Hidetaka Yamaoka

Multiscale modeling and mechanics of filamentous actin cytoskeleton

The adaptive structure and functional changes of the actin cytoskeleton are induced by its mechanical behavior at various temporal and spatial scales. In particular, the mechanical behaviors at different scales play important roles in the mechanical functions of various cells, and these multiscale phenomena require clarification. To establish a milestone toward achieving multiscale modeling and simulation, this paper reviews mathematical analyses and simulation methods applied to the mechanics of the filamentous actin cytoskeleton. The actin cytoskeleton demonstrates characteristic behaviors at every temporal and spatial scale, and mathematical models and simulation methods can be applied to each level of actin cytoskeletal structure ranging from the molecular to the network level. This paper considers studies on mathematical models and simulation methods based on the molecular dynamics, coarse-graining, and continuum dynamics approaches. Every temporal and spatial scale of actin cytoskeletal structure is considered, and it is expected that discrete and continuum dynamics ranging from functional expression at the molecular level to macroscopic functional expression at the whole cell level will be developed and applied to multiscale modeling and simulation.

Stratified reduction of classical many-body systems with symmetry

The centre-of-mass system for N bodies admits a natural SO(3) action, and thereby is stratified into strata according to the orbit types for the SO(3) action. The principal stratum consists of non-singular configurations for which the isotropy subgroup is trivial, so that it is made into an SO(3) principal fibre bundle. The strata of lower dimension consist of singular configurations at each of which the isotropy subgroup is not trivial. Practically, singular configurations are collinear ones and simultaneous multiple collision. Classical Lagrangian and Hamiltonian systems are defined on the tangent and the cotangent bundles over each stratum, respectively. The Euler–Lagrange and the Hamilton equations for the N-body system are derived on the variational principle, according to the stratification of the centre-of-mass system. The reduction procedure will be accordingly performed for the Lagrangian and the Hamiltonian systems with symmetry, respectively. By the rotational symmetry, the Euler–Lagrange and the Hamilton equations are reduced to those defined on reduced bundles from the tangent and the cotangent bundles, respectively.

Stratified reduction of many-body kinetic energy operators

The center-of-mass system of many bodies admits a natural action of the rotation group SO(3). According to the orbit types for the SO(3) action, the center-of-mass system is stratified into three types of strata. The principal stratum consists of nonsingular configurations for which the isotropy subgroup is trivial, and the other two types of strata consist of singular configurations for which the isotropy subgroup is isomorphic with either SO(2) or SO(3). Depending on whether the isotropy subgroup is isomorphic with SO(2) or SO(3), the stratum in question consists of collinear configurations or of a single configuration of the multiple collision. It is shown that the kinetic energy operator is expressed as the sum of rotational and vibrational energy operators on each stratum except for the stratum of multiple collision. The energy operator for nonsingular configurations has singularity at singular configurations. However, the singularity is not essential in the sense that both of the rotational and vibr...

Stratified dynamical systems and their boundary behaviour for three bodies in space, with insight into small vibrations

The centre-of-mass system for many particles is stratified into strata by the rotation group action. The principal stratum consists of nonlinear configurations. The collinear configurations form a lower dimensional stratum. Classical mechanics for many particles with nonlinear configurations and for those with collinear configurations are set up on the tangent or cotangent bundles over respective strata and can be reduced by the use of rotational symmetry. A question arises as to how a many-body system behaves in a neighbourhood of a collinear configuration. The system may make a vibration to bend its collinear configuration, which is a motion taking place across the boundary of the principal stratum. This paper deals with the behaviour of those boundaries for three bodies in space. The equations of motion for small vibrations as boundary behaviour at a collinear configuration will be given as a limit of those equations of motion for nonlinear configurations by means of two key facts: that the isotropy subgroup may act non-trivially on the tangent space at the collinear configuration and that vibrations take place in a constant plane in space. Further, the perturbation of small vibrations is studied by applying Mosers averaging method to show that small vibrations may give rise to finite rotations after a sufficiently large number of periods.

Continuum dynamics on a vector bundle for a directed medium

We develop a dynamics on a vector bundle that accurately describes the mechanical behavior of a directed medium. The directed medium is a continuum with microstructures, which is described by a deformable vector, called a director. In geometric continuum mechanics, an elastic body is viewed as a differentiable manifold, while a directed medium is viewed as a vector bundle whose fiber denotes a collection of the deformable directors. In this study, we begin with geometrical settings of the continuum dynamics on a tangent bundle of a vector bundle, and derive a weak form and equations of motion for the directed medium. Moreover, we apply our resultant equations to a Cosserat rod, as an example, and find that the derived equations of motion coincide with the balance laws of large deformable rods. Additionally, the equations of motion for the Cosserat rod are reduced to those for the special case of the Cosserat rod under undeformed cross-sectional conditions. Finally, we discuss an application of our Cosserat rod results to biopolymers and comment on the analyses of their smaller microstructures. This is important for future development because the microstructural changes of the biopolymers are deeply related to their macro conformations and, eventually, to the biological functions depending on both the macro and micro conformations.

Rotational-vibrational energy spectra of triatomic molecules near relative equilibria

Complete Hamiltonian operators have been obtained in terms of internal coordinates on the basis of the fiber bundle theory in geometry [J. Math Phys. 44, 4411 (2003)]. In this article, the full Hamiltonian is specialized for a rigid and for a semirigid molecule. For the rigid molecule, all internal coordinates are fixed at constants, so that the Hamiltonian operator comes to take an ordinary matrix form, and accordingly, the Schrodinger equation becomes an algebraic eigenvalue equation. The eigenvalues then provide rotational energy spectra of the rigid molecule. For the semirigid molecule, the full Hamiltonian is expanded in the vicinity of an equilibrium position into a power series in an infinitesimal parameter, to which the perturbation method is applied to obtain energy spectra in the form of a power series in the infinitesimal parameter. Indeed, the energy spectra are calculated to the second order term in the infinitesimal parameter in both the cases where the unperturbed energy spectra are nondege...