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In sports and daily life, we often experience situations where we should carry an object by pushing it. In this study, we analyse sumo, a Japanese traditional sport, in which one can be a winner by pushing the opponent to the outside of dohyo, the wrestling ring. The optimal strategy for the lighter wrestler to carry the heavier opponent by sliding on dohyo is explored in terms of physics. Although the lighter wrestler can never slide the opponent by pushing forward, this can be achieved by exerting the force diagonally upward. As a result of analysis, we obtain the magnitude and direction of the force that should be applied to initiate the sliding motion and ensure its fastest movement. The result reveals the existence of a critical weight ratio of the wrestlers; if the ratio is upper than a certain value, the lighter wrestler should push to a specific direction, while otherwise the optimal force direction depends on the weight. Due to the generality of physics and mathematics, the application is not limited to sumo; the result provides the most effective way to carry an object on a floor in all activities in sports, exercises, and daily life.

During cell migration, forces applied to a cell from its environment influence the motion. When the cell is placed on a substrate, such a force is provided by the cell-substrate adhesion. Modulation of adhesivity, often performed by the modulation of the substrate stiffness, tends to cause common responses for cell spreading, cell speed, persistence, and random motility coefficient. Although the reasons for the response of cell spreading and cell speed have been suggested, other responses are not well understood. In this study, we develop a simple toy model for cell migration driven by the relation of two forces: the adhesive force and the plasma membrane tension. The simplicity of the model allows us to perform the calculation not only numerically but also analytically, and the analysis provides formulas directly relating the adhesivity to cell spreading, persistence, and the random motility coefficient. Accordingly, the results offer a unified picture on the causal relations between those multiple cellular responses. In addition, cellular properties that would influence the migratory behavior are suggested.

How the internal degree of freedom of particles influences self-organization is explored by considering cluster formation in many-particle systems. We analyze a general class of dynamical systems in which the interactions between particles depend on their spatial distance and the difference of their internal states. In particular, we analyze a three-particle system in which two types of steady patterns exist, namely, (i) a regular triangle (two-dimensional cluster) and (ii) a straight line (one-dimensional cluster). The results show that the linear pattern can be stable when the internal degree of freedom exists, while it is always unstable when the dynamics depend only on the spatial distance. Based on this analysis, we can understand why this difference occurs. If the internal states can cause asymmetry of the interactions, this can enable the particles to remain in a one-dimensional cluster.

The swarm oscillator model describes the long-time behavior of interacting chemotactic particles, and it shows numerous types of macroscopic patterns. However, the reason why so many kinds of patterns emerge is not clear. In this study, we elucidate the mechanism underlying the diversity of the pattens by analyzing the model for two particles. Focusing on the behavior when the two particles are spatially close, we find that the dynamics is classified into eight types, which explain most of the observed 13 types of patterns.

Metastatic cancer cells differ from their non-metastatic counterparts not only in terms of molecular composition and genetics, but also by the very strategy they employ for locomotion. Here, we analyzed large-scale statistics for cells migrating on linear microtracks to show that metastatic cancer cells follow a qualitatively different movement strategy than their non-invasive counterparts. The trajectories of metastatic cells display clusters of small steps that are interspersed with long “flights”. Such movements are characterized by heavy-tailed, truncated power law distributions of persistence times and are consistent with the Lévy walks that are also often employed by animal predators searching for scarce prey or food sources. In contrast, non-metastatic cancerous cells perform simple diffusive movements. These findings are supported by preliminary experiments with cancer cells migrating away from primary tumors in vivo. The use of chemical inhibitors targeting actin-binding proteins allows for “reprogramming” the Lévy walks into either diffusive or ballistic movements.

We numerically investigate the swarm oscillator model, which describes a long-time asymptotic behavior of chemotactic elements [D. Tanaka, Phys. Rev. Lett. 99 (2007) 134103]. As many as thirteen kinds of collective behavior are observed for various sets of specific two parameters in the model. Those two parameters imply the time delays from the stimulation on an element by the field that intermediates the interaction between elements to the two kinds of responses of the element. The analysis of the two-element system clarifies why some phases and phase boundaries emerge.

