The spectral analysis of thermal fluctuations, or flickering, is a simple and non-invasive method widely used to determine the mechanical properties of artificial and biological lipid membranes. In its most common implementation, the position of the edge of a cell or vesicle is tracked from optical microscopy videos. However, a systematic disagreement with X-ray scattering and micromechanical manipulation data has brought into question the validity of the method. We present an improved analysis protocol that resolves these discrepancies by accounting for the finite vertical resolution of the optics used to image fluctuations.

We study the effect of transmembrane proteins on the shape, composition and thermodynamic stability of the surrounding membrane. When the coupling between membrane composition and curvature is strong enough the nearby membrane composition and shape both undergo a transition from over-damped to under-damped spatial variation, well before the membrane becomes unstable in the bulk. This transition is associated with a change in the sign of the thermodynamic energy and hence favors the early stages of coat assembly necessary for vesiculation (budding) and may suppress the activity of mechanosensitive membrane channels and transporters. Our results suggest an approach to obtain physical parameters of the membrane that are otherwise difficult to measure. [4][5][6][7]. Here, we are primarily interested in the non-specific lipid-protein interactions that arise from the coupling of their hydrophobic regions [8][9][10][11][12][13][14], although we can also allow for selective enrichment of membrane component(s) near the protein.We employ a continuum theory in which small deformations of the lipid environment near a rigid inclusion can be described by a number of local field variables, such as the profile of the mid-plane of the bilayer, its composition and membrane thickness . Furthermore, the free-energy cost associated with thickness deformation is completely decoupled at lowest order [21], and it can be independently analyzed although we do not do so here.We allow for selective enrichment/depletion of curvature sensitive inclusions in the vicinity of a membrane protein or, equivalently, lipid asymmetry between leaflets that is characterized by a local spontaneous curvature, the preferred mean curvature in the absence of any mechanical stresses on the membrane [33][34][35][36][37][38][39][40][41]. This local variation may be relatively large near a membrane protein if its geometry is such that it bends or deforms the surrounding membrane (see Fig 1). Our approach leads to a real-space description of the membrane around an inclusion of arbitrary symmetry.We consider a two-component membrane in which the local compositional asymmetry between the different layers and/or the density of curvature-sensitive inclusions is phenomenologically coupled to the local mean curvature of the membrane [33,34]. When the compositional variation is weak and the membrane displacement is small, the free-energy can be written as a Landau-Ginzburg ex- The surface variation of the rigid inclusion in theẑ direction is coarse-grained out so that the geometry is defined by its radius r0 and two functions describing the height U(θ) and contact angle U (θ) of the hydrophobic belt. These parameterize the protein-membrane interface (red line), where u(r0, θ) = U(θ) andn · ∇u(r0, θ) = U (θ), withn as the inward unit normal vector. We require both the normal force and the torque on the inclusion to vanish. The latter can lead to an equilibrium tilt angle ψ about the axis labeled by ε.pansion [33][34][35][42][43][44],where only the lowest-order terms ar...

We study the leaf-to-leaf distances on one-dimensionally ordered, full and complete m-ary tree graphs using a recursive approach. In our formulation, unlike in traditional graph theory approaches, leaves are ordered along a line emulating a one-dimensional lattice. We find explicit analytical formulas for the sum of all paths for arbitrary leaf separation r as well as the average distances and the moments thereof. We show that the resulting explicit expressions can be recast in terms of Hurwitz-Lerch transcendants. Results for periodic trees are also given. For incomplete random binary trees, we provide first results by numerical techniques; we find a rapid drop of leaf-to-leaf distances for large r.

The role of recycling in the control of membrane domains is a contentious issue and currently an open research question. In this context, we study the coarsening of strongly microphase-separated membrane domains in the presence of recycling of material. The dynamics of cluster size distribution is studied under both scale-free and size-dependent recycling. Closed-form solutions to the steady-state distribution and its associated moments are found in both cases. For the size-independent case, the time evolution of the moments is analytically obtained, providing exact results for their corresponding relaxation times. Since these moments and relaxation times are measurable quantities that may be determined by comparison with experimental data, our results provide a framework with which to understand and assess the interplay between membrane recycling and domain formation.