Numerical study of anisotropic diffusion in Turing patterns based on Finsler geometry modeling.

We numerically study the anisotropic Turing patterns (TPs) of an activator-inhibitor system described by the reaction-diffusion (RD) equation of Turing, focusing on anisotropic diffusion using the Finsler geometry (FG) modeling technique. In FG modeling, the diffusion coefficients are dynamically generated to be direction dependent owing to an internal degree of freedom (IDOF) and its interaction with the activator and inhibitor. Because of this dynamical diffusion coefficient, FG modeling of the RD equation sharply contrasts with the standard numerical technique in which direction-dependent coefficients are manually assumed. To find the solution of the RD equations in FG modeling, we use a hybrid numerical technique combining the Metropolis Monte Carlo method for IDOF updates and discrete RD equations for steady-state configurations of the activator-inhibitor variables. We find that the newly introduced IDOF and its interaction are a possible origin of spontaneously emergent anisotropic patterns of living organisms, such as zebra and fishes. Moreover, the IDOF makes TPs controllable by external conditions if the IDOF is identified with the direction of cell diffusion accompanied by thermal fluctuations.

Coarse-Grained Lattice Modeling and Monte Carlo Simulations of Stress Relaxation in Strain-Induced Crystallization of Rubbers.

Two-dimensional triangulated surface models for membranes and their three-dimensional (3D) extensions are proposed and studied to understand the strain-induced crystallization (SIC) of rubbers. It is well known that SIC is an origin of stress relaxation, which appears as a plateau in the intermediate strain region of stress-strain curves. However, this SIC is very hard to implement in models because SIC is directly connected to a solid state, which is mechanically very different from the amorphous state. In this paper, we show that the crystalline state can be quite simply implemented in the Gaussian elastic bond model, which is a straightforward extension of the Gaussian chain model for polymers, by replacing bonds with rigid bodies or eliminating bonds. We find that the results of Monte Carlo simulations for stress-strain curves are in good agreement with the reported experimental data of large strains of up to 1200%. This approach allows us to intuitively understand the stress relaxation caused by SIC.

Finsler geometry modeling and Monte Carlo study of liquid crystal elastomers under electric fields.

The shape transformation of liquid crystal elastomers (LCEs) under external electric fields is studied through Monte Carlo simulations of models constructed on the basis of Finsler geometry (FG). For polydomain side-chain-type LCEs, it is well known that the external-field-driven alignment of the director is accompanied by an anisotropic shape deformation. However, the mechanism of this deformation is quantitatively still unclear in some part and should be studied further from the microscopic perspective. In this paper, we evaluate whether this shape deformation is successfully simulated, or in other words, quantitatively understood, by the FG model. The FG assumed inside the material is closely connected to the directional degrees of freedom of LC molecules and plays an essential role in the anisotropic transformation. We find that the existing experimental data on the deformations of polydomain LCEs are in good agreement with the Monte Carlo results. It is also found that experimental diagrams of strain versus external voltage of a monodomain LCE in the nematic phase are well described by the FG model.

Mathematical Modeling and Simulations for Large-Strain J-Shaped Diagrams of Soft Biological Materials.

Herein, we study stress⁻strain diagrams of soft biological materials such as animal skin, muscles, and arteries by Finsler geometry (FG) modeling. The stress⁻strain diagram of these biological materials is always J-shaped and is composed of toe, heel, linear, and failure regions. In the toe region, the stress is almost zero, and the length of this zero-stress region becomes very large (≃150%) in, for example, certain arteries. In this paper, we study long-toe diagrams using two-dimensional (2D) and 3D FG modeling techniques and Monte Carlo (MC) simulations. We find that, except for the failure region, large-strain J-shaped diagrams are successfully reproduced by the FG models. This implies that the complex J-shaped curves originate from the interaction between the directional and positional degrees of freedom of polymeric molecules, as implemented in the FG model.

Bending of Thin Liquid Crystal Elastomer under Irradiation of Visible Light: Finsler Geometry Modeling.

In this paper, we show that the 3D Finsler geometry (FG) modeling technique successfully explains a reported experimental result: a thin liquid crystal elastomer (LCE) disk floating on the water surface deforms under light irradiation. In the reported experiment, the upper surface is illuminated by a light spot, and the nematic ordering of directors is influenced, but the nematic ordering remains unchanged on the lower surface contacting the water. This inhomogeneity of the director orientation on/inside the LCE is considered as the origin of the shape change that drives the disk on the water in the direction opposite the movement of the light spot. However, the mechanism of the shape change is still insufficiently understood because to date, the positional variable for the polymer has not been directly included in the interaction energy of the models for this system. We find that this shape change of the disk can be reproduced using the FG model. In this FG model, the interaction between σ, which represents the director field corresponding to the directional degrees of LC, and the polymer position is introduced via the Finsler metric. This interaction, which is a direct consequence of the geometry deformation, provides a good description of the shape deformation of the LCE disk under light irradiation.

