My area of research is mathematical biology. The main objective
of my research program is to understand and explain physiological
processes through the development and analysis of mathematical models.

Spatio-temporal dynamics of proteins in the cellular nucleus

We use mathematical models to interpret
FRAP (Fluorescence Recovery
After Photobleaching) data obtained in the nucleus. The
experimental data
contains
information about the mobility of nuclear proteins. Conventional
models assume that the dynamics of the proteins are governed by
diffusion
alone, and are used to determine estimates of an effective diffusion
coefficient
for the proteins. For many proteins, the effective diffusion
coefficient
is orders of magnitude smaller than the diffusion coefficient expected
based on the molecular weight of the proteins. We have
hypothesized
that the mobility of the proteins is reduced because of their temporary
association with immobile or slowly moving nuclear structures.
Incorporating
such hypothesis in a PDE model leads to systems of reaction-diffusion
equations,
solutions of which are non-trivial. We have proposed an alternate
modelling approach, namely to use compartmental modelling, leading to a
system of ODEs much easier to analyze, and simplifying the task of
parameter
estimation. Using perturbation analysis, we have characterized
fluorescence recovery
curves. The analysis has led to a clear explanation of two
important
limiting dynamical types of behaviour exhibited by experimental
recovery
curves, namely, (1) a reduced diffusive recovery, and (2) a biphasic
recovery
characterized by a fast phase and a slow phase. The perturbation
analysis also led to a relationship of the results of the two different
modelling approaches (reaction-diffusion equations versus compartmental
modelling).

We have applied our results in the
analysis of FRAP data of both
nuclear
actin and histone H1. We showed that the FRAP data for actin is
consistent with the hypothesis that nuclear actin exists in both
monomeric
and filamentous forms (this is a highly controversial topic in cell
biology,
and although many researchers believe this to be the case, there is not
yet definitive proof), and obtained estimates of binding and unbinding
rates (of monomeric actin to and from filamentous actin). The
second application was in
the context of histone H1 dynamics. Histone H1 molecules are
either
bound to a spatially homogeneously distributed chromatin structure, or
unbound and free to diffuse. Fluorescence recovery data suggest
that
almost all of the histone H1 population is bound to the chromatin
structure,
and only a very small proportion of the histone H1 molecules are free
to
diffuse. This small proportion allows histone molecules to move
randomly
from one binding site to another, which we believe is crucial for the
functional
dynamics of histone (in particular, if histone H1 proteins were
permanently
associated with chromatin, it would be more difficult for chromatin
remodelling
factors to gain access to chromatin).

This
research answers questions about
modelling methodology in interpreting experiments
on protein dynamics. Correct determination of kinetic parameters
underlying protein dynamics is essential for further modelling work,
and
contributes to the understanding of nuclear processes.

Origin of nuclear
compartmentsAn interesting and open problem in cell
biology concerns the dynamic
nature of nuclear
architecture. Under certain conditions, such as viral infection
and
transcriptional inhibition, the nuclear architecture undergoes profound
changes, with compartments either being disassembled or enlarged.
Furthermore, most nuclear compartments are observed to disassemble
during
the M-phase of the cell cycle, and reassemble in the daughter
cells. The current focus in our research is to understand the
dynamical organization of the eukaryotic cell nucleus. In
particular,
we are addressing the mechanism responsible for the formation,
maintenance and
disappearance of speckles, which are heterogeneously distributed
nuclear compartments enriched with pre-mRNA splicing factors. It
has
been hypothesized that self-organization of
dephosphorylated splicing factors, modulated by a
phosphorylation-dephosphorylation cycle, is responsible for the origin
and disappearance of speckles. Also, it is thought that the
existence
of an underlying nuclear structure plays a major role in the
organization of splicing factors. Based on these hypotheses, we
have
derived a fourth-order aggregation-diffusion model that describes a
possible
mechanism underlying the organization of splicing factors in
speckles.
A linear stability analysis about homogeneous steady-state solutions
has shown how the self-interaction among dephosphorylated splicing
factors can result in the onset of spatial patterns. Also, a
bifurcation analysis of the model can describe how the processes of
phosphorylation and dephosphorylation modulate the onset of the
compartmentalization of splicing factors.

This research advances knowledge in the area of spatial pattern formation. Specifically, it addresses the origin of nuclear architecture, which ultimately affects nuclear function. Successful outcomes of this research puts us in a better position to predict outcomes of changes in nuclear architecture, such as seen with cancer.

This research is in collaboration with PhD student Gustavo Carrero and Dr. Michael Hendzel from the Cross Cancer Institute, Department of Oncology, University of Alberta.In my earlier research, I have focussed attention on mathematical
models of electrical activity in pancreatic beta-cells. These
cells produce and secrete the hormone insulin, which is the principal
hormone regulating the blood glucose level. In the presence of
glucose, these cells exhibit a complex pattern of oscillations called
bursting. Pancreatic beta-cells are organized in clusters, called
islets of Langerhans, and bursting is synchronized within islets.
The synchronization is due to coupling between neighbouring cells,
through low-resistance electrical pathways called gap junctions.

Modelling efforts related to the electrical activity of beta-cells can
be divided broadly into two categories, single-cell models and
coupled-cell models. Single-cell
models typically include much biophysical detail, and are used
to explain specific experimental observations. These models
represent bursting as observed in an islet, and a single-cell model can
be viewed as a model of the
behaviour of an average cell within an islet. Coupled-cell models typically
include less biophysical detail, so that attention can be focussed on
the role of coupling in
the generation of synchronized oscillations.

During the last few years, my attention has been concentrated on
coupled-cell models for pancreatic beta cells. Through
mathematical and computational analyses of these models, I have
obtained a broad and deep understanding of the role of coupling, and
its interaction with system noise and parameter variability, in the
generation of network behaviour. A recent discovery is bursting
as an emergent phenomenon (bursting obtained in a network of cells,
each of which is incapable of bursting individually).

Research on bursting oscillations continues within a broadened
context. In particular, attention is focussed on a comparison of
coupling mechanisms, and their effect on different classes of bursting
models, as well as on the behaviour of discrete-time models that
generate bursting oscillations.

Bursting oscillations are observed in many electrically excitable cells
other than pancreatic beta cells, such as thalamic neurons and
hippocampal
pyramidal neurons. This research therefore has strong reciprocal
connections with mathematical neurophysiology.

Dynamics of group formation in self-organizing collectives of individuals

(research description contributed by PhD student Raluca Eftimie; research is in collaboration with Dr. Mark Lewis)

"I am studying the dynamics of group formation in self-organizing collectives of individuals. The spatial interaction between individuals is the result of 3 types of social forces: attraction towards other members of the group, repulsion from them and a tendency to align with neighbors.

My aim is to develop a mathematical model (using a nonlinear system of partial differential equations) that takes into consideration these aspects and addresses two problems: group formation and activity patterns.

Depending on the amplitudes of these social forces, the model may have applications, for example, to the study of insect swarms (high attraction and low or no alignment), bird flocks or schools of fish (high attraction, high alignment).

The mathematical tools that I am using include: analytical methods for PDE's, bifurcation theory, perturbation theory, numerical simulations."

Cancer

Development and analysis of an integro-differential equation that
describes the re-seeding of ovarian cancer in the peritoneal cavity.

This research is in collaboration with Dr. Thomas Hillen.

Bistability in the firing patterns of motoneurons

Motoneurons are the
neurons controlling muscle contractions. The electrical activity
of these neurons is characterized by tonic firing, the frequency of
which depends on synaptic activation or the strength of current applied
to the soma. Under certain experimental conditions, vertebrate
motoneurons exhibit bistability between firing patterns. It is thought
that the bistability is the result of an interaction between L-type
calcium currents in the dendrites and spike-generating currents in the
soma. Experimentally and theoretically, it has been observed that
the attenuation of a signal travelling from the soma to the dendrites
is less than the attenuation of a signal travelling in the opposite
direction. The Booth-Rinzel model (a popular mathematical model
for the electrical activity in motoneurons exhibiting bistability) does
not reproduce this property, possibly because it uses a constant
coupling conductance between the somatic and dendritic compartments.
This means that the predictions of the model on the experimental
conditions that give rise to bistability likely are inaccurate.
The goal of this project is to determine modifications to the model so
that it exhibits the correct voltage attenuation properties while
maintaining bistability. Successful completion of this project
will provide insight into the relationship between voltage attenuation
and bistability.

This research is carried
out in collaboration with Dr. Kelvin Jones
from the Department of Biomedical Engineering at the University of
Alberta.

Vasomotion

Vasomotion refers to the spontaneous oscillatory constriction of
small
blood vessels. Vasomotion is observed to be enhanced in states of
high resistance such as hypertension, and decreased in states of low
resistance
such as pregnancy. However, the functional significance of
vasomotion
remains controversial. In collaboration with Dr.
Sandra Davidge from the Departments of Obstetrics/Gynaecology and
Physiology,
MSc student Chris Meyer and I have resolved a controversy existing in
the
literature regarding the effect of vasomotion on vascular
resistance.
We currently are using mathematical models to investigate mechanisms
thought
to underlie the phenomenon of vasomotion. Another issue of
interest
is to determine the influence of the stiffness of the blood vessels on
the amplitude of the vasomotion.

Diabetes

Diabetes is a disease of the glucose regulatory
system characterized
by hyperglycemia. Type 2 diabetes (also known as adult-onset or
non-insulin-dependent
diabetes) is associated with a deficit in the mass of beta cells,
reduced
insulin secretion, and resistance to the action of insulin.
However,
the relative contribution and interaction of these defects remains to
be
clarified. Models are used to study the dynamics of the
interaction between beta cell
mass, insulin and glucose.

This research is in collaboration with Dr. Diane Finegood from Simon Fraser University and her PhD student Brian Topp.