Dietmar Oelz

MODEL HIERARCHIES FOR CELL AGGREGATION BY CHEMOTAXIS

We present partial differential equation (PDE) model hierarchies for the chemotactically driven motion of biological cells. Starting from stochastic differential models, we derive a kinetic formulation of cell motion coupled to diffusion equations for the chemoattractants. We also derive a fluid dynamic (macroscopic) Keller–Segel type chemotaxis model by scaling limit procedures. We review rigorous convergence results and discuss finite-time blow-up of Keller–Segel type systems. Finally, recently developed PDE-models for the motion of leukocytes in the presence of multiple chemoattractants and of the slime mold Dictyostelium Discoideum are reviewed.

Size distribution dependence of prion aggregates infectivity

We consider a model for the polymerization (fragmentation) process involved in infectious prion self-replication and study both its dynamics and non-zero steady state. We address several issues. Firstly, we extend a previous study of the nucleated polymerization model [M.L. Greer, L. Pujo-Menjouet, G.F. Webb, A mathematical analysis of the dynamics of prion proliferation, J. Theoret. Biol. 242 (2006) 598; H. Engler, J. Pruss, G.F. Webb, Analysis of a model for the dynamics of prions II, J. Math. Anal. Appl. 324 (2006) 98] to take into account size dependent replicative properties of prion aggregates. This is achieved by a choice of coefficients in the model that are not constant. Secondly, we show stability results for this steady state for general coefficients where reduction to a system of differential equations is not possible. We use a duality method based on recent ideas developed for population models. These results confirm the potential influence of the amyloid precursor production rate in promoting amyloidogenic diseases. Finally, we investigate how the converting factor may depend upon the aggregate size. Besides the confirmation that size-independent parameters are unlikely to occur, the present study suggests that the PrPsc aggregate size repartition is amongst the most relevant experimental data in order to investigate this dependence. In terms of prion strain, our results indicate that the PrPsc aggregate repartition could be a constraint during the adaptation mechanism of the species barrier overcoming, that opens experimental perspectives for prion amyloid polymerization and prion strain investigation.

Modeling of the actin-cytoskeleton in symmetric lamellipodial fragments.

The pushing structures of cells include laminar sheets, termed lamellipodia, made up of a meshwork of actin filaments that grow at the front and depolymerise at the rear, in a treadmilling mode. We here develop a mathematical model to describe the turnover and the mechanical properties of this network. Our basic modeling assumptions are that the lamellipodium is idealized as a two-dimensional structure, and that the actin network consists of two families of possibly bent, but locally parallel filaments. Instead of dealing with individual polymers, the filaments are assumed to be continuously distributed. The model has the potential to include the effects of (de)polymerization, of the mechanical effects of cross-linking, bundling, and motor proteins, of cell-substrate adhesion, as well as of the leading edge of the membrane. In the first version presented here, the total amount of F-actin is prescribed by assuming a constant polymerisation speed at the leading edge and a fixed total number and length distribution of filaments. We assume that cross-links at filament crossing points as well as integrin linkages with the matrix break and reform in response to incremental changes in network organisation. In this first treatment, the model successfully simulates the persistence of the treadmilling network in radially spread cells.

A Combination of Actin Treadmilling and Cross-Linking Drives Contraction of Random Actomyosin Arrays

We investigate computationally the self-organization and contraction of an initially random actomyosin ring. In the framework of a detailed physical model for a ring of cross-linked actin filaments and myosin-II clusters, we derive the force balance equations and solve them numerically. We find that to contract, actin filaments have to treadmill and to be sufficiently cross linked, and myosin has to be processive. The simulations reveal how contraction scales with mechanochemical parameters. For example, they show that the ring made of longer filaments generates greater force but contracts slower. The model predicts that the ring contracts with a constant rate proportional to the initial ring radius if either myosin is released from the ring during contraction and actin filaments shorten, or if myosin is retained in the ring, while the actin filament number decreases. We demonstrate that a balance of actin nucleation and compression-dependent disassembly can also sustain contraction. Finally, the model demonstrates that with time pattern formation takes place in the ring, worsening the contractile process. The more random the actin dynamics are, the higher the contractility will be.

