12 research outputs found
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
time and 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.Comment: 40 pages 15 figure
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
into bicellular maps of genus 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 , where are the lengths of the
respective backbones, while the folding of the original structure has
time complexity, where is the length of the longer sequence.Comment: 18 pages, 13 figure
Shapes of interacting RNA complexes
Shapes of interacting RNA complexes are studied using a filtration via their
topological genus. A shape of an RNA complex is obtained by (iteratively)
collapsing stacks and eliminating hairpin loops. This shape-projection
preserves the topological core of the RNA complex and for fixed topological
genus there are only finitely many such shapes.Our main result is a new
bijection that relates the shapes of RNA complexes with shapes of RNA
structures.This allows to compute the shape polynomial of RNA complexes via the
shape polynomial of RNA structures. We furthermore present a linear time
uniform sampling algorithm for shapes of RNA complexes of fixed topological
genus.Comment: 38 pages 24 figure
The boundary length and point spectrum enumeration of partial chord diagrams using cut and join recursion
We introduce the boundary length and point spectrum, as a joint
generalization of the boundary length spectrum and boundary point spectrum in
arXiv:1307.0967. We establish by cut-and-join methods that the number of
partial chord diagrams filtered by the boundary length and point spectrum
satisfies a recursion relation, which combined with an initial condition
determines these numbers uniquely. This recursion relation is equivalent to a
second order, non-linear, algebraic partial differential equation for the
generating function of the numbers of partial chord diagrams filtered by the
boundary length and point spectrum.Comment: 16 pages, 6 figure
Enumeration of chord diagrams on many intervals and their non-orientable analogs
Two types of connected chord diagrams with chord endpoints lying in a
collection of ordered and oriented real segments are considered here: the real
segments may contain additional bivalent vertices in one model but not in the
other. In the former case, we record in a generating function the number of
fatgraph boundary cycles containing a fixed number of bivalent vertices while
in the latter, we instead record the number of boundary cycles of each fixed
length. Second order, non-linear, algebraic partial differential equations are
derived which are satisfied by these generating functions in each case giving
efficient enumerative schemes. Moreover, these generating functions provide
multi-parameter families of solutions to the KP hierarchy. For each model,
there is furthermore a non-orientable analog, and each such model likewise has
its own associated differential equation. The enumerative problems we solve are
interpreted in terms of certain polygon gluings. As specific applications, we
discuss models of several interacting RNA molecules. We also study a matrix
integral which computes numbers of chord diagrams in both orientable and
non-orientable cases in the model with bivalent vertices, and the large-N limit
is computed using techniques of free probability.Comment: 23 pages, 7 figures; revised and extended versio
Structural relation matching: an algorithm to identify structural patterns into RNAs and their interactions
RNA molecules play crucial roles in various biological processes. Their three-dimensional configurations determine the functions and, in turn, influences the interaction with other molecules. RNAs and their interaction structures, the so-called RNA-RNA interactions, can be abstracted in terms of secondary structures, i.e., a list of the nucleotide bases paired by hydrogen bonding within its nucleotide sequence. Each secondary structure, in turn, can be abstracted into cores and shadows. Both are determined by collapsing nucleotides and arcs properly. We formalize all of these abstractions as arc diagrams, whose arcs determine loops. A secondary structure, represented by an arc diagram, is pseudoknot-free if its arc diagram does not present any crossing among arcs otherwise, it is said pseudoknotted. In this study, we face the problem of identifying a given structural pattern into secondary structures or the associated cores or shadow of both RNAs and RNA-RNA interactions, characterized by arbitrary pseudoknots. These abstractions are mapped into a matrix, whose elements represent the relations among loops. Therefore, we face the problem of taking advantage of matrices and submatrices. The algorithms, implemented in Python, work in polynomial time. We test our approach on a set of 16S ribosomal RNAs with inhibitors of Thermus thermophilus, and we quantify the structural effect of the inhibitors
Partial chord diagrams and matrix models
In this article, the enumeration of partial chord diagrams is discussed via
matrix model techniques. In addition to the basic data such as the number of
backbones and chords, we also consider the Euler characteristic, the backbone
spectrum, the boundary point spectrum, and the boundary length spectrum.
Furthermore, we consider the boundary length and point spectrum that unifies
the last two types of spectra. We introduce matrix models that encode
generating functions of partial chord diagrams filtered by each of these
spectra. Using these matrix models, we derive partial differential equations -
obtained independently by cut-and-join arguments in an earlier work - for the
corresponding generating functions.Comment: 42 pages, 14 figure