We consider a Gierer-Meinhardt system on a surface coupled with a parabolic PDE in the bulk, the domain confined by this surface. Such a model was recently proposed and analyzed for two-dimensional bulk domains by Gomez, Ward and Wei (SIAM J. Appl. Dyn. Syst. 18, 2019). We prove the well-posedness of the bulk-surface system in arbitrary space dimensions and show that solutions remain uniformly bounded in parabolic Hölder spaces for all times. The cytosolic diffusion is typically much larger than the lateral diffusion on the membrane. This motivates to a corresponding asymptotic reduction, which consists of a nonlocal system on the membrane. We prove the convergence of solutions of the full system towards unique solutions of the reduction. © 2022 American Institute of Mathematical Sciences. All rights reserved.

Standard diffuse approximations of the Willmore flow often lead to intersecting phase boundaries that in many cases do not correspond to the intended sharp interface evolution. Here we introduce a new two-variable diffuse approximation that includes a rather simple but efficient penalization of the deviation from a quasi-one dimensional structure of the phase fields. We justify the approximation property by a Gamma convergence result for the energies and a matched asymptotic expansion for the flow. Ground states of the energy are shown to be one-dimensional, in contrast to the presence of saddle solutions for the usual diffuse approximation. Finally we present numerical simulations that illustrate the approximation property and apply our new approach to problems where the usual approach leads to an undesired behavior. © EDP Sciences, SMAI 2021.

The amplification of an external signal is a key step in direction sensing of biological cells. We consider a simple model for the response to a time-depending signal, which was previously proposed by the last three authors. The model consists of a bulk-surface reaction-diffusion model. We prove that in a suitable asymptotic limit the system converges to a bulk-surface parabolic obstacle-type problem. For this model and a reduction to a nonlocal surface equation we show an L1contraction property and, in the case of time-constant signals, the stability of stationary states. © 2021 Society for Industrial and Applied Mathematics.

We propose a model for cell polarization as a response to an external signal which results in a system of PDEs for different variants of a protein on the cell surface and interior respectively. We study stationary states of this model in certain parameter regimes in which several reaction rates on the membrane as well as the diffusion coefficient within the cell are large. It turns out that in suitable scaling limits steady states converge to solutions of some obstacle type problems. For these limiting problems we prove the onset of polarization if the total mass of protein is sufficiently small. For some variants we can even characterize precisely the critical mass for which polarization occurs. © European Mathematical Society 2020.

We are concerned with a stochastic mean curvature flow of graphs over a periodic domain of any space dimension. For the first time, we are able to construct martingale solutions which satisfy the equation pointwise and not only in a generalized (distributional or viscosity) sense. Moreover, we study their large-time behavior. Our analysis is based on a viscous approximation and new global bounds, namely, an Lω,x,t∞ estimate for the gradient and an Lω,x,t2 bound for the Hessian. The proof makes essential use of the delicate interplay between the deterministic mean curvature part and the stochastic perturbation, which permits to show that certain gradient-dependent energies are supermartingales. Our energy bounds in particular imply that solutions become asymptotically spatially homogeneous and approach a Brownian motion perturbed by a random constant. © 2020, The Author(s).

We discuss a variational model, given by a weighted sum of perimeter, bending and Riesz interaction energies, that could be considered as a toy model for charged elastic drops. The different contributions have competing preferences for strongly localized and maximally dispersed structures. We investigate the energy landscape in dependence of the size of the 'charge', that is, the weight of the Riesz interaction energy. In the two-dimensional case, we first prove that for simply connected sets of small elastica energy, the elastica deficit controls the isoperimetric deficit. Building on this result, we show that for small charge the only minimizers of the full variational model are either balls or centred annuli. We complement these statements by a non-existence result for large charge. In three dimensions, we prove area and diameter bounds for configurations with small Willmore energy and show that balls are the unique minimizers of our variational model for sufficiently small charge. © Copyright 2019 Royal Society of Edinburgh.

We study a scalar elliptic problem in the data driven context. Our interest is to study the relaxation of a data set that consists of the union of a linear relation and single outlier. The data driven relaxation is given by the union of the linear relation and a truncated cone that connects the outlier with the linear subspace. © 2020, The Author(s).

