arXiv subjects

Christian M. Reidys

Publications and source records attributed to Christian M. Reidys.

At least 73 records · Page 4Linked to original sources

arXiv2010

Inverse Folding of RNA Pseudoknot Structures

Background: RNA exhibits a variety of structural configurations. Here we consider a structure to be tantamount to the noncrossing Watson-Crick and \pairGU-base pairings (secondary structure) and additional cross-serial base pairs. These interactions are called pseudoknots and are observed across the whole spectrum of RNA functionalities. In the context of studying natural RNA structures, searching for new ribozymes and designing artificial RNA, it is of interest to find RNA sequences folding into a specific structure and to analyze their induced neutral networks. Since the established inverse folding algorithms, {\tt RNAinverse}, {\tt RNA-SSD} as well as {\tt INFO-RNA} are limited to RNA secondary structures, we present in this paper the inverse folding algorithm {\tt Inv} which can deal with 3-noncrossing, canonical pseudoknot structures. Results: In this paper we present the inverse folding algorithm {\tt Inv}. We give a detailed analysis of {\tt Inv}, including pseudocodes. We show that {\tt Inv} allows to design in particular 3-noncrossing nonplanar RNA pseudoknot 3-noncrossing RNA structures--a class which is difficult to construct via dynamic programming routines. {\tt Inv} is freely available at \url{http://www.combinatorics.cn/cbpc/inv.html}. Conclusions: The algorithm {\tt Inv} extends inverse folding capabilities to RNA pseudoknot structures. In comparison with {\tt RNAinverse} it uses new ideas, for instance by considering sets of competing structures. As a result, {\tt Inv} is not only able to find novel sequences even for RNA secondary structures, it does so in the context of competing structures that potentially exhibit cross-serial interactions.

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arXiv2010

The evolution of random reversal graph

The random reversal graph offers new perspectives, allowing to study the connectivity of genomes as well as their most likely distance as a function of the reversal rate. Our main result shows that the structure of the random reversal graph changes dramatically at $λ_n=1/\binom{n+1}{2}$. For $λ_n=(1-ε)/\binom{n+1}{2}$, the random graph consists of components of size at most $O(n\ln(n))$ a.s. and for $(1+ε)/\binom{n+1}{2}$, there emerges a unique largest component of size $\sim \wp(ε) \cdot 2^n\cdot n$!$ a.s.. This "giant" component is furthermore dense in the reversal graph.

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arXiv2009

Loops in canonical RNA pseudoknot structures

In this paper we compute the limit distributions of the numbers of hairpin-loops, interior-loops and bulges in k-noncrossing RNA structures. The latter are coarse grained RNA structures allowing for cross-serial interactions, subject to the constraint that there are at most k-1 mutually crossing arcs in the diagram representation of the molecule. We prove central limit theorems by means of studying the corresponding bivariate generating functions. These generating functions are obtained by symbolic inflation of Ik5-shapes.

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arXiv2009

Random $k$-noncrossing partitions

In this paper, we introduce polynomial time algorithms that generate random $k$-noncrossing partitions and 2-regular, $k$-noncrossing partitions with uniform probability. A $k$-noncrossing partition does not contain any $k$ mutually crossing arcs in its canonical representation and is 2-regular if the latter does not contain arcs of the form $(i,i+1)$. Using a bijection of Chen {\it et al.} \cite{Chen,Reidys:08tan}, we interpret $k$-noncrossing partitions and 2-regular, $k$-noncrossing partitions as restricted generalized vacillating tableaux. Furthermore, we interpret the tableaux as sampling paths of a Markov-processes over shapes and derive their transition probabilities.

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arXiv2009

Random 3-noncrossing partitions

In this paper, we introduce polynomial time algorithms that generate random 3-noncrossing partitions and 2-regular, 3-noncrossing partitions with uniform probability. A 3-noncrossing partition does not contain any three mutually crossing arcs in its canonical representation and is 2-regular if the latter does not contain arcs of the form $(i,i+1)$. Using a bijection of Chen {\it et al.} \cite{Chen,Reidys:08tan}, we interpret 3-noncrossing partitions and 2-regular, 3-noncrossing partitions as restricted generalized vacillating tableaux. Furthermore, we interpret the tableaux as sampling paths of Markov-processes over shapes and derive their transition probabilities.

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arXiv2009

Shapes of RNA pseudoknot structures

In this paper we study abstract shapes of $k$-noncrossing, $σ$-canonical RNA pseudoknot structures. We consider ${\sf lv}_k^{\sf 1}$- and ${\sf lv}_k^{\sf 5}$-shapes, which represent a generalization of the abstract $π'$- and $π$-shapes of RNA secondary structures introduced by \citet{Giegerich:04ashape}. Using a novel approach we compute the generating functions of ${\sf lv}_k^{\sf 1}$- and ${\sf lv}_k^{\sf 5}$-shapes as well as the generating functions of all ${\sf lv}_k^{\sf 1}$- and ${\sf lv}_k^{\sf 5}$-shapes induced by all $k$-noncrossing, $σ$-canonical RNA structures for fixed $n$. By means of singularity analysis of the generating functions, we derive explicit asymptotic expressions.

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arXiv2009

Random $k$-noncrossing RNA Structures

In this paper we derive polynomial time algorithms that generate random $k$-noncrossing matchings and $k$-noncrossing RNA structures with uniform probability. Our approach employs the bijection between $k$-noncrossing matchings and oscillating tableaux and the $P$-recursiveness of the cardinalities of $k$-noncrossing matchings. The main idea is to consider the tableaux sequences as paths of stochastic processes over shapes and to derive their transition probabilities.

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arXiv2009

A generalization of the brauer algebra

We study two variations of the Brauer algebra $B_n(x)$. The first is the algebra $A_n(x)$, which generalizes the Brauer algebra by considering loops. The second is the algebra $L_n(x)$, the $A_n(x)$-subalgebra generated by diagrams without horizontal arcs. $A_n(x)$ and $L_n(x)$ have for $x \neq 0$ an hereditary-chain indexed by all integers. Following the ideas of Martin in the context of the partition algebra, and Doran et al. for the Brauer algebra, we study semisimplicity of $A_n(x)$ using restriction and induction in $A_n(x)$ and $L_n(x)$. Our main result is that $A_n(x)$ is semisimple if $x \notin Z$ and that $L_n(x)$ is semisimple if $x \neq 0$.

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arXiv2009

RNA-RNA interaction prediction: partition function and base pair pairing probabilities

In this paper, we study the interaction of an antisense RNA and its target mRNA, based on the model introduced by Alkan {\it et al.} (Alkan {\it et al.}, J. Comput. Biol., Vol:267--282, 2006). Our main results are the derivation of the partition function \cite{Backhofen} (Chitsaz {\it et al.}, Bioinformatics, to appear, 2009), based on the concept of tight-structure and the computation of the base pairing probabilities. This paper contains the folding algorithm {\sf rip} which computes the partition function as well as the base pairing probabilities in $O(N^4M^2)+O(N^2M^4)$ time and $O(N^2M^2)$ space, where $N,M$ denote the lengths of the interacting sequences.

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arXiv2009

On the decomposition of $k$-noncrossing RNA structures

An $k$-noncrossing RNA structure can be identified with an $k$-noncrossing diagram over $[n]$, which in turn corresponds to a vacillating tableaux having at most $(k-1)$ rows. In this paper we derive the limit distribution of irreducible substructures via studying their corresponding vacillating tableaux. Our main result proves, that the limit distribution of the numbers of irreducible substructures in $k$-noncrossing, $σ$-canonical RNA structures is determined by the density function of a $Γ(-\lnτ_k,2)$-distribution for some $τ_k<1$.

q-bio.BM↗

arXiv2009

Irreducibility in RNA structures

In this paper we study irreducibility in RNA structures. By RNA structure we mean RNA secondary as well as RNA pseudoknot structures. In our analysis we shall contrast random and minimum free energy (mfe) configurations. We compute various distributions: of the numbers of irreducible substructures, their locations and sizes, parameterized in terms of the maximal number of mutually crossing arcs, $k-1$, and the minimal size of stacks $σ$. In particular, we analyze the size of the largest irreducible substructure for random and mfe structures, which is the key factor for the folding time of mfe configurations.

q-bio.BM↗

arXiv2008

Folding 3-noncrossing RNA pseudoknot structures

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, $σ$-canonical RNA structure. Here an RNA structure is 3-noncrossing if it does not contain more than three mutually crossing arcs and $σ$-canonical, if each of its stacks has size greater or equal than $σ$. 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.

