# Scientific

## Introduction to polymerization kinetics

To understand the cytoskeleton, it helps to also gain some background in simple polymer assembly, and the mathematics used to describe it. Here I review a succession of elementary models for polymers of various types starting from a mixture consisting only of subunits, called monomers. I point out that the accumulated polymer mass over time depends on the type of underlying assembly reaction. The idea of critical monomer concentration is introduced, and shown to arise as a consequence of scaling the models. We then consider the specific case of actin polymers and show that treadmilling (growth of one end and shrinkage of the other) can occur at a particular concentration. Growth of actin filaments at their tips in discussed in the context of a transcritical bifurcation. I introduce the Mogilner-Oster thermal ratchet and its relation to cell protrusion caused by actin filament polymerization against a load force.

## Mathematical Cell Biology Summer Course Lecture 12

Cell Mechanics #2: Biopolymer mechanics. The energy

functional, Young~Rs modulus, Euler-Lagrange equations; Microtubules

buckling in vesicles [Elbaum et al 1996 Phys Rev Lett] and in cells

[Brangwynne et al 2006 J Cell Sci].

## An Excitable Contractile Cell

In 1980, Gary Odell, George Oster and coworkers published papers on a mechanochemical model for epithelial invagination (folding of a sheet of cells) in the early stages of formation of an embryo. An attractive feature of this model is that it combines a chemical switch with a simple mechanical element (a spring with variable rest-length). I discuss this model, relate it to our previous experience with biochemical switches, and to the mechanical spring-based systems described in Jun Allard's first lecture. This model also anticipates a later lecture on models for 2D cell motion based on springs and dashpot elements.

## Mathematical Cell Biology Summer Course Lecture 9

Cell Mechanics #1: Bonds, springs, dashpots and motors.

Wobbling keratocytes [Barnhart et al 2010 Biophys J]; Slip-clutch in

nerve growth cones and fixed-timestep stochastic simulation [Chan and

Odde 2008 Science]

## Small GTPases and cell polarization

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

- 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

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)

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

- 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)

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).