Rab GTPases (>60 members in humans) function as master regulators of intracellular membrane trafficking. Correct and specific localization of Rab proteins is required for their function. How the distinct spatial distribution of Rab GTPases in the cell is regulated remains elusive. To globally assess the subcellular localization of Rab1, we determined kinetic parameters of two pathways that control the spatial cycles of Rab1, i.e., vesicular transport and GDP dissociation inhibitor (GDI)-mediated recycling. We demonstrate that the switching between GTP and GDP binding states, which is governed by guanine nucleotide exchange factors (GEFs), GTPase-activating proteins (GAPs), GDI, and GDI displacement factor (GDF), is a major determinant of Rab1's ability to effectively cycle between cellular compartments and eventually its subcellular distribution. In silico perturbations of vesicular transport, GEFs, GAPs, GDI, and GDF using a mathematical model with simplified cellular geometries showed that these regulators play an important role in the subcellular distribution and activity of Rab1. © 2019 American Chemical Society.

The Allen–Cahn equation is a prototype model for phase separation processes, a fundamental example of a nonlinear spatial dynamic and an important approximation of a geometric evolution equation by a reaction–diffusion equation. Stochastic perturbations, especially in the case of additive noise, to the Allen–Cahn equation have attracted considerable attention. We consider here an alternative random perturbation determined by a Brownian flow of spatial diffeomorphism that was introduced by Röger and Weber (Stoch Partial Differ Equ Anal Comput 1(1):175–203, 2013). We first provide a large deviation principle for stochastic flows in spaces of functions that are Hölder continuous in time, which extends results by Budhiraja et al. (Ann Probab 36(4):1390–1420, 2008). From this result and a continuity argument we deduce a large deviation principle for the Allen–Cahn equation perturbed by a Brownian flow in the limit of small noise. Finally, we present two asymptotic reductions of the large deviation functional. © 2016 Springer Science+Business Media New York

We analyze a certain class of coupled bulk–surface reaction–drift–diffusion systems arising in the modeling of signalling networks in biological cells. The coupling is by a nonlinear Robin-type boundary condition for the bulk variable and a corresponding source term on the cell boundary. For reaction terms with at most linear growth and under different regularity assumptions on the data we prove the existence of weak and classical solutions. In particular, we show that solutions grow at most exponentially with time. Furthermore, we rigorously derive an asymptotic reduction to a non-local reaction–drift–diffusion system on the membrane in the fast-diffusion limit. © 2018, Springer International Publishing AG, part of Springer Nature.

We consider a coupled bulk/surface model for advection and diffusion of interacting chemical species in biological cells. Specifically, we consider a signalling protein that can exist in both a cytosolic and a membrane-bound state, along with a variable that gives a coarse-grained description of the cytoskeleton. The main focus of our work is on the well-posedness of the model, whereby the coupling at the boundary is the main source of analytical difficulty. A priori Lp-estimates, together with classical Schauder theory, deliver global existence of classical solutions for small data on bounded, Lipschitz domains. For two physically reasonable regularised versions of the boundary coupling, we are able to prove global existence of solutions for arbitrary data. In addition, we prove the existence of a family of steady-state solutions of the main model which are parametrised by the total mass of the membrane-bound signal molecule. © 2016 Elsevier Inc.

For a bounded smooth domain in the plane and smooth boundary data we consider the minimisation of the Willmore functional for graphs subject to Dirichlet or Navier boundary conditions. For H2- regular graphs we show that bounds for theWillmore energy imply bounds on the surface area and on the height of the graph. We then consider the L1-lower semicontinuous relaxation of the Willmore functional, which is shown to be indeed its largest possible extension, and characterise properties of functions with finite relaxed energy. In particular, we deduce compactness and lower-bound estimates for energy-bounded sequences. The lower bound is given by a functional that describes the contribution by the regular part of the graph and is defined for a suitable subset of BV(ω). We further show that finite relaxed Willmore energy implies the attainment of the Dirichlet boundary data in an appropriate sense, and obtain the existence of a minimiser in L∞ ∩ BV for the relaxed energy. Finally, we extend our results to Navier boundary conditions and more general curvature energies of Canham-Helfrich type. © European Mathematical Society 2017.

We consider the energetic description of a visco-plastic evolution and derive an existence result. The energies are convex, but not necessarily quadratic. Our model is a strain gradient model in which the curl of the plastic strain contributes to the energy. Our existence results are based on a time-discretization, the limit procedure relies on Helmholtz decompositions and compensated compactness. © 2017 World Scientific Publishing Company.

