Actuation of magnetoelastic membranes in precessing magnetic fields.

Superparamagnetic nanoparticles incorporated into elastic media offer the possibility of creating actuators driven by external fields in a multitude of environments. Here, magnetoelastic membranes are studied through a combination of continuum mechanics and molecular dynamics simulations. We show how induced magnetic interactions affect the buckling and the configuration of magnetoelastic membranes in rapidly precessing magnetic fields. The field, in competition with the bending and stretching of the membrane, transmits forces and torques that drives the membrane to expand, contract, or twist. We identify critical field values that induce spontaneous symmetry breaking as well as field regimes where multiple membrane configurations may be observed. Our insights into buckling mechanisms provide the bases to develop soft, autonomous robotic systems that can be used at micro- and macroscopic length scales.

Contractile actuation and dynamical gel assembly of paramagnetic filaments in fast precessing fields.

Flexible superparamagnetic filaments are studied under the influence of fast precessing magnetic fields using simulations and a continuum approximation analysis. We find that individual filaments can be made to exert controllable tensile forces along the precession axis. These forces are exploited for microscopic actuation. In bulk, the filaments can be rapidly assembled into different configurations whose material properties depend on the field parameters. The precession frequency affects filament aggregation and conformation by changing the net torques on the filament ends. Using a time-dependent precession angle allows considerable freedom in choosing properties for filament aggregates. As an example, we design a field that twists chains together to dynamically assemble a self-healing gel.

Constriction by Dynamin: Elasticity versus Adhesion.

Any cellular fission process is completed when the neck connecting almost-separate membrane compartments is severed. This crucial step is somehow accomplished by proteins from the dynamin family, which polymerize into helical spirals around such necks. Much research has been devoted to elucidating the specifics of that somehow, but despite no shortage of ideas, the question is not settled. Pictorially obvious notions of strangling or pushing are difficult to render in mechanically precise terms. Moreover, because dynamin is a GTPase, it is tempting to speculate that it has a motor activity that assists the necessary severing action, but again the underlying mechanics is not obvious. We believe the difficulty to be the mechanically nontrivial nature of confining elastic filaments onto curved surfaces, for which efficient methods to conceptualize the associated forces and torques have only recently appeared. Here we investigate the implications of a conceptually simple yet mechanically challenging model: consider an elastic helical filament confined to a surface mimicking the neck between two membrane compartments, which we assume to take the shape of a catenoid. What can we say about the expected length of such adsorbed filaments, their shapes, and the forces they exert, as a function of the key parameters in the model? While real dynamin is surely more complex, we consider such a minimal model to be the indispensable baseline. Without knowing what such a model can and cannot explain, it is difficult to justify more complex mechanisms, or understand the constraints under which this machinery evolved in the first place.

Cylindrical confinement of semiflexible polymers.

Equilibrium states of a closed semiflexible polymer binding to a cylinder are described. This may be either by confinement or by constriction. Closed completely bound states are labeled by two integers: the number of oscillations, n, and the number of times it winds the cylinder, p, the latter being a topological invariant. We examine the behavior of these states as the length of the loop is increased by evaluating the energy, the conserved axial torque, and the contact force. The ground state for a given p is the state with n=1; a short loop with p=1 is an elliptic deformation of a parallel circle; as its length increases it elongates along the cylinder axis with two hairpin ends. Excited states with n≥2 and p=1 possess n-fold axial symmetry. Short (long) loops possess energies ≈pE(0)(nE(0)), with E(0) the energy of a circular loop with same radius as the cylinder; in long loops the axial torque vanishes. Confined bound excited states are initially unstable; however, above a critical length each n-fold state becomes stable: The folded hairpin cannot be unfolded. The ground state for each p is also initially unstable with respect to deformations rotating the loop off the surface into the interior. A closed planar elastic curve aligned along the cylinder axis making contact with the cylinder on its two sides is identified as the ground state of a confined loop. Exterior bound states behave very differently, if free to unbind, as signaled by the reversal in the sign of the contact force. If p=1, all such states are unstable. If p≥2, however, a topological obstruction to complete unbinding exists. If the loop is short, the bound state with p=2 and n=1 provides a stable constriction of the cylinder, partially unbinding as the length is increased. This motif could be relevant to an understanding of the process of membrane fission mediated by dynamin rings.

