Applied Mathematics & Computational Biology

Ph.D. Mathematics
M.S. Mathematics
M.A. Conceptual Foundations of Science
B.A. Physics and Philosophy

Statement of Research Interests

My research focuses on  complex adaptive behavior
arising in biological systems ranging from biochemical networks
within living cells to networks of neurons in the central nervous
systems.  I use numerical and analytical techniques to
investigate principles of biological computation in a
wide variety of model systems.

As a graduate student in mathematics under Prof. Jack Cowan at
the University of Chicago I used equivariant bifurcation theory
and Monte Carlo simulations to examine large-scale pattern
formation in the neural networks of the mammalian visual
system, both during embryonic development and during geometric
visual hallucinations.

As a postdoctoral research associate in the computational
neurobiology laboratory of Prof. Terrence Sejnowski at the Salk
Institute for Biological Studies I have pursued two lines of
work.  In the first project, I investigated how neurons
encode information in their patterns of electric discharge.
This work drew on nonlinear oscillator theory and simulations of
low-dimensional stochastic dynamical systems to elucidate the
role of dynamical attractors and bifurcations in neural coding.

In the second project, which lays the foundation for my future
work, I constructed a computational framework for simulating a
spatially heterogeneous network of biochemical reactions.  Biochemical
networks are of fundamental importance in the control and behavior of
individual cells; understanding their function holds the key to
synthesizing bioinformatics data into working models of
living cells.

Analysis and Simulation of Signal-Transduction Networks

All cells, from nerve cells in the brain to single-celled
microorganisms, communicate with other cells and with their
environments via networks of biochemical interactions known as
signal-transduction networks.  Discovering the properties
and functions of biochemical networks is fundamental to
understanding biological systems ranging from the immune system
to learning and memory.

In collaboration with Prof. William Loomis (UCSD Biology) and
Prof.  Herbert Levine and Wouter-Jan Rappel (UCSD Center for
Theoretical Biological Physics) I have constructed a numerical
platform based on finite-element analysis for solving
general boundary-coupled reaction-diffusion partial differential
equations in arbitrary two-dimensional geometries.  This platform
is applicable to any signal-transduction system; we have used it
to understand the mechanism by which eukaryotic cells such as the
social amoeba Dictyostelium discoideum choose a direction
of movement in a chemoattractant gradient.

Building on this numerical platform, my future work will address
two fundamental problems in modeling biochemical networks:
stochasticity and principled reduction of complexity.

Spatially Distributed Stochastic Biochemical Networks

Although biochemical networks involve familiar laws of chemical
kinetics, it has become clear that classical analytical methods,
based on ordinary differential equations, are inadequate for
understanding their function.  Unlike reagents in a test tube,
reactants in living cells are often localized to small subvolumes
in a cell.  The finite-element method naturally accommodates
spatially heterogeneous chemical reactions; such heterogeneities
play a key role in establishing a eukaryotic cell's orientation
for directed motility.

In addition to nonuniform spatial distribution of reactants,
cellular biochemical systems may contain as few as dozens of
interacting particles, leading to significant statistical
fluctuations and randomness in the chemical dynamics.  I am
now extending the finite-element method to include the effects of
fluctuations.  I use a master-equation formulation in which
transitions between finite elements and transitions between
chemical states have an equal footing.  This approach to
stochastic partial differential equations bridges the classical
ODE analysis of chemical systems and the molecular dynamics codes
that track each ion and molecule individually but are too
computationally costly for modeling an entire cell.
 

Principled Dimension-Reduction of Complex Networks

The rapidly growing data characterizing biochemical signaling
pathways reveal an enormous complexity of chemical components,
many of which can exist in multiple active or inactive states.
By viewing a stochastic chemical reaction system as a flow of
probability on a graph, one can reduce the descriptional
complexity of the system by aggregating multiple chemically
distinct states into a smaller number of effective or
``coarse-grained'' chemical states.  I am developing a principled
approach to simplifying complex reaction networks, based on
error-minimization techniques borrowed from communication theory.
In analogy with the application of information theory to the
behavior of neurons and neural networks, I am developing the
analysis of biochemical signal-transduction in
information-theoretic terms.  Establishing principles of
computation in biological systems should shed light on the
mechanisms governing eukaryotic chemotaxis, plasticity of
synaptic connections in the brain, and other important signaling
networks.

In order to harness the volumes of quantitative data becoming
available through high-throughput genomic and proteomic analyses,
practitioners of bioinformatics will have to rely on
quantitative biophysical models of working cells and
subcellular processes.  My research provides the numerical and
analytical tools that will be essential to understanding the
complex biochemical interactions controlling cell function.