arXiv subjects

Christian M. Reidys

Publications and source records attributed to Christian M. Reidys.

At least 55 records · Page 3Linked to original sources

arXiv2013

A bijection for tri-cellular maps

In this paper we give a bijective proof for a relation between uni- bi- and tricellular maps of certain topological genus. While this relation can formally be obtained using Matrix-theory as a result of the Schwinger-Dyson equation, we here present a bijection for the corresponding coefficient equation. Our construction is facilitated by repeated application of a certain cutting, the contraction of edges, incident to two vertices and the deletion of certain edges.

math.CO↗

arXiv2013

Uniform generation of RNA pseudoknot structures with genus filtration

In this paper we present a sampling framework for RNA structures of fixed topological genus. We introduce a novel, linear time, uniform sampling algorithm for RNA structures of fixed topological genus $g$, for arbitrary $g>0$. Furthermore we develop a linear time sampling algorithm for RNA structures of fixed topological genus $g$ that are weighted by a simplified, loop-based energy functional. For this process the partition function of the energy functional has to be computed once, which has $O(n^2)$ time complexity.

cs.CE↗

arXiv2013

Enumeration of RNA complexes via random matrix theory

We review a derivation of the numbers of RNA complexes of an arbitrary topology. These numbers are encoded in the free energy of the hermitian matrix model with potential V(x)=x^2/2-stx/(1-tx), where s and t are respective generating parameters for the number of RNA molecules and hydrogen bonds in a given complex. The free energies of this matrix model are computed using the so-called topological recursion, which is a powerful new formalism arising from random matrix theory. These numbers of RNA complexes also have profound meaning in mathematics: they provide the number of chord diagrams of fixed genus with specified numbers of backbones and chords as well as the number of cells in Riemann's moduli spaces for bordered surfaces of fixed topological type.

q-bio.QM↗

arXiv2013

A bijection between unicellular and bicellular maps

In this paper we present a combinatorial proof of a relation between the generating functions of unicellular and bicellular maps. This relation is a consequence of the Schwinger-Dyson equation of matrix theory. Alternatively it can be proved using representation theory of the symmetric group. Here we give a bijective proof by rewiring unicellular maps of topological genus $(g+1)$ into bicellular maps of genus $g$ and pairs of unicellular maps of lower topological genera. Our result has immediate consequences for the folding of RNA interaction structures, since the time complexity of folding the transformed structure is $O((n+m)^5)$, where $n,m$ are the lengths of the respective backbones, while the folding of the original structure has $O(n^6)$ time complexity, where $n$ is the length of the longer sequence.

math.CO↗

arXiv2012

A phase transition in energy-filtered RNA secondary structures

In this paper we study the effect of energy parameters on minimum free energy (mfe) RNA secondary structures. Employing a simplified combinatorial energy model, that is only dependent on the diagram representation and that is not sequence specific, we prove the following dichotomy result. Mfe structures derived via the Turner energy parameters contain only finitely many complex irreducible substructures and just minor parameter changes produce a class of mfe-structures that contain a large number of small irreducibles. We localize the exact point where the distribution of irreducibles experiences this phase transition from a discrete limit to a central limit distribution and subsequently put our result into the context of quantifying the effect of sparsification of the folding of these respective mfe-structures. We show that the sparsification of realistic mfe-structures leads to a constant time and space reduction and that the sparsifcation of the folding of structures with modified parameters leads to a linear time and space reduction. We furthermore identify the limit distribution at the phase transition as a Rayleigh distribution.

math.CO↗

arXiv2012

Topological recursion for chord diagrams, RNA complexes, and cells in moduli spaces

We introduce and study the Hermitian matrix model with potential V(x)=x^2/2-stx/(1-tx), which enumerates the number of linear chord diagrams of fixed genus with specified numbers of backbones generated by s and chords generated by t. For the one-cut solution, the partition function, correlators and free energies are convergent for small t and all s as a perturbation of the Gaussian potential, which arises for st=0. This perturbation is computed using the formalism of the topological recursion. The corresponding enumeration of chord diagrams gives at once the number of RNA complexes of a given topology as well as the number of cells in Riemann's moduli spaces for bordered surfaces. The free energies are computed here in principle for all genera and explicitly for genera less than four.

hep-th↗

arXiv2012

On the combinatorics of sparsification

Background: We study the sparsification of dynamic programming folding algorithms of RNA structures. Sparsification applies to the mfe-folding of RNA structures and can lead to a significant reduction of time complexity. Results: We analyze the sparsification of a particular decomposition rule, $Λ^*$, that splits an interval for RNA secondary and pseudoknot structures of fixed topological genus. Essential for quantifying the sparsification is the size of its so called candidate set. We present a combinatorial framework which allows by means of probabilities of irreducible substructures to obtain the expected size of the set of $Λ^*$-candidates. We compute these expectations for arc-based energy models via energy-filtered generating functions (GF) for RNA secondary structures as well as RNA pseudoknot structures. For RNA secondary structures we also consider a simplified loop-energy model. This combinatorial analysis is then compared to the expected number of $Λ^*$-candidates obtained from folding mfe-structures. In case of the mfe-folding of RNA secondary structures with a simplified loop energy model our results imply that sparsification provides a reduction of time complexity by a constant factor of 91% (theory) versus a 96% reduction (experiment). For the "full" loop-energy model there is a reduction of 98% (experiment).

