Computational Bio-Systems

Introduction
Recently I have extended my research to study problems in the biological sciences. I started my research by investigating the dynamics of multicomponent lipid bilayer vesicle membranes. As the principal components of living cells, bilayer membranes play an important role in many cell functions.  Typically, membranes contain multiple lipid components, transmembrane proteins, etc, and hence have complex structures and shapes. As in most biological problems, the shape of these membranes is intimately connected with their biological function. Recent experimental results on vesicle membranes have shown that phase transitions of lipid components may occur and these can lead to dramatic morphology changes of the vesicle (see the recent experiment results on giant unilamellar vesicles (GUVs) by T. Baumgart, S.T. Hess and W.W. Webb, Nature, 425, Oct 23, 2003). This is thought to occur because the different surface lipid components have different surface energies and bending stiffnesses. While there have been several studies of the equilibrium of multicomponent vesicles, the theoretical and computational modeling of vesicle dynamics is still at an early stage.
Giant unilamellar vesicles (GUV)) with initial ternary mixtures of lipid component and cholesterol may separate into binary components, ordered (blue) and disordered(red). For experiment details, see Baumgart et al, Nature (2003).	Initial configuration: uniform distribution. Red curve: curvature; blue curve: concentration of a phase. Later time: the membrane experiences phase decompositions under a shear flow applied at the top and bottom.
A bud forms on and eventually separates from the mother-membrane. For details, See McMahon, Gallop. Nature (2005).	Bud formation due to surface tension variation along a biomembrane. Phase on left part of the interface has lower surface energy. Length and area are conserved. [a]: initial phase configuration and velocity streamlines; [b],[c] and [d]: phase configuration and velocity streamlines at later times.
Mathematical Model and Numerical Methods
Our group has focused on the development of theoretical and computational models of vesicle dynamics. The energy of the system includes the surface energy of the vesicle that depends on the concentration of the surface components, the normal bending energy that depends on the mean and Gaussian curvatures of the surface, and the potential energy that describes the line tension between surface phases. Using an energy variation approach, we have derived a set of thermodynamically consistent governing equations that couple the surface phase transitions to the interface morphology and evolution through generalized surface tension forces imparted to the fluid contained in and surrounding the vesicle.  On the surface, the phases evolve according to a high-order, advection-reaction-diffusion equation of Cahn-Hilliard type. Because of the presence of high-order nonlinear interactions in these physical processes, it is very challenging to analyze and simulate the phase decomposition and the motion of phase boundaries on a moving active surface in a viscous fluid. To solve the system of equations numerically, we have developed a method that combines the immersed interface method to solve the flow equations and the generalized Laplace-Young surface tension jump conditions, the level-set method to represent and evolve the surface, and a non-stiff Eulerian phase field algorithm to update the concentration on the surface.
Publications
v S. Li, J. Lowengrub, “Surface phase separation and morphological transition of a multicomponent vesicle”, in preparation. Ø An extended abstract accepted by the 8th International Conference on Systems Biology can be found at here.