186 research outputs found
Eigenvector Synchronization, Graph Rigidity and the Molecule Problem
The graph realization problem has received a great deal of attention in
recent years, due to its importance in applications such as wireless sensor
networks and structural biology. In this paper, we extend on previous work and
propose the 3D-ASAP algorithm, for the graph realization problem in
, given a sparse and noisy set of distance measurements. 3D-ASAP
is a divide and conquer, non-incremental and non-iterative algorithm, which
integrates local distance information into a global structure determination.
Our approach starts with identifying, for every node, a subgraph of its 1-hop
neighborhood graph, which can be accurately embedded in its own coordinate
system. In the noise-free case, the computed coordinates of the sensors in each
patch must agree with their global positioning up to some unknown rigid motion,
that is, up to translation, rotation and possibly reflection. In other words,
to every patch there corresponds an element of the Euclidean group Euc(3) of
rigid transformations in , and the goal is to estimate the group
elements that will properly align all the patches in a globally consistent way.
Furthermore, 3D-ASAP successfully incorporates information specific to the
molecule problem in structural biology, in particular information on known
substructures and their orientation. In addition, we also propose 3D-SP-ASAP, a
faster version of 3D-ASAP, which uses a spectral partitioning algorithm as a
preprocessing step for dividing the initial graph into smaller subgraphs. Our
extensive numerical simulations show that 3D-ASAP and 3D-SP-ASAP are very
robust to high levels of noise in the measured distances and to sparse
connectivity in the measurement graph, and compare favorably to similar
state-of-the art localization algorithms.Comment: 49 pages, 8 figure
Results on geometric networks and data structures
This thesis discusses four problems in computational geometry.
In traditional colored range-searching problems, one wants to store a set
of n objects with m distinct colors for the following queries: report all
colors such that there is at least one object of that color intersecting
the query range. Such an object, however, could be an `outlier' in its
color class. We consider a variant of this problem where one has to report
only those colors such that at least a fraction t of the objects of that
color intersects the query range, for some parameter t. Our main results
are on an approximate version of this problem, where we are also allowed to
report those colors for which a fraction (1-epsilon)t intersects the query
range, for some fixed epsilon > 0. We present efficient data structures for
such queries with orthogonal query ranges in sets of colored points, and
for point stabbing queries in sets of colored rectangles.
A box-tree is a bounding-volume hierarchy that uses axis-aligned boxes as
bounding volumes. R-trees are box-trees with nodes of high degree. The
query complexity of a box-tree with respect to a given type of query is the
maximum number of nodes visited when answering such a query. We describe
several new algorithms for constructing box-trees with small worst-case
query complexity with respect to queries with axis-parallel boxes and with
points. We also prove lower bounds on the worst-case query complexity for
box-trees, which show that our results are optimal or close to optimal.
The geometric minimum-diameter spanning tree (MDST) of a set of n points is
a tree that spans the set and minimizes the Euclidian length of the longest
path in the tree. So far, the MDST can only be found in slightly subcubic
time. We give two fast approximation schemes for the MDST, i.e.
factor-(1+epsilon) approximation algorithms. One algorithm uses a grid and
takes time O*(1/epsilon^(5 2/3) + n), where the O*-notation hides terms of
type O(log^O(1) 1/epsilon). The other uses the well-separated pair
decomposition and takes O(1/epsilon^3 n + (1/epsilon) n log n) time. A
combination of the two approaches runs in O*(1/epsilon^5 + n) time.
The dilation of a geometric graph is the maximum, over all pairs of points
in the graph, of the ratio of the Euclidean length of the shortest path
between them in the graph and their Euclidean distance. We consider a
generalized version of this notion, where the nodes of the graph are not
points but axis-parallel rectangles in the plane. The arcs in the graph are
horizontal or vertical segments connecting a pair of rectangles, and the
distance measure we use is the L1-distance. We study the following problem:
given n non-intersecting rectangles and a graph describing which pairs of
rectangles are to be connected, we wish to place the connecting segments
such that the dilation is minimized. We obtain the following results: for
arbitrary graphs, the problem is NP-hard; for trees, we can solve the
problem by linear programming on O(n^2) variables and constraints; for
paths, we can solve the problem in time O(n^3 log n); for rectangles sorted
vertically along a path, the problem can be solved in O(n^2) time
Spin Ice, Fractionalization and Topological Order
The spin ice compounds {\dys} and {\holm} are highly unusual magnets which
epitomize a set of concepts of great interest in modern condensed matter
physics: their low-energy physics exhibits an emergent gauge field and their
excitations are magnetic monopoles which arise from the fractionalization of
the microscopic magnetic spin degrees of freedom. In this review, we provide an
elementary introduction to these concepts and we survey the thermodynamics,
statics and dynamics---in and out of equilibrium---of spin ice from these
vantage points. Along the way, we touch on topics such as emergent Coulomb
plasmas, observable "Dirac strings", and irrational charges. We close with the
outlook for these unique materials.Comment: (15 pages, 9 figures) see
http://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-020911-125058
for the published versio
Rational Design of Small-Molecule Inhibitors of Protein-Protein Interactions: Application to the Oncogenic c-Myc/Max Interaction
Protein-protein interactions (PPIs) constitute an emerging class of targets for pharmaceutical intervention pursued by both industry and academia. Despite their fundamental role in many biological processes and diseases such as cancer, PPIs are still largely underrepresented in today's drug discovery. This dissertation describes novel computational approaches developed to facilitate the discovery/design of small-molecule inhibitors of PPIs, using the oncogenic c-Myc/Max interaction as a case study.First, we critically review current approaches and limitations to the discovery of small-molecule inhibitors of PPIs and we provide examples from the literature.Second, we examine the role of protein flexibility in molecular recognition and binding, and we review recent advances in the application of Elastic Network Models (ENMs) to modeling the global conformational changes of proteins observed upon ligand binding. The agreement between predicted soft modes of motions and structural changes experimentally observed upon ligand binding supports the view that ligand binding is facilitated, if not enabled, by the intrinsic (pre-existing) motions thermally accessible to the protein in the unliganded form.Third, we develop a new method for generating models of the bioactive conformations of molecules in the absence of protein structure, by identifying a set of conformations (from different molecules) that are most mutually similar in terms of both their shape and chemical features. We show how to solve the problem using an Integer Linear Programming formulation of the maximum-edge weight clique problem. In addition, we present the application of the method to known c-Myc/Max inhibitors.Fourth, we propose an innovative methodology for molecular mimicry design. We show how the structure of the c-Myc/Max complex was exploited to designing compounds that mimic the binding interactions that Max makes with the leucine zipper domain of c-Myc.In summary, the approaches described in this dissertation constitute important contributions to the fields of computational biology and computer-aided drug discovery, which combine biophysical insights and computational methods to expedite the discovery of novel inhibitors of PPIs
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