math.CO↗

arXiv2012

The topological filtration of $γ$-structures

In this paper we study $γ$-structures filtered by topological genus. $γ$-structures are a class of RNA pseudoknot structures that plays a key role in the context of polynomial time folding of RNA pseudoknot structures. A $γ$-structure is composed by specific building blocks, that have topological genus less than or equal to $γ$, where composition means concatenation and nesting of such blocks. Our main results are the derivation of a new bivariate generating function for $γ$-structures via symbolic methods, the singularity analysis of the solutions and a central limit theorem for the distribution of topological genus in $γ$-structures of given length. In our derivation specific bivariate polynomials play a central role. Their coefficients count particular motifs of fixed topological genus and they are of relevance in the context of genus recursion and novel folding algorithms.

math.CO↗

arXiv2011

The 5'-3' distance of RNA secondary structures

Recently Yoffe {\it et al.} observed that the average distances between $5'-3'$ ends of RNA molecules are very small and largely independent of sequence length. This observation is based on numerical computations as well as theoretical arguments maximizing certain entropy functionals. In this paper we compute the exact distribution of $5'-3'$ distances of RNA secondary structures for any finite $n$. We furthermore compute the limit distribution and show that already for $n=30$ the exact distribution and the limit distribution are very close. Our results show that the distances of random RNA secondary structures are distinctively lower than those of minimum free energy structures of random RNA sequences.

math.CO↗

arXiv2011

Topology of RNA-RNA interaction structures

The topological filtration of interacting RNA complexes is studied and the role is analyzed of certain diagrams called irreducible shadows, which form suitable building blocks for more general structures. We prove that for two interacting RNAs, called interaction structures, there exist for fixed genus only finitely many irreducible shadows. This implies that for fixed genus there are only finitely many classes of interaction structures. In particular the simplest case of genus zero already provides the formalism for certain types of structures that occur in nature and are not covered by other filtrations. This case of genus zero interaction structures is already of practical interest, is studied here in detail and found to be expressed by a multiple context-free grammar extending the usual one for RNA secondary structures. We show that in $O(n^6)$ time and $O(n^4)$ space complexity, this grammar for genus zero interaction structures provides not only minimum free energy solutions but also the complete partition function and base pairing probabilities.

math.CO↗

arXiv2011

Random induced subgraphs of Cayley graphs induced by transpositions

In this paper we study random induced subgraphs of Cayley graphs of the symmetric group induced by an arbitrary minimal generating set of transpositions. A random induced subgraph of this Cayley graph is obtained by selecting permutations with independent probability, $λ_n$. Our main result is that for any minimal generating set of transpositions, for probabilities $λ_n=\frac{1+ε_n}{n-1}$ where $n^{-{1/3}+δ}\le ε_n<1$ and $δ>0$, a random induced subgraph has a.s. a unique largest component of size $\wp(ε_n)\frac{1+ε_n}{n-1}n!$, where $\wp(ε_n)$ is the survival probability of a specific branching process.

math.PR↗

arXiv2010

Linear chord diagrams on two intervals

Consider all possible ways of attaching disjoint chords to two ordered and oriented disjoint intervals so as to produce a connected graph. Taking the intervals to lie in the real axis with the induced orientation and the chords to lie in the upper half plane canonically determines a corresponding fatgraph which has some associated genus $g\geq 0$, and we consider the natural generating function ${\bf C}_g^{[2]}(z)=\sum_{n\geq 0} {\bf c}^{[2]}_g(n)z^n$ for the number ${\bf c}^{[2]}_g(n)$ of distinct such chord diagrams of fixed genus $g\geq 0$ with a given number $n\geq 0$ of chords. We prove here the surprising fact that ${\bf C}^{[2]}_g(z)=z^{2g+1} R_g^{[2]}(z)/(1-4z)^{3g+2} $ is a rational function, for $g\geq 0$, where the polynomial $R^{[2]}_g(z)$ with degree at most $g$ has integer coefficients and satisfies $R_g^{[2]}({1\over 4})\neq 0$. Earlier work had already determined that the analogous generating function ${\bf C}_g(z)=z^{2g}R_g(z)/(1-4z)^{3g-{1\over 2}}$ for chords attached to a single interval is algebraic, for $g\geq 1$, where the polynomial $R_g(z)$ with degree at most $g-1$ has integer coefficients and satisfies $R_g(1/4)\neq 0$ in analogy to the generating function ${\bf C}_0(z)$ for the Catalan numbers. The new results here on ${\bf C}_g^{[2]}(z)$ rely on this earlier work, and indeed, we find that $R_g^{[2]}(z)=R_{g+1}(z) -z\sum_{g_1=1}^g R_{g_1}(z) R_{g+1-g_1}(z)$, for $g\geq 1$.

