Dan Tanaka

Large-scale vortex lattice emerging from collectively moving microtubules

Spontaneous collective motion, as in some flocks of bird and schools of fish, is an example of an emergent phenomenon. Such phenomena are at present of great interest and physicists have put forward a number of theoretical results that so far lack experimental verification. In animal behaviour studies, large-scale data collection is now technologically possible, but data are still scarce and arise from observations rather than controlled experiments. Multicellular biological systems, such as bacterial colonies or tissues, allow more control, but may have many hidden variables and interactions, hindering proper tests of theoretical ideas. However, in systems on the subcellular scale such tests may be possible, particularly in in vitro experiments with only few purified components. Motility assays, in which protein filaments are driven by molecular motors grafted to a substrate in the presence of ATP, can show collective motion for high densities of motors and attached filaments. This was demonstrated recently for the actomyosin system, but a complete understanding of the mechanisms at work is still lacking. Here we report experiments in which microtubules are propelled by surface-bound dyneins. In this system it is possible to study the local interaction: we find that colliding microtubules align with each other with high probability. At high densities, this alignment results in self-organization of the microtubules, which are on average 15 µm long, into vortices with diameters of around 400 µm. Inside the vortices, the microtubules circulate both clockwise and anticlockwise. On longer timescales, the vortices form a lattice structure. The emergence of these structures, as verified by a mathematical model, is the result of the smooth, reptation-like motion of single microtubules in combination with local interactions (the nematic alignment due to collisions)—there is no need for long-range interactions. Apart from its potential relevance to cortical arrays in plant cells and other biological situations, our study provides evidence for the existence of previously unsuspected universality classes of collective motion phenomena.

Robustness of the noise-induced phase synchronization in a general class of limit cycle oscillators.

We show that a wide class of uncoupled limit-cycle oscillators can be in-phase synchronized by common weak additive noise. An expression of the Lyapunov exponent is analytically derived to study the stability of the noise-driven synchronizing state. The result shows that such a synchronization can be achieved in a broad class of oscillators with little constraint on their intrinsic property. On the other hand, the leaky integrate-and-fire neuron oscillators do not belong to this class, generating intermittent phase slips according to a power law distribution of their intervals.

Noise Induced Phase Synchronization of a General Class of Limit Cycle Oscillators

We consider the effect of weak additive noise to a general class of limit cycle oscillators. Employing phase reduction method, the largest Lyapunov exponent of the noise driven oscillators are shown to be negative for a broad class of oscillators, which implies that identical limit cycle oscillators driven by a common additive noise can achieve phase synchronization without direct mutual interactions. Generalization of the results including the effect of colored noise and the effect of noise to spatially extended systems are also discussed.

Turing Instability Leads Oscillatory Systems to Spatiotemporal Chaos

We present that Turing instability can lead oscillatory reaction-diffusion (RD) systems to spatiotemporal chaos instead of spatially periodic steady states. Similar onset of spatiotemporal chaos was discovered in an equation describing seismic waves. We demonstrate that the seismic equation can be derived from a certain class of oscillatory RD systems in the neighborhood of a codimension-two Turing-Benjamin-Feir point. Also, we show numerical studies of reduced equations and discuss robustness of this spatiotemporal chaos.

Bifurcation Scenario to Nikolaevskii Turbulence in Small Systems

First, we confirm that the chaos in the Kuramoto–Sivashinsky equation occurs through period-doubling cascade (Feigenbaum scenario). Then, we show that the chaos in the Nikolaevskii equation occurs via the torus-doubling bifurcation (Ruelle–Takens–Newhouse scenario).

Modal and Total Power Spectra of Nikolaevskii Turbulence

We numerically investigate the modal and total time correlation functions and the corresponding power spectra for the Nikolaevskii equation. The modal power spectrum exhibits Lorentzian for low-frequency ranges, while the total power spectrum diverges in the low-frequency limit. The dynamic exponent is equal to 2 and 3/2 for smaller and larger system sizes, respectively. The relationship between the Nikolaevskii turbulence and the soft-mode turbulence is also discussed.