Direct and remote constriction of membrane necks.

The physical properties of membrane necks are relevant in vesiculation, a process that plays an essential role in cellular physiology. Because the neck's radius is, in general, finite, membrane scission and the consequent pinching off of the vesicle can only occur if it is narrowed to permit the necessary membrane topological reformation. Here we examine, in a simple single phase lipid vesicle, how external forces can promote neck constriction not only by direct compression at the neck but also, counterintuitively, by dilation at remote locations. These results provide a new perspective on the role played by actin polymerization in the process of endocytosis.

Force dipoles and stable local defects on fluid vesicles.

An exact description is provided of an almost spherical fluid vesicle with a fixed area and a fixed enclosed volume locally deformed by external normal forces bringing two nearby points on the surface together symmetrically. The conformal invariance of the two-dimensional bending energy is used to identify the distribution of energy as well as the stress established in the vesicle. While these states are local minima of the energy, this energy is degenerate; there is a zero mode in the energy fluctuation spectrum, associated with area- and volume-preserving conformal transformations, which breaks the symmetry between the two points. The volume constraint fixes the distance S, measured along the surface, between the two points; if it is relaxed, a second zero mode appears, reflecting the independence of the energy on S; in the absence of this constraint a pathway opens for the membrane to slip out of the defect. Logarithmic curvature singularities in the surface geometry at the points of contact signal the presence of external forces. The magnitude of these forces varies inversely with S and so diverges as the points merge; the corresponding torques vanish in these defects. The geometry behaves near each of the singularities as a biharmonic monopole, in the region between them as a surface of constant mean curvature, and in distant regions as a biharmonic quadrupole. Comparison of the distribution of stress with the quadratic approximation in the height functions points to shortcomings of the latter representation. Radial tension is accompanied by lateral compression, both near the singularities and far away, with a crossover from tension to compression occurring in the region between them.

Confinement of semiflexible polymers.

A variational framework is developed to examine the equilibrium states of a semiflexible polymer that is constrained to lie on a fixed surface. As an application the confinement of a closed polymer loop of fixed length 2π R within a spherical cavity of smaller radius, R(0), is considered. It is shown that an infinite number of distinct periodic completely attached equilibrium states exist, labeled by two integers: n = 2,3,4,... and p = 1,2,3,..., the number of periods of the polar and azimuthal angles, respectively. Small loops oscillate about a geodesic circle: n = 2, p = 1 is the stable ground state; states with higher n exhibit instabilities. If R ≥ 2R(0) new states appear as oscillations about a doubly covered geodesic circle; the state n = 3,p = 2 replaces the twofold as the ground state in a finite band of values of R. With increasing R, loop states make a transition from oscillatory and orbital behavior on crossing the poles, returning to oscillation upon collapse to a multiple cover of a geodesic circle (signaled, respectively, by an increase in p and an increase in n). The force transmitted to the surface does not increase monotonically with loop size, but does asymptotically. It behaves discontinuously where n changes. The contribution to energy from geodesic curvature is bounded. In large loops, the energy becomes dominated by a state independent contribution proportional to the loop size; the energy gap between the ground state and excited states disappears.

Spinor representation of surfaces and complex stresses on membranes and interfaces.

Variational principles are developed within the framework of a spinor representation of the surface geometry to examine the equilibrium properties of a membrane or interface. This is a far-reaching generalization of the Weierstrass-Enneper representation for minimal surfaces, introduced by mathematicians in the 1990s, permitting the relaxation of the vanishing mean curvature constraint. In this representation the surface geometry is described by a spinor field, satisfying a two-dimensional Dirac equation, coupled through a potential associated with the mean curvature. As an application, the mesoscopic model for a fluid membrane as a surface described by the Canham-Helfrich energy quadratic in the mean curvature is examined. An explicit construction is provided of the conserved complex-valued stress tensor characterizing this surface.