Mathematical Biology

Fragments of fish pigment cells can form and center aggregates of
pigment granules by dynein-motor-driven transport along a
self-organized radial array of microtubules (MTs). I will present a
quantitative model that describes pigment aggregation and MT-aster
self-organization and the subsequent centering of both structures.
The model is based on the observations that MTs are immobile and
treadmill, while dynein-motor-covered granules have the ability to
nucleate MTs. From assumptions based on experimental observations,
I'll derive partial integro-differential equations describing the
coupled granule-MT interaction. Analysis explains the mechanism of
aster self-organization as a positive feedback loop between motor
aggregation at the MT minus ends and MT nucleation by motors.
Furthermore, the centering mechanism is explained as a global
geometric bias in the cell established by spontaneously-nucleated
microtubules. Numerical simulations lend additional support to the
analysis. The model sheds light on role of polymer dynamics and
polymer-motor interactions in cytoskeletal organization.

• Local Pulse Analysis for RD equations
• Actin Waves
• Matlab examples and exercises

Here I return the pattern-formation discussion to the context of symmetry breaking and chemical patterns of GTPases such as Cdc42, Rac and Rho in cell polarization. I explain the spatial context of GTPase distributions in a stimulated chemotactic cell and derive the system of partial differential equations (PDEs) for these. These examples motivate the usefulness of LPA as an additional tool in testing for pattern formation and polarizability.

This is a brief summary of a collaborative project with William Bement on patterns of Cdc42 and Rho that arise when the surface of a cell (frog egg, Xenopus oocyte) is wounded. Based on experimental observations and previous discoveries about proteins such as Abr (a GEF/GAP that activates Rho and inactivates Cdc42) were were able to derive a model that captures aspects the phenomena. The model, in turn, inspired several new experiments, notably those where wounds in close proximity were studied. The observed patterns of overlap of influence from neighbouring wounds can be explained by the model.

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