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Register a new userCombinatorial Analysis for Pseudoknot RNA with Complex Structure
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Yangyang Zhao
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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.
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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.
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 $k$-noncrossing RNA structures with minimum arc-length 4 and at most $k-1$ mutually crossing bonds. Let ${sf T}_{k}^{[4]}(n)$ denote the number of $k$-noncrossing RNA structures with arc-length $ge 4$ over $n$ vertices. We prove (a) a functional equation for the generating function $sum_{nge 0}{sf T}_{k}^{[4]}(n)z^n$ and (b) derive for $kle 9$ the asymptotic formula ${sf T}_{k}^{[4]}(n)sim c_k n^{-((k-1)^2+(k-1)/2)} gamma_k^{-n}$. Furthermore we explicitly compute the exponential growth rates $gamma_k^{-1}$ and asymptotic formulas for $4le kle 9$.
Recently several minimum free energy (MFE) folding algorithms for predicting the joint structure of two interacting RNA molecules have been proposed. Their folding targets are interaction structures, that can be represented as diagrams with two backbones drawn horizontally on top of each other such that (1) intramolecular and intermolecular bonds are noncrossing and (2) there is no zig-zag configuration. This paper studies joint structures with arc-length at least four in which both, interior and exterior stack-lengths are at least two (no isolated arcs). The key idea in this paper is to consider a new type of shape, based on which joint structures can be derived via symbolic enumeration. Our results imply simple asymptotic formulas for the number of joint structures with surprisingly small exponential growth rates. They are of interest in the context of designing prediction algorithms for RNA-RNA interactions.
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.
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