Extracting Essential Features of Biological NetworksAbstract
Because biological signaling networks have many components, it has
become common to model such networks using large systems of coupled
ordinary differential equations. However, there is as yet no simple way
of determining how solutions to large systems depend on their
parameters. In contrast, large systems of differential equations
describing electronic circuits are routinely reduced to simpler systems
that quantitatively capture circuit behavior using lumped parameters for
resistance, capacitance, and inductance. We show that biological
signaling networks can similarly be reduced to systems involving a few
equations and effective parameters. The effective parameters lump the
system's many components together, yielding a simplified system that
contains within it information on all of the many components. We apply
this method to 2 examples from recent literature: 1) A model of heat
shock response in e. Coli consisting of 31 equations and 48 parameters.
We reduce this model to just 1 equation and 3 effective parameters.
The reduced system quantitatively agrees with the original, and
demonstrates that feedback loops do not necessarily confer a faster heat
shock response at lower cost, as had been claimed. 2) Beta-catenin
degradation in the Wnt signaling network - a model of 14 equations is
reduced to 1-3 equations, and the features that determine the rate of
beta-catenin degradation are extracted.
Princeton University, Biophysics Seminar, December 5^{th}, 2008
NYU, Mostly Biomathematics Lunchtime Seminar, November 11^{th}, 2008

Self-Assembly of Spherical Colliodal Particles at Low N Abstract
The number of rigid structures that a system of N particles
can form grows exponentially with N. Stabilizing any one structure over
all others is thus a challenging problem. We consider a system of N
spherical colloidal particles that cannot deform or overlap, and which
exhibit a short-range attractive force. We present a method, using
graph theory and geometry, that solves for all possible rigid packings
of N particles - the resultant set of packings is provably
complete. We then present a mechanism that is capable of stabilizing
any one structure over all others (in the zero temperature limit), and
which is experimentally realizable - thereby, potentially allowing us to
direct the self-assembly of a desired structure. We compare to
preliminary experimental results.
Harvard University, WAM Seminar, December 16^{th}, 2008
NYU, AML Seminar, October 9^{th}, 2008