Esa Kuusela

Continuum model of cell adhesion and migration

The motility of cells crawling on a substratum has its origin in a thin cell organ called lamella. We present a 2-dimensional continuum model for the lamella dynamics of a slowly migrating cell, such as a human keratinocyte. The central components of the model are the dynamics of a viscous cytoskeleton capable to produce contractile and swelling stresses, and the formation of adhesive bonds in the plasma cell membrane between the lamella cytoskeleton and adhesion sites at the substratum. We will demonstrate that a simple mechanistic model, neglecting the complicated signaling pathways and regulation processes of a living cell, is able to capture the most prominent aspects of the lamella dynamics, such as quasi-periodic protrusions and retractions of the moving tip, retrograde flow of the cytoskeleton and the related accumulation of focal adhesion complexes in the leading edge of a migrating cell. The developed modeling framework consists of a nonlinearly coupled system of hyperbolic, parabolic and ordinary differential equations for the various molecular concentrations, two elliptic equations for cytoskeleton velocity and hydrodynamic pressure in a highly viscous two-phase flow, with appropriate boundary conditions including equalities and inequalities at the moving boundary. In order to analyse this hybrid continuum model by numerical simulations for different biophysical scenarios, we use suitable finite element and finite volume schemes on a fixed triangulation in combination with an adaptive level set method describing the free boundary dynamics.

Computation of particle settling speed and orientation distribution in suspensions of prolate spheroids

A numerical technique for the dynamical simulation of three-dimensional rigid particles in a Newtonian fluid is presented. The key idea is to satisfy the no-slip boundary condition on the particle surface by a localized force-density distribution in an otherwise force-free suspending fluid. The technique is used to model the sedimentation of prolate spheroids of aspect ratio b/a=5 at Reynolds number 0⋅3. For a periodic lattice of single spheroids, the ideas of Hasimoto are extended to obtain an estimate for the finite-size correction to the sedimentation velocity. For a system of several spheroids in periodic arrangement, a maximum of the settling speed is found at the effective volume fraction φ(b/a)2≈0⋅4, where φ is the solid-volume fraction. The occurence of a maximum of the settling speed is partially explained by the competition of two effects: (i) a change in the orientation distribution of the prolate spheroids whose major axes shift from a mostly horizontal orientation (corresponding to small sedimentation speeds) at small φ to a more uniform orientation at larger φ, and (ii) a monotonic decrease of the the settling speed with increasing solid-volume fraction similar to that predicted by the Richardson–Zaki law ∝(1−φ)5⋅5 for suspensions of spheres.

Container Size Dependence of the Velocity Fluctuations in Suspensions of Monodisperse Spheres

We show that a constraint force method can be used in an advantageous fashion to replace the no-slip boundary conditions in a Navier-Stokes simulation of rigid particles suspended in Newtonian fluids. The constraint forces mimic the presence of particles in the fluid. We use the method to study the container-size dependence of the velocity fluctuations in batch sedimentation under periodic boundary conditions and in quadrilateral containers with fixed walls.

Memory Effects and Memory Functions in Surface Diffusion

We consider the diffusive dynamics of particles in various model systems with strong interactions. We study the temporal dependence of the single-particle velocity autocorrelation function o(t), and its corresponding memory function. We find o(t) to decay non-exponentially and in most cases follow a power-law o(t) t- x at intermediate times i, while at long times there is a crossover to an exponential decay. We characterize the possible values of the decay exponent x, and show that x correlates with interaction and ordering effects. In many cases, the memory function follows behavior similar to that of o(t). These results suggest that o(t) can be used to obtain information about the ordering of the system and about the nature of predominant interactions between adparticles using experimental techniques such as scanning tunneling microscopy, in which o(t) can be measured in terms of discrete adparticle displacements. Finally, our studies suggest that the decay of velocity correlations in collective diffusion is qualitatively similar to the case of tracer diffusion.