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# Research

My primary areas of research are mathematical biology and computational mathematics. I have been working on the phase field or diffuse interface approach for the modeling and simulations of some interfacial problems, in particular, those related to fluid lipid bilayer vesicle membranes. The specific issues I have studied include the adhesion of multi-component vesicle membranes onto both flat and curved substrates, and the fusion process occurring between two lipid bilayer membranes. I am currently working on phase-field based variational implicit-solvent models for biomolecular interactions. See my research statement here.

## Phase field based variational implicit-solvent models

Details will be coming soon.

## Adhesion of multi-component vesicle

Lipid bi-layer membrane has both fluidity and elasticity properties. To model the equilibrium shapes of the lipid bilayer membrane, Helfrich introduced the functional energy

E = \int_A [κ(H-a)^2 + G] dA

where 'A' represents the membrane surface (close or open); 'H' and 'G' are the mean curvature and Gaussian curvature of the surface; 'κ' and '' represent the bending and Gaussian rigidities of the membrane surface; a is the spontaneous curvature. Then the equilibrium shape of the lipid bilayer vesicle membrane is determined by minimizing the total energy E. For the multi-component vesicle membrane, we  can add few more terms in total energy E as follows:

E = \int_A κ(η)[H-a(η)]^2 dA+σ\int_A ξ/2|η'|^2+1/(4ξ )(η^2-1)^2 dA+\int_A W(η)P(x)dA

Here a phase field function 'η' is incorporated into the total energy functional, and 'κ', 'a' depend on the phase field function. Gaussian curvature contribution disappears due to the fact that Gaussian energy are fixed as a constant for the closed surface case. The second term with 'σ' involved is the Ginzburg-Landau formulation of the line tension energy at the interfacial  region of the multi-component vesicle membrane. An additional term involving 'w' represents the interaction between the multi-component vesicle membrane and the substrates. And 'P' is the potential of the interaction which depends on the distance of the point 'x' from the substrate. Doing the calculus of variation yields the Euler-Lagrange equations of equilibrium. We can numerically solve the equations and find a few solutions from different branches. We can also test that if a free vesicle membrane with two phases mixing is adhered onto a substrate, then the two mixing phases in the free states are separated due to the substrate geometry. To certain degree this numerical result confirms the experimental finding by Gordon and Deserno (EPL 2008). Details can be found at PRE 81, 041919, 2010. This result indicates that there may be a new mechanism for the formation of raft-like domains.

In fact, Gordon and Deserno (EPL 2008) believe that thermo-fluctuation effect is another effect promoting the phase separation. In our model, we only consider the deterministic case where the thermo effect is not incorporated yet. In the future work of the adhesion of multi-component vesicle membrane, the thermo-fluctuation effect will be the main consideration.

## Phase field model on the vesicle fusion process

Vesicle fusion is an unbelievably complicated  process in biology and is still a mystery to the biologist. The difficulty of the fusion process lies in the transient intermediate state which currently is mostly call the stalk state. As an first attempt, we ignore the complexity of the stalk state, and rather only focus on the pre-fusion and post-fusion states of the fusion process. These two states can be viewed as the two equilibrium states of the two vesicles before fusion and after fusion where they are interacting with each other by some interacting potential.

The model we use here differ from the adhesion of multi-component vesicle membrane in the sense that, point-wise adhesion effect is incorporated into the interaction between vesicles and the substrates; while global adhesion effect is considered in the fusion process between the two vesicles. By carrying out simulations based on the gradient flow of the associated energy functional, we are also able to elucidate the dynamic transitions between the prefusion and postfusion states. Details can be found at PRE 84, 011903, 2011.

## Others

See my publication page.