Aggregation of Red Blood Cells: From Rouleaux to Clot Formation

Red blood cells are known to form aggregates in the form of rouleaux. This aggregation process is believed to be reversible, but there is still no full understanding on the binding mechanism. There are at least two competing models, based either on bridging or on depletion. We review recent experimental results on the single cell level and theoretical analyses of the depletion model and of the influence of the cell shape on the binding strength. Another important aggregation mechanism is caused by activation of platelets. This leads to clot formation which is life saving in the case of wound healing but also a major cause of death in the case of a thrombus induced stroke. We review historical and recent results on the participation of red blood cells in clot formation

Parameterization invariance and shape equations of elastic axisymmetric vesicles

The issue of different parameterizations of the axisymmetric vesicle shape addressed by Hu Jian-Guo and Ou-Yang Zhong-Can [ Phys.Rev. E {\bf 47} (1993) 461 ] is reassesed, especially as it transpires through the corresponding Euler - Lagrange equations of the associated elastic energy functional. It is argued that for regular, smooth contours of vesicles with spherical topology, different parameterizations of the surface are equivalent and that the corresponding Euler - Lagrange equations are in essence the same. If, however, one allows for discontinuous (higher) derivatives of the contour line at the pole, the differently parameterized Euler - Lagrange equations cease to be equivalent and describe different physical problems. It nevertheless appears to be true that the elastic energy corresponding to smooth contours remains a global minimum.Comment: 10 pages, latex, one figure include

Second variation of the Helfrich-Canham Hamiltonian and reparametrization invariance

A covariant approach towards a theory of deformations is developed to examine both the first and second variation of the Helfrich-Canham Hamiltonian -- quadratic in the extrinsic curvature -- which describes fluid vesicles at mesoscopic scales. Deformations are decomposed into tangential and normal components; At first order, tangential deformations may always be identified with a reparametrization; at second order, they differ. The relationship between tangential deformations and reparametrizations, as well as the coupling between tangential and normal deformations, is examined at this order for both the metric and the extrinsic curvature tensors. Expressions for the expansion to second order in deformations of geometrical invariants constructed with these tensors are obtained; in particular, the expansion of the Hamiltonian to this order about an equilibrium is considered. Our approach applies as well to any geometrical model for membranes.Comment: 20 page

Viscous regularization and r-adaptive remeshing for finite element analysis of lipid membrane mechanics

As two-dimensional fluid shells, lipid bilayer membranes resist bending and stretching but are unable to sustain shear stresses. This property gives membranes the ability to adopt dramatic shape changes. In this paper, a finite element model is developed to study static equilibrium mechanics of membranes. In particular, a viscous regularization method is proposed to stabilize tangential mesh deformations and improve the convergence rate of nonlinear solvers. The Augmented Lagrangian method is used to enforce global constraints on area and volume during membrane deformations. As a validation of the method, equilibrium shapes for a shape-phase diagram of lipid bilayer vesicle are calculated. These numerical techniques are also shown to be useful for simulations of three-dimensional large-deformation problems: the formation of tethers (long tube-like exetensions); and Ginzburg-Landau phase separation of a two-lipid-component vesicle. To deal with the large mesh distortions of the two-phase model, modification of vicous regularization is explored to achieve r-adaptive mesh optimization

Electrical Double Layer and Phospholipid Membranes. Aspects of Interface Water Structure

It is inferred that the formation of an electrical double layer in phospholipid membranes is affected by the modified water. structure at the membrane/water interface. Theoretical evidence is presented indicating a decrease in the number density of water hydrogen bonds and the appearance of a layer of spontaneously polarized water at the solid/water interface which are the consequence of the orientational properties of the water structure and the asymmetric structure of the water molecule respectively. Consequences of such modifications of water structure are discussed on the basis of simple models regarding ion distribution at the interface, effect on the phospholipid chain melting phase transition temperature and the membrane-membrane interaction

Vesicle shape, molecular tilt, and the suppression of necks

Can the presence of molecular-tilt order significantly affect the shapes of lipid bilayer membranes, particularly membrane shapes with narrow necks? Motivated by the propensity for tilt order and the common occurrence of narrow necks in the intermediate stages of biological processes such as endocytosis and vesicle trafficking, we examine how tilt order inhibits the formation of necks in the equilibrium shapes of vesicles. For vesicles with a spherical topology, point defects in the molecular order with a total strength of $+2$ are required. We study axisymmetric shapes and suppose that there is a unit-strength defect at each pole of the vesicle. The model is further simplified by the assumption of tilt isotropy: invariance of the energy with respect to rotations of the molecules about the local membrane normal. This isotropy condition leads to a minimal coupling of tilt order and curvature, giving a high energetic cost to regions with Gaussian curvature and tilt order. Minimizing the elastic free energy with constraints of fixed area and fixed enclosed volume determines the allowed shapes. Using numerical calculations, we find several branches of solutions and identify them with the branches previously known for fluid membranes. We find that tilt order changes the relative energy of the branches, suppressing thin necks by making them costly, leading to elongated prolate vesicles as a generic family of tilt-ordered membrane shapes.Comment: 10 pages, 7 figures, submitted to Phy. Rew.

Budding transition for bilayer fluid vesicles with area-difference elasticity

We consider a curvature model for bilayer vesicles with an area-difference elasticity or non-local bending-energy term. Such a model interpolates between the bilayer-couple and spontaneous-curvature models. We report preliminary results for the budding transition. The shape transformation between the dumbbell and the pear phases can be continuous or discontinuous depending on the ratio of the non-local to the local bending rigidities

Thermodynamics of vesicle growth and instability

We describe the growth of vesicles, due to the accretion of lipid molecules to their surface, in terms of linear irreversible thermodynamics. Our treatment differs from those previously put forward by consistently including the energy of the membrane in the thermodynamic description. We calculate the critical radius at which the spherical vesicle becomes unstable to a change of shape in terms of the parameters of the model. The analysis is carried out both for the case when the increase in volume is due to the absorption of water and when a solute is also absorbed through the walls of the vesicle.Comment: To be published in Phys. Rev.

Stresses in lipid membranes

The stresses in a closed lipid membrane described by the Helfrich hamiltonian, quadratic in the extrinsic curvature, are identified using Noether's theorem. Three equations describe the conservation of the stress tensor: the normal projection is identified as the shape equation describing equilibrium configurations; the tangential projections are consistency conditions on the stresses which capture the fluid character of such membranes. The corresponding torque tensor is also identified. The use of the stress tensor as a basis for perturbation theory is discussed. The conservation laws are cast in terms of the forces and torques on closed curves. As an application, the first integral of the shape equation for axially symmetric configurations is derived by examining the forces which are balanced along circles of constant latitude.Comment: 16 pages, introduction rewritten, other minor changes, new references added, version to appear in Journal of Physics