math.CO↗

arXiv2010

RNA-RNA interaction prediction based on multiple sequence alignments

Many computerized methods for RNA-RNA interaction structure prediction have been developed. Recently, $O(N^6)$ time and $O(N^4)$ space dynamic programming algorithms have become available that compute the partition function of RNA-RNA interaction complexes. However, few of these methods incorporate the knowledge concerning related sequences, thus relevant evolutionary information is often neglected from the structure determination. Therefore, it is of considerable practical interest to introduce a method taking into consideration both thermodynamic stability and sequence covariation. We present the \emph{a priori} folding algorithm \texttt{ripalign}, whose input consists of two (given) multiple sequence alignments (MSA). \texttt{ripalign} outputs (1) the partition function, (2) base-pairing probabilities, (3) hybrid probabilities and (4) a set of Boltzmann-sampled suboptimal structures consisting of canonical joint structures that are compatible to the alignments. Compared to the single sequence-pair folding algorithm \texttt{rip}, \texttt{ripalign} requires negligible additional memory resource. Furthermore, we incorporate possible structure constraints as input parameters into our algorithm. The algorithm described here is implemented in C as part of the \texttt{rip} package. The supplemental material, source code and input/output files can freely be downloaded from \url{http://www.combinatorics.cn/cbpc/ripalign.html}. \section{Contact} Christian Reidys \texttt{duck@santafe.edu}

math-ph↗

arXiv2010

Combinatorial analysis of interacting RNA molecules

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.

math.CO↗

arXiv2010

On the uniform generation of modular diagrams

In this paper we present an algorithm that generates $k$-noncrossing, $σ$-modular diagrams with uniform probability. A diagram is a labeled graph of degree $\le 1$ over $n$ vertices drawn in a horizontal line with arcs $(i,j)$ in the upper half-plane. A $k$-crossing in a diagram is a set of $k$ distinct arcs $(i_1, j_1), (i_2, j_2),\ldots,(i_k, j_k)$ with the property $i_1 < i_2 < \ldots < i_k < j_1 < j_2 < \ldots< j_k$. A diagram without any $k$-crossings is called a $k$-noncrossing diagram and a stack of length $σ$ is a maximal sequence $((i,j),(i+1,j-1),\dots,(i+(σ-1),j-(σ-1)))$. A diagram is $σ$-modular if any arc is contained in a stack of length at least $σ$. Our algorithm generates after $O(n^k)$ preprocessing time, $k$-noncrossing, $σ$-modular diagrams in $O(n)$ time and space complexity.

math.CO↗

arXiv2010

Combinatorics of RNA-RNA interaction

RNA-RNA binding is an important phenomenon observed for many classes of non-coding RNAs and plays a crucial role in a number of regulatory processes. Recently several MFE folding algorithms for predicting the joint structure of two interacting RNA molecules have been proposed. Here joint structure means that in a diagram representation the intramolecular bonds of each partner are pseudoknot-free, that the intermolecular binding pairs are noncrossing, and that there is no so-called ``zig-zag'' configuration. This paper presents the combinatorics of RNA interaction structures including their generating function, singularity analysis as well as explicit recurrence relations. In particular, our results imply simple asymptotic formulas for the number of joint structures.

math.CO↗

arXiv2010

Modular, $k$-noncrossing diagrams

In this paper we compute the generating function of modular, $k$-noncrossing diagrams. A $k$-noncrossing diagram is called modular if it does not contains any isolated arcs and any arc has length at least four. Modular diagrams represent the deformation retracts of RNA pseudoknot structures \cite{Stadler:99,Reidys:07pseu,Reidys:07lego} and their properties reflect basic features of these bio-molecules. The particular case of modular noncrossing diagrams has been extensively studied \cite{Waterman:78b, Waterman:79,Waterman:93, Schuster:98}. Let ${\sf Q}_k(n)$ denote the number of modular $k$-noncrossing diagrams over $n$ vertices. We derive exact enumeration results as well as the asymptotic formula ${\sf Q}_k(n)\sim c_k n^{-(k-1)^2-\frac{k-1}{2}}γ_{k}^{-n}$ for $k=3,..., 9$ and derive a new proof of the formula ${\sf Q}_2(n)\sim 1.4848\, n^{-3/2}\,1.8489^{-n}$ \cite{Schuster:98}.

math.CO↗

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.

math.CO↗