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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.

The Filament Based Lamellipodium Model (FBLM) is a two-phase two-dimensional continuum model, describing the dynamcis of two interacting families of locally parallel actin filaments [31]. It contains accounts of the filaments' bending stiffness, of adhesion to the substrate, and of cross-links connecting the two families.An extension of the model is presented with contributions from nucleation of filaments by branching, from capping, from contraction by actin-myosin interaction, and from a pressure-like repulsion between parallel filaments due to Coulomb interaction. The effect of a chemoattractant is described by a simple signal transduction model influencing the polymerization speed. Simulations with the extended model show its potential for describing various moving cell shapes, depending on the signal transduction procedure, and for predicting transients between nonmoving and moving states as well as changes of direction.

Kinetic transport equations with a given confining potential and non-linear relaxation type collision operators are considered. General (monotone) energy dependent equilibrium distributions are allowed with a chemical potential ensuring mass conservation. Existence and uniqueness of solutions is proven for initial data bounded by equilibrium distributions. The diffusive macroscopic limit is carried out using compensated compactness theory. The result are drift-diffusion equations with nonlinear diffusion. The most notable examples are of the form ∂ t ρ = ∇ · (∇ρ m + ρ ∇V ), ranging from porous medium equations to fast diffusion, with the exponent satisfying 0 < m < 5/3 in R 3 .

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 idealised 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 includes (de)polymerization, of the mechanical effects of cross-linking, 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 organization. In this first treatment, the model successfully simulates the persistence of the treadmilling network in radially spread cells.

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 1173 Math. Models Methods Appl. Sci. 2006.16:1173-1197. Downloaded from www.worldscientific.com by UNIVERSITY OF PITTSBURGH on 03/10/15. For personal use only. 1174 F. Chalub et al. by scaling limit procedures. We review rigorous convergence results and discuss finitetime 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.

We consider a microscopic model for friction mediated by transient elastic linkages introduced in [8, 10]. In this study we extend results and the general approach employed in [5]. 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 [5] concerning the convergence of the system to a macroscopic friction law in the semi-coupled case, but under weaker assumptions. Furthermore we consider the fully coupled case and prove existence and uniqueness of the solution.