Conformational Space of the Translocation Domain of Botulinum Toxin: Atomistic Modeling and Mesoscopic Description of the Coiled-Coil Helix Bundle.

The toxicity of botulinum multi-domain neurotoxins (BoNTs) arises from a sequence of molecular events, in which the translocation of the catalytic domain through the membrane of a neurotransmitter vesicle plays a key role. A recent structural study of the translocation domain of BoNTs suggests that the interaction with the membrane is driven by the transition of an α helical switch towards a ß hairpin. Atomistic simulations in conjunction with the mesoscopic Twister model are used to investigate the consequences of this proposition for the toxin-membrane interaction. The conformational mobilities of the domain, as well as the effect of the membrane, implicitly examined by comparing water and water-ethanol solvents, lead to the conclusion that the transition of the switch modifies the internal dynamics and the effect of membrane hydrophobicity on the whole protein. The central two α helices, helix 1 and helix 2, forming two coiled-coil motifs, are analyzed using the Twister model, in which the initial deformation of the membrane by the protein is caused by the presence of local torques arising from asymmetric positions of hydrophobic residues. Different torque distributions are observed depending on the switch conformations and permit an origin for the mechanism opening the membrane to be proposed.

Squeezed helical elastica.

We theoretically study the conformations of a helical semi-flexible filament confined to a flat surface. This squeezed helix exhibits a variety of unexpected shapes resembling circles, waves or spirals depending on the material parameters. We explore the conformation space in detail and show that the shapes can be understood as the mutual elastic interaction of conformational quasi-particles. Our theoretical results are potentially useful to determine the material parameters of such helical filaments in an experimental setting.

How bio-filaments twist membranes.

We study the deformations of a fluid membrane imposed by adhering stiff bio-filaments due to the torques they apply. In the limit of small deformations, we derive a general expression for the energy and the deformation field of the membrane. This expression is specialised to different important cases including closed and helical bio-filaments. In particular, we analyse interface-mediated interactions and membrane wrapping when the filaments apply a local torque distribution on a tubular membrane.

Toroidal membrane vesicles in spherical confinement.

We investigate the morphology of a toroidal fluid membrane vesicle confined inside a spherical container. The equilibrium shapes are assembled in a geometrical phase diagram as a function of scaled area and reduced volume of the membrane. For small area the vesicle can adopt its free form. When increasing the area, the membrane cannot avoid contact and touches the confining sphere along a circular contact line, which extends to a zone of contact for higher area. The elastic energies of the equilibrium shapes are compared to those of their confined counterparts of spherical topology to predict under which conditions a topology change is favored energetically.

Confotronic dynamics of tubular filaments.

Tubular lattices are ubiquitous in nature and technology. Microtubules and nanotubes of all kinds act as important pillars of biological cells and the man-made nano-world. We show that when prestress is introduced in such structures, localized conformational quasiparticles emerge and govern the collective shape dynamics of the lattice. When coupled via cooperative interactions these quasiparticles form larger-scale quasipolymer superstructures exhibiting collective dynamic modes and giving rise to a hallmark behavior radically different from semiflexible beams.

Dipoles in thin sheets.

A flat elastic sheet may contain pointlike conical singularities that carry a metrical "charge" of Gaussian curvature. Adding such elementary defects to a sheet allows one to make many shapes, in a manner broadly analogous to the familiar multipole construction in electrostatics. However, here the underlying field theory is non-linear, and superposition of intrinsic defects is non-trivial as it must respect the immersion of the resulting surface in three dimensions. We consider a "charge-neutral" dipole composed of two conical singularities of opposite sign. Unlike the relatively simple electrostatic case, here there are two distinct stable minima and an infinity of unstable equilibria. We determine the shapes of the minima and evaluate their energies in the thin-sheet regime where bending dominates over stretching. Our predictions are in surprisingly good agreement with experiments on paper sheets.

Myotubularin and PtdIns3P remodel the sarcoplasmic reticulum in muscle in vivo.

