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Register a new user$k$-noncrossing RNA structures with arc-length $ge 3$
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In this paper we enumerate $k$-noncrossing RNA pseudoknot structures with given minimum arc- and stack-length. That is, we study the numbers of RNA pseudoknot structures with arc-length $ge 3$, stack-length $ge sigma$ and in which there are at most $k-1$ mutually crossing bonds, denoted by ${sf T}_{k,sigma}^{[3]}(n)$. In particular we prove that the numbers of 3, 4 and 5-noncrossing RNA structures with arc-length $ge 3$ and stack-length $ge 2$ satisfy ${sf T}_{3,2}^{[3]}(n)^{}sim K_3 n^{-5} 2.5723^n$, ${sf T}^{[3]}_{4,2}(n)sim K_4 n^{-{21/2}} 3.0306^n$, and ${sf T}^{[3]}_{5,2}(n)sim K_5 n^{-18} 3.4092^n$, respectively, where $K_3,K_4,K_5$ are constants. Our results are of importance for prediction algorithms for RNA pseudoknot structures.
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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 $4le kle 9$.
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.
We enumerate the number of RNA contact structures according to their genus, i.e. the topological character of their pseudoknots. By using a recently proposed matrix model formulation for the RNA folding problem, we obtain exact results for the simple case of an RNA molecule with an infinitely flexible backbone, in which any arbitrary pair of bases is allowed. We analyze the distribution of the genus of pseudoknots as a function of the total number of nucleotides along the phosphate-sugar backbone.
Our work is concerned with the generation and targeted design of RNA, a type of genetic macromolecule that can adopt complex structures which influence their cellular activities and functions. The design of large scale and complex biological structures spurs dedicated graph-based deep generative modeling techniques, which represents a key but underappreciated aspect of computational drug discovery. In this work, we investigate the principles behind representing and generating different RNA structural modalities, and propose a flexible framework to jointly embed and generate these molecular structures along with their sequence in a meaningful latent space. Equipped with a deep understanding of RNA molecular structures, our most sophisticated encoding and decoding methods operate on the molecular graph as well as the junction tree hierarchy, integrating strong inductive bias about RNA structural regularity and folding mechanism such that high structural validity, stability and diversity of generated RNAs are achieved. Also, we seek to adequately organize the latent space of RNA molecular embeddings with regard to the interaction with proteins, and targeted optimization is used to navigate in this latent space to search for desired novel RNA molecules.
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