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Graduate Student Supervision
Doctoral Student Supervision (Jan 2008 - Nov 2019)
Individually and collectively, cells are organized systems with many interacting parts. Mathematical models allow us to infer behaviour at one level of organization from information at another level. In this thesis, I explore two biological questions that are answered through the development of new mathematical approaches and novel models.(1) Molecular motors are responsible for transporting material along molecular tracks (microtubules) in cells. Typically, transport is described by a system of reaction-advection-diffusion partial differential equations (PDEs). Recently, quasi-steady-state (QSS) methods have been applied to models with linear reactions to approximate the behaviour of the PDE system. To understand how nonlinear reactions affect the overall transport process at the cellular level, I extend the QSS approach to certain nonlinear reaction models, reducing the full PDE system to a single nonlinear PDE. I find that the approximating PDE is a conservation law for the total density of motors within the cell, with effective diffusion and velocity that depend nonlinearly on the motor densities and model parameters. Cell-scale predictions about the organization and distribution of motors can be drawn from these effective parameters.(2) Rho GTPases are a family of protein regulators that modulate cell shape and forces exerted by cells. Meanwhile, cells sense forces such as tension. The implications of this two-way feedback on cell behaviour is of interest to biologists. I explore this question by developing a simple mathematical model for GTPase signalling and cell mechanics. The model explains a spectrum of behaviours, including relaxed or contracted cells and cells that oscillate between these extremes. Through bifurcation analysis, I find that changes in single cell behaviour can be explained by the strength of feedback from tension to signalling. When such model cells are connected to one another in a row or in a 2D sheet, waves of contraction/relaxation propagate through the tissue. Model predictions are qualitatively consistent with developmental-biology observations such as the volume fluctuations in a cellular monolayer. The model suggests a mechanism for the organization of tissue-scale behaviours from signalling and mechanics, which could be extended to specific experimental systems.
Cell migration is a complex interplay of biochemical and biophysical mechanisms. I investigate the link between individual and collective cell behaviour using mathematical and computational modelling. Specifically, I study: (1) cell-cell interactions in a discrete framework with a spatial sensing range, (2) migration of a cluster of cells during zebrafish (Danio rerio) development, and (3) collective migration of cancer cells and their interactions with the extracellular-matrix (ECM).My 1D model (1), is approximated by a continuum equation and investigated using asymptotic approximations, steady-state analysis, and linear stability analysis. Analysis and computations characterize regimes corresponding to cell clustering, and provide a link between micro and macro-scale parameters. Results suggest that drift (i.e. due to chemotaxis), can disrupt the formation of cellular aggregates.In (2), I investigate spontaneous polarization of a cell-cluster (the posterior lateral line primordium, PLLP) in zebrafish development. I use a cell-based computational framework (hybrid discrete cell model, HyDiCell3D) coupled with differential equation model to track the segregation and migration of the PLLP. My model includes mutual inhibition between the diffusible growth factors Wnt and FGF. I find that a non-uniform degradation of an extracellular chemokine (CXCL12a) and chemotaxis is essential for long-range cohesive migration. Results compare favourably with data from the Chitnis lab (NIH).I continue using HyDiCell3D in (3) to elucidate mechanisms that facilitate cancer invasion. I focus on: wound healing in a cell-sheet (2D epithelium), and cell-clusters (3D spheroids) embedded in ECM with internal signalling mediated by podocalyxin, a trans-membrane molecule. Experimental data from the Roskelley lab (UBC) motivates the model derivation. I use the models to investigate the role of cell-cell and cell-ECM adhesion in collective migration as well as the emergence of a distinct phenotype (leader-cells) that guide the migration. ECM induced disruption in the localization of podocalyxin on the cell membrane is captured in the model along with morphological changes of spheroids. The model predicts that cell polarity and cell division axis influence the invasive potential. Lastly, I developed quantitative methods for image analysis and automated tracking of cells in a densely packed environment to compare modelling results and biological data.
In response to chemical stimulation, many eukaryotic cells are able to sense the direction of the stimulus and initiate movement. To do so, the cell must break symmetry and develop a front and back in a process known as polarization.During polarization, members of the Rho GTPase family (Cdc42, Rac, and Rho) are recruited to the plasma membrane and localize to form a front and a back of the polarizing cell. These proteins exist in both active forms (on the inner surface of the membrane of the cell), and inactive forms (in the cytosol).In earlier work, I have shown that the property of membrane-cytosol interconversion, together with appropriate feedbacks, endows the Rho proteins with the ability to initiate cell polarity, resulting in a high Cdc42/Rac region, which will become the front, and a high Rho region, which will become the back of the cell.Here I show that this property of polarizability can be explained using a simplified model system comprising of a single active/inactive protein pair with positive feedback.In this model, a travelling wave of GTPase activation is initiated at one end of the domain, moves across the cell, and eventually stops inside the domain, resulting in a stable polar distribution. The key requirements for the mechanism to work include conservation of total amount of protein, a sufficiently large difference in diffusion rates of the two forms, and nonlinear positive feedback that allows for multiple homogeneous steady states to exist.Using singular perturbation theory, I explain the mathematical basis of wave-pinning behaviour, and discuss its biological and mathematical implications. I show that this mechanism for generating a chemical pattern is distinct from Turing pattern formation. I also analyze the transition from a spatially heterogeneous solution to a spatially homogeneous solution as the diffusion coefficient of the active form is increased, and show the existence of other unstable stationary wavefronts. Finally, I argue that this wave-pinning mechanism can account for a number of features of cell polarization such as spatial amplification, maintenance of polarity, and the sensitivity to new stimuli that is typical of polarization of eukaryotic cells.