Parallel Tempering Monte Carlo Studies of Phase Transition of Free Boundary Planar Surfaces.

We numerically study surface models defined on hexagonal disks with a free boundary. 2D surface models for planar surfaces have recently attracted interest due to the engineering applications of functional materials such as graphene and its composite with polymers. These 2D composite meta-materials are strongly influenced by external stimuli such as thermal fluctuations if they are sufficiently thin. For this reason, it is very interesting to study the shape stability/instability of thin 2D materials against thermal fluctuations. In this paper, we study three types of surface models including Landau-Ginzburg (LG) and Helfirch-Polyakov models defined on triangulated hexagonal disks using the parallel tempering Monte Carlo simulation technique. We find that the planar surfaces undergo a first-order transition between the smooth and crumpled phases in the LG model and continuous transitions in the other two models. The first-order transition is relatively weak compared to the transition on spherical surfaces already reported. The continuous nature of the transition is consistent with the reported results, although the transitions are stronger than that of the reported ones.

J-shaped stress-strain diagram of collagen fibers: Frame tension of triangulated surfaces with fixed boundaries.

We present Monte Carlo data of the stress-strain diagrams obtained using two different triangulated surface models. The first is the canonical surface model of Helfrich and Polyakov (HP), and the second is a Finsler geometry (FG) model. The shape of the experimentally observed stress-strain diagram is called J-shaped. Indeed, the diagram has a plateau for the small strain region and becomes linear in the relatively large strain region. Because of this highly nonlinear behavior, the J-shaped diagram is far beyond the scope of the ordinary theory of elasticity. Therefore, the mechanism behind the J-shaped diagram still remains to be clarified, although it is commonly believed that the collagen degrees of freedom play an essential role. We find that the FG modeling technique provides a coarse-grained picture for the interaction between the collagen and the bulk material. The role of the directional degrees of freedom of collagen molecules or fibers can be understood in the context of FG modeling. We also discuss the reason for why the J-shaped diagram cannot (can) be explained by the HP (FG) model.

Finsler Geometry Modeling of Phase Separation in Multi-Component Membranes.

A Finsler geometric surface model is studied as a coarse-grained model for membranes of three components, such as zwitterionic phospholipid (DOPC), lipid (DPPC) and an organic molecule (cholesterol). To understand the phase separation of liquid-ordered (DPPC rich) L o and liquid-disordered (DOPC rich) L d , we introduce a binary variable σ ( = ± 1 ) into the triangulated surface model. We numerically determine that two circular and stripe domains appear on the surface. The dependence of the morphological change on the area fraction of L o is consistent with existing experimental results. This provides us with a clear understanding of the origin of the line tension energy, which has been used to understand these morphological changes in three-component membranes. In addition to these two circular and stripe domains, a raft-like domain and budding domain are also observed, and the several corresponding phase diagrams are obtained.

Possible effects of tilt order on phase transitions of a fixed connectivity surface model.

We study the phase structure of a phantom tethered surface model shedding light on the internal degrees of freedom (IDOF), which correspond to the three-dimensional rodlike structure of the lipid molecules. The so-called tilt order is assumed as IDOF on the surface model. The model is defined by combining the conventional spherical surface model and the XY model, which describes not only the interaction between lipids but also the interaction between the lipids and the surface. The interaction strength between IDOF and the surface varies depending on the interaction strength between the variables of IDOF. We know that the model without IDOF undergoes a first-order transition of surface fluctuations and a first order collapsing transition. We observe in this paper that the order of the surface fluctuation transition changes from first order to second order and to higher order with increasing strength of the interaction between IDOF variables. On the contrary, the order of collapsing transition remains first order and is not influenced by the presence of IDOF.

Phase structure of a surface model on dynamically triangulated spheres with elastic skeletons.

We find three distinct phases; a tubular phase, a planar phase, and the spherical phase, in a triangulated fluid surface model. It is also found that these phases are separated by discontinuous transitions. The fluid surface model is investigated within the framework of the conventional curvature model by using the canonical Monte Carlo simulations with dynamical triangulations. The mechanical strength of the surface is given only by skeletons, and no two-dimensional bending energy is assumed in the Hamiltonian. The skeletons are composed of elastic linear chains and rigid junctions and form a compartmentalized structure on the surface, and for this reason the vertices of triangles can diffuse freely only inside the compartments. As a consequence, an inhomogeneous structure is introduced in the model; the surface strength inside the compartments is different from the surface strength on the compartments. However, the rotational symmetry is not influenced by the elastic skeletons; there is no specific direction on the surface. In addition to the three phases mentioned above, a collapsed phase is expected to exist in the low bending rigidity regime that was not studied here. The inhomogeneous structure and the fluidity of vertices are considered to be the origin of such a variety of phases.

Development of neurons expressing estrogen receptor alpha transiently in facial nucleus of prenatal and postnatal rat brains.

