Numerical simulations for biological membranes

The study of the physics of fluid biological membranes has broad applications, especially for human health. Recently, we begin to systematically study the shape deformation of vesicle membranes by numerical simulations, using a unified energetic variational formulation with phase field methods. Our final goal is to understand such deformations and its relation to the external fields and internal bulk and surface properties of the vesicle, in collaboration with biophysicists and biologists.

The usual single component biological vesicles are formed by certain amphophilic molecules assembled in water to build bilayer structure that makes vesicles simple models for the study of some of the physico-chemical properties of cell as well as of their shapes. One simplistic model is based on the principle that the equilibrium shape of such a membrane is determined by minimizing the elastic bending energy:
      
where H is the mean curvature of the membrane surface, a is the surface tension, b the bending rigidity and c the stretching rigidity,  is the spontaneous curvature that describes the asymmetry effect of the membrane or its environment, K is the Gaussian curvature. With two constant valued a and c, the first term and the last term may be neglected as they remain constants for vesicles with a given surface area and preserving a topological structure. The problem is to study the vesicle shapes minimizing the bending energy with prescribed cell volume and surface area. One classical method for simulating a changing interface is to employ a mesh that has grid points on the interfaces, and deforms according to the motion of the boundary, such as the boundary integral and boundary element methods. Keeping track of the moving mesh may entail computational difficulties and large displacement in internal domains may cause mesh entanglement.

We turned to the energetic phase field models that offer many advantages including the easy treatment of topological changes of the interface. In our phase field model, a phase field function is introduced on a computational domain  to label the inside/outside of the vesicle, where the membrane  is the zero level set  while the inside is represented by the positive part and the outside by the negative part. As a typical diffusive interface description, an ideal phase field function , where  represent the diffusive interface width. As ,  describes a sharp interface.

And we can formulate the mean curvature H by  and so as the elastic bending energy
 ,

Volume
  
and surface area
   [1, 2].

Theoretically, with the help of a special ansatz, we proved that minimizing the bending energy with prescribed surface area and bulk volume is exactly minimizing  with prescribed  and  [4].

3-D view of some equilibrium configurations

equilibrium shape transformations by changing the surface area while fixing the volume (axis-symmetric case).
(Star pathways involve topological changes)

Movie

A movie for the shape transformation by decreasing the surface area (axis-symmetric case, cut view)

Symmetric torus and non-symmetric torus vs tori found in experiment

Formation of large coherent structures from small specimen

         2. Effect of Spontaneous curvature [2]

With a large constant spontaneous curvature, an ellipsoid finally splits to three same sized sphere-like vesicles

Variable spontaneous curvature: on some points, the surface bulks

3. Shape transformation in fluid fields [5]

The transformation of Vesicles may include complex topological changes such as splitting and merging which can be easily handled within the phase field model. On the other hand, topological information may be needed in application and/or the membrane topology needs to be controlled. There is a lack of discussion in the literature on how to effectively recover the topological information of the membrane/interface within the phase field formulation.

We showed how to retrieve the Euler number of the vesicles by the phase field function based on the classical Gauss-Bonnet formula. We presented the formula of the Euler number by a general phase field function as well as a number of simplified Euler number formulae which can be applied under different conditions [3]. Those formulae were successfully applied in detecting the topological changes in our experiments.

1. A Phase Field Approach in the Numerical Study of the Elastic Bending Energy for Vesicle Membranes, with Qiang Du and Chun Liu, Journal of Computational Physics, 198, pp. 450-468, 2004
    
2. Modeling the Spontaneous Curvature Effects in Static Cell Membrane Deformations by a Phase Field Formulation, with Qiang Du, Chun Liu and Rolf Ryham, Communications in Pure and Applied Analysis, 4, pp. 537-548, 2005
    
3. Retrieving topological information for phase field models, with Qiang Du and Chun Liu, SIAM Journal on Applied Mathematics 65, pp. 1913-1932, 2005
    
4. A phase field formulation of the Willmore problem, with Qiang Du, Chun Liu and Rolf Ryham, Nonlinearity, 18, pp. 1249-1267, 2005
    
5. Simulating the Deformation of Vesicle Membranes under Elastic Bending Energy in Three Dimensions, with Qiang Du and Chun Liu, Journal of Computational Physics, 212, pp. 757-777, 2006
    
6. Centroidal Voronoi tessellation algorithms for image compression and segmentation, with Qiang Du, Max Gunzburger and Lili Ju,  Journal of Mathematical Imaging and Vision, DOI: 10.1007/s10851-005-3620-4, Jan. 2006
    
7. Convergence of numerical approximations to a phase field bending elasticity model of membrane deformations, with Qiang Du, International Journal of Numerical Analysis and Modelling, accepted for publication, 2006
    
8. Modeling Vesicle Deformations in Flow Fields via Energetic Variational Approaches, with Qiang Du, Chun Liu and Rolf Ryham, submitted to Nonlinearity, 2006.
    
9. Modelling and Simulations of Multi-component Lipid Membranes and Open Membranes via Diffusive Interface Approaches, with Qiang Du, submitted to Journal of Mathematical Biology, 2006
    
10. Asymptotic Analysis of Phase Field Formulations of Bending Elasticity Models, submitted to SIAM Mathematical Analysis, 2006
     
11. A Modified Interfacial Energy for Capturing the Euler Number, with Qiang Du, Chun Liu and Rolf Ryham, preprint, 2006