John Pelesko

Department of Mathematical
Sciences

University of Delaware

The
membranes of biological cells are complex structures consisting primarily of
lipids, but also contain cholesterol, carbohydrates, proteins, and other
complex macromolecules.

In
an effort to better understand biological membranes, researchers often study
artificially self-assembled vesicles or planar membranes. Typically, these
structures are built from a simple lipid bi-layer. Recently, in an effort to
tackle the question of *domain*
or *raft* formation, researchers
have begun studying lipid bi-layers, where each leaflet of the layer consists of
a different mixture of lipids and cholesterol. In these experiments,
researchers have observed phase separation within the bi-layer and the
phenomenon of induction of domain formation across the leaflets of a lipid
bi-layer. In this project, IĠd like to construct a mathematical model of phase
separation and domain induction in a lipid bi-layer. Roughly speaking, weĠd
like to understand how and why lipids of different types move and cluster
together in a self-assembled lipid bi-layer.

To
attack this problem, IĠd like to follow two lines of thought. In the first, IĠd
like to attempt to construct an Ising-like model of phase separation in a lipid
bi-layer. While the Ising model was originally proposed to study the phenomenon
of spontaneous magnetization, it is readily adaptable to modeling other systems
such as ours. In the second line of thought, IĠd like to explore continuum
approaches to the problem. This will lead us into the realm of
reaction-diffusion equations.

To model and analyze this
problem we will need to understand basic elements of biology, the mathematics
of the Ising Model, and elements of continuum mechanics. Our work will build on
the following key papers:

- Collins and Keller: "Tuning lipid mixtures to induce or
suppress domain formation across leaflets of unsupported asymmetric
bilayers," PNAS, v. 105, 1, pp. 124-128.
- Cipra: "An Introduction to the Ising Model," American
Mathematical Monthly, v. 94,10, pp. 937-959.
- Hayes: "The World in a Spin,"
American Scientist, v. 88, 5, pp. 384-388.
- Yethiraj and Weisshaar: "Why are lipid rafts not observed
in vivo?" Biophysical Journal, v. 93, pp. 3113-3119, 2007.