Rolf J. Ryham

A phase field formulation of the Willmore problem

In this paper, we demonstrate, through asymptotic expansions, the convergence of a phase field formulation to model surfaces minimizing the mean curvature energy with volume and surface area constraints. Under the assumption of the existence of a smooth limiting surface, it is shown that the interface of a phase field, which is a critical point of the elastic bending energy, converges to a critical point of the surface energy. Further, the elastic bending energy of the phase field converges to the surface energy and the Lagrange multipliers associated with the volume and surface area constraints remain uniformly bounded. This paper is a first step to analytically justify the numerical simulations performed by Du, Liu and Wang in 2004 to model equilibrium configurations of vesicle membranes.

The hemifusion structure induced by Influenza virus haemagglutinin is determined by physical properties of the target membranes

Influenza A virus haemagglutinin conformational change drives the membrane fusion of viral and endosomal membranes at low pH. Membrane fusion proceeds through an intermediate called hemifusion1,2. For viral fusion, the hemifusion structures are not determined3. Here, influenza virus-like particles4 carrying wild-type haemagglutinin or haemagglutinin hemifusion mutant G1S5 and liposome mixtures were studied at low pH by Volta phase plate cryo-electron tomography, which improves the signal-to-noise ratio close to focus. We determined two distinct hemifusion structures: a hemifusion diaphragm and a novel structure termed a ‘lipidic junction’. Liposomes with lipidic junctions were ruptured with membrane edges stabilized by haemagglutinin. The rupture frequency and hemifusion diaphragm diameter were not affected by G1S mutation, but decreased when the cholesterol level in the liposomes was close to physiological concentrations. We propose that haemagglutinin induces a merger between the viral and target membranes by one of two independent pathways: a rupture–insertion pathway leading to the lipidic junction and a hemifusion-stalk pathway leading to a fusion pore. The latter is relevant under the conditions of influenza virus infection of cells. Cholesterol concentration functions as a pathway switch because of its negative spontaneous curvature in the target bilayer, as determined by continuum analysis.

Calculating Transition Energy Barriers and Characterizing Activation States for Steps of Fusion

We use continuum mechanics to calculate an entire least energy pathway of membrane fusion, from stalk formation, to pore creation, and through fusion pore enlargement. The model assumes that each structure in the pathway is axially symmetric. The static continuum stalk structure agrees quantitatively with experimental stalk architecture. Calculations show that in a stalk, the distal monolayer is stretched and the stored stretching energy is significantly less than the tilt energy of an unstretched distal monolayer. The string method is used to determine the energy of the transition barriers that separate intermediate states and the dynamics of two bilayers as they pass through them. Hemifusion requires a small amount of energy independently of lipid composition, while direct transition from a stalk to a fusion pore without a hemifusion intermediate is highly improbable. Hemifusion diaphragm expansion is spontaneous for distal monolayers containing at least two lipid components, given sufficiently negative diaphragm spontaneous curvature. Conversely, diaphragms formed from single-component distal monolayers do not expand without the continual injection of energy. We identify a diaphragm radius, below which central pore expansion is spontaneous. For larger diaphragms, prior studies have shown that pore expansion is not axisymmetric, and here our calculations supply an upper bound for the energy of the barrier against pore formation. The major energy-requiring deformations in the steps of fusion are: widening of a hydrophobic fissure in bilayers for stalk formation, splay within the expanding hemifusion diaphragm, and fissure widening initiating pore formation in a hemifusion diaphragm.

Aqueous Viscosity Is the Primary Source of Friction in Lipidic Pore Dynamics

A new theory, to our knowledge, is developed that describes the dynamics of a lipidic pore in a liposome. The equations of the theory capture the experimentally observed three-stage functional form of pore radius over time--stage 1, rapid pore enlargement; stage 2, slow pore shrinkage; and stage 3, rapid pore closure. They also show that lipid flow is kinetically limited by the values of both membrane and aqueous viscosity; therefore, pore evolution is affected by both viscosities. The theory predicts that for a giant liposome, tens of microns in radius, water viscosity dominates over the effects of membrane viscosity. The edge tension of a lipidic pore is calculated by using the theory to quantitatively account for pore kinetics in stage 3, rapid pore closing. This value of edge tension agrees with the value as standardly calculated from the stage of slow pore closure, stage 2. For small, submicron liposomes, membrane viscosity affects pore kinetics, but only if the viscosity of the aqueous solution is comparable to that of distilled water. A first-principle fluid-mechanics calculation of the friction due to aqueous viscosity is in excellent agreement with the friction obtained by applying the new theory to data of previously published experimental results.

The aqueous viscous drag of a contracting open surface.

A problem for fluid flow around an axisymmetric spherical surface with a hole is presented to characterize pore dynamics in liposomes. A rotational stream function for the contraction of a punctured plane region is obtained and is used in the perturbation expansion for a stream function in the case of a spherical surface with a hole of small radius compared to the spherical radius. The Rayleigh dissipation function is calculated and used to infer the aqueous friction induced by the contraction of the hole. The theoretical aqueous friction coefficient is compared with one derived from experimental data, and they are in agreement.