Analytical and numerical solutions for shapes of quiescent two-dimensional vesicles (original) (raw)
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Using a mathematical model, we investigate the role of hydrodynamic forces on three-dimensional axisymmetric multicomponent vesicles. The equations are derived using an energy variation approach that accounts for different surface phases, the excess energy associated with surface domain boundaries, bending energy and inextensibility. The equations are high-order (fourth order) nonlinear and nonlocal. To solve the equations numerically, we use a nonstiff, pseudo-spectral boundary integral method that relies on an analysis of the equations at small scales. We also derive equations governing the dynamics of inextensible vesicles evolving in the absence of hydrodynamic forces and simulate numerically the evolution of this geometric model. We find that compared with the geometric model, hydrodynamic forces provide additional paths for relaxing inextensible vesicles. The presence of hydrodynamic forces may enable the dynamics to overcome local energy barriers and reach lower energy states than those accessible by geometric motion or energy minimization algorithms. Because of the intimate connection between morphology, surface phase distribution and biological function, these results have important consequences in understanding the role vesicles play in biological processes.
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Journal of Physics: Condensed Matter, 2011
We examined the dynamics of the deformation and phase separation of two-component vesicles. First, we numerically investigated the effects of (i) thermal noise, (ii) hydrodynamic flow induced by the line tension of the domain boundary and (iii) composition-dependent bending rigidity on the coarsening dynamics of a phase-separated pattern on the surfaces of vesicles with fixed shapes. The dynamical exponent z (NDB ∼ t −z , the total length of the domain boundaries) of the coarsening of phase-separated pattern was found to decrease from z = 1/3 under no thermal noise to 1/5 < z < 1/4 when including the effects of thermal noise. We also found that the hydrodynamic effect enhances the coarsening in a bicontinuous phase separation for a spherical vesicle. In phase separations of a shape-fixed tubular vesicle, a band-like phase separation with periodicity along the longer axis of the tube occurs because of the composition-dependent bending rigidity and the higher curvatures at the tube end-caps. Second, we also explored the dynamics of shape deformation coupled with phase separation through the bending rigidity of the membrane which depends on the local composition in lipid, and found that the composition-dependent bending rigidity crucially influences the phase separation and deformation of the vesicle. The results of simulations are in good agreement with experimentally observed behaviors known as "shape convergence" [
The Journal of Chemical Physics, 2013
A numerical method is proposed for evaluating the curvature dependency of elastic parameters of a spherical vesicle based on a calculation of the pressure profile across the membrane. The proposed method is particularly useful for small unilamellar vesicles (SUVs), in which the internal structure of the membrane is asymmetric owing to the high curvature. In this case, the elastic energy is insufficiently described as a perturbation from a planar membrane. The calculated saddle-splay curvature modulus of SUVs, which is about 16 nm in diameter, is found to be much higher than that of a planar membrane. A comparison of the free energy change in the initial stage of vesicle-tobicelle transformation with the Fromherz theory demonstrates that the elastic parameters estimated for SUVs provide better estimation of the free energy than those estimated for a planar membrane.
Dynamic model and stationary shapes of fluid vesicles
The European Physical Journal E, 2006
A phase-field model that takes into account the bending energy of fluid vesicles is presented. The Canham-Helfrich model is derived in the sharp-interface limit. A dynamic equation for the phasefield has been solved numerically to find stationary shapes of vesicles with different topologies and the dynamic evolution towards them. The results are in agreement with those found by minimization of the Canham-Helfrich free energy. This fact shows that our phase-field model could be applied to more complex problems of instabilities.
Shape instabilities in vesicles: A phase-field model
The European Physical Journal Special Topics, 2007
A phase field model for dealing with shape instabilities in fluid membrane vesicles is presented. This model takes into account the Canham-Helfrich bending energy with spontaneous curvature. A dynamic equation for the phase-field is also derived. With this model it is possible to see the vesicle shape deformation dynamically, when some external agent instabilizes the membrane, for instance, inducing an inhomogeneous spontaneous curvature. The numerical scheme used is detailed and some stationary shapes are shown together with a shape diagram for vesicles of spherical topology and no spontaneous curvature, in agreement with known results.
Dynamic shape transformations of fluid vesicles
Soft Matter, 2010
We incorporate a volume-control algorithm into a recently developed one-particle-thick mesoscopic fluid membrane model to study vesicle shape transformation under osmotic conditions. Each coarsegrained particle in the model represents a cluster of lipid molecules and the inter-particle interaction potential effectively captures the dual character of fluid membranes as elastic shells with out-of-plane bending rigidity and 2D viscous fluids with in-plane viscosity. The osmotic pressure across the membrane is accounted for by an external potential, where the instantaneous volume of the vesicles is calculated via a local triangulation algorithm. Through coarse-grained molecular dynamics simulations, we mapped out a phase diagram of the equilibrium vesicle shapes in the space of spontaneous curvature and reduced vesicle volume. The produced equilibrium vesicle shapes agree strikingly well with previous experimental data. We further found that the vesicle shape transformation pathways depend on the volume change rate of the vesicle, which manifests the role of dynamic relaxation of internal stresses in vesicle shape transformations. Besides providing an efficient numerical tool for the study of membrane deformations, our simulations shed light on eliciting desired cellular functions via experimental control of membrane configurations.
Vesicle Shapes from Molecular Dynamics Simulations
The Journal of Physical Chemistry B, 2006
Lipid bilayer membranes are known to form various structures such as large sheets or vesicles. When the two leaflets of the bilayer have an equal composition, the membrane preferentially forms a flat sheet or a spherical vesicle. However, a difference in the composition of the two leaflets may result in a curved bilayer or in a wide variety of vesicle shapes. Vesicles with different shapes have already been shown in experiments and diverse vesicle shapes have been predicted theoretically from energy minimization of continuous curves. Here we present a molecular dynamics study of the effect of small changes in the phospholipid headgroups on the spontaneous curvature of the bilayer and on the resulting vesicle shape transformations. Small asymmetries in the bilayers already result in high spontaneous curvature and large vesicle deformations. Vesicle shapes that are formed include ellipsoids, discoids, pear-shaped vesicles, cup-shaped vesicles, as well as budded vesicles. Comparison of these vesicles with theoretically derived vesicle shapes shows both resemblances and differences.
Morphological stability analysis of vesicles with mechanical–electrical coupling effects
Acta Mechanica Sinica, 2010
Using a recently established liquid crystal model for vesicles, we present a theoretical method to analyze the morphological stability of liquid crystal vesicles in an electric field. The coupled mechanical-electrical effects associated with elastic bending, osmotic pressure, surface tension, Maxwell pressure, as well as flexoelectric and dielectric properties of the membrane are taken into account. The first and second variations of the free energy are derived in a compact form by virtue of the surface variational principle. The former leads to the shape equation of a vesicle embedded in an electric field, and the latter allows us to examine the stability of a given vesicle morphology. As an illustrative example, we analyze the stability of a spherical vesicle under a uniform electric field. This study is helpful for understanding and revealing the morphological evolution mechanisms of vesicles in electric fields and some associated phenomena of cells.