The sarcoplasmic reticulum (SR) is a specialized form of endoplasmic reticulum (ER) in skeletal muscle and is essential for calcium homeostasis. The mechanisms involved in SR remodeling and maintenance of SR subdomains are elusive. In this study, we identified myotubularin (MTM1), a phosphoinositide phosphatase mutated in X-linked centronuclear myopathy (XLCNM, or myotubular myopathy), as a key regulator of phosphatidylinositol 3-monophosphate (PtdIns3P) levels at the SR. MTM1 is predominantly located at the SR cisternae of the muscle triads, and Mtm1-deficient mouse muscles and myoblasts from XLCNM patients exhibit abnormal SR/ER networks. In vivo modulation of MTM1 enzymatic activity in skeletal muscle using ectopic expression of wild-type or a dead-phosphatase MTM1 protein leads to differential SR remodeling. Active MTM1 is associated with flat membrane stacks, whereas dead-phosphatase MTM1 mutant promotes highly curved cubic membranes originating from the SR and enriched in PtdIns3P. Overexpression of a tandem FYVE domain with high affinity for PtdIns3P alters the shape of the SR cisternae at the triad. Our findings, supported by the parallel analysis of the Mtm1-null mouse and an in vivo study, reveal a direct function of MTM1 enzymatic activity in SR remodeling and a key role for PtdIns3P in promoting SR membrane curvature in skeletal muscle. We propose that alteration in SR remodeling is a primary cause of X-linked centronuclear myopathy. The tight regulation of PtdIns3P on specific membrane subdomains may be a general mechanism to control membrane curvature.

Interface-mediated interactions: entropic forces of curved membranes.

Particles embedded in a fluctuating interface experience forces and torques mediated by the deformations and by the thermal fluctuations of the medium. Considering a system of two cylinders bound to a fluid membrane, we show that the entropic contribution enhances the curvature-mediated repulsion between the two cylinders. This is contrary to the usual attractive Casimir force in the absence of curvature-mediated interactions. For a large distance between the cylinders, we retrieve the renormalization of the surface tension of a flat membrane due to thermal fluctuations.

Self-contact and instabilities in the anisotropic growth of elastic membranes.

We investigate the morphology of thin discs and rings growing in the circumferential direction. Recent analytical results suggest that this growth produces symmetric excess cones (e cones). We study the stability of such solutions considering self-contact and bending stress. We show that, contrary to what was assumed in previous analytical solutions, beyond a critical growth factor, no symmetric e cone solution is energetically minimal any more. Instead, we obtain skewed e cone solutions having lower energy, characterized by a skewness angle and repetitive spiral winding with increasing growth. These results are generalized to discs with varying thickness and rings with holes of different radii.

Local membrane mechanics of pore-spanning bilayers.

The mechanical behavior of lipid bilayers spanning the pores of highly ordered porous silicon substrates was scrutinized by local indentation experiments as a function of surface functionalization, lipid composition, solvent content, indentation velocity, and pore radius. Solvent-containing nano black lipid membranes (nano-BLMs) as well as solvent-free pore-spanning bilayers were imaged by fluorescence and atomic force microscopy prior to force curve acquisition, which allows distinguishing between membrane-covered and uncovered pores. Force indentation curves on pore-spanning bilayers attached to functionalized hydrophobic porous silicon substrates reveal a predominately linear response that is mainly attributed to prestress in the membranes. This is in agreement with the observation that indentation leads to membrane lysis well below 5% area dilatation. However, membrane bending and lateral tension dominate over prestress and stretching if solvent-free supported membranes obtained from spreading giant liposomes on hydrophilic porous silicon are indented. An elastic regime diagram is presented that readily allows determining the dominant contribution to the mechanical response upon indentation as a function of load and pore radius.

Elasticity mapping of pore-suspending native cell membranes.

The mechanics of cellular membranes are governed by a non-equilibrium composite framework consisting of the semiflexible filamentous cytoskeleton and extracellular matrix proteins linked to the lipid bilayer. While elasticity information of plasma membranes has mainly been obtained from whole cell analysis, techniques that allow addressing local mechanical properties of cell membranes are desirable to learn how their lipid and protein composition is reflected in the elastic behavior on local length scales. Introduced here is an approach based on basolateral membranes of polar epithelial Madin-Darby canine kidney (MDCK) II cells, prepared on a highly ordered porous substrate that allows elastic mapping on a submicrometer-length scale. A strong correlation between the density of actin filaments and the measured membrane elasticity is found. Spatially resolved indentation experiments carried out with atomic force and fluorescence microscope permit relation of the supramolecular structure to the elasticity of cellular membranes. It is shown that the elastic response of the pore spanning cell membranes is governed by local bending modules rather than lateral tension.

Conical defects in growing sheets.

A growing or shrinking disc will adopt a conical shape, its intrinsic geometry characterized by a surplus angle phi(e) at the apex. If growth is slow, the cone will find its equilibrium. Whereas this is trivial if phi(e)0. We construct these states in the regime where bending dominates and determine their energies and how stress is distributed in them. For each state a critical value of phi(e) is identified beyond which the cone touches itself. Before this occurs, all states are stable; the ground state has twofold symmetry.

