Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions

Abstract

In this paper, we study the three-dimensional deformation of a vesicle membrane under the elastic bending energy, with prescribed bulk volume and surface area. Both static and dynamic deformations are considered. A newly developed energetic variational formulation is employed to give an effective Eulerian description. Efficient time and spatial discretizations are considered and implemented. Numerical experiments illustrate some fascinating phenomena that are of interests in real applications.

Introduction

Biological vesicle membranes are fascinating subjects widely studied in biology, biophysics and bioengineering. Being ubiquitous in biological systems, they not only are essential to the function of cells but also exhibit rich physical properties and complex behavior. Vesicle membranes are formed by certain amphiphilic molecules assembled in water to build bilayers. The molecules in the fluid-like membrane are free to wander and are allowed to shear at no cost. The bilayer structure makes vesicle membranes simple models for studying the physico-chemical properties of cell as well as of their shape deformations.

As shear stresses lead to no energetic cost, the equilibrium shape of such a membrane may be determined by the resulting elastic bending energy, which was first studied by Canham, Evans and Helfrich. The elastic bending energy is formulated in the form of a surface integral of different curvature terms on the membrane in the isotropic case [9], [10]. The general elastic bending energy is obtained from the Hooke’s Law [6], [23], [30], [31], [32], [34], [35]:

$$E=\underset{\Gamma }{\int }\{a_1+a_2(H-c_0{)}^2+a_{3}G\}\mathrm{d}s\text{,}$$

where a₁ represents the surface tension, which demonstrate the interaction effects between the vesicle material and the ambient fluid material. H = (k₁ + k₂)/2 is the mean curvature of the membrane surface, with k₁ and k₂ as the principle curvatures, and G = k₁k₂ is the Gaussian curvature. a₂ is the bending rigidity and a₃ the stretching rigidity. They are determined by the interaction and properties of the materials that form the membrane. c₀ is the spontaneous curvature that describes the asymmetry effect of the membrane or its environment. We note that for the asymmetry effect of the lipid bilayer, one may attribute to the intrinsic spontaneous curvature effect of a monolayer the area difference of a bilayer or even the presence of the protein molecules within the lipid bilayer (when c₀ is a function on the membrane surface) [2], [11], [12], [23], [24], [26], [28], [32], [34].

This paper constitutes one step in our attempt to systematically study the shape deformation of vesicle membranes and its relation to the external fields and internal bulk and surface properties of the vesicle. In earlier works, we have laid ground work for the study of membrane deformations using a unified energetic variational approach via the diffusive interface approximation (EVADIA) [14]. Our approach was further analytically justified in [16] with rigorous consistency checks and convergence proofs, and computationally substantiated through numerical experiments with or without the spontaneous curvature effects [14], [15]. Most of the computational experiments, however, were limited to the three-dimensional axis-symmetric cases. In this paper, we present full three-dimensional simulations that reveal both the effectiveness of the energetic variational formulation and also interesting membrane shapes and deformations previously not explored in the literature.

For a closed surface, the last term in (1.1), which is equal to the Euler–Poincaré index, represents the topological structure of the membrane which has also been formulated and studied in the context of the energetic variational formulation [13]. With a constant a₁, the first term can be neglected as it remains constant for vesicles with a given surface area. By taking the bending rigidity k as a constant, we consider in this paper the simplified form of the bending energy given by

$$E=\underset{\Gamma }{\int }(H-c_0{)}^2\mathrm{d}s\text{.}$$

The effect of the spontaneous curvature is retained in the above energy. Thus, the central theme of this paper is to study the equilibrium configurations and the dynamic deformations of vesicle shapes under the bending energy (1.2), with fixed cell volume and surface area in the full three-dimensional cases.

Comparing with the numerical studies given in [13], the removal of the axis-symmetry constraint allows us to systematically analyze the shape transformations of the vesicles in the real physical three-dimensional cases. We can re-evaluate the presence and the stability results that was previously established for the axial symmetric situations [13]. We can also explore the non-axis-symmetrical shapes and discover new configurations. Furthermore, this added numerical capability sets us up in a position to simulate and study some more complex shape transformations, including intricate merging and splitting of the vesicle membranes.

The paper is organized as follows: in Section 2, we present the energetic variational formulation, and address the approach of Lagrange multipliers. This is different from the penalty constraint method used in [13] and the difference and connection of these two methods are examined. In Section 3, we discuss the discretization schemes and some detailed implementation issues. Spectral methods are used due to their high resolution feature. In Section 4, we first present some convergence tests for our method, then we assemble a number of interesting experiments together to explore the true world of three-dimensional vesicle membrane deformations and to illustrate the application potential of our approach. Both three-dimensional axis-symmetrical and non-axis-symmetrical vesicles are observed. We also present some experiments showing the merging and splitting of the vesicle membranes. We then make some conclusion remarks and present the direction of our future work in Section 5.