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.
In this paper we derive the generating function of RNA structures with pseudoknots. We enumerate all $k$-noncrossing RNA pseudoknot structures categorized by their maximal sets of mutually intersecting arcs. In addition we enumerate pseudoknot structures over circular RNA. For 3-noncrossing RNA structures and RNA secondary structures we present a novel 4-term recursion formula and a 2-term recursion, respectively. Furthermore we enumerate for arbitrary $k$ all $k$-noncrossing, restricted RNA structures i.e. $k$-noncrossing RNA structures without 2-arcs i.e. arcs of the form $(i,i+2)$, for $1\l…
In this paper we present the asymptotic enumeration of RNA structures with pseudoknots. We develop a general framework for the computation of exponential growth rate and the sub exponential factors for $k$-noncrossing RNA structures. Our results are based on the generating function for the number of $k$-noncrossing RNA pseudoknot structures, ${\sf S}_k(n)$, derived in \cite{Reidys:07pseu}, where $k-1$ denotes the maximal size of sets of mutually intersecting bonds. We prove a functional equation for the generating function $\sum_{n\ge 0}{\sf S}_k(n)z^n$ and obtain for $k=2$ and $k=3$ the analy…
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 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.
In this paper we study $k$-noncrossing RNA structures with arc-length $\ge 3$, i.e. RNA molecules in which for any $i$, the nucleotides labeled $i$ and $i+j$ ($j=1,2$) cannot form a bond and in which there are at most $k-1$ mutually crossing arcs. Let ${\sf S}_{k,3}(n)$ denote their number. Based on a novel functional equation for the generating function $\sum_{n\ge 0}{\sf S}_{k,3}(n)z^n$, we derive for arbitrary $k\ge 3$ exponential growth factors and for $k=3$ the subexponential factor. Our main result is the derivation of the formula ${\sf S}_{3,3}(n) \sim \frac{6.11170\cdot 4!}{n(n-1)...(n…
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 m…
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.
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 …
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 pseudokn…
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 …
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 $4\le k\le 9$.
In this paper we analyze the length-spectrum of blocks in $γ$-structures. $γ$-structures are a class of RNA pseudoknot structures that plays a key role in the context of polynomial time RNA folding. A $γ$-structure is constructed by nesting and concatenating specific building components having topological genus at most $γ$. A block is a substructure enclosed by crossing maximal arcs with respect to the partial order induced by nesting. We show that, in uniformly generated $γ$-structures, there is a significant gap in this length-spectrum, i.e., there asymptotically almost surely exists a uniqu…
In this paper we study the distribution of stacks in $k$-noncrossing, $τ$-canonical RNA pseudoknot structures ($<k,τ> $-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 $<k,τ>$-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 $<k,τ>$-structures ha…
A k-noncrossing RNA pseudoknot structure is a graph over $\{1,...,n\}$ without 1-arcs, i.e. arcs of the form (i,i+1) and in which there exists no k-set of mutually intersecting arcs. In particular, RNA secondary structures are 2-noncrossing RNA structures. In this paper we prove a central and a local limit theorem for the distribution of the numbers of 3-noncrossing RNA structures over n nucleotides with exactly h bonds. We will build on the results of \cite{Reidys:07rna1} and \cite{Reidys:07rna2}, where the generating function of k-noncrossing RNA pseudoknot structures and the asymptotics for…
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 gen…
Ribonucleic Acid (RNA) can fold into shapes that perform functions in the cell. These foldings are governed by Watson-Crick base pairing, namely Adenine to Uracil and Cytosine to Guanine (A-U and G-C). The properties of the H-P (hydrophobic-hydrophilic) model of protein folding has been well studied in the two dimensional orthogonal case, and we attempt to achieve similar results. We prove that (1) there is an infinite family of even-length sequences with unique optimal foldings, (2) there are two ideal foldings for an even length sequence and given a sequence is is quickly verifiable if both,…
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 …
We extend recent methods for parametric sequence alignment to the parameter space for scoring RNA folds. This involves the construction of an RNA polytope. A vertex of this polytope corresponds to RNA secondary structures with common branching. We use this polytope and its normal fan to study the effect of varying three parameters in the free energy model that are not determined experimentally. Our results indicate that variation of these specific parameters does not have a dramatic effect on the structures predicted by the free energy model. We additionally map a collection of known RNA secon…
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 obta…
We analyze the distribution of RNA secondary structures given by the Knudsen-Hein stochastic context-free grammar used in the prediction program Pfold. We prove that the distribution of base pairs, helices and various types of loops in RNA secondary structures in this probabilistic model is asymptotically Gaussian, for a generic choice of the grammar probabilities. Our proofs are based on singularity analysis of probability generating functions. Finally, we use our results to discuss how this model reflects the properties of some known ribosomal secondary structures.
