||Forbidden Families of
The general area of investigation is extremal combinatorics. The goal is to expand on the M.Sc. work of Christina Koch and also continue the work of Maxwell Allman and myself on bounds for a special family of configurations. The following is a typical problem given in matrix language. Let m be given and let F be a given kxt (0,1)-matrix. Let A be an mxn (0,1)-matrix with no repeated columns and no submatrix that is a row and column permutation of F. We seek bounds on n in terms of m,F. (so called problem of Forbidden Configurations). It is perhaps surprising that n < cm^k, for some c, but we can do even better for many F.
Interested students could contact my previous USRA students: Foster Tom and Maxwell Allman.
||Cluster Analysis of Super Resolution
This project is to work on novel methods for interpreting data from a very modern microscopic imaging technique, dSTORM. The images, which are taken at UBC, are large - each contains ~1-4 x 10 6 data points. The points (each representing light emitted by a single fluorescent molecule attached to a defined protein in a cell) are, as expected, distributed in discrete clusters, but the shapes of the clusters are highly irregular. The goal of this project will be to learn about existing methods, understand why they fail for automatic detection and classification of the observed clusters, and then develop new approaches to this problem. The lab produces novel code for displaying and analysing data acquired from both light and electron microscopes and we would prefer an individual who is familiar with the programs we are using; C++, Matlab and OpenGL. This project will be joint supervised by Edwin Moore (Cell and Physiological Sciences) and Dan Coombs (Mathematics).
Molecular-scale simulation of calcium ions within cardiac tissue.
We want to model calcium ion movement and interactions with the resident proteins within the dyadic cleft (which is within cardiac cells). Input data would be the geometry of the cleft, the position of the relevant proteins and their assumed behaviour in response to calcium ions as well as to other intracellular signaling molecules. The output of the model would be a ‘calcium spark’, which is the calcium transient produced by a single dyadic cleft. Since the volume of the dyadic cleft is measured in femtolitres, the model will be constructed using stochastic approaches. Experimental results show that the positions of the relevant molecules within the cleft are subject to changes in response to both physiological and pathological factors. Changes in the molecules’ positions are also correlated with changes in both the magnitude and kinetics of the calcium spark. The goal of this project is to duplicate the experimental results and to make testable predictions. This project will be joint supervised by Edwin Moore (Cell and Physiological Sciences) and Dan Coombs (Mathematics).
|| Modeling collective migration of cells
during embryonic development.
Biologists have discovered remarkable patterns of collective cell migration during early development of animal embryos. For example, the so-called neural crest cells (NCC) migrate in streams along the spine of the embryos of chicks, frogs and zebrafish. The migration is very rapid, and resembles metastasis of cancer cells so much that NCC migration has been used as a model for the latter. Moreover, NCCs from different sources manage to stay unmixed while migrating side by side. Later, they seem to be directed to different destinations along the spine, and then toward the front of the body, where they form various tissues and organs.
There are several mysteries about the collective migration. How do cells interact with each other to maintain cohesion among those from the same source, while keeping a boundary between cell clusters from difference sources? How do the cells decide where to stop or turn into a different route? The intensive efforts by biologists have produced some hypotheses. But as these questions involve the intimate coupling between biochemical signaling and cell mechanics, answering them requires the help of quantitative analysis.
In collaboration with developmental biologists, we have been developing mathematical models on various morphogenetic processes that test the existing hypotheses and strive for a clear in-depth quantitative understanding. These models typically involve ODEs describing the dynamics of the signaling molecules and molecular motors, as well as ODEs or PDEs governing the mechanical behaviour of the cells and tissues. This USRA project will study the signaling pathways controlling the cell-cell communication during collective cell migration, and explore how the chemo-mechanical coupling leads to different patterns. The student will help build the models and carry out computations to explore their predictions. See more background information on my webpage http://www.math.ubc.ca/~jfeng/ under "Research".