Cellular inertia.

It has been experimentally reported that chemotactic cells exhibit cellular memory, that is, a tendency to maintain the migration direction despite changes in the chemoattractant gradient. In this study, we analyzed a phenomenological model assuming the presence of cellular inertia, as well as a response time in motility, resulting in the reproduction of the cellular memory observed in the previous experiments. According to the analysis, the cellular motion is described by the superposition of multiple oscillative functions induced by the multiplication of the oscillative polarity and motility. The cellular intertia generates cellular memory by regulating phase differences between those oscillative functions. By applying the theory to the experimental data, the cellular inertia was estimated at [Formula: see text] min. In addition, physiological parameters, such as response time in motility and intracellular processing speed, were also evaluated. The agreement between the experiemental data and theory suggests the possibility of the presence of the response time in motility, which has never been biologically verified and should be explored in the future.

Role of the internal degree of freedom of particles in cluster formation.

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.

A mechanical toy model linking cell-substrate adhesion to multiple cellular migratory responses.

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.

Lévy-like movement patterns of metastatic cancer cells revealed in microfabricated systems and implicated in vivo.

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.

Universal area distributions in the monolayers of confluent mammalian cells.

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.

Juggling motion in a system of motile coupled oscillators.

A particular dynamic steady state emerging in the swarm oscillator model--a system of interacting motile elements with an internal degree of freedom--is presented. In the state, elements form a rotating triangle whose corners appear to catch and throw elements. This motion is referred to as "juggling motion" in this paper. How this motion is maintained is studied theoretically. In particular, the five-element system, the minimum system that exhibits the motion, is investigated. With the help of the analysis of the two-element system, several factors essential to maintaining this motion are obtained.

Dimensionality of clusters in a swarm oscillator model.

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.

Hierarchical cluster structures in a one-dimensional swarm oscillator model.

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.

Solution of reduced equations derived with singular perturbation methods.

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, and the renormalization group method are well known. It is shown that all of the solutions of the reduced equations constructed with those methods are exactly equal to the 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 the Lie symmetry group.