Mathematical Biology

Here I connect some of the background already discussed to the concepts of GTPase activity and bistability in the context of cell polarization. I explain in detail how the ideas of bistability were used to reject some competing hypotheses for mutual interactions of Cdc42 and Rho (two GTPases implicated in cell motility and polarization), and how mathematical models were then gradually assembled based on the remaining hypotheses. I discuss both mutual inhibition and positive feedback as
possible mechanisms. I then introduce the evidence for Cdc42-Rho interactions based on a collaboration with William Bement. This is further explained in a lecture by Cory Simon, former UBC PhD student.

In this talk, I will describe a smoothed particle hydrodynamics method for simulating the motion and deformation of red blood cells. After validating the model and numerical method using the dynamics of healthy red cells in shear and channel flows, we focus on the loss of red cell deformability as a result of malaria infection. The current understanding ascribes the loss of RBC deformability to a 10-fold increase in membrane stiffness caused by extra cross-linking in the spectrin network. Local measurements by micropipette aspiration, however, have reported only an increase of about 3-fold in the shear modulus. We believe the discrepancy stems from the rigid parasite particles inside infected cells, and have carried out 3D numerical simulations of RBC stretching tests by optical tweezers to demonstrate this mechanism. Our results show that the presence of a sizeable parasite greatly reduces the ability of RBCs to deform under stretching. Thus, the previous interpretation of RBC-deformation data in terms of membrane stiffness alone is flawed. With the solid inclusion, the apparently contradictory data can be reconciled, and the observed loss of deformability can be predicted quantitatively using the local membrane elasticity measured by micropipettes.

• Cell biology imaging techniques
  • 1. Introduction: Basic optics | Phase contrast | DIC | Mechanism of fluorescence | Fluorophores
  • 2. Fluorescence microscopy: Fluorescent labelling biological samples |
    Epifluorescence microscopy |
    Confocal fluorescence microscopy
  • 3. Advanced techniques: FRAP | FRET | TIRF | Super-resolution imaging
    (time permitting)
  • 4. FRAP data and modelling integrin dynamics

• Cell biology imaging techniques
  • 1. Introduction: Basic optics | Phase contrast | DIC | Mechanism of fluorescence | Fluorophores
  • 2. Fluorescence microscopy: Fluorescent labelling biological samples |
    Epifluorescence microscopy |
    Confocal fluorescence microscopy
  • 3. Advanced techniques: FRAP | FRET | TIRF | Super-resolution imaging
    (time permitting)
  • 4. FRAP data and modelling integrin dynamics

This opening lecture lists some of the questions and issues propelling current research in Cell Biology and modelling in this field. I introduce basic features of eukaryotic cells that can crawl, and explain briefly the role of the actin cytoskeleton in cell motility. I also introduce the biochemical signalling that regulates the cytoskeleton and the concept of cell polarization. By simplifying the
enormously complex signalling networks, and applying tools of mathematics (nonlinear dynamics, scaling, bifurcations), we can hope to get some understanding of a few of the basic mechanisms that areresponsible for symmetry breaking, robustness, pattern formation, self-assembly, and other cell-level phenomena.