arXiv subjects

Christian Schmeiser

Publications and source records attributed to Christian Schmeiser.

At least 19 recordsLinked to original sources

arXiv2026

Pitchfork bifurcation and traveling waves for a planar ensemble of rigid filaments with repulsive interaction

The so-called Filament Based Lamellipodium Model is a complex modeling framework for a very heterogeneous chemo-mechanical system of cell biology. It contains a model for Coulomb repulsion between filaments, whose effect on the stability of the system has been unclear. In this work, a strongly simplified version of the model is considered, showing a destabilizing effect of the repulsion. This instability results in a pitchfork bifurcation with an additional rotational symmetry, leading to a two-dimensional bifurcating manifold of traveling wave solutions. The simplified model is derived, its linearization around the trivial steady state is analyzed, and a formal bifurcation analysis is carried out. It is shown that the pitchfork bifurcation maybe super- or sub-critical. Time dependent numerical simulations illustrate these results and provide additional, more global information on the emergence of periodic and chaotic dynamics by secondary bifurcations.

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arXiv2026

Steady States of Transport-Coagulation-Nucleation Models

To model the dynamics of polymers formed through nucleation, elongated by polymerisation, shortened by depolymerisation and subject to aggregation reactions, we study a nonlinear integro-differential equation. Growth and shrinkage are described by transport terms, nucleation by a positive boundary condition, and aggregation by a Smoluchowski coagulation kernel. Our main result is the existence of steady states for the multiplicative coagulation kernel despite this kernel producing gelation in finite time for the pure coagulation equation. This is made possible by a sufficiently strong decay rate for large polymers. Beyond the existence result, the qualitative properties of the steady states are illustrated through explicit examples and numerical experiments. The analytical results connect the growth behaviour of the transport velocity and of the coagulation kernel to the decay properties of steady states.

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arXiv2025

Modeling Protein Diffusion Across ER-Nuclear Envelope Junctions Reveals Efficient Transport via Simple Diffusion

The endoplasmic reticulum (ER) is the largest continuous membrane-bound organelle in the cell and plays a central role in the synthesis and turnover of many lipids and proteins. It connects directly to the nucleus through specialized contact points known as ER-nuclear envelope (NE) junctions. In our recent study, we found that these ER-NE junctions are both narrow and infrequent, measuring less than 20 nanometers in diameter and occurring at a frequency of approximately 0.1 per square micrometer. However, it remains unclear whether such limited and narrow connections are sufficient to support efficient transport between the ER and NE. Here, we built a mathematical model of ER-to-NE protein diffusion, incorporating ultrastructural parameters, the frequency of ER-NE junctions, and the diffusion coefficient of proteins within the ER lumen. To validate the model, we experimentally quantified the transport rate of ER luminal proteins to the NE using fluorescence recovery after photobleaching (FRAP). Our model and experimental data demonstrate that simple diffusion is sufficient to account for the rapid transport of proteins from the ER to the NE, despite the limited and narrow nature of the connecting junctions. Together, these findings offer mechanistic insight into how ER-NE connectivity enables rapid protein transport and lay the groundwork for future studies on ER-nucleus communication.

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arXiv2025

First order non-instantaneous corrections in collisional kinetic alignment models

In this work the standard kinetic theory assumption of instantaneous collisions is lifted. As a continuation of of a previous paper by Kanzler, Schmeiser, and Tora [KRM, 2024], a model for higher order non-instantaneous alignment collisions is presented and studied in the asymptotic regime of short collision duration. A first order accurate approximative model is derived as a correction to the instantaneous limit. Rigorous results on its well-posedness and on the instantaneous limit are proven. The approximative model is a system of two equations. An equally accurate scalar approximation is suggested.

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arXiv2024

Stability of equilibria of the spatially inhomogeneous Vicsek-BGK equation across a bifurcation

The Vicsek-BGK equation is a kinetic model for alignment of particles moving with constant speed between stochastic reorientation events with sampling from a von Mises distribution. The spatially homogeneous model shows a steady state bifurcation with exchange of stability. The main result of this work is an extension of the bifurcation result to the spatially inhomogeneous problem under the additional assumption of a sufficiently large Knudsen number. The mathematical core is the proof of linearized stability, which employs a new hypocoercivity approach based on Laplace-Fourier transformation. The bifurcation result includes global existence of smooth solutions for close-to-equilibrium initial data. For large data smooth solutions might blow up in finite time whereas weak solutions with bounded Boltzmann entropy are shown to exist globally.

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arXiv2024

Special macroscopic modes and hypocoercivity

We study linear inhomogeneous kinetic equations with an external confining potential and a collision operator admitting several local conservation laws (local density, momentum and energy). We classify all special macroscopic modes (stationary solutions and time-periodic solutions). We also prove the convergence of all solutions of the evolution equation to such non-trivial modes, with a quantitative exponential rate. This is the first hypocoercivity result with multiple special macroscopic modes with constructive estimates depending on the geometry of the potential.

