Mathematics

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].

• 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

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.

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.

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.