Current Research Projects1. Models of "Vulnerable" Atherosclerotic PlaqueCardiovascular disease (CVD) affected about 80 million Americans in 2006 and was responsible for 800,000 deaths. One common form of CVD is atherosclerosis which is a ``furring'' of the artery walls leading to a narrowing of the vessel lumen. It was originally thought that plaque buildup would make patients more susceptible to Myocardial Infarction (MI) or heart attacks. However, doctors now think that the rupture of certain plaques called Thin Capped Fibroatheromas or ''vulnerable plaques'' are responsible for most fatalities. In one study, 73% of all deaths from MI were caused by plaque rupture. This has led to the definition and characterization of vulnerable plaque which is thought to be the major culprit behind MIs. The formation of plaque is a very complicated process. Some of the important
factors are shown in the figure above. Our group is
trying to quantify plaque growth in order to aid in the detection and diagnosis
of plaque. Our research aims to answer some
important basic questions concerning plaque growth. For example, how
do shear stresses affect the leukocyte adhesion cascade (LAC) and, subsequently,
how does the LAC affect plaque formation? Which factors determine
whether a plaque remodels inwardly or outwardly? Why do necrotic
cores form inside vulnerable plaques and what determines their shape?
Movie A Movie B Movie C Movie D Movie E 2. Stochastic Inverse Problems The study of stochastic processes is important in many branches of mathematics, science and engineering. One example of a stochastic process is a (persistant) brownian motion where changes in a walker's position arise from two components: a deterministic drift and a random noise. Now consider a situation where one only has knowledge of the exit times of a random walk, i.e. one only knows the times at which the random walker reaches a certain value. Is it possible to infer the drift from these times? This problem is a simple example of a Stochastic Inverse Problem . Such problems have great importance in biology. For example, under certain modeling assumptions, they can be used to interpret Dynamic Force Spectroscopy (DFS) data in the context of protein folding and chemical bond rupture. In DFS experiments, an Atomic Force Microscope is used to slowly unfold a macromolecule. Because thermal fluctuations are important at such small scales, the data generated by such experiments consists not of a single rupture force, but a distribution of such forces. One active area of research is to interpret these data sets. What can the distribution of rupture forces tell us about how macromolecules fold and unfold? The extent to which a macromolecule has unfolded can be described in terms
its bond coordinate which, qualitatively, can be thought of as a measure
of its endtoend distance. Often, the unfolding can be modeled as a random
walk for its bond coordinate. A DFS experiment can be thought of
as an activated escape of a brownian particle from a potential well, which
accounts for all the
intramolecular forces that hold the molecule in a
particular configuration. The goal then
is to infer the shape of the potential well from rupture time/force data.
Previous Research Projects3. DNA Search and RepairDNA repair enzymes are responsible for locating and repairing damaged bases on DNA. Without them, mutations in the DNA can occur, ultimately giving rise to diseases like cancer. An important problem that has puzzled biophysicists since the 70s is how enzymes are able to find small targets (such as damaged bases) on a long DNA strand so rapidly. For example the time to perform a 1D random search along a DNA strand, with the enzyme permanently associated with the DNA, is so long that it exceeds even the life cycle of the cell! Likewise, a 3D random search within the volume of a cell nucleus (or within the cell itself for prokaryotes) yields search times that are inconsistent with experimental reaction rates. In the 1980s, a theory called facilitated diffusion was proposed
to explain fast search times for searching enzymes. The theory
posited that up to a 100 fold speedup in the search could occur
if a combination of 1D and 3D searches was used. However
facilitated diffusion requires certain conditions to be met.
For example, the enzyme must spend half the search time in 1D and half
in 3D. These conditions are seldom met in practice. Subsequently
many researchers have tried to refine and extend
the theory to real biological systems.
A vicinal surface is one that results from a slight miscut from a high symmetry plane of a crystal. As a result, the evolution of the surface is mediated by the motion of discrete atomicheight steps. These steps change their position in response to attachmentdetachment kinetics at step edges and diffusion on terraces (the region between steps). For relaxing axisymmetric nanostructures under nearequilibrium conditions, the circular steps of these nanostructures are subject to two kinds of physical effect. First, the radii of steps shrink because of step line tension. Second, stepstep repulsions (stemming from entropic and/or elastic interactions) ensure that steps cannot get too close to each other. As relaxation proceeds, a circular facet develops on the top of the structure and expands as the inner most steps repeatedly collapse. It has been known for some time (e.g. see publication #4 below) that within a macroscopic treatment, nanostructure relaxation below the roughening temperature is described by nonlinear PDEs that are singular at facet edges. These singularities are intimately related to the nonanalyticity of the surface free energy below the roughening temperature. One way to to solve these problems is to regard the facet edge as a free boundary whose position is time dependent and unknown a priori (see publication #3). The question that naturally arises is what the boundary conditions should be at the facet edge. In publication #1, we propose a boundary condition that depends on the motion of the inner most step: if the step collapse frequency is large, the facet expands quickly. On the other hand, if the inner most step does not move at all, the facet is frozen in time. The boundary condition bridges two length scales, coupling together the downward macroscopic translation of the facet with microscopic step collapses. Application of this boundary condition yields solutions that are in excellent agreement with simulation results. In publication #2, we study finite and semiinfinite collections of
straight and circular steps,
comparing the discrete equations to their respective continuum PDEs.
Boundary conditions
are derived from simple physical considerations at the facet edges.
For straight steps,
simple similarity solutions exist which  in some sense  provide an exact
description of the step motion.
The solutions have different scalings in time depending on whether
the system is semiinfinite or strictly finite.
For circular steps where stepstep interactions are weak compared to step line tension,
the continuum PDE reduces to a kinematic wave equation.
Shockwave theories and the method of characteristics can be used to treat facet translation
in this case.
Multirate integration of a set of Ordinary Differential Equations (ODEs) involves taking time steps of different size for different components in the solution. This strategy is especially useful when a small fraction of the components evolves on faster time scales than the rest. One of the main difficulties with implementing multirate
methods is that slower components in the ODE system
have to be interpolated accurately. Another difficulty
is that it is unclear what order of interpolant is needed
if the multirate method is to be of nth order.
Can this interpolation be implemented without using any
extra function evaluations from the ODE?
Our group is currently developing
a 4th order multirate method for locally coupled ODEs.
Our aim is to apply this
method to a wide variety of PDEs that have been discretized
using the method of lines.
