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Research overview
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| My research combines
the areas of knot theory, differential geometry, data mining,
and computer science to
produce applications to neuroscience and biopolymers
such as RNA, DNA, and
Proteins. Part of my work has focused on simplifying
and generalizing measures of entanglement
such as the writhe, the linking number and average
crossing number. I applied these results
to define and compute a number of geometric shape
descriptors to characterize the
structure of biopolymers such as proteins and RNA
tertiary structures. Analysis
of these shape descriptors uses data mining techniques;
I have also adapted these
concepts to differentiate human brain anatomical
characteristics. I also work on problems related
to RNA structural genomics. In particular, apply
mathematical and computational biology methods to
study RNA 3D structure patterns (motifs),
and design models to predict
RNA 3D structures.
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RNA tertiary structures
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RNA folding is recognized as hierarchical.
An RNA sequence forms secondary structural elements
(helices and single strands), followed by recurrent
tertiary interactions, and then it folds into a native
structure. RNA tertiary motifs are recurrent interactions
connecting between secondary structural elements.
Understanding the role of RNA tertiary motifs in
RNA folding will help to understand RNA 3D prediction.
In collaboration with Tamar Schlick,
I have focused on studying RNA structural genomics.
In particular, we applied mathematical and computational
biology methods to understand 3D structure patterns
(motifs) of RNA molecules. By using network analysis
and statistics, we studied complex interaction networks
formed by a number of known RNA tertiary motifs,
showing the existence of higher order motifs built
by a combination of smaller sub-motifs. We studied
and classified RNA junctions into families, and showed
that RNA junctions are composed of recurrent helical
configurations. Interestingly, helical elements in
junctions are found to align in parallel and perpendicular
configurations. We also discovered new RNA motifs,
and showed that larger junctions are composed of
smaller sub-junction elements. By defining 2D/3D
motif restraints and guidelines for the prediction
of RNA helical conformations, this analysis can ultimately
help in the difficult task of RNA 3D structure prediction.
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Classification of RNA four-way junctions into nine
families according to their coaxial stacking properties and flexible
helical arms.
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Biopolymer entanglement
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Lattice models of self-avoiding random walks have been extensively
used to simulate polymerchains with volume exclusion. It is of interest
to quantify microscopic entanglement, and if possible, relate it to
macroscopic physical properties of the polymer ensemble, such as the
stress-strain curve, rubber elasticity, and various phase change phenomena.
The writhe is a measure of non-planarity that can indicate chirality
of knots. The presence of knots in a closed circular DNA plasmid can
give information about the binding and mechanism of enzymes acting
on the DNA molecule. Approximating the writhe of a space curve in general
requires choosing a number of planar projections, computing the projected
writhe for each of these projections, and averaging the results. For
polygons on lattices and space groups, the restricted geometry of the
lattice or space group leads to simplification of the writhe calculation,
and provides an exact calculation for the writhe.
Polymer structures and characteristic curves on surfaces in nature
can take many form circles, open arcs, branched structures, and a mix
of these. So it is important to generalize the writhe formulae for
linear and branched complexes on lattices or space groups. Important
special cases are the calculation of writhe of biopolymers like DNA,
RNA, and proteins.
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Diamond lattice structure with its corresponding writhe
formula. The choice of the vectors are the unit orthogonal
vectors (1,0,0), (0,1,0), and (0,0,1). |
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Geometric measures applied to neuroscience
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Geometric measures such as the writhe, average crossing number, ropelength
and thickness can also be defined
in a natural way for oriented graph embeddings
and apply to the characterization of sulcus paths
along the surface of the human brain.
Brain
surfaces can be obtained from triangulated meshes
extracted from MRI
data. A set of sulcus paths can be traced to
describe the anatomy of the brain. Such approach can
provide an automatic way to distinguish sulcus
paths coming from either the left or right
hemisphere, as well as gender. |
Red line demarcates calcarine sulcus on the brain surface. |
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