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: