Tatsuzo Nagai

Vertex models for two-dimensional grain growth

Abstract We develop two-dimensional vertex models which are expected to capture the essential ingredients of the long-time behaviour of two-dimensional grain growth and permit efficient large-scale computer simulations. Such simulations are carried out starting with the initial 24 000 randomly chosen Voronoi cells. We confirm the 1/2 power law for the growth of average grain size. We also find the normalized grain-size distribution and the edge number distribution which exhibit stationary behaviour at long times except for small-amplitude oscillatory behaviour for the latter.

A dynamic cell model for the formation of epithelial tissues

Abstract A dynamic model is proposed for monolayered epithelial tissue, in which the monolayer is assumed to consist of prismatic cells, so that the system is described as a two-dimensional polygonal pattern. Its dynamic behaviour is determined by equations of motion for vertices in the polygonal pattern and elementary change in the system (change in the connection of vertex pairs). The vertices are driven by the sum of interfacial tension on cell boundaries and the resistance force against the deformation of cells. It is shown by computer simulations that our model possesses the characteristics of epithelial tissue, that is it has a mechanism which makes the edge number and size of a cell uniform and the shape symmetric. This mechanism finally gives rise to a regular polygonal pattern similar to a honeycomb pattern, even though initial patterns have large variations. Local equilibrium dynamics under which the size of each cell is flexibly adapted to its local environment are compared with non-local equilibrium dynamics. The local equilibrium dynamics produce more natural cellular patterns.

Computer simulation of emerging asymmetry in the mouse blastocyst

The mechanism of embryonic polarity establishment in mammals has long been controversial. Whereas some claim prepatterning in the egg, we recently presented evidence that mouse embryonic polarity is not established until blastocyst and proposed the mechanical constraint model. Here we apply computer simulation to clarify the minimal cellular properties required for this morphology. The simulation is based on three assumptions: (1) behavior of cell aggregates is simulated by a 3D vertex dynamics model; (2) all cells have equivalent mechanical properties; (3) an inner cavity with equivalent surface properties is gradually enlarged. However, an initial attempt reveals a requirement for an additional assumption: (4) the surface of the cavity is firmer than intercellular surfaces, suggesting the presence of a basement membrane lining the blastocyst cavity, which is indeed confirmed by published data. The simulation thus successfully produces a structure recapitulating the mouse blastocyst. The axis of the blastocyst, however, remains variable, leading us to an additional assumption: (5) the aggregate is enclosed by a capsule, equivalent to the zona pellucida in vivo. Whereas a spherical capsule does not stabilize the blastocyst axis, an ellipsoidal capsule eventually orients the axis in accordance with its longest diameter. These predictions are experimentally verified by time-lapse recordings of mouse embryos. During simulation, equivalent cells form two distinct populations composed of smaller inner cells and larger outer cells. These results reveal a unique feature of early mammalian development: an asymmetry may emerge autonomously in an equivalent population with no need for a priori intrinsic differences.

Two different mechanisms of planar cell intercalation leading to tissue elongation.

During development, certain cells intercalate with each other towards tissue‐elongation, exemplified in sea‐urchin gut‐elongation, amphibian gastrulation, and Drosophila germ‐band extension. Their mechanism is not universal among intercalation events. To clarify the minimal cellular properties required for cell‐intercalation, we computer‐simulated the process using three‐dimensional geometrical cell‐models. We identified two different mechanisms: (1) cell‐junction‐remodeling by cell‐junction contraction along a specific direction, as observed in Drosophila germ‐band extension, and (2) cell‐shuffling by orientated cell‐extension of bipolar cells, as observed in amphibian gastrulation. The cell‐junction‐remodeling was characterized by well‐defined accumulation of contractile molecules along a specific direction of cell‐junctions. Length contraction of approximately one cell‐junction per cell is enough for the entire tissue‐elongation. The cell‐shuffling was characterised by rhythmic cell‐extension and orientated movement of cytoskeleton within the elongated cells. Furthermore, tissue‐elongation along a polarized axis was limited to a 2.5‐fold increase in the cell‐junction‐remodeling, while no limit was defined for the cell‐shuffling. Developmental Dynamics 237:1826–1836, 2008.

Vertex Dynamics of Two-Dimensional Cellular Patterns

Time evolution of a two-dimensional cellular structure is studied by computer for a deterministic dissipative model system of interacting vertices. Despite its completely deterministic nature, our model gives rise to fully-developed random cellular structures which abound in nature. Both the growth law and the shape distribution for cells are obtained and are compared with other existing studies.

Molecular dynamics of interacting kinks. I

A system consisting of a large number (up to 40 000) of kinks and antikinks moving under attractive interactions, which are annihilated on contacting each other, is studied using the method of molecular dynamics computer simulation. The average distance between neighboring kinks increases logarithmically in time after a short initial transient period. The size distribution function of domains between neighboring kinks is also computed and found to develop a characteristic cut-off structure. Interpretation of the results in terms of simple kinetic models is given. The results are compared with the recent neutron scattering experiments on layered antiferromagnets by Ikeda.

Computer simulation of cellular pattern growth in two and three dimensions

Abstract Computer simulations of cellular pattern growth (domain growth) have been carried out for two- and three-dimensional vertex model. Geometrical properties beyond the edge-number distribution have been studied for a two-dimensional cellular pattern in the scaling regime. The method of simulation is shown in three dimensions and simulation results are presented for the self-similarity of the cellular pattern growth, the growth law and the topological properties. The results are compared with existing simulations and experiments. It is concluded that the three-dimensional vertex models provide us with efficient simulation models of systems with large numbers of cells.

Rheology of random foams

The computational study of the rheology of two‐dimensional random cellular systems such as foam or concentrated emulsion systems is presented. The numerical simulations are performed under homogeneous shear using a computationally efficient model in which effects of viscous dissipation inside the liquid films are taken into consideration. It is shown that the system possesses a finite yield stress and behaves like Bingham plastics. The role of the topological change of the cellular structure is also examined.

Scaling behavior of two-dimensional domain growth: Computer simulation of vertex models

Computer simulations are performed for vertex models which are coarse-grained models for dynamical cellular patterns in two dimensions. By simulating large systems, we obtain conclusive evidence of scaling behavior, that is, a power law for the growth of the average cell size and the scaling properties for the distribution functions of edge number and size of cells. Several versions of the vertex models are obtained by making some approximations for the equation of motion of a vertex, and we compare the statistical properties of the patterns in the scaling regime.

Excitonic Instability and Ultrasonic Attenuations in Strong Magnetic Fields

Anomalous behaviors of the ultrasonic attenuations in Bi under strong magnetic fields found by Mase et al. are analyzed by taking account of the excitonic instability. Characteristic features of anomalous growth of the peak value of the lower field side, in the case where the two peaks of the attenuation coefficient due to electrons and holes lie close as a function of magnetic field, can be attributed to the fluctuation of the order parameter of excitons above the critical point T c , which is around 1 K. Critical exponent of the excess contributions from the fluctuation to the attenuation coefficient is one half, which is in accordance with the experiments.