Phase field model for cell spreading dynamics.

We suggest a 3D phase field model to describe 3D cell spreading on a flat substrate. The model is a simplified version of a minimal model that was developed in Winkler (Commun Phys 2:82, 2019). Our model couples the order parameter u with 3D polarization (orientation) vector field [Formula: see text] of the actin network. We derive a closed integro-differential equation governing the 3D cell spreading dynamics on a flat substrate, which includes the normal velocity of the membrane, curvature, volume relaxation rate, a function determined by the molecular effects of the subcell level, and the adhesion effect. This equation is easily solved numerically. The results are in agreement with the early fast phase observed experimentally in Dobereiner (Phys Rev Lett 93:108105, 2004). Also we find agreement with the universal power law (Cuvelier in Curr Biol 17:694-699, 2007) which suggest that cell adhesion or contact area versus time behave as [Formula: see text] in the early stage of cell spreading dynamics, and slow down at the next stages.

Transport on intermediate time scales in flows with cat's eye patterns.

We consider the advection-diffusion transport of tracers in a one-parameter family of plane periodic flows where the patterns of streamlines feature regions of confined circulation in the shape of "cat's eyes," separated by meandering jets with ballistic motion inside them. By varying the parameter, we proceed from the regular two-dimensional lattice of eddies without jets to the sinusoidally modulated shear flow without eddies. When a weak thermal noise is added, i.e., at large Péclet numbers, several intermediate time scales arise, with qualitatively and quantitatively different transport properties: depending on the parameter of the flow, the initial position of a tracer, and the aging time, motion of the tracers ranges from subdiffusive to superballistic. We report on results of extensive numerical simulations of the mean-squared displacement for different initial conditions in ordinary and aged situations. These results are compared with a theory based on a Lévy walk that describes the intermediate-time ballistic regime and gives a reasonable description of the behavior for a certain class of initial conditions. The interplay of the walk process with internal circulation dynamics in the trapped state results at intermediate time scales in nonmonotonic characteristics of aging not captured by the Lévy walk model.

Anomalous transport in cellular flows: The role of initial conditions and aging.

We consider the diffusion-advection problem in two simple cellular flow models (often invoked as examples of subdiffusive tracer motion) and concentrate on the intermediate time range, in which the tracer motion indeed may show subdiffusion. We perform extensive numerical simulations of the systems under different initial conditions and show that the pure intermediate-time subdiffusion regime is only evident when the particles start at the border between different cells, i.e., at the separatrix, and is less pronounced or absent for other initial conditions. The motion moreover shows quite peculiar aging properties, which are also mirrored in the behavior of the time-averaged mean squared displacement for single trajectories. This kind of behavior is due to the complex motion of tracers trapped inside the cell and is absent in classical models based on continuous-time random walks with no dynamics in the trapped state.

Stability of a directional solidification front in subdiffusive media.

The efficiency of crystal growth in alloys is limited by the morphological instability, which is caused by a positive feedback between the interface deformation and the diffusive flux of solute at the front of the phase transition. Usually this phenomenon is described in the framework of the normal diffusion equation, which stems from the linear relation between time and the mean squared displacement of molecules 〈x2(t)〉∼K1t (K1 is the classical diffusion coefficient) that is characteristic of Brownian motion. However, in some media (e.g., in gels and porous media) the random walk of molecules is hindered by obstacles, which leads to another power law, 〈x2(t)〉∼Kαtα, where 0<α≤1. As a result, the diffusion is anomalous, and it is governed by an integro-differential equation including a fractional derivative in time variable, i.e., a memory. In the present work, we investigate the stability of a directional solidification front in the case of an anomalous diffusion. Linear stability of a moving planar directional solidification front is studied, and a generalization of the Mullins-Sekerka stability criterion is obtained. Also, an asymptotic nonlinear long-wave evolution equation of Sivashinsky's type, which governs the cellular structures at the interface, is derived.