# Mathematical Biology

## Mathematical Cell Biology Summer Course Lecture 21

Data Analysis Methods

## Diffusion, Reaction, and Biological pattern formation

We first consider the topic of biological patterns and then place it in the context of developmental biology and positional information. The example of the fruit fly (Drosophilla) development is used to motivate the basic questions. We next consider how chemical interaction coupled to diffusion can give rise to pattern formation. We discuss Turing's (1952) theory for pattern formation and derive the conditions for this to happen in a system of two interacting chemicals. Returning to the fruit-fly example, we observe that the mechanism for development (based on reading the level of bicoid protein) has been shown to be distinct from a Turing pattern

## Mathematical Cell Biology Summer Course Lecture 19

Data Analysis Methods

## 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].