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Large-scale vortex lattice emerging from collectively moving microtubules

Abstract

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 interest1,2,3,4,5 and physicists have put forward a number of theoretical results that so far lack experimental verification6,7,8. 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 tissues9,10, 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 components11,12,13. 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 system14,15, 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 cells16,17 and other biological situations, our study provides evidence for the existence of previously unsuspected universality classes of collective motion phenomena.

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Figure 1: Emergence of vortices of microtubules.
Figure 2: Nematic order inside a vortex stream.
Figure 3: Collision between microtubules.
Figure 4: Motion of isolated microtubules.
Figure 5: Collective dynamics of mathematical model of interacting, self-propelled particles.

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Acknowledgements

We thank H. Sakakibara, Y. Uchida and R. Nakamori for the preparation of the material and observations. They as well as H. Kitahata are also thanked for discussions. Y. Sumino is supported by a Grant-in-Aid for Research Activity Start-up (no. 23840019), Grant-in-Aid for Scientific Research (B) (no. 21340023) and Grant-in-Aid for Scientific Research (A) (no. 20244067). K.H.N. would like to acknowledge the support of a fellowship from the JSPS (no. 23-1819).

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Authors

Contributions

K.O. discovered the phenomenon of vortex lattice formation described here and conceived and performed the experiments with Y. Shitaka. Y. Sumino, K.H.N., D.T. and H.C. conceived and designed the simulations. Y. Sumino, K.H.N. and H.C. performed and analysed the simulations. In vitro motility assays were carried out under the directions of K.O. Y. Sumino, K.H.N., Y. Shitaka, H.C., K.Y. and K.O. interpreted the data. Y. Sumino, K.H.N., H.C. and K.O. wrote the paper.

Corresponding author

Correspondence to Kazuhiro Oiwa.

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The authors declare no competing financial interests.

Supplementary information

Supplementary Information

This file contains Supplementary Methods and Supplementary Figures 1-3. The Supplementary Methods describe methods for experimental procedure, data analysis and numerical simulation. Figures display, density dependence of vortex pattern, phase diagram of mathematical model, and collision behavior on different dynein c density (and kinesin density). (PDF 618 kb)

Supplementary Movie 1

This movie shows formation process of vortex pattern, corresponding to figure 1a. The density of microtubule was 0.05 µm-2. Microtubules were homogeneously distributed initially. They formed stream with time and finally the stream made closed loop, vortex. (MOV 12020 kb)

Supplementary Movie 2

This movie shows internal structure a vortex pattern, corresponding to figure 2a. Only 0.1% of microtubules were prepared to be more fluorescent. In a stream, microtubules moved both directions, clearly indicating a vortex has nematic order. (MOV 20042 kb)

Supplementary Movie 3

This movie shows collision behavior of microtubules. The densities of dynein c used was 2500 μm-2. Density of the microtubules on the surface was approximately 0.1 microtubules per 100 μm2 suitable for observations of individual collisions. (MOV 16186 kb)

Supplementary Movie 4

This movie shows result of numerical simulation of our mathematical model in T=0 to T=2000, corresponding to Figure 5a. 2621440 particles were used in 512 × 512 area with periodic boundary condition. Other parameters are written in the main text and supplementary information1. The brightest region corresponds to the 15 particles in a unit area. Green and red colors corresponds to the density of particles moving clockwise (ω<0) and counterclockwise (ω>0), respectively. (MOV 12549 kb)

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Sumino, Y., Nagai, K., Shitaka, Y. et al. Large-scale vortex lattice emerging from collectively moving microtubules. Nature 483, 448–452 (2012). https://doi.org/10.1038/nature10874

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