In this paper we study properties of topological RNA structures, i.e.~RNA contact structures with cross-serial interactions that are filtered by their topological genus. RNA secondary structures within this framework are topological structures having genus zero. We derive a new bivariate generating function whose singular expansion allows us to analyze the distributions of arcs, stacks, hairpin- , interior- and multi-loops. We then extend this analysis to H-type pseudoknots, kissing hairpins as well as $3$-knots and compute their respective expectation values. Finally we discuss our results an…
In this paper we study canonical $γ$-structures, 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 result is the derivation of the generating function of $γ$-structures via symbolic enumeration using so called irreducible shadows. We furthermore recursively compute the generating polynomials of irreducible shadows of genus $\le γ$…
An RNA sequence is a word over an alphabet on four elements $\{A,C,G,U\}$ called bases. RNA sequences fold into secondary structures where some bases match one another while others remain unpaired. Pseudoknot-free secondary structures can be represented as well-parenthesized expressions with additional dots, where pairs of matching parentheses symbolize paired bases and dots, unpaired bases. The two fundamental problems in RNA algorithmic are to predict how sequences fold within some model of energy and to design sequences of bases which will fold into targeted secondary structures. Predicting…
Motivation: Predicting the secondary structure of an RNA sequence is useful in many applications. Existing algorithms (based on dynamic programming) suffer from a major limitation: their runtimes scale cubically with the RNA length, and this slowness limits their use in genome-wide applications. Results: We present a novel alternative $O(n^3)$-time dynamic programming algorithm for RNA folding that is amenable to heuristics that make it run in $O(n)$ time and $O(n)$ space, while producing a high-quality approximation to the optimal solution. Inspired by incremental parsing for context-free gra…
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 distribut…
In this paper we show how to express RNA tertiary interactions via the concepts of tangled diagrams. Tangled diagrams allow to formulate RNA base triples and pseudoknot-interactions and to control the maximum number of mutually crossing arcs. In particular we study two subsets of tangled diagrams: 3-noncrossing tangled-diagrams with $\ell$ vertices of degree two and 2-regular, 3-noncrossing partitions (i.e. without arcs of the form $(i,i+1)$). Our main result is an asymptotic formula for the number of 2-regular, 3-noncrossing partitions, denoted by $p_{3,2}(n)$, 3-noncrossing partitions over $…
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 con…
The combinatorics of RNA plays a central role in biology. Mathematical biologists have several commonly-used models for RNA: words in a fixed alphabet (representing the primary sequence of nucleotides) and plane trees (representing the secondary structure, or folding of the RNA sequence). This paper considers an augmented version of the standard model of plane trees, one that incorporates some observed constraints on how the folding can occur. In particular we assume the alphabet consists of complementary pairs, for instance the Watson-Crick pairs A-U and C-G of RNA. Given a word in the alphab…
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 experien…
In this paper we analyze the length-spectrum of rainbows in RNA secondary structures. A rainbow in a secondary structure is a maximal arc with respect to the partial order induced by nesting. We show that there is a significant gap in this length-spectrum. We shall prove that there asymptotically almost surely exists a unique longest rainbow of length at least $n-O(n^{1/2})$ and that with high probability any other rainbow has finite length. We show that the distribution of the length of the longest rainbow converges to a discrete limit law and that, for finite $k$, the distribution of rainbow…