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arXiv2008

Pseudoknot RNA structures with arc-length $\ge 4$

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 $\sum_{n\ge 0}{\sf T}_{k}^{[4]}(n)z^n$ and (b) derive for $k\le 9$ the asymptotic formula ${\sf T}_{k}^{[4]}(n)\sim c_k n^{-((k-1)^2+(k-1)/2)} γ_k^{-n}$. Furthermore we explicitly compute the exponential growth rates $γ_k^{-1}$ and asymptotic formulas for $4\le k\le 9$.

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arXiv2008

Stacks in canonical RNA pseudoknot structures

In this paper we study the distribution of stacks in $k$-noncrossing, $τ$-canonical RNA pseudoknot structures ($ $-structures). An RNA structure is called $k$-noncrossing if it has no more than $k-1$ mutually crossing arcs and $τ$-canonical if each arc is contained in a stack of length at least $τ$. Based on the ordinary generating function of $ $-structures \cite{Reidys:08ma} we derive the bivariate generating function ${\bf T}_{k,τ}(x,u)=\sum_{n \geq 0} \sum_{0\leq t \leq \frac{n}{2}} {\sf T}_{k, τ}^{} (n,t) u^t x^n$, where ${\sf T}_{k,τ}(n,t)$ is the number of $ $-structures having exactly $t$ stacks and study its singularities. We show that for a certain parametrization of the variable $u$, ${\bf T}_{k,τ}(x,u)$ 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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arXiv2008

Canonical RNA pseudoknot structures

In this paper we study $k$-noncrossing, canonical RNA pseudoknot structures with minimum arc-length $\ge 4$. Let ${\sf T}_{k,σ}^{[4]} (n)$ denote the number of these structures. We derive exact enumeration results by computing the generating function ${\bf T}_{k,σ}^{[4]}(z)= \sum_n{\sf T}_{k,σ}^{[4]}(n)z^n$ and derive the asymptotic formulas ${\sf T}_{k,3}^{[4]}(n)^{}\sim c_k n^{-(k-1)^2-\frac{k-1}{2}} (γ_{k,3}^{[4]})^{-n}$ 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.

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arXiv2008

Large components in random induced subgraphs of n-cubes

In this paper we study random induced subgraphs of the binary $n$-cube, $Q_2^n$. This random graph is obtained by selecting each $Q_2^n$-vertex with independent probability $λ_n$. Using a novel construction of subcomponents we study the largest component for $λ_n=\frac{1+χ_n}{n}$, where $ε\ge χ_n\ge n^{-{1/3}+ δ}$, $δ>0$. We prove that there exists a.s. a unique largest component $C_n^{(1)}$. We furthermore show that $χ_n=ε$, $| C_n^{(1)}|\sim α(ε) \frac{1+χ_n}{n} 2^n$ and for $o(1)=χ_n\ge n^{-{1/3}+δ}$, $| C_n^{(1)}| \sim 2 χ_n \frac{1+χ_n}{n} 2^n$ holds. This improves the result of \cite{Bollobas:91} where constant $χ_n=χ$ is considered. In particular, in case of $λ_n=\frac{1+ε} {n}$, our analysis implies that a.s. a unique giant component exists.

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arXiv2008

Asymptotic analysis of $k$-noncrossing matchings

In this paper we study $k$-noncrossing matchings. A $k$-noncrossing matching is a labeled graph with vertex set $\{1,...,2n\}$ arranged in increasing order in a horizontal line and vertex-degree 1. The $n$ arcs are drawn in the upper halfplane subject to the condition that there exist no $k$ arcs that mutually intersect. We derive: (a) for arbitrary $k$, an asymptotic approximation of the exponential generating function of $k$-noncrossing matchings $F_k(z)$. (b) the asymptotic formula for the number of $k$-noncrossing matchings $f_{k}(n) \sim c_k n^{-((k-1)^2+(k-1)/2)} (2(k-1))^{2n}$ for some $c_k>0$.

math.CO↗

arXiv2008

Efficient Counting and Asymptotics of $k$-noncrossing tangled-diagrams

In this paper we enumerate $k$-noncrossing tangled-diagrams. A tangled-diagram is a labeled graph whose vertices are $1,...,n$ have degree $\le 2$, and are arranged in increasing order in a horizontal line. Its arcs are drawn in the upper halfplane with a particular notion of crossings and nestings. Our main result is the asymptotic formula for the number of $k$-noncrossing tangled-diagrams $T_{k}(n) \sim c_k n^{-((k-1)^2+(k-1)/2)} (4(k-1)^2+2(k-1)+1)^n$ for some $c_k>0$.

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