Shape equations and curvature bifurcations induced by inhomogeneous rigidities in cell membranes

The effects of internal forces and membrane heterogeneity on three-dimensional cell shapes

The shape of cells and the control thereof plays a central role in a variety of cellular processes, including endo- and exocytosis, cell division and cell movement. Intra- and extracellular forces control the shapes, and while the shape changes in some processes such as exocytosis are intracellularly-controlled and localized in the cell, movement requires force transmission to the environment, and the feedback from it can affect the cell shape and mode of movement used. The shape of a cell is determined by its cytoskeleton (CSK), and thus shape changes involved in various processes involve controlled remodeling of the CSK. While much is known about individual components involved in these processes, an integrated understanding of how intra- and extracellular signals are coupled to the control of the mechanical changes involved is not at hand for any of them. As a first step toward understanding the interaction between intracellular forces imposed on the membrane and cell shape, we investigate the role of distributed surrogates for cortical forces in producing the observed three-dimensional shapes. We show how different balances of applied forces lead to such shapes, that there are different routes to the same end state, and that state transitions between axisymmetric shapes need not all be axisymmetric. Examples of the force distributions that lead to protrusions are given, and the shape changes induced by adhesion of a cell to a surface are studied. The results provide a reference framework for developing detailed models of intracellular force distributions observed experimentally, and provide a basis for studying how movement of a cell in a tissue or fluid is influenced by its shape.

Post-buckling of a pressured biopolymer spherical shell with the mode interaction

Imperfection sensitivity is essential for mechanical behaviour of biopolymer shells characterized by high geometric heterogeneity. The present work studies initial post-buckling and imperfection sensitivity of a pressured biopolymer spherical shell based on non-axisymmetric buckling modes and associated mode interaction. Our results indicate that for biopolymer spherical shells with moderate radius-to-thickness ratio (say, less than 30) and smaller effective bending thickness (say, less than 0.2 times average shell thickness), the imperfection sensitivity predicted based on the axisymmetric mode without the mode interaction is close to the present results based on non-axisymmetric modes with the mode interaction with a small (typically, less than 10%) relative errors. However, for biopolymer spherical shells with larger effective bending thickness, the maximum load an imperfect shell can sustain predicted by the present non-axisymmetric analysis can be significantly (typically, around 30%) lower than those predicted based on the axisymmetric mode without the mode interaction. In such cases, a more accurate non-axisymmetric analysis with the mode interaction, as given in the present work, is required for imperfection sensitivity of pressured buckling of biopolymer spherical shells. Finally, the implications of the present study to two specific types of biopolymer spherical shells (viral capsids and ultrasound contrast agents) are discussed.

A nanoscale linear-to-linear motion converter of graphene

Motion conversion plays an irreplaceable role in a variety of machinery. Although many macroscopic motion converters have been widely used, it remains a challenge to convert motion at the nanoscale. Here we propose a nanoscale linear-to-linear motion converter, made of a flake-substrate system of graphene, which can convert the out-of-plane motion of the substrate into the in-plane motion of the flake. The curvature gradient induced van der Waals potential gradient between the flake and the substrate provides the driving force to achieve motion conversion. The proposed motion converter may have general implications for the design of nanomachinery and nanosensors.

Equilibrium shape equation and geometrically permissible condition for two-component lipid bilayer vesicles

Equilibrium shapes of vesicles composed of a mixture of partially miscible amphiphiles are investigated. To take into account the influences of the composition, a simple phenomenological coupling between the co mposition and the curvatures, including the mean curvature and the Gauss curvature of the membrane surface, is suggested. By minimizing the potential functional, the general shape equation is obtained and solved analytically for vesicles with simple shapes. Besides, the geometrical constraint equation and geometrically permissible condition for the two-component lipid vesicles are put forward. The influences of physical parameters on the geometrically permissible phase diagrams are predicted. The close relations between the predictions and existing experimental phenomena published recently are shown.

Equilibrium theory and geometrical constraint equation for two-component lipid bilayer vesicles

This paper aims at the general mathematical framework for the equilibrium theory of two-component lipid bilayer vesicles. To take into account the influences of the local compositions together with the mean curvature and Gaussian curvature of the membrane surface, a general potential functional is constructed. We introduce two kinds of virtual displacement modes: the normal one and the tangential one. By minimizing the potential functional, the equilibrium differential equations and the boundary conditions of two-component lipid vesicles are derived. Additionally, the geometrical constraint equation and geometrically permissible condition for the two-component lipid vesicles are presented. The physical, mathematical, and biological meanings of the equilibrium differential equations and the geometrical constraint equations are discussed. The influences of physical parameters on the geometrically permissible phase diagrams are predicted. Numerical results can be used to explain recent experiments.

