Computational
BioSystems
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
mothermembrane. 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
highorder, advectionreactiondiffusion equation of CahnHilliard type.
Because of the presence of highorder 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 LaplaceYoung surface tension
jump conditions, the levelset method to represent and evolve the
surface, and a nonstiff Eulerian phase field algorithm to update the
concentration on the surface. 



Publications 

v
S. Li, J. Lowengrub, ÒSurface phase separation and
morphological transition of a multicomponent vesicleÓ, in preparation. ¯ An
extended abstract accepted by the 8th International Conference on Systems
Biology can be found at here. 

