Summary of Research Interests
 Molecular Biology
 Polymer Conformations
 Human Brain Project
Statement of Research Interests
I am interested in the applications of topology to molecular biology
and polymer configuration, both in theory development and computational
simulation. Another interest is the mathematical analysis of human brain
functional data.
Molecular Biology
The DNA of all organisms has a complex and fascinating topology. It
can be viewed as two very long, closed curves that are intertwined millions
of times, linked to other closed curves, tied into knots, and subjected
to four or five successive orders of coiling to convert it into a compact
form for information storage. For information retrieval and cell viability,
some geometric and topological features must be introduced, and others
quickly removed. Some enzymes maintain the proper geometry and topology
by passing one strand of DNA through another via an enzymebridged transient
break in the DNA; this enzyme action plays a crucial role in cell metabolism,
including segregation of daughter chromosomes at the termination of replication
and in maintaining proper in vivo (in the cell) DNA topology. Other enzymes
break the DNA apart and recombine the ends by exchanging them. These enzymes
regulate the expression of specific genes, mediate viral integration into
and excision from the host genome, mediate transposition and repair of
DNA, and generate antibody and genetic diversity. These enzymes perform
incredible feats of topology at the molecular level; the description and
quantization of such enzyme action absolutely requires the language and
computational machinery of topology.
The longrange goal of this project is to develop a complete set of
experimentally observable topological parameters with which to describe
and compute enzyme mechanism and the structure of the active enzymeDNA
synaptic intermediate. One of the important unsolved problems in biology
is the threedimensional structure of proteins, DNA and active proteinDNA
complexes in solution (in the cell), and the relationship between structure
and function. It is the 3dimensional shape in solution which is biologically
important, but difficult to determine. The topological approach to enzymology
is an indirect method in which the descriptive and analytical powers of
topology and geometry are employed in an effort to infer the structure
of active enzymeDNA complexes in vitro (in a test tube) and in vivo. In
the topological approach to enzymology experimental protocol, molecular
biologists react circular DNA substrate with enzyme and capture enzyme
signature in the form of changes in the geometry (supercoiling) and topology
(knotting and linking) of the circular substrate. The mathematical problem
is then to deduce enzyme mechanism and synaptic complex structure from
these observations.
Polymer Conformations
Polymers in dilute solution can be modeled by means of selfavoiding
walks on a lattice, the lattice spacing serving to simulate volume exclusion.
Topological entanglement (knotting and linking) restricts the number of
configurations available to a macromolecule, and is thus a measure of configurational
entropy. A linear polymer can be modeled as a selfavoiding walk (SAW)
on the simple cubic lattice; a ring polymer can be modeled as a selfavoiding
polygon (SAP) on that same lattice. Microscopic topological entanglement
of polymer strands is believed to effect macroscopic physical characteristics
of polymer systems, such as the stressstrain curve, rubber elasticity,
and various phase change phenomena. Physical properties of semicrystalline
polymers are believed to strongly depend on entanglement of polymer strands
in the amorphous region. Knots in linear polymers may be trapped as "tight
knots" by the crystallization procedure. The dependence of knotting probability
on chain thickness can be exploited in a cyclization reaction on linear
DNA to determine the effect of salt concentration on chain diameter due
to Coulomb shielding. Mathematical models include discrete models on regular
lattices, and continuum models in 3space. In the asymptotic regime (lengths
going to infinity) on the simple cubic lattice and in the continuum,one
can prove that almost all sufficiently long chains are knotted, almost
all sufficiently long circles are chiral, and that the topological entanglement
complexity (measure in many ways) goes to infinity at least linearly with
the length. For short chains, the pivot algorithm is known to be ergodic
on selfavoiding walks and polygons in the simple cubic lattice, and it
provides a computationally efficient method for computing entanglement
statistics for short chains. Energetic terms can be introduced into the
in the Metropolis Monte Carlo computations for knot probability to simulate
both solvent quality and Coulomb shielding, producing knot probability
curves that qualitatively agree with laboratory random knotting results
and other Monte Carlo models (the wormlike model) which include volume
exclusion.
Human Brain Project
As a member of an interdisciplinary Human
Brain Project research team, I am interested in the mathematical analysis
and visualization of human brain functional data. We use cerebral blood
flow as a marker for neural activity, obtaining data on blood flow using
the modalities of positron emission tomography (PET) and functional magnetic
resonance imaging (fMRI). My student Ivo
Dinov and I are investigating the use of fractal and wavelet encoding
of brain architecture (high resolution magnetic resonance imaging (MRI)
scans) and 3D images of activation foci produced by PET and fMRI scans.
The signaltonoise ratio is low in these human brain functional modalities,
and there are serious difficulties inherent in comparing functional data
across scans, subjects, groups and modalities. We intend to use these
new encoding algorithms, plus other geometrical and topological ideas to
aid the group effort in the study of human brain functional data.
