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Register a new userRNA-LEGO: Combinatorial Design of Pseudoknot RNA
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In this paper we enumerate $k$-noncrossing RNA pseudoknot structures with given minimum stack-length. We show that the numbers of $k$-noncrossing structures without isolated base pairs are significantly smaller than the number of all $k$-noncrossing structures. In particular we prove that the number of 3- and 4-noncrossing RNA structures with stack-length $ge 2$ is for large $n$ given by $311.2470 frac{4!}{n(n-1)...(n-4)}2.5881^n$ and $1.217cdot 10^{7} n^{-{21/2}} 3.0382^n$, respectively. We furthermore show that for $k$-noncrossing RNA structures the drop in exponential growth rates between the number of all structures and the number of all structures with stack-size $ge 2$ increases significantly. Our results are of importance for prediction algorithms for pseudoknot-RNA and provide evidence that there exist neutral networks of RNA pseudoknot structures.
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There exists many complicated $k$-noncrossing pseudoknot RNA structures in nature based on some special conditions. The special characteristic of RNA structures gives us great challenges in researching the enumeration, prediction and the analysis of prediction algorithm. We will study two kinds of typical $k$-noncrossing pseudoknot RNAs with complex structures separately.
In this paper we study $k$-noncrossing, canonical RNA pseudoknot structures with minimum arc-length $ge 4$. Let ${sf T}_{k,sigma}^{[4]} (n)$ denote the number of these structures. We derive exact enumeration results by computing the generating function ${bf T}_{k,sigma}^{[4]}(z)= sum_n{sf T}_{k,sigma}^{[4]}(n)z^n$ and derive the asymptotic formulas ${sf T}_{k,3}^{[4]}(n)^{}sim c_k n^{-(k-1)^2-frac{k-1}{2}} (gamma_{k,3}^{[4]})^{-n}$ for $k=3,...,9$. In particular we have for $k=3$, ${sf T}_{3,3}^{[4]}(n)^{}sim c_3 n^{-5} 2.0348^n$. Our results prove that the set of biophysically relevant RNA pseudoknot structures is surprisingly small and suggest a new structure class as target for prediction algorithms.
In this paper we study the distribution of stacks in $k$-noncrossing, $tau$-canonical RNA pseudoknot structures ($<k,tau> $-structures). An RNA structure is called $k$-noncrossing if it has no more than $k-1$ mutually crossing arcs and $tau$-canonical if each arc is contained in a stack of length at least $tau$. Based on the ordinary generating function of $<k,tau>$-structures cite{Reidys:08ma} we derive the bivariate generating function ${bf T}_{k,tau}(x,u)=sum_{n geq 0} sum_{0leq t leq frac{n}{2}} {sf T}_{k, tau}^{} (n,t) u^t x^n$, where ${sf T}_{k,tau}(n,t)$ is the number of $<k,tau>$-structures having exactly $t$ stacks and study its singularities. We show that for a certain parametrization of the variable $u$, ${bf T}_{k,tau}(x,u)$ has a unique, dominant singularity. The particular shift of this singularity parametrized by $u$ implies a central limit theorem for the distribution of stack-numbers. Our results are of importance for understanding the ``language of minimum-free energy RNA pseudoknot structures, generated by computer folding algorithms.
In this paper we present a selfcontained analysis and description of the novel {it ab initio} folding algorithm {sf cross}, which generates the minimum free energy (mfe), 3-noncrossing, $sigma$-canonical RNA structure. Here an RNA structure is 3-noncrossing if it does not contain more than three mutually crossing arcs and $sigma$-canonical, if each of its stacks has size greater or equal than $sigma$. Our notion of mfe-structure is based on a specific concept of pseudoknots and respective loop-based energy parameters. The algorithm decomposes into three parts: the first is the inductive construction of motifs and shadows, the second is the generation of the skeleta-trees rooted in irreducible shadows and the third is the saturation of skeleta via context dependent dynamic programming routines.
We present a novel topological classification of RNA secondary structures with pseudoknots. It is based on the topological genus of the circular diagram associated to the RNA base-pair structure. The genus is a positive integer number, whose value quantifies the topological complexity of the folded RNA structure. In such a representation, planar diagrams correspond to pure RNA secondary structures and have zero genus, whereas non planar diagrams correspond to pseudoknotted structures and have higher genus. We analyze real RNA structures from the databases wwPDB and Pseudobase, and classify them according to their topological genus. We compare the results of our statistical survey with existing theoretical and numerical models. We also discuss possible applications of this classification and show how it can be used for identifying new RNA structural motifs.
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