Elastic properties of biological membranes influenced by attached proteins

Positively charged proteins can attach themselves to the negatively charged outer surface of biological cell membranes and liposomes. In this work, the influence of the intrinsic shape of the membrane-attached proteins on the elastic properties of the membrane is considered theoretically. It is shown that attachment of anisotropic proteins to the outer surface of biological membranes may induce tubulation of the membrane. The attachment of semi-flexible rod-like proteins increases the local bending constant, while the attachment of semi-flexible plate-like anisotropic proteins may also reduce the local bending constant of the membrane. The role of the hydrophobic protrusion of the attached protein which is embedded in the outer membrane layer is also discussed.

Stability of biphasic vesicles with membrane embedded proteins

The basic physical properties of homogeneous membranes are relatively well known, while the effects of inhomogeneities with membranes are very much an active field of study. In this paper, a biphasic lipid vesicle with membrane embedded proteins is investigated. To take into account the influences of the proteins, a simple phenomenological coupling between the local fraction of proteins and the mean curvature square is suggested. By minimizing the energy of system, the E-L equations and boundary conditions are obtained and solved analytically for vesicle with a simple shape. Besides, stability phase diagrams and stability factor are put forward by linear perturbation analysis. Our results show two different situations which are strongly dependent on the nature of the proteins: a regime of easy instability when the proteins are strongly coupled to the membrane and a regime of difficult instability.

Morphology and phase behavior of two-component lipid membranes

The stability and shapes of domains with different bending rigidities in lipid membranes are investigated. These domains can be formed from the inclusion of an impurity in a lipid membrane or from the phase separation within the membrane. We show that, for weak line tensions, surface tensions and finite spontaneous curvatures, an equilibrium phase of protruding circular domains or striped domains may be obtained. We also predict a possible phase transition between the investigated morphologies.

Effect of 2,6-di-O-methyl-alpha-cyclodextrin on hemolysis and morphological change in rabbit's red blood cells

The effects of 2,6-di-O-methyl-alpha-cyclodextrin (DM-alpha-CyD) on hemolysis and morphological changes in rabbit's red blood cells (RBC) were examined, compared with those of alpha-cyclodextrin (CyD) and 2-hydoxypropyl-alpha-cyclodextrin (HP-alpha-CyD). The hemolytic activity of alpha-CyDs increased in the order of HP-alpha-CyD<alpha-CyD<DM-alpha-CyD. The three alpha-CyDs induced morphological changes of RBC from discocyte to stomatocyte. At the same concentration (3mM) of alpha-CyDs, DM-alpha-CyD and alpha-CyD released phospholipids, rather than cholesterol, and DM-alpha-CyD markedly released proteins from RBC membranes, compared to alpha-CyD and HP-alpha-CyD. The treatment of RBC with DM-alpha-CyD lowered the extent of a fluorescent sphingomyelin analogue from lipid rafts of RBC membranes in a concentration-dependent manner. These results suggest that DM-alpha-CyD has higher hemolytic and morphological change activity than alpha-CyD and HP-alpha-CyD through more extraction of phospholipids including sphingomyelin and proteins, not cholesterol, from RBC membranes than alpha-CyD and HP-alpha-CyD.

On the role of anisotropy of membrane constituents in formation of a membrane neck during budding of a multicomponent membrane

The expression for the isotropic membrane bending energy was generalized for the case of a multicomponent membrane where the membrane constituents (single molecules or small complexes of molecules-membrane inclusions) were assumed to be anisotropic. Using this generalized expression for the membrane energy it was shown that the change of intrinsic shape of membrane components may induce first-order-like shape transitions leading to the formation of a membrane neck. The predicted discontinuous membrane shape transition and the concomitant lateral segregation of membrane components were applied to study membrane budding. Based on the results presented we conclude that the budding process might be driven by accumulation of anisotropic membrane components in the necks connecting the bud and the parent membrane, and by accumulation of isotropic (conical) membrane components on the bud. Both processes may strongly depend on the intrinsic shape of membrane components and on the direct interactions between them.

Theoretical analysis of adhering lipid vesicles with free edges

A theoretical model for describing the adhesion of lipid vesicle with free edges is developed. For adhesion in contact potential or in finite-range potential, the total energy functional is defined as the sum of elastic free energy, the surface energy, the line tension energy and the contact potential or the long-ranged potential. The equilibrium differential equation and boundary conditions for opening-up lipid vesicles are derived through minimizing the total energy functional. Numerical solutions to these equations are obtained under the axial symmetric condition. These numerical solutions can be used to qualitatively explain the influence of the substrate on the open-up lipid vesicles.

General Mathematical Frame for Open or Closed Biomembranes (Part I): Equilibrium Theory and Geometrically Constraint Equation

This paper aims at constructing a general mathematical frame for the equilibrium theory of open or closed biomembranes. Based on the generalized potential functional, the equilibrium differential equation for open biomembrane (with free edge) or closed one (without boundary) is derived. The boundary conditions for open biomembranes are obtained. Besides, the geometrically constraint equation for the existence, formation and disintegration of open or closed biomembranes is revealed. The physical and biological meanings of the equilibrium differential equation and the geometrically constraint equation are discussed. Numerical simulation results for axisymmetric open biomembranes show the effectiveness and convenience of the present theory.