexpNumNbors.
We describe four novel algorithms, RNAhairpin, RNAmloopNum, RNAmloopOrder, and RNAmloopHP, which compute the Boltzmann partition function for global structural constraints-respectively for the number of hairpins, the number of multiloops, maximum order (or depth) of multiloops, and the simultaneous number of hairpins and multiloops. Given an RNA sequence of length n and a user-specified integer 0 £ K £ n, RNAhairpin (resp. RNAmloopNum and RNAmloopOrder) computes the partition functions RNAmloopHP) sample from the low-energy ensemble of structures having h hairpins (resp. m multiloops and h hairpins), for given h, m. Moreover, by using the fast Fourier transform (FFT), RNAhairpin and RNAmloopNum have been improved to run in time O(n 4 ) and space O(n 2 ), although this improvement is not possible for RNAmloopOrder. We present two applications of the novel algorithms. First, we show that for many Rfam families of RNA, structures sampled from RNAmloopHP are more accurate than the minimum free-energy structure; for instance, sensitivity improves by almost 24% for transfer RNA, while for certain ribozyme families, there is an improvement of around 5%. Second, we show that the probabilities p(k) = Z(k)/Z of forming k hairpins (resp. multiloops) provide discriminating novel features for a support vector machine or relevance vector machine binary classifier for Rfam families of RNA. Our data suggests that multiloop order does not provide any significant discriminatory power over that of hairpin and multiloop number, and since these probabilities can be efficiently computed using the FFT, hairpin and multiloop formation probabilities could be added to other features in existent noncoding RNA gene finders. Our programs, written in C/C + + , are publicly available online at: http://bioinformatics.bc.edu/clotelab/RNAparametric.
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