# Mathematics

## Mathematical Cell Biology Summer Course Lecture 15

Cell Mechanics #4: Applications of thermal forces. Elastic

Brownian ratchet [Mogilner and Oster 1996 Biophys J]; Pulling by a

depolymerizing microtubule, master equations in discrete state space

[Peskin and Oster 1995 Biophys J]; Gel symmetry breaking [van der

Gucht et al 2005 Proc Natl Acad Sci].

## Microtubules, - polymer size distribution - and other balance equation models

I introduce the differences between microtubules and actin biopolymers, and describe the growing and shrinking phases (with catastrophe and rescue transitions). The equations for polymer size distributions are explained and related to balance equations in a more general setting. The generic 1D balance equation is derived, and special cases of transport and diffusion are explained in both continuous and discrete settings.

## Mathematical Cell Biology Summer Course Lecture 13

Cell Mechanics #3: Thermal forces. Z-rings in a liposome

[Cytrynbaum et al 2012 Phys Rev E]; Fokker-Planck equations, the

Einstein relation and the principle of detailed balance;

Diffusion-limited attachment, Kramer rate theory, Bell~Rs Law;

Dimer-level microtubule assembly and Gillespie stochastic simulation

[vanBuren et al 2002 Proc Natl Acad Sci].

## 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.