Contact lines for fluid surface adhesion.

When a fluid surface adheres to a substrate, the location of the contact line adjusts in order to minimize the overall energy. This implies boundary conditions which depend on the characteristic surface deformation energies. We develop a general geometrical framework within which these conditions can be derived in a completely systematic way. We treat both adhesion to a rigid substrate and adhesion between two fluid surfaces, and illustrate our general results for several important Hamiltonians involving both curvature and curvature gradients. Some of these have previously been studied using very different techniques. With the exception of capillary phenomena, the Hamiltonian will not only be sensitive to boundary translations, but may also respond to changes in slope and even in curvature. The functional form of the additional contributions will follow readily from our treatment. We will show that the boundary conditions describing adhesion between two fluid surfaces express the balance of stresses and torques, as one would expect. At a rigid substrate, however, this simple identification will generally fail. This is because local rotations of the surface normal will be entirely "enslaved" to translations on the substrate. As a consequence, stresses and torques enter a single balance condition and cannot be disentangled.

Balancing torques in membrane-mediated interactions: exact results and numerical illustrations.

Torques on interfaces can be described by a divergence-free tensor which is fully encoded in the geometry. This tensor consists of two terms, one originating in the couple of the stress, the other capturing an intrinsic contribution due to curvature. In analogy to the description of forces in terms of a stress tensor, the torque on a particle can be expressed as a line integral along a contour surrounding the particle. Interactions between particles mediated by a fluid membrane are studied within this framework. In particular, torque balance places a strong constraint on the shape of the membrane. Symmetric two-particle configurations admit simple analytical expressions which are valid in the fully nonlinear regime; in particular, the problem may be solved exactly in the case of two membrane-bound parallel cylinders. This apparently simple system provides some flavor of the remarkably subtle nonlinear behavior associated with membrane-mediated interactions.

Mechanical properties of pore-spanning lipid bilayers probed by atomic force microscopy.

We measure the elastic response of a free-standing lipid membrane to a local indentation by using an atomic force microscope. Starting point is a planar gold-coated alumina substrate with a chemisorbed 3-mercaptopropionic acid monolayer displaying circular pores of very well defined and tunable size, over which bilayers composed of N,N,-dimethyl-N,N,-dioctadecylammonium bromide or 1,2-dioleoyl-3-trimethylammonium-propane chloride were spread. Centrally indenting these "nanodrums" with an atomic force microscope tip yields force-indentation curves, which we quantitatively analyze by solving the corresponding shape equations of continuum curvature elasticity. Since the measured response depends in a known way on the system geometry (pore size, tip radius) and on material parameters (bending modulus, lateral tension), this opens the possibility to monitor local elastic properties of lipid membranes in a well-controlled setting.

How to determine local elastic properties of lipid bilayer membranes from atomic-force-microscope measurements: a theoretical analysis.

Measurements with an atomic force microscope (AFM) offer a direct way to probe elastic properties of lipid bilayer membranes locally: provided the underlying stress-strain relation is known, material parameters such as surface tension or bending rigidity may be deduced. In a recent experiment a pore-spanning membrane was poked with an AFM tip, yielding a linear behavior of the force-indentation curves. A theoretical model for this case is presented here which describes these curves in the framework of Helfrich theory. The linear behavior of the measurements is reproduced if one neglects the influence of adhesion between tip and membrane. Including it via an adhesion balance changes the situation significantly: force-distance curves cease to be linear, hysteresis and nonzero detachment forces can show up. The characteristics of this rich scenario are discussed in detail in this paper.

Interface-mediated interactions between particles: a geometrical approach.

Particles bound to an interface interact because they deform its shape. The stresses that result are fully encoded in the geometry and described by a divergence-free surface stress tensor. This stress tensor can be used to express the force on a particle as a line integral along any conveniently chosen closed contour that surrounds the particle. The resulting expression is exact (i.e., free of any "smallness" assumptions) and independent of the chosen surface parametrization. Additional surface degrees of freedom, such as vector fields describing lipid tilt, are readily included in this formalism. As an illustration, we derive the exact force for several important surface Hamiltonians in various symmetric two-particle configurations in terms of the midplane geometry; its sign is evident in certain interesting limits. Specializing to the linear regime, where the shape can be analytically determined, these general expressions yield force-distance relations, several of which have originally been derived by using an energy-based approach.