Mathematical Biology

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

• 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

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.

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].

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].