Animals moving together cohesively is a commonly observed phenomenon in biology, with bird flocks and fish schools as familiar examples. Mathematical models have been developed in order to understand the mechanisms that lead to such coordinated motion. The Lagrangian framework of modeling, wherein individuals within the group are modeled as point particles with position and velocity, permits construction of inter-individual interactions via `social forces' of attraction, repulsion and alignment. Although such models have been studied extensively via numerical simulation, analytical conclusions have been difficult to obtain, owing to the large size of the associated system of differential equations. In this thesis, I contribute to the modeling of collective motion in two ways. First, I develop a simplified model of motion and, by focusing on simple, regular solutions, am able to connect group properties to individual characteristics in a concrete manner via derivations of existence and stability conditions for a number of solution types. I show that existence of particular solutions depends on the attraction-repulsion function, while stability depends on the derivative of this function.Second, to establish validity and motivate construction of specific models for collective motion, actual data is required. I describe work gathering and analyzing dynamic data on group motion of surf scoters, a type of diving duck. This data represents, to our knowledge, the largest animal group size (by almost an order of magnitude) for which the trajectory of each group member is reconstructed. By constructing spatial distributions of neighbour density and mean deviation, I show that frontal neighbour preference and angular deviation are important features in such groups. I show that the observed spatial distribution of neighbors can be obtained in a model incorporating a topological frontal interaction, and I find an optimal parameter set to match simulated data to empirical data.
Master's Student Supervision (2010 - 2018)
Individual cells adapt their morphology as a function of their differentiation status and in response to environmental cues and selective pressures. While it known that the great majority of these cues and pressures are mediated by changes in intracellular signal transduction, the precise regulatory mechanisms that govern cell shape, size and polarity are not well understood. Systematic investigation of cell morphology involves experimentally perturbing biochemical pathways and observing changes in phenotype. In order to facilitate this work, experimental biologists need software capable of analyzing a large number of microscopic images to classify cells and recognize cell types. Furthermore, automatic cell classification enables pathologists to rapidly diagnose diseases like leukemia that are marked by cell shape deformation.This thesis describes a methodology to identify cells in microscopy images and compute quantitative descriptors that characterize their morphology. Phase-contrast microscopy data is used for the purpose of demonstration. Cells are identified with minimal user input using advanced image segmentation methods. Features (e.g. area, perimeter, curvature, circularity, convexity, etc.) are extracted from segmented cell boundary to quantify cell morphology. Correlated features are combined to reduce dimensionality and the resulting feature set is clustered to identify distinct cell morphologies. Clustering results obtained from different combinations of features are compared to identify a minimal set of features without compromising classification accuracy.
Patterns are ubiquitous in nature, but the underlying mechanisms giving rise to them are not always understood. One such pattern is the inhomogeneous domain of a cell that is established during polarization. Cell polarization is a way in which cells respond to and interact with their environment. For example, white blood cells locate and destroy bacteria, and yeast cells create buds for reproduction. Signalling proteins such as GTPases are redistributed throughout the cell and, through downstream effects, rearrangement of the actin cytoskeleton follows. This redistribution can occur in response to a stimulus, such a chemoattractant, or it may be spontaneous. Because many biological details are unknown, mathematical models are developed to recreate features of cell polarization and determine the minimal modules or characteristics for these features. Polarization models are often simple, conceptual reaction-diffusion equations for one or more signalling molecules. But comparing these models is often difficult, and there are many models in the literature for different cell types or behaviour.A new method of nonlinear stability analysis, called local perturbation analysis (LPA), was developed by Stan Maree, and later Bill Holmes, to take advantage of models with substantial diffusion disparities. This method recapitulates the dynamics of a pulse applied to a reaction-diffusion system using a system of ordinary differential equations. Bifurcation analysis of these equations is relatively easy, and LPA detects pattern formation through threshold and Turing dynamics and provides bifurcation maps of these regimes for any parameter. LPA is well-suited to cell polarization models, because the signalling proteins we model often have both fast-diffusing inactive and slow-diffusing active forms. In this thesis, I introduce LPA and its use through a wave pinning model and its extension, a model for actin waves. I then review and analyze five additional cell polarization models using combinations of LPA, simulations, and Turing analysis. In many cases, I discovered new dynamics of the models. LPA helps us to map patterning regimes and their robustness to changes in parameters, and provides a new avenue for us to compare many current models for cell polarization.
A single cell, such as a frog egg, is able to repair injuries by orchestrating a localized signalling response on the plasma membrane. Proteins called Rho GTPases are recruited to, and form patterns around, the wound site. Patterning allows testable hypotheses to be made about the structure of the signalling network. Here, we extend a Rho GTPase signalling model from Simon et al. (2013) to test how a family of enzymes, protein kinase C (PKC), plays a role in cell repair signalling. Our models let PKCs affect basal Rho GTPase activation and/or inactivation rates, with increasing spatial detail. Ultimately, the model variants do not account for Rho GTPase patterning in all experiments. We suggest a new round of modelling and experiments to correct these issues.