This report presents a simple model that describes the motion of a single Dictyostelium discoideum cell exposed to a traveling wave of cyclic adenosine monophosphate (cAMP). The model incorporates two types of responses to stimulation by cAMP: the changes in the polarity and motility of the cell. The periodic change in motility is assumed to be induced by periodic cAMP stimulation on the basis of previous experimental studies. Consequently, the net migration of the cell occurs in a particular direction with respect to wave propagation, which explains the migration of D. discoideum cells in aggregation. The wave period and the difference between the two response times are important parameters that determine the direction of migration. The theoretical prediction compared with experiments presented in another study. The transition from the single-cell state of the population of D. discoideum cells to the aggregation state is understood to be a specific example of spontaneous breakage of symmetry in biology.

When mammalian cells form confluent monolayers completely filling a plane, these apparently random “tilings” show regularity in the statistics of cell areas for various types of epithelial and endothelial cells. The observed distributions are reproduced by a model which accounts for cell growth and division, with the latter treated stochastically both in terms of the sizes of the dividing cells as well as the sizes of the “newborn” ones – remarkably, the modeled and experimental distributions fit well when all free parameters are estimated directly from experiments.

This paper is mainly a review concerning singular perturbation methods by means of Lie group analysis which has been presented by the author. We make use of a particular type of approximate Lie symmetries in those methods in order to construct reduced systems which describe the long-time behavior of the original dynamical system. Those methods can be used in analyzing not only ordinary differential equations but also difference equations. Although this method has been mainly used in order to derive asymptotic behavior, when we can find exact Lie symmetries, we succeed in construction of exact solutions.

We investigate what is called swarm oscillator model where interacting motile oscillators form various kinds of ordered structures. We particularly focus on the dimensionality of clusters which oscillators form. In two-dimensional space, oscillators spontaneously form one-dimensional clusters or two-dimensional clusters. By studying the three-oscillator system, we analytically find the conditions of the appearance of those patterns. The validity of those conditions in applying to systems of more oscillators is demonstrated by numerically investigating a system of twenty oscillators.

Swarm oscillator model derived by one of the authors (Tanaka), where interacting motile elements form various kinds of patterns, is investigated. We particularly focus on the cluster patterns in one-dimensional space. We mathematically derive all static and stable configurations in final states for a particular but a large set of parameters. In the derivation, we introduce renormalized expression of this model. We find that the static final states are hierarchical cluster structures in which a cluster consists of smaller clusters in a nesting manner.

For singular perturbation problems in dynamical systems, various appropriate singular perturbation methods have been proposed to eliminate secular terms appearing in the naive expansion. For example, the method of multiple time scales, the normal form method, center manifold theory, the renormalization group method are well known. In this paper, it is shown that all of the solutions of the reduced equations constructed with those methods are exactly equal to sum of the most divergent secular terms appearing in the naive expansion. For the proof, a method to construct a perturbation solution which differs from the conventional one is presented, where we make use of the theory of Lie symmetry group., Comment: To be published in Phys. Rev. E

A new singular perturbation method based on the Lie symmetry group is presented to a system of difference equations. This method yields consistent derivation of a renormalization group equation which gives an asymptotic solution of the difference equation. The renormalization group equation is a Lie differential equation of a Lie group which leaves the system approximately invariant. For a 2-D symplectic map, the renormalization group equation becomes a Hamiltonian system and a long-time behaviour of the symplectic map is described by the Hamiltonian. We study the Poincar\'e-Birkoff bifurcation in the 2-D symplectic map by means of the Hamiltonian and give a condition for the bifurcation., Comment: Accepted to J. Phys. A, 7 pages

A renormalization group method with the Lie symmetry is presented for the singular perturbation problems. Asymptotic solutions are obtained as group-invariant solutions under approximate Lie group admitted by perturbed differential equations., 9 pages, 1 figure

Lie theory is applied to perturbation problems of ordinary differential equations to construct approximate solutions and invariant manifolds according to the renormalization group approach of Iwasa and Nozaki [“A method to construct asymptotic solutions invariant under the renormalization group,” Prog. Theor. Phys. 116, 605 (2006)]. It is proved that asymptotic behavior of solutions is obtained from the Lie equations even if original equations have no symmetries. Normal forms of the Lie equations are introduced to investigate the existence of invariant manifolds.