Matthias Röger

Turing instabilities in a mathematical model for signaling networks

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.

The Allen–Cahn action functional in higher dimensions

The Allen‐Cahn action functional is related to the probability of rare events in the stochastically perturbed Allen‐Cahn equation. Formal calculations suggest a reduced action functionalin the sharp interface limit. We prove the corresponding lower bound in two and three space dimensions. One difficulty is that diffuse interfaces may collapse in the limit. We therefore consider the limit of diffuse surface area measures and introduce a generalized velocity and generalized reduced action functional in a class of evolving measures.

Solutions for the Stefan problem with Gibbs-Thomson law by a local minimisation

A new construction scheme for a time-discrete version of the Stefan problem with Gibbs–Thomson law is introduced. Extending a scheme due to Luckhaus [11] our approach uses a local minimisation of certain penalised functionals instead of minimising these functionals globally. The main difference is that local minimisation allows for surface loss of approximate phase interfaces in the limit. The theory of varifolds is used to obtain the convergence of approximate Gibbs–Thomson equations. A particular situation exhibits that local minimisation provides more physically appealing solutions than those constructed by global minimisation.

Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids

Abstract We introduce a new sharp interface model for the flow of two immiscible, viscous, incompressible fluids. In contrast to classical models for two-phase flows we prescribe an evolution law for the interfaces that takes diffusional effects into account. This leads to a coupled system of Navier–Stokes and Mullins–Sekerka type parts that coincides with the asymptotic limit of a diffuse interface model. We prove the long-time existence of weak solutions, which is an open problem for the classical two-phase model. We show that the phase interfaces have in almost all points a generalized mean curvature.

Symmetry breaking in a bulk–surface reaction–diffusion model for signalling networks

Signaling 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. The second possibility on the other hand 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 to investigate the evolution beyond the regime where the linearization applies.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.

Tightness for a stochastic Allen-Cahn equation

We study an Allen–Cahn equation perturbed by a multiplicative stochastic noise that is white in time and correlated in space. Formally this equation approximates a stochastically forced mean curvature flow. We derive a uniform bound for the diffuse surface area, prove the tightness of solutions in the sharp interface limit, and show the convergence to phase-indicator functions.

Existence of Weak Solutions for the Mullins-Sekerka Flow

We prove the long-time existence of solutions for the Mullins- Sekerka flow. We use a time discrete approximation which was introduced by Luckhaus and Sturzenhecker [Calc. Var. PDE 3 (1995)] and pass in a new weak formulation to the limit.

Confined Elastic Curves

We consider the problem of minimizing Eulers 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

Buckling Instability of Viral Capsids—A Continuum Approach

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Weak solutions for a stochastic mean curvature flow of two-dimensional graphs

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