A viscous two-phase model for contractile actomyosin bundles

A mathematical model in one dimension for a non-sarcomeric actomyosin bundle featuring anti-parallel flows of anti-parallel F-actin is introduced. The model is able to relate these flows to the effect of cross-linking and bundling proteins, to the forces due to myosin-II filaments and to external forces at the extreme tips of the bundle. The modeling is based on a coarse graining approach starting with a microscopic model which includes the description of chemical bonds as elastic springs and the force contribution of myosin filaments. In a second step we consider the asymptotic regime where the filament lengths are small compared to the overall bundle length and restrict to the lowest order contributions. There it becomes apparent that myosin filaments generate forces which are partly compensated by drag forces due to cross-linking proteins. The remaining local contractile forces are then propagated to the tips of the bundle by the viscosity effect of bundling proteins in the filament gel. The model is able to explain how a disordered bundle of comparatively short actin filaments interspersed with myosin filaments can effectively contract the two tips of the actomyosin bundle. It gives a quantitative description of these forces and of the anti-parallel flows of the two phases of anti-parallel F-actin. An asymptotic version of the model with infinite viscosity can be solved explicitly and yields an upper bound to the contractile force of the bundle.

Simulation of lamellipodial fragments

A steepest descent approximation scheme is derived for a recently developed model for the dynamics of the actin cytoskeleton in the lamellipodia of living cells. The scheme is used as a numerical method for the simulation of thought experiments, where a lamellipodial fragment is pushed by a pipette, and subsequently changes its shape and position.

Analysis of a Relaxation Scheme for a Nonlinear Schrödinger Equation Occurring in Plasma Physics

AbstractThis paper is devoted to the analysis of a relaxation-type numerical scheme for a nonlinear Schrodinger equation arising in plasma physics. The scheme is shown to be preservative in the sense that it preserves mass and energy. We prove the well-posedness of the semidiscretized system and prove convergence to the solution of the time-continuous model.

Numerical Treatment of the Filament-Based Lamellipodium Model (FBLM)

We describe in this work the numerical treatment of the Filament-Based Lamellipodium Model (FBLM). This model is a two-phase two-dimensional continuum model, describing the dynamics of two interacting families of locally parallel F-actin filaments. It includes, among others, the bending stiffness of the filaments, adhesion to the substrate, and the cross-links connecting the two families. The numerical method proposed is a Finite Element Method (FEM) developed specifically for the needs of this problem. It is comprised of composite Lagrange–Hermite two-dimensional elements defined over a two-dimensional space. We present some elements of the FEM and emphasize in the numerical treatment of the more complex terms. We also present novel numerical simulations and compare to in-vitro experiments of moving cells.

ON A STRUCTURED MODEL FOR LOAD-DEPENDENT REACTION KINETICS OF TRANSIENT ELASTIC LINKAGES MEDIATING NONLINEAR FRICTION ∗

We consider a microscopic model for friction mediated by transient elastic linkages introduced in [D. Oelz and C. Schmeiser, Arch. Ration. Mech. Anal., 198 (2010), pp. 963--980; D. Oelz, C. Schmeiser, and V. Small, Cell Adh. Migr., 2 (2008), pp. 117--126]. In this study we extend results and the general approach employed in [V. Milisic and D. Oelz, J. Math. Pures Appl. (9), 96 (2011), pp. 484--501]. We introduce a new unknown and reformulate the model. Based on this framework, we derive new a priori estimates. In a first step this approach allows us to reproduce results of our earlier paper concerning the convergence of the system to a macroscopic friction law in the semicoupled case, but under weaker assumptions. Furthermore, we consider the fully coupled case and prove existence and uniqueness of the solution.

A quasilinear parabolic singular perturbation problem

We study the following singular perturbation problem for a quasilinear uniformly parabolic operator of interest in combustion theory: div F(∇u)-∂u = β(u), where u ≥ 0, β(s) = (1/e)β(s/e), e > 0, β is Lipschitz continuous, supp β = [0, 1] and β > 0 in (0, 1). We obtain uniform estimates, we pass to the limit (e → 0) and we show that, under suitable assumptions, the limit function u is a solution to the free boundary problem div F(∇) - ∂u = 0 in {u > 0}, u = α(υ, M) on ∂{u > 0}, in a pointwise sense and in a viscosity sense. Here u denotes the derivative of u with respect to the inward unit spatial normal υ to the free boundary ∂{u > 0}, M = ∫ β(s) ds, α(υ, M) := Φ (M) and Φ(α) := - A(αυ) +αυ · F(αυ), where A(p) is such that F(p) = ∇A(p) with A(0) = 0. Some of the results obtained are new even when the operator under consideration is linear.