# Mathematics

## Mathematical Cell Biology Summer Course Lecture 18

- Combining mechanics and biochemistry
- Application of scaling to deciphering a molecular mechanism
- Actin and cytoskeleton assembly
- Actin dynamics in the (1D) cell lamellipod
- Continuity (Balance) eqs and Reaction-Diffusion eqs (PDEs)
- Bicoid gradients

## Mathematical Cell Biology Summer Course Lecture 17

Cell Mechanics #5: Membranes. Canham-Helfrich energies, the

Monge representation, Metropolis-Hastings simulation for thermal

fluctuations. Antigen bonds in T cells [Allard et al 2012 Biophys J].

## Models of T cell activation based on TCR-pMHC bond kinetics

In order for an immune cell, such as a T-cell to do its job (kill virus infected cells) it must first undergo an activation event. Activation requires the encounter of the cell surface T-cell receptors (TCRs) with bits of protein that are displayed in special complexes (peptide-MHC complexes) on the surface of a target cell. all cells of the body display such p-MHC complexes, but in normal circumstances only those perceived as infected will be destroyed by T-cells in the process of immune surveillance. In this seminar I will describe both theoretical and experimental work aiming to understand the events that culminate in the activation of the T-cell.

## Mathematical Cell Biology Summer Course Lecture 16

- 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

## 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.