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arXiv2023

One-dimensional short-range nearest-neighbor interaction and its nonlinear diffusion limit

Repulsion between individuals within a finite radius is encountered in numerous applications, including cell exclusion, i.e. avoidance of overlapping cells, bird flocks, or microscopic pedestrian models. We define such individual based particle dynamics in one spatial dimension with minimal assumptions of the repulsion force f as well as their external velocity v and prove their characteristic properties. Moreover, we are able to perform a rigorous limit from the microscopic to the macroscopic scale, where we could recover the finite interaction radius as a density threshold. Specific choices for the repulsion force f lead to well known nonlinear diffusion equations on the macroscopic scale, as e.g. the porous medium equation. At both scaling levels numerical simulations are presented and compared to underline the analytical results.

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arXiv2023

Two kinetic models for non-instantaneous binary alignment collisions

A new type of kinetic models with non-instantaneous binary collisions is considered. Collisions are described by a transport process in the joint state space of a pair of particles. The interactions are of alignment type, where the states of the particles approach each other. For two spatially homogeneous models with deterministic or stochastic collision times existence and uniqueness of solutions, the long time behavior, and the instantaneous limit are considered, where the latter leads to standard kinetic models of Boltzmann type.

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arXiv2023

Reversal Collision Dynamics

Motivated by the study of reversal behaviour of myxobacteria, in this article we are interested in a kinetic model for reversal dynamics, in which particles with directions close to be opposite undergo binary collision resulting in reversing their orientations. To this aim, a generic model for binary collisions between particles with states in a general metric space exhibiting specific symmetry properties is proposed and investigated. The reversal process is given by an involution on the space, and the rate of collision is only supposed to be bounded and lower semi-continuous. We prove existence and uniqueness of measure solutions as well as their convergence to equilibrium, using the graph-theoretical notion of connectivity. We first characterise the shape of equilibria in terms of connected components of a graph on the state space, which can be associated to the initial data of the problem. Strengthening the notion of connectivity on subsets for which the rate of convergence is bounded below, we then show exponential convergence towards the unique steady-state associated to the initial condition. The article is concluded with numerical simulations set on the one-dimensional torus giving evidence to the analytical results.

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arXiv2022

The influence of nucleus mechanics in modelling adhesion-independent cell migration in structured and confined environments

Recent biological experiments have shown that certain types of cells are able to move in structured and confined environment even without the activation of focal adhesion. Focusing on this particular phenomenon and based on previous works, we derive a novel two-dimensional mechanical model, which relies on the following physical ingredients: the asymmetrical renewal of the actin cortex supporting the membrane, resulting in a backward flow of material; the mechanical description of the nucleus membrane and the inner nuclear material; the microtubule network guiding nucleus location; the contact interactions between the cell and the external environment. The resulting fourth order system of partial differential equations is then solved numerically to conduct a study of the qualitative effects of the model parameters, mainly those governing the mechanical properties of the nucleus and the geometry of the confining structure. Coherently with biological observations, we find that cells characterized by a stiff nucleus are unable to migrate in channels that can be crossed by cells with a softer nucleus. Regarding the geometry, cell velocity and ability to migrate are influenced by the width of the channel and the wavelength of the external structure. Even though still preliminary, these results can be potentially useful in determining the physical limit of cell migration in confined environment and in designing scaffold for tissue engineering.

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arXiv2022

Hypocoercivity and reaction-diffusion limit for a nonlinear generation-recombination model

A reaction-kinetic model for a two-species gas mixture undergoing pair generation and recombination reactions is considered on a flat torus. For dominant scattering with a non-moving constant-temperature background the macroscopic limit to a reaction-diffusion system is carried out. Exponential decay to equilibrium is proven for the kinetic model by hypocoercivity estimates. This seems to be the first rigorous derivation of a nonlinear reaction-diffusion system from a kinetic model as well as the first hypocoercivity result for a nonlinear kinetic problem without smallness assumptions. The analysis profits from uniform bounds of the solution in terms of the equilibrium velocity distribution.

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arXiv2021

Bifurcation of elastic curves with modulated stiffness

We investigate the equilibrium configurations of closed planar elastic curves of fixed length, whose stiffness, also known as the bending rigidity, depends on an additional density variable. The underlying variational model relies on the minimization of a bending energy with respect to shape and density and can be considered as a one-dimensional analogue of the Canham-Helfrich model for heterogeneous biological membranes. We present a generalized Euler-Bernoulli elastica functional featuring a density-dependent stiffness coefficient. In order to treat the inherent nonconvexity of the problem we introduce an additional length scale in the model by means of a density gradient term. We derive the system of Euler-Lagrange equations and study the bifurcation structure of solutions with respect to the model parameters. Both analytical and numerical results are presented.

