Structure preserving discretisations of gradient flows for axisymmetric two-phase biomembranes

The form and evolution of multi-phase biomembranes is of fundamental importance in order to understand living systems. In order to describe these membranes, we consider a mathematical model based on a Canham--Helfrich--Evans two-phase elastic energy, which will lead to fourth order geometric evolution problems involving highly nonlinear boundary conditions. We develop a parametric finite element method in an axisymmetric setting. Using a variational approach, it is possible to derive weak formulations for the highly nonlinear boundary value problems such that energy decay laws, as well as conservation properties, hold for spatially discretised problems. We will prove these properties and show that the fully discretised schemes are well-posed. Finally, several numerical computations demonstrate that the numerical method can be used to compute complex, experimentally observed two-phase biomembranes

Finite element methods for fourth order axisymmetric geometric evolution equations

Fourth order curvature driven interface evolution equations frequently appear in the natural sciences. Often axisymmetric geometries are of interest, and in this situation numerical computations are much more efficient. We will introduce and analyze several new finite element schemes for fourth order geometric evolution equations in an axisymmetric setting, and for selected schemes we will show existence, uniqueness and stability results. The presented schemes have very good mesh and stability properties, as will be demonstrated by several numerical examples

Numerical approximation of curve evolutions in Riemannian manifolds

We introduce variational approximations for curve evolutions in two-dimensional Riemannian manifolds that are conformally flat, i.e.\ conformally equivalent to the Euclidean space. Examples include the hyperbolic plane, the hyperbolic disk, the elliptic plane as well as any conformal parameterization of a two-dimensional surface in ${\mathbb R}^d$, $d\geq 3$. In these spaces we introduce stable numerical schemes for curvature flow and curve diffusion, and we also formulate a scheme for elastic flow. Variants of the schemes can also be applied to geometric evolution equations for axisymmetric hypersurfaces in ${\mathbb R}^d$. Some of the schemes have very good properties with respect to the distribution of mesh points, which is demonstrated with the help of several numerical computations

Cahn-Hilliard-Brinkman systems for tumour growth

A phase field model for tumour growth is introduced that is based on a Brinkman law for convective velocity fields. The model couples a convective Cahn-Hilliard equation for the evolution of the tumour to a reaction-diffusion-advection equation for a nutrient and to a Brinkman-Stokes type law for the fluid velocity. The model is derived from basic thermodynamical principles, sharp interface limits are derived by matched asymptotics and an existence theory is presented for the case of a mobility which degenerates in one phase leading to a degenerate parabolic equation of fourth order. Finally numerical results describe qualitative features of the solutions and illustrate instabilities in certain situations

Finite Element Approximation for the Dynamics of Fluidic Two-Phase Biomembranes

Biomembranes and vesicles consisting of multiple phases can attain a multitude of shapes, undergoing complex shape transitions. We study a Cahn--Hilliard model on an evolving hypersurface coupled to Navier--Stokes equations on the surface and in the surrounding medium to model these phenomena. The evolution is driven by a curvature energy, modelling the elasticity of the membrane, and by a Cahn--Hilliard type energy, modelling line energy effects. A stable semidiscrete finite element approximation is introduced and, with the help of a fully discrete method, several phenomena occurring for two-phase membranes are computed

Numerical computations of facetted pattern formation in snow crystal growth

Facetted growth of snow crystals leads to a rich diversity of forms, and exhibits a remarkable sixfold symmetry. Snow crystal structures result from diffusion limited crystal growth in the presence of anisotropic surface energy and anisotropic attachment kinetics. It is by now well understood that the morphological stability of ice crystals strongly depends on supersaturation, crystal size and temperature. Until very recently it was very difficult to perform numerical simulations of this highly anisotropic crystal growth. In particular, obtaining facet growth in combination with dendritic branching is a challenging task. We present numerical simulations of snow crystal growth in two and three space dimensions using a new computational method recently introduced by the authors. We present both qualitative and quantitative computations. In particular, a linear relationship between tip velocity and supersaturation is observed. The computations also suggest that surface energy effects, although small, have a larger effect on crystal growth than previously expected. We compute solid plates, solid prisms, hollow columns, needles, dendrites, capped columns and scrolls on plates. Although all these forms appear in nature, most of these forms are computed here for the first time in numerical simulations for a continuum model.Comment: 12 pages, 28 figure

Mathematical modeling and numerical simulation of semiconductor detectors

We report on a system of nonlinear partial differential equations describing signal conversion and amplification in semiconductor detectors. We explain the main ideas governing the numerical treatment of this system as they are implemented in our code WIAS-TeSCA. This software package has been used by the MPI Semiconductor Laboratory for numerical simulation of innovative radiation detectors. We present some simulation results focussing on three-dimensional effects in X-ray detectors for satellite missions

Pulses of enhanced North Pacific Intermediate Water ventilation from the Okhotsk Sea and Bering Sea during the last deglaciation

Under modern conditions only North Pacific Intermediate Water is formed in the Northwest Pacific Ocean. This situation might have changed in the past. Recent studies with General Circulation Models indicate a switch to deep-water formation in the Northwest Pacific during Heinrich Stadial 1 (17.5–15.0 kyr) of the last glacial termination. Reconstructions of past ventilation changes based on paleoceanographic proxy records are still insufficient to test whether a deglacial mode of deep-water formation in the North Pacific Ocean existed. Here we present deglacial ventilation records based on radiocarbon-derived ventilation ages in combination with epibenthic stable carbon isotopes from the Northwest Pacific including the Okhotsk Sea and Bering Sea, the two potential source regions for past North Pacific ventilation changes. Evidence for most rigorous ventilation of the mid-depth North Pacific occurred during Heinrich Stadial 1 and the Younger Dryas, simultaneous to significant reductions in Atlantic Meridional Overturning Circulation. Concurrent changes in δ13C and ventilation ages point to the Okhotsk Sea as driver of millennial-scale changes in North Pacific Intermediate Water ventilation during the last deglaciation. Our records additionally indicate that changes in the δ13C intermediate water (700–1750 m water depth) signature and radiocarbon-derived ventilation ages are in antiphase to those of the deep North Pacific Ocean (>2100 m water depth) during the last glacial termination. Thus, intermediate and deep-water masses of the Northwest Pacific have a differing ventilation history during the last deglaciation

The genome of Romanomermis culicivorax:revealing fundamental changes in the core developmental genetic toolkit in Nematoda

Background: The genetics of development in the nematode Caenorhabditis elegans has been described in exquisite detail. The phylum Nematoda has two classes: Chromadorea (which includes C. elegans) and the Enoplea. While the development of many chromadorean species resembles closely that of C. elegans, enoplean nematodes show markedly different patterns of early cell division and cell fate assignment. Embryogenesis of the enoplean Romanomermis culicivorax has been studied in detail, but the genetic circuitry underpinning development in this species has not been explored. Results: We generated a draft genome for R. culicivorax and compared its gene content with that of C. elegans, a second enoplean, the vertebrate parasite Trichinella spiralis, and a representative arthropod, Tribolium castaneum. This comparison revealed that R. culicivorax has retained components of the conserved ecdysozoan developmental gene toolkit lost in C. elegans. T. spiralis has independently lost even more of this toolkit than has C. elegans. However, the C. elegans toolkit is not simply depauperate, as many novel genes essential for embryogenesis in C. elegans are not found in, or have only extremely divergent homologues in R. culicivorax and T. spiralis. Our data imply fundamental differences in the genetic programmes not only for early cell specification but also others such as vulva formation and sex determination. Conclusions: Despite the apparent morphological conservatism, major differences in the molecular logic of development have evolved within the phylum Nematoda. R. culicivorax serves as a tractable system to contrast C. elegans and understand how divergent genomic and thus regulatory backgrounds nevertheless generate a conserved phenotype. The R. culicivorax draft genome will promote use of this species as a research model