433 research outputs found
The Combinatorics of the Foldings of RNA
RNA, much like DNA, is made up of four building blocks called nucleotides, Adenine, Guanine, Cytosine, and Uracil. These nucleotides form words that like to fold in on itself and bond together, each type of nucleotide bonding with only one other type of nucleotide. Therefore, order and number of nucleotides present will determine how many times the strand of RNA can fold. Using these guidelines, we considered what happens when we have only one bonding pair. Expanding on what was proven in k-Foldability of Words” (2017), we were able to expand on the number of ways a word can fold by adding to the list of ways any word of length 2n can fold. We also approached the problem from a different view by looking at how words with the same length and foldability compare to each other and defining operations between these words
When Can You Fold a Map?
We explore the following problem: given a collection of creases on a piece of
paper, each assigned a folding direction of mountain or valley, is there a flat
folding by a sequence of simple folds? There are several models of simple
folds; the simplest one-layer simple fold rotates a portion of paper about a
crease in the paper by +-180 degrees. We first consider the analogous questions
in one dimension lower -- bending a segment into a flat object -- which lead to
interesting problems on strings. We develop efficient algorithms for the
recognition of simply foldable 1D crease patterns, and reconstruction of a
sequence of simple folds. Indeed, we prove that a 1D crease pattern is
flat-foldable by any means precisely if it is by a sequence of one-layer simple
folds.
Next we explore simple foldability in two dimensions, and find a surprising
contrast: ``map'' folding and variants are polynomial, but slight
generalizations are NP-complete. Specifically, we develop a linear-time
algorithm for deciding foldability of an orthogonal crease pattern on a
rectangular piece of paper, and prove that it is (weakly) NP-complete to decide
foldability of (1) an orthogonal crease pattern on a orthogonal piece of paper,
(2) a crease pattern of axis-parallel and diagonal (45-degree) creases on a
square piece of paper, and (3) crease patterns without a mountain/valley
assignment.Comment: 24 pages, 19 figures. Version 3 includes several improvements thanks
to referees, including formal definitions of simple folds, more figures,
table summarizing results, new open problems, and additional reference
Emergence of stable and fast folding protein structures
The number of protein structures is far less than the number of sequences. By
imposing simple generic features of proteins (low energy and compaction) on all
possible sequences we show that the structure space is sparse compared to the
sequence space. Even though the sequence space grows exponentially with N (the
number of amino acids) we conjecture that the number of low energy compact
structures only scales as ln N. This implies that many sequences must map onto
countable number of basins in the structure space. The number of sequences for
which a given fold emerges as a native structure is further reduced by the dual
requirements of stability and kinetic accessibility. The factor that determines
the dual requirement is related to the sequence dependent temperatures,
T_\theta (collapse transition temperature) and T_F (folding transition
temperature). Sequences, for which \sigma =(T_\theta-T_F)/T_\theta is small,
typically fold fast by generically collapsing to the native-like structures and
then rapidly assembling to the native state. Such sequences satisfy the dual
requirements over a wide temperature range. We also suggest that the functional
requirement may further reduce the number of sequences that are biologically
competent. The scheme developed here for thinning of the sequence space that
leads to foldable structures arises naturally using simple physical
characteristics of proteins. The reduction in sequence space leading to the
emergence of foldable structures is demonstrated using lattice models of
proteins.Comment: latex, 18 pages, 8 figures, to be published in the conference
proceedings "Stochastic Dynamics and Pattern Formation in Biological Systems
Dynamical chaos and power spectra in toy models of heteropolymers and proteins
The dynamical chaos in Lennard-Jones toy models of heteropolymers is studied
by molecular dynamics simulations. It is shown that two nearby trajectories
quickly diverge from each other if the heteropolymer corresponds to a random
sequence. For good folders, on the other hand, two nearby trajectories may
initially move apart but eventually they come together. Thus good folders are
intrinsically non-chaotic. A choice of a distance of the initial conformation
from the native state affects the way in which a separation between the twin
trajectories behaves in time. This observation allows one to determine the size
of a folding funnel in good folders. We study the energy landscapes of the toy
models by determining the power spectra and fractal characteristics of the
dependence of the potential energy on time. For good folders, folding and
unfolding trajectories have distinctly different correlated behaviors at low
frequencies.Comment: 8 pages, 9 EPS figures, Phys. Rev. E (in press
Rigid Origami Vertices: Conditions and Forcing Sets
We develop an intrinsic necessary and sufficient condition for single-vertex
origami crease patterns to be able to fold rigidly. We classify such patterns
in the case where the creases are pre-assigned to be mountains and valleys as
well as in the unassigned case. We also illustrate the utility of this result
by applying it to the new concept of minimal forcing sets for rigid origami
models, which are the smallest collection of creases that, when folded, will
force all the other creases to fold in a prescribed way
Modeling study on the validity of a possibly simplified representation of proteins
The folding characteristics of sequences reduced with a possibly simplified
representation of five types of residues are shown to be similar to their
original ones with the natural set of residues (20 types or 20 letters). The
reduced sequences have a good foldability and fold to the same native structure
of their optimized original ones. A large ground state gap for the native
structure shows the thermodynamic stability of the reduced sequences. The
general validity of such a five-letter reduction is further studied via the
correlation between the reduced sequences and the original ones. As a
comparison, a reduction with two letters is found not to reproduce the native
structure of the original sequences due to its homopolymeric features.Comment: 6 pages with 4 figure
An Algorithmic Study of Manufacturing Paperclips and Other Folded Structures
We study algorithmic aspects of bending wires and sheet metal into a
specified structure. Problems of this type are closely related to the question
of deciding whether a simple non-self-intersecting wire structure (a
carpenter's ruler) can be straightened, a problem that was open for several
years and has only recently been solved in the affirmative.
If we impose some of the constraints that are imposed by the manufacturing
process, we obtain quite different results. In particular, we study the variant
of the carpenter's ruler problem in which there is a restriction that only one
joint can be modified at a time. For a linkage that does not self-intersect or
self-touch, the recent results of Connelly et al. and Streinu imply that it can
always be straightened, modifying one joint at a time. However, we show that
for a linkage with even a single vertex degeneracy, it becomes NP-hard to
decide if it can be straightened while altering only one joint at a time. If we
add the restriction that each joint can be altered at most once, we show that
the problem is NP-complete even without vertex degeneracies.
In the special case, arising in wire forming manufacturing, that each joint
can be altered at most once, and must be done sequentially from one or both
ends of the linkage, we give an efficient algorithm to determine if a linkage
can be straightened.Comment: 28 pages, 14 figures, Latex, to appear in Computational Geometry -
Theory and Application
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