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arXiv2021

Asymptotic limits for a non-linear integro-differential equation modelling leukocytes' rolling on arterial walls

We consider a non-linear integro-differential model describing $z$, the position of the cell center on the real line presented in [Grec et al., J. Theo. Bio. 2018]. We introduce a new $\varepsilon$-scaling and we prove rigorously the asymptotics when $\varepsilon$ goes to zero. We show that this scaling characterizes the long-time behavior of the solutions of our problem in the cinematic regime (the velocity $\dot{z}$ tends to a limit). The convergence results are first given when $ψ$, the elastic energy associated to linkages, is convex and regular (the second order derivative of $ψ$ is bounded). In the absence of blood flow, when $ψ$, is quadratic, we compute the final position $z_\infty$ to which we prove that $z$ tends. We then build a rigorous mathematical framework for $ψ$ being convex but only Lipschitz. We extend convergence results with respect to $\varepsilon$ to this case when $ψ'$ admits a finite number of jumps. In the last part, we show that in the constant force case (see Model 3 in [Grec et al], $ψ$ is the absolute value), we solve explicitly the problem and recover the above asymptotic results.

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arXiv2021

Kinetic Model for Myxobacteria with Directional Diffusion

In this article a kinetic model for the dynamics of myxobacteria colonies on flat surfaces is investigated. The model is based on the kinetic equation for collective bacteria dynamics introduced in arXiv:2001.02711, which is based on the assumption of hard binary collisions of two different types: alignment and reversal, but extended by additional Brownian forcing in the free flight phase of single bacteria. This results in a diffusion term in velocity direction at the level of the kinetic equation, which opposes the concentrating effect of the alignment operator. A global existence and uniqueness result as well as exponential decay to uniform equilibrium is proved in the case where the diffusion is large enough compared to the total bacteria mass. Further, the question wether in a small diffusion regime nonuniform stable equilibria exist is positively answered by performing a formal bifurcation analysis, which revealed the occurrence of a pitchfork bifurcation. These results are illustrated by numerical simulations.

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arXiv2021

Thermalization of a rarefied gas with total energy conservation: existence, hypocoercivity, macroscopic limit

The thermalization of a gas towards a Maxwellian velocity distribution with the background temperature is described by a kinetic relaxation model. The sum of the kinetic energy of the gas and the thermal energy of the background are conserved, and the heat flow in the background is governed by the Fourier law. For the coupled nonlinear system of the kinetic and the heat equation, existence of solutions is proved on the one-dimensional torus. Spectral stability of the equilibrium is shown on the torus in arbitrary dimensions by hypocoercivity methods. The macroscopic limit towards a nonlinear cross-diffusion problem is carried out formally.

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arXiv2021

Propagator norm and sharp decay estimates for Fokker-Planck equations with linear drift

We are concerned with the short- and large-time behavior of the $L^2$-propagator norm of Fokker-Planck equations with linear drift, i.e. $\partial_t f=\mathrm{div}_{x}{(D \nabla_x f+Cxf)}$. With a coordinate transformation these equations can be normalized such that the diffusion and drift matrices are linked as $D=C_S$, the symmetric part of $C$. The main result of this paper (Theorem 3.4) is the connection between normalized Fokker-Planck equations and their drift-ODE $\dot x=-Cx$: Their $L^2$-propagator norms actually coincide. This implies that optimal decay estimates on the drift-ODE (w.r.t. both the maximum exponential decay rate and the minimum multiplicative constant) carry over to sharp exponential decay estimates of the Fokker-Planck solution towards the steady state. A second application of the theorem regards the short time behaviour of the solution: The short time regularization (in some weighted Sobolev space) is determined by its hypocoercivity index, which has recently been introduced for Fokker-Planck equations and ODEs (see [5, 1, 2]). In the proof we realize that the evolution in each invariant spectral subspace can be represented as an explicitly given, tensored version of the corresponding drift-ODE. In fact, the Fokker-Planck equation can even be considered as the second quantization of $\dot x=-Cx$.

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arXiv2021

Sharpening of decay rates in Fourier based hypocoercivity methods

This paper is dealing with two $L^2$ hypocoercivity methods based on Fourier decomposition and mode-by-mode estimates, with applications to rates of convergence or decay in kinetic equations on the torus and on the whole Euclidean space. The main idea is to perturb the standard $L^2$ norm by a twist obtained either by a nonlocal perturbation build upon diffusive macroscopic dynamics, or by a change of the scalar product based on Lyapunov matrix inequalities. We explore various estimates for equations involving a Fokker-Planck and a linear relaxation operator. We review existing results in simple cases and focus on the accuracy of the estimates of the rates. The two methods are compared in the case of the Goldstein-Taylor model in one-dimension.

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arXiv2020

Long-time behaviour of a model for p62-ubiquitin aggregation in cellular autophagy

The qualitative behavior of a recently formulated ODE model for the dynamics of heterogenous aggregates is analyzed. Aggregates contain two types of particles, oligomers and cross-linkers. The motivation is a preparatory step of cellular autophagy, the aggregation of oligomers of the protein p62 in the presence of ubiquitin cross-linkers. A combination of explicit computations, formal asymptotics, and numerical simulations has led to conjectures on the bifurcation behavior, certain aspects of which are proven rigorously in this work. In particular, the stability of the zero state, where the model has a smoothness deficit is analyzed by a combination of regularizing transformations and blow-up techniques. On the other hand, in a different parameter regime, the existence of polynomially growing solutions is shown by Poincaré compactification, combined with a singular perturbation analysis .

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