The transient expression of estrogen receptor alpha (ERalpha) in the facial nucleus of rats during development was already reported. However, how and whether the receptor functions physiologically in the nucleus of developing rats are as yet unclear. In this study, we applied a retrograde tracer into one of the possible target muscles of the motoneurons in the nucleus, that is, the transverse auricular muscle (Mta), and examined whether ERalpha-immunopositive neurons take up the tracer. Because it is probable that neurogenesis, apoptosis, and maturation may be associated with the transient expression of ERalpha, we attempted to analyze the neurons expressing the receptor in the nucleus. We found that ERalpha-immunopositive neurons in the medial facial subnucleus innervate mostly the Mta. Quantitative analyses showed that the number of motoneurons projecting to the Mta remained the same throughout the ages examined, whereas that of ERalpha-immunopositive neurons decreased between postnatal days 6 and 11. Apoptosis and neurogenesis in the nucleus were not affected by the expression of ERalpha during development. ERalpha expression coincided with the maturation of neurons in the nucleus. Thus, it is possible that ERalpha expression in the facial nucleus during development plays important roles in the development of motoneurons and/or external pinna muscles.

Collapsing transition of spherical tethered surfaces with many holes.

We investigate a tethered (i.e., fixed connectivity) surface model on spherical surfaces with many holes by using the canonical Monte Carlo simulations. Our result in this paper reveals that the model has only a collapsing transition at finite bending rigidity, where no surface fluctuation transition can be seen. The first-order collapsing transition separates the smooth phase from the collapsed phase. Both smooth and collapsed phases are characterized by Hausdorff dimension H approximately 2 , consequently, the surface becomes smooth in both phases. The difference between these two phases can be seen only in the size of surface. This is consistent with the fact that we can see no surface fluctuation transition at the collapsing transition point. These two types of transitions are well known to occur at the same transition point in the conventional surface models defined on the fixed connectivity surfaces without holes.

Shape transformations of a compartmentalized fluid surface.

A surface model on compartmentalized spheres is studied by using the Monte Carlo simulation technique with dynamical triangulations. We found that the model exhibits a variety of phases: The spherical phase, the tubular phase, the planar phase, the wormlike planar phase, the wormlike long phase, the wormlike short phase, and the collapsed phase. We also demonstrated that almost all phases are separated from their neighboring phases by first-order transitions. Mechanical strength of the surface is given only by elastic skeletons, which are the compartment boundaries, and therefore vertices diffuse freely inside the compartments. We confirm that the cytoskeletal structure and the lateral diffusion of vertices are origins of such a variety of phases.

First-order phase transition in the tethered surface model on a sphere.

We show that the tethered surface model of Helfrich and Polyakov-Kleinert undergoes a first-order phase transition separating the smooth phase from the crumpled one. The model is investigated by the canonical Monte Carlo simulations on spherical and fixed connectivity surfaces of size up to N = 15 212. The first-order transition is observed when N > 7000, which is larger than those in previous numerical studies, and a continuous transition can also be observed on the smaller surfaces. Our results are therefore consistent with those obtained in previous studies on the phase structure of the model.

First-order phase transition of fixed connectivity surfaces.

We report numerical evidence of the discontinuous transition of a tethered membrane model which is defined within a framework of the membrane elasticity of Helfrich. Two kinds of phantom tethered membrane models are studied via the canonical Monte Carlo simulation on triangulated fixed connectivity surfaces of spherical topology. One surface model is defined by the Gaussian term and the bending energy term, and the other, which is tensionless, is defined by the bending energy term and a hard wall potential. The bending energy is defined by using the normal vector at each vertex. Both models undergo the first-order phase transition characterized by a gap of the bending energy. The phase structure of the models depends on the choice of discrete bending energy.

Phase transitions of a tethered surface model with a deficit angle term.

The Nambu-Goto model is investigated by using the canonical Monte Carlo simulations on fixed connectivity surfaces of spherical topology. Three distinct phases are found: crumpled, tubular, and smooth. The crumpled and the tubular phases are smoothly connected, and the tubular and the smooth phases are connected by a discontinuous transition. The surface in the tubular phase forms an oblong and one-dimensional object similar to a one-dimensional linear subspace in the Euclidean three-dimensional space R3 . This indicates that the rotational symmetry inherent in the model is spontaneously broken in the tubular phase, and it is restored in the smooth and the crumpled phases.

Monte Carlo simulations of branched polymer surfaces without bending elasticity.

We study a model of elastic surfaces that was first constructed by Baillie et al. for an interpolation between the models of fluid and crystalline membranes. The Hamiltonian of the model is a linear combination of the Gaussian energy and a squared scalar curvature energy. These energy terms are discretized on dynamically triangulated surfaces that are allowed to self-intersect. We confirm that the model has not only crumpled phases but also a branched polymer phase, and find that the model undergoes a first-order phase transition between the branched polymer phase and one of the crumpled phases. We find also that the model undergoes a second- (or higher-) order phase transition between the branched polymer phase and another crumpled phase.