9,474 research outputs found
Bounds on the maximum multiplicity of some common geometric graphs
We obtain new lower and upper bounds for the maximum multiplicity of some
weighted and, respectively, non-weighted common geometric graphs drawn on n
points in the plane in general position (with no three points collinear):
perfect matchings, spanning trees, spanning cycles (tours), and triangulations.
(i) We present a new lower bound construction for the maximum number of
triangulations a set of n points in general position can have. In particular,
we show that a generalized double chain formed by two almost convex chains
admits {\Omega}(8.65^n) different triangulations. This improves the bound
{\Omega}(8.48^n) achieved by the double zig-zag chain configuration studied by
Aichholzer et al.
(ii) We present a new lower bound of {\Omega}(12.00^n) for the number of
non-crossing spanning trees of the double chain composed of two convex chains.
The previous bound, {\Omega}(10.42^n), stood unchanged for more than 10 years.
(iii) Using a recent upper bound of 30^n for the number of triangulations,
due to Sharir and Sheffer, we show that n points in the plane in general
position admit at most O(68.62^n) non-crossing spanning cycles.
(iv) We derive lower bounds for the number of maximum and minimum weighted
geometric graphs (matchings, spanning trees, and tours). We show that the
number of shortest non-crossing tours can be exponential in n. Likewise, we
show that both the number of longest non-crossing tours and the number of
longest non-crossing perfect matchings can be exponential in n. Moreover, we
show that there are sets of n points in convex position with an exponential
number of longest non-crossing spanning trees. For points in convex position we
obtain tight bounds for the number of longest and shortest tours. We give a
combinatorial characterization of the longest tours, which leads to an O(nlog
n) time algorithm for computing them
Markov modeling of peptide folding in the presence of protein crowders
We use Markov state models (MSMs) to analyze the dynamics of a
-hairpin-forming peptide in Monte Carlo (MC) simulations with
interacting protein crowders, for two different types of crowder proteins
[bovine pancreatic trypsin inhibitor (BPTI) and GB1]. In these systems, at the
temperature used, the peptide can be folded or unfolded and bound or unbound to
crowder molecules. Four or five major free-energy minima can be identified. To
estimate the dominant MC relaxation times of the peptide, we build MSMs using a
range of different time resolutions or lag times. We show that stable
relaxation-time estimates can be obtained from the MSM eigenfunctions through
fits to autocorrelation data. The eigenfunctions remain sufficiently accurate
to permit stable relaxation-time estimation down to small lag times, at which
point simple estimates based on the corresponding eigenvalues have large
systematic uncertainties. The presence of the crowders have a stabilizing
effect on the peptide, especially with BPTI crowders, which can be attributed
to a reduced unfolding rate , while the folding rate
is left largely unchanged.Comment: 18 pages, 6 figure
Molecular dynamics simulations of lead clusters
Molecular dynamics simulations of nanometer-sized lead clusters have been
performed using the Lim, Ong and Ercolessi glue potential (Surf. Sci. {\bf
269/270}, 1109 (1992)). The binding energies of clusters forming crystalline
(fcc), decahedron and icosahedron structures are compared, showing that fcc
cuboctahedra are the most energetically favoured of these polyhedral model
structures. However, simulations of the freezing of liquid droplets produced a
characteristic form of ``shaved'' icosahedron, in which atoms are absent at the
edges and apexes of the polyhedron. This arrangement is energetically favoured
for 600-4000 atom clusters. Larger clusters favour crystalline structures.
Indeed, simulated freezing of a 6525-atom liquid droplet produced an imperfect
fcc Wulff particle, containing a number of parallel stacking faults. The
effects of temperature on the preferred structure of crystalline clusters below
the melting point have been considered. The implications of these results for
the interpretation of experimental data is discussed.Comment: 11 pages, 18 figues, new section added and one figure added, other
minor changes for publicatio
Calibrated Langevin dynamics simulations of intrinsically disordered proteins
We perform extensive coarse-grained (CG) Langevin dynamics simulations of
intrinsically disordered proteins (IDPs), which possess fluctuating
conformational statistics between that for excluded volume random walks and
collapsed globules. Our CG model includes repulsive steric, attractive
hydrophobic, and electrostatic interactions between residues and is calibrated
to a large collection of single-molecule fluorescence resonance energy transfer
data on the inter-residue separations for 36 pairs of residues in five IDPs:
-, -, and -synuclein, the microtubule-associated protein
, and prothymosin . We find that our CG model is able to
recapitulate the average inter-residue separations regardless of the choice of
the hydrophobicity scale, which shows that our calibrated model can robustly
capture the conformational dynamics of IDPs. We then employ our model to study
the scaling of the radius of gyration with chemical distance in 11 known IDPs.
We identify a strong correlation between the distance to the dividing line
between folded proteins and IDPs in the mean charge and hydrophobicity space
and the scaling exponent of the radius of gyration with chemical distance along
the protein.Comment: 16 pages, 10 figure
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