Swarming behavior of
microtubles driven by dyneins
Ken H. Nagai (Sano lab. JSPS PD (Univ. Tokyo)),Self-propelled particlewater
180 mmpentanoloil

aqueous phase (2.3 vol%)•K. N., et al., Phys. Rev. E (2005).

•F. Takabatake, et al., J. Chem. Phys.(2009)•K. H. N., et al. Phys. Rev. E (2013).

Main questions

How to gain momentum without external force
How to break symmetry
Collective phenomena of active matters

Swarming behavior of active
particles

T. Vicsek, Phys. Rep. (2012).

Swarming behavior througha short range interactionThere is only short-range
interaction about the directionof motion.
Univesal Characteristcs!
Nonequilibrium phase transition
Giant Fluctuation•T. Vicsek, et al., Phys. Rev. Lett. (1995).
•G. Grégoire and H. Chaté, Phys. Rev. Lett. (2004).

Experimental evidence

J. Deseigne, et al., 2010.

V. Schaller, A.Bausch, 2010.

Vicsek model does NOT
have memory
No memory (correlation)

Deformable self-propelled particles should have memory.

• What is the role of memory of motion?
T. Ohta, T. Ohkuma, Phys. Rev. Lett. (2009).
A. B. Verkhovsky, et al., Curr. Biol. (1999).

Mathematical model
parallel alignment oranti-parallel alignment(short-range nematicinteraction)
ω has a finite correlation time, τ.

v0=8.75 μm/s, ω0=-0.00624 rad/s, l=15.0 μm, α=0.0583 rad/s,<ω2>=2.48×10-4 rad2/s2Order parameter
1. Nematic order parameter
If all particles align parallel or anti-parallel (nematic order), S=1.
If there is no global nematic order, S=0.

2. Density fluctuation
u2 is the variance of density in a 60 μm2×60 μm2 region. na is the average
number of particles in the same region.
If the density is not uniform, this value is large.

Phase diagram

Phase diagram

Phase diagram
F. Ginelli, et al, 2010.
Same!
RandomGlobal nematicPhase diagram

Vortex latticeMicrotubule• No polymerization and depolymerization (stabilized withtaxol)• Microtubules were labeled with Cy3.

Dynein-c

Inner-arm dynein purified from Chlamydomonas flagella

Experimental setup
10-20 m
25nm
50nm

•ATP concentration was 1 mM.
•In vitro motility assay consisted of dynein grafted to glass surface andto microtubules.

•Density of dyneins was 750-2,500/µm2.
•Average speed of a microtubule was 8.75 μm/s.

Interaction of microtubules

Microtubule aligns parallel or anti-parallel through collision(Short-range nematic interaction)Difference between dyneinsand kinesins
DyneinKinesin• Most microtubules does not change the direction.

Trajectory of an isolated
microtubule•Smooth trajectory (finite correlation time of direction of motion)•ωi/v0 is the curvature of trajectory.

Analysis of trajectory
Trajectory

Autocorrelation ofDistribution ofcurvaturecurvature•Correlation length along the trajectory was 542±4 μm(>>microtubule length (10 μm))!
•Average curvature was -7.1×10-4 μm-1 (clockwise rotation).
•Standard deviation of curvature was 1.8×10-3 μm-1.

Collective behavior of
microtubules

100 μm•Density was 40 μg/ml (5 MT in 100 μm2).
•The size of a vortex was 443±64 μm (major axis)(>>microtubule length (10 μm))!

Whole view

10000 μm
10 mm x 18 mm

Density Dependence
14.3 µg/ml

28.6 µg/ml

57.2 µg/ml

114.5 µg/ml

100 μmSituation corresponding to
the experiments
τ=171 s, L=7.68 mm, ρ=4.44×10-2 μm2, N=2621440

Vortex lattice!

Conclusion and remarks• Using a mathematical model, we found long correlation timeof motion and high density leads to vortex lattice of activeparticles. Only short-range nematic interaction is needed forvortex lattice (not hydrodynamical one (I. H. Riedel, et al.,Science (2005))).

• We experimentally confirmed this type of swarming behavior
using motility assay consisted of dynein proteins grafted to
glass surface and to microtubules.

• The macroscopical meaning of the finite correlation time of
the curvature of microtubule’s trajectory are being analyzed.

• Y. S, K. H. N., et al., Nature (2012).