# Mathematics

## Small GTPases and cell polarization

Speaker:

Leah Edelstein-Keshet
Date:

Fri, May 4, 2012
Location:

PIMS, University of British Columbia
Conference:

Mathematical Cell Biology Summer Course Abstract:

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.

## Mathematical Cell Biology Summer Course Lecture 7

Speaker:

Raibatak (Dodo) Das
Date:

Fri, May 4, 2012
Location:

PIMS, University of British Columbia
Conference:

Mathematical Cell Biology Summer Course Abstract:

- 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

## A Particle Based Model for Healthy and Malaria Infected Red Blood Cells

Speaker:

James J. Feng
Date:

Thu, May 3, 2012
Location:

PIMS, University of British Columbia
Conference:

Mathematical Cell Biology Summer Course Abstract:

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.

## Switches, Oscillators (and the Cell Cycle)

Speaker:

Leah Edelstein-Keshet
Date:

Thu, May 3, 2012
Location:

PIMS, University of British Columbia
Conference:

Mathematical Cell Biology Summer Course Abstract:

By incorporating positive and negative feedback into phosphorylation cycles of proteins such as GTPases (described in Lecture 2), one arrives at a biochemical mini-circuits with properties of a switch or an oscillator. I provide examples from papers by Boris Kholodenko. I also show the connection between bistability and hysteresis and relaxation oscillations (e.g. in the Fitzhugh-Nagumo model). I briefly
discuss applications of such ideas to models of the cell cycle proposed over the years by John Tyson.

## Mathematical Cell Biology Summer Course Lecture 5

Speaker:

Raibatak (Dodo) Das
Date:

Thu, May 3, 2012
Location:

PIMS, University of British Columbia
Conference:

Mathematical Cell Biology Summer Course Abstract:

- 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

## Simple biochemical motifs (1, 2, & 3)

Speaker:

Leah Edelstein-Keshet
Date:

Wed, May 2, 2012
Location:

PIMS, University of British Columbia
Conference:

Mathematical Cell Biology Summer Course Abstract:

This triplet of introductory lectures summarizes a few of the most basic biochemical models with the simple rate equations that they satisfy. I describe production-decay, with Michaelis-Menten and sigmoidal terms, showing how the latter can lead to bistable behaviour and hysteresis. I describe two bistable genetic circuits: the toggle switch by Gardner et al (2000) Nature 403, and the phage-lambda gene by Hasty et al (2000) PNAs 97. The idea of bifurcations is discussed. Finally, I introduce
phosphorylation cycles, and show that sharp responses can arise when the enzymes responsible (kinase and phosphatase) operate near saturation. (This is the so called Goldbeter-Koshland ultrasensitivity).

## Mathematical Cell Biology Summer Course Lecture 3

Speaker:

Raibatak (Dodo) Das
Date:

Wed, May 2, 2012
Location:

PIMS, University of British Columbia
Conference:

Mathematical Cell Biology Summer Course Abstract:

- 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

## Small Number and the Basketball Tournament

Speaker:

Veselin Jungic
Speaker:

Mark Maclean
Speaker:

Rena Sinclair
Date:

Tue, May 1, 2012
Location:

Simon Fraser University, Burnaby, Canada
Location:

University of British Columbia, Vancouver, Canada
Conference:

BIRS First Nations Math Education Workshop Abstract:

The mathematical context of the third story, Small Number and the Basketball Tournament, contains some basic principles of combinatorics. The plot of the story and the closing question are structured in a manner that allows the moderator to introduce the notion of permutations and combinations. Since the numbers used in the story are relatively small, this can be used to encourage the young audience to explore on their own. Mathematics is also present in the background. Small Number and his friends do mathematics after school in the Aboriginal Friendship Centre. He loves playing the game of Set and when he comes home his sister is just finishing her math homework. Small Number and his friend would like to participate in a big half-court tournament, and so on.

For more details see http://mathcatcher.irmacs.sfu.ca/content/small-number

For more details see http://mathcatcher.irmacs.sfu.ca/content/small-number

## Mathematical Cell Biology Summer Course Lecture 2

Speaker:

Raibatak (Dodo) Das
Date:

Tue, May 1, 2012
Location:

PIMS, University of British Columbia
Conference:

Mathematical Cell Biology Summer Course Abstract:

- 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

## Introduction

Speaker:

Leah Edelstein-Keshet
Date:

Tue, May 1, 2012
Location:

PIMS, University of British Columbia
Conference:

Mathematical Cell Biology Summer Course Abstract:

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.