4,720 research outputs found
Reduction of -Regular Noncrossing Partitions
In this paper, we present a reduction algorithm which transforms -regular
partitions of to -regular partitions of .
We show that this algorithm preserves the noncrossing property. This yields a
simple explanation of an identity due to Simion-Ullman and Klazar in connection
with enumeration problems on noncrossing partitions and RNA secondary
structures. For ordinary noncrossing partitions, the reduction algorithm leads
to a representation of noncrossing partitions in terms of independent arcs and
loops, as well as an identity of Simion and Ullman which expresses the Narayana
numbers in terms of the Catalan numbers
Watson-Crick pairing, the Heisenberg group and Milnor invariants
We study the secondary structure of RNA determined by Watson-Crick pairing
without pseudo-knots using Milnor invariants of links. We focus on the first
non-trivial invariant, which we call the Heisenberg invariant. The Heisenberg
invariant, which is an integer, can be interpreted in terms of the Heisenberg
group as well as in terms of lattice paths.
We show that the Heisenberg invariant gives a lower bound on the number of
unpaired bases in an RNA secondary structure. We also show that the Heisenberg
invariant can predict \emph{allosteric structures} for RNA. Namely, if the
Heisenberg invariant is large, then there are widely separated local maxima
(i.e., allosteric structures) for the number of Watson-Crick pairs found.Comment: 18 pages; to appear in Journal of Mathematical Biolog
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, ,
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 -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 -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).Comment: 27 pages, 12 figure
Funnels in Energy Landscapes
Local minima and the saddle points separating them in the energy landscape
are known to dominate the dynamics of biopolymer folding. Here we introduce a
notion of a "folding funnel" that is concisely defined in terms of energy
minima and saddle points, while at the same time conforming to a notion of a
"folding funnel" as it is discussed in the protein folding literature.Comment: 6 pages, 3 figures, submitted to European Conference on Complex
Systems 200
Statistical mechanics of RNA folding: a lattice approach
We propose a lattice model for RNA based on a self-interacting two-tolerant
trail. Self-avoidance and elements of tertiary structure are taken into
account. We investigate a simple version of the model in which the native state
of RNA consists of just one hairpin. Using exact arguments and Monte Carlo
simulations we determine the phase diagram for this case. We show that the
denaturation transition is first order and can either occur directly or through
an intermediate molten phase.Comment: 8 pages, 9 figure
Phase behaviour of DNA in presence of DNA-binding proteins
To characterize the thermodynamical equilibrium of DNA chains interacting
with a solution of non-specific binding proteins, a Flory-Huggins free energy
model was implemented. We explored the dependence on DNA and protein
concentrations of the DNA collapse. For physiologically relevant values of the
DNA-protein affinity, this collapse gives rise to a biphasic regime with a
dense and a dilute phase; the corresponding phase diagram was computed. Using
an approach based on Hamiltonian paths, we show that the dense phase has either
a molten globule or a crystalline structure, depending on the DNA bending
rigidity, which is influenced by the ionic strength. These results are valid at
the thermodynamical equilibrium and should therefore be consistent with many
biological processes, whose characteristic timescales range typically from 1 ms
to 10 s. Our model may thus be applied to biological phenomena that involve
DNA-binding proteins, such as DNA condensation with crystalline order, which
occurs in some bacteria to protect their chromosome from detrimental factors;
or transcription initiation, which occurs in clusters called transcription
factories that are reminiscent of the dense phase characterized in this study.Comment: 20 pages, 9 figures, accepted for publication at The Biophysical
Journa
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