On the stress and torque tensors in fluid membranes

We derive the membrane elastic stress and torque tensors using the standard Helfrich model and a direct variational method in which the edges of a membrane are infinitesimally translated and rotated. We give simple expressions of the stress and torque tensors both in the local tangent frame and in projection onto a fixed frame. We recover and extend the results of Capovilla and Guven [J. Phys. A, 2002, \textbf{35}, 6233], which were obtained using covariant geometry and Noether's theorem: we show that the Gaussian rigidity contributes to the torque tensor and we include the effect of a surface potential in the stress tensor. Many interesting situations may be investigated directly using force and torque balances instead of full energy minimization. As examples, we consider the force exerted at the end of a membrane tubule, membrane adhesion and domain contact conditions.Comment: 7 pages, 5 figure

Elastic interaction between "hard'' or "soft" pointwise inclusions on biological membranes

We calculate the induced elastic-interaction between pointwise membrane inclusions that locally interact up to quadratic order with the membrane curvature tensor. For isotropic inclusions, we recover the usual interaction proportional to the inverse fourth power of the separation, however with a prefactor showing a non-trivial dependence on the rigidity $\Gamma$ of the quadratic potential. In the large $\Gamma$ limit, corresponding to ``hard'' inclusions, we recover the standard prefactor first obtained by Goulian et al. [Europhys. Lett. \textbf{22}, 145 (1993)]. In the small $\Gamma$ limit, corresponding to "soft" inclusions, we recover the recent result of Marchenko and Misbah [Eur. Phys. J. E \textbf{8}, 477 (2002)]. This shows that the latter result bears no fundamental discrepancy with previous works, but simply corresponds to the limit of soft inclusions. We discuss how the same inclusion can be depicted as hard or soft according to the degree of coarse-graining of the pointwise description. Finally, we argue that conical transmembrane proteins should be fundamentally considered as hard inclusions.Comment: 6 page

Membrane properties revealed by spatiotemporal response to a local inhomogeneity

We study theoretically the spatiotemporal response of a lipid membrane submitted to a local chemical change of its environment, taking into account the time-dependent profile of the reagent concentration due to diffusion in the solution above the membrane. We show that the effect of the evolution of the reagent concentration profile becomes negligible after some time. It then becomes possible to extract interesting properties of the membrane response to the chemical modification. We find that a local density asymmetry between the two monolayers relaxes by spreading diffusively in the whole membrane. This behavior is driven by intermonolayer friction. Moreover, we show how the ratio of the spontaneous curvature change to the equilibrium density change induced by the chemical modification can be extracted from the dynamics of the local membrane deformation. Such information cannot be obtained by analyzing the equilibrium vesicle shapes that exist in different membrane environments in light of the area-difference elasticity model.Comment: 11 pages, 4 figure

General up to next-nearest neighbour elasticity of triangular lattices in three dimensions

We establish the most general form of the discrete elasticity of a 2D triangular lattice embedded in three dimensions, taking into account up to next-nearest neighbour interactions. Besides crystalline system, this is relevant to biological physics (e.g., red blood cell cytoskeleton) and soft matter (e.g., percolating gels, etc.). In order to correctly impose the rotational invariance of the bulk terms, it turns out to be necessary to take into account explicitly the elasticity associated with the vertices located at the edges of the lattice. We find that some terms that were suspected in the litterature to violate rotational symmetry are in fact admissibl

Bilayer elasticity at the nanoscale: the need for new terms

Continuum elastic models that account for membrane thickness variations are especially useful in the description of nanoscale deformations due to the presence of membrane proteins with hydrophobic mismatch. We show that terms involving the gradient and the Laplacian of the area per lipid are significant and must be retained in the effective Hamiltonian of the membrane. We reanalyze recent numerical data, as well as experimental data on gramicidin channels, in light of our model. This analysis yields consistent results for the term stemming from the gradient of the area per molecule. The order of magnitude we find for the associated amplitude, namely 13-60 mN/m, is in good agreement with the 25 mN/m contribution of the interfacial tension between water and the hydrophobic part of the membrane. The presence of this term explains a systematic variation in previously published numerical data.Comment: 34 pages, 9 figure

A Gaussian model for the membrane of red blood cells with cytoskeletal defects

We study a Gaussian model of the membrane of red blood cells: a ``phantom" triangular network of springs attached at its vertices to a fluid bilayer with curvature elasticity and tension. We calculate its fluctuation spectrum and we discuss the different regimes and non-monotonic features, including the precise crossover at the mesh size between the already known limits with two different tensions and the renormalisation of the bending rigidity at low wavevectors. We also show that the non-diagonal correlations reveal, in ``dark field", the cytoskeletal defects. As a first step toward a non-invasive defect spectroscopy, the specific case of lacking bonds is studied numerically and analytically

Universal amplitudes of the Casimir-like interactions between four types of rods in fluid membranes

The fluctuation-induced, Casimir-like interaction between two parallel rods of length L adsorbed on a fluid membrane is calculated analytically at short separations d<<L. The rods are modeled as constraints imposed on the membrane curvature along a straight line. This allows to define four types of rods, according to whether the membrane can twist along the rod and/or curve across it. For stiff constraints, all the interaction potentials between the different types of rods are attractive and proportional to L/d. Two of the four types of rods are then equivalent, which yields six universal Casimir amplitudes. Repulsion can occur between different rods for soft constraints. Numerical results obtained for all ranges of d/L show that the attraction potential reaches kT for d/L\simeq0.2. At separations smaller than d_c \approx L(L/l_p)^(1/3), where l_p is the rod persistence length, two rods with fixed ends will bend toward each other and finally come into contact because of the Casimir interaction.Comment: 6 pages, 3 figure