ICMMA 2014, 'Crowd Dynamics', Meiji University, 2015/01/08

Collective Motion of Self-Propelled Objects:
From Molecule to Colloid
Masaki Sano
Department of Physics
The University of TokyoNon-equilibrium Phase Transition & Turbulence

Information Thermodynamics

Non-equilibrium Statisical Mechanics
Active Colloids

Cell Motility

Active Soft Matter
Traction Force Microscopy

Synergy of Fluctuation and Structure— Quest for Universal Laws in Non-Equilibrium Systems
Grant-in-Aid for Scientific Research on Innovative Areas of MEXT, JAPAN(FY2013-2017)

Quest for Universal Laws in Non-Equilibrium Systems

StatisticalMechanicsNonlinear Dynamics

Meso
New development

Self-organization

Theory ofFluctuation

New Development

Generalized 2ndMicro
Lawof Thermodynamics
4

Experimental
technique forsmall systems

Objectives and Research Contents of Our Project How to extract useful information from fluctuations
 How do self-organized structures arise from fluctuations?
 Are there universal laws in nonequilibrium systems?
 How are nonequilibrium fluctuation and structureinteracting in soft/solid condensed matter, active matter,and biomatter?
 How can a collection of mere matter exert basic functionsof life, such as self-production, self-propulsion,information processing?

Bridging gaps between matter and life

Living Matter

Information transferprocessing

Order, Disorder, Entropy

1. Information Thermodynamics

Information AspectMetabolism

3. Protocell

2. Active Matter

Energy Aspect

Material Aspect

Nonequilibrium States

How to create cell from matter ?
How to select useful information?
How to decrease entropy?

MatterCollective Motion of Active Matter
Biological Systems

Artificial Systems
 Self-Propelled Colloid Molecular Motor + Track Single Particle Motion

 Single Cell Migration

 Dense Suspension

 Multicellular Migration

Directed Movement・DirectedMovement Interaction
・Interaction

Noise(Fluctuation)(Fluctuation)
・Noise

Polar

Polar

Thermal Fluctuation

Nematic

Nematic

Deterministic Chaos, etc.

Collective Motion of Microtubules on Kinesin Bed
Motility assay of kinesin motors
and microtubules

Poster by Sakurako Tanida“Pattern Frmation of Microtuble”