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

Determining kinetic parameters underlying protein dynamics

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 compartments

An 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.



Bursting oscillations

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.