We study a stochastically perturbed mean curvature flow for graphs in (Formula presented.) over the two-dimensional unit-cube subject to periodic boundary conditions. The stochastic perturbation is a one dimensional white noise acting uniformly in all points of the surface in normal direction. We establish the existence of a weak martingale solution. The proof is based on energy methods and therefore presents an alternative to the stochastic viscosity solution approach. To overcome difficulties induced by the degeneracy of the mean curvature operator and the multiplicative gradient noise present in the model we employ a three step approximation scheme together with refined stochastic compactness and martingale identification methods. © 2016 Springer-Verlag Berlin Heidelberg

We propose and investigate a model for lipid raft formation and dynamics in biological membranes. The model describes the lipid composition of the membrane and an interaction with cholesterol. To account for cholesterol exchange between cytosol and cell membrane we couple a bulk-diffusion to an evolution equation on the membrane. The latter describes the relaxation dynamics for an energy which takes lipid-phase separation and lipid-cholesterol interaction energy into account. It takes the form of an (extended) Cahn-Hilliard equation. Different laws for the exchange term represent equilibrium and nonequilibrium models. We present a thermodynamic justification, analyze the respective qualitative behavior and derive asymptotic reductions of the model. In particular we present a formal asymptotic expansion near the sharp interface limit, where the membrane is separated into two pure phases of saturated and unsaturated lipids, respectively. Finally we perform numerical simulations and investigate the long-time behavior of the model and its parameter dependence. Both the mathematical analysis and the numerical simulations show the emergence of raft-like structures in the nonequilibrium case whereas in the equilibrium case only macrodomains survive in the long-time evolution.

We prove a Gamma-convergence result for a family of bending energies defined on smooth surfaces in R3 equipped with a director field. The energies strongly penalize the deviation of the director from the surface unit normal and control the derivatives of the director. Such types of energies arise, for example, in a model for bilayer membranes introduced by Peletier and Röger (Arch Ration Mech Anal 193(3), 475–537, 2009). Here we prove in three space dimensions in the vanishing-tilt limit a Gamma-liminf estimate with respect to a specific curvature energy. In order to obtain appropriate compactness and lower semi-continuity properties we use tools from geometric measure theory, in particular the concept of generalized Gauss graphs and curvature varifolds. © 2015, Springer-Verlag Berlin Heidelberg.

We study the minimum energy configuration of a uniform distribution of negative charge subject to Coulomb repulsive self-interaction and attractive interaction with a fixed positively charged domain. After having established existence and uniqueness of a minimizing configuration, we prove charge neutrality and the complete screening of the Coulomb potential exerted by the positive charge, and we discuss the regularity properties of the solution. We also determine, in the variational sense of Γ-convergence, the limit model when the charge density of the negative phase is much higher than the positive one. © 2016 Society for Industrial and Applied Mathematics.

We consider the reduced Allen-Cahn action functional, which arises as the sharp interface limit of the Allen-Cahn action functional and can be understood as a formal action functional for a stochastically perturbed mean curvature flow. For suitable evolutions of (generalized) hypersurfaces this functional consists of the integral over time and space of the sum of the squares of the mean curvature and of the velocity vectors. Given initial and final conditions, we investigate the associated action minimization problem, for which we propose a weak formulation, and within the latter we prove compactness and lower-semicontinuity properties of a suitably generalized action functional. Furthermore, we derive the Euler-Lagrange equation for smooth stationary trajectories and investigate some related conserved quantities. To conclude, we analyze the explicit case in which the initial and final data are concentric spheres. In this particular situation we characterize the properties of the minimizing rotationally symmetric trajectory in dependence of the given time span. © 2014 Springer-Verlag Berlin Heidelberg.

We consider the problem of minimizing the Willmore energy connected surfaces with prescribed surface area which are confined to a finite container. To this end, we approximate the surface by a phase field function u taking values close to +1 on the inside of the surface and -1 on its outside. The confinement of the surface is now simply given by the domain of definition of u. A diffuse interface approximation for the area functional, as well as for the Willmore energy, are well known. We address the topological constraint of connectedness by a nested minimization of two phase fields, the second one being used to identify connected components of the surface. In this article, we provide a proof of Gamma-convergence of our model to the sharp interface limit. © 2014 Society for Industrial and Applied Mathematics.

Signalling molecules play an important role for many cellular functions. We investigate here a general system of two membrane reaction-diffusion equations coupled to a diffusion equation inside the cell by a Robin-type boundary condition and a flux term in the membrane equations. A specific model of this form was recently proposed by the authors for the GTPase cycle in cells. We investigate here a putative role of diffusive instabilities in cell polarization. By a linearized stability analysis, we identify two different mechanisms. The first resembles a classical Turing instability for the membrane subsystem and requires (unrealistically) large differences in the lateral diffusion of activator and substrate. On the other hand, the second possibility is induced by the difference in cytosolic and lateral diffusion and appears much more realistic. We complement our theoretical analysis by numerical simulations that confirm the new stability mechanism and allow us to investigate the evolution beyond the regime where the linearization applies. © 2014 IOP Publishing Ltd & London Mathematical Society.

We present an asymptotic analysis of a mesoscale energy for bilayer membranes that has been introduced and analyzed in two space dimensions by the second and third authors [Arch. Ration. Mech. Anal. 193 (2009), 475–537]. The energy is both nonlocal and nonconvex. It combines a surface area and a Monge–Kantorovich-distance term, leading to a competition between preferences for maximally concentrated and maximally dispersed configurations. Here we extend key results of our previous analysis to the three-dimensional case. First we prove a general lower estimate and formally identify a curvature energy in the zerothickness limit. Secondly we construct a recovery sequence and prove a matching upper-bound estimate. © 2014, Springer Basel.

This paper is concerned with diffuse-interface approximations of the Willmore ow.We first present numerical results of standard diffuse-interface models for colliding one dimensional interfaces. In such a scenario evolutions towards interfaces with corners can occur that do not necessarily describe the adequate sharp-interface dynamics. We therefore propose and investigate alternative diffuse-interface approximations that lead to a different and more regular behavior if interfaces collide. These dynamics are derived from approximate energies that converge to the L1-lower-semicontinuous envelope of the Willmore energy, which is in general not true for the more standard Willmore approximation.

The crystallographic structure of spherical viruses is modeled using a multiscale approach combining a macroscopic Helfrich model for morphology evolution with a microscopic approximation of a classical density functional theory for the protein interactions. The derivation of the model is based on energy dissipation and conservation of protein number density. The resulting set of equations is solved within a diffuse domain approach using finite elements and shows buckling transitions of spherical shapes into faceted viral shapes. © 2012 Society for Industrial and Applied Mathematics.

GTPase molecules are important regulators in cells that continuously run through an activation/deactivation and membrane-attachment/membrane-detachment cycle. Activated GTPase is able to localize in parts of the membranes and to induce cell polarity. As feedback loops contribute to the GTPase cycle and as the coupling between membrane-bound and cytoplasmic processes introduces different diffusion coefficients a Turing mechanism is a natural candidate for this symmetry breaking. We formulate a mathematical model that couples a reaction-diffusion system in the inner volume to a reaction-diffusion system on the membrane via a flux condition and an attachment/detachment law at the membrane. We present a reduction to a simpler non-local reaction-diffusion model and perform a stability analysis and numerical simulations for this reduction. Our model in principle does support Turing instabilities but only if the lateral diffusion of inactivated GTPase is much faster than the diffusion of activated GTPase. © 2011 Springer-Verlag.

We consider the problem of minimizing Euler's elastica energy for simple closed curves confined to the unit disk. We approximate a simple closed curve by the zero level set of a function with values +1 on the inside and-1 on the outside of the curve. The outer container now becomes just the domain of the phase field. Diffuse approximations of the elastica energy and the curve length are well known; implementing the topological constraint thus becomes the main difficulty here. We propose a solution based on a diffuse approximation of the winding number, present a proof that one can approximate a given sharp interface using a sequence of phase fields, and show some numerical results using finite elements based on subdivision surfaces. © 2011 Society for Industrial and Applied Mathematics.

We study perturbations of the Allen-Cahn equation and prove the convergence to forced mean curvature flow in the sharp interface limit. We allow for perturbations that are square-integrable with respect to the diffuse surface area measure. We give a suitable generalized formulation for forced mean curvature flow and apply previous results for the Allen-Cahn action functional. Finally, we discuss some applications. © 2011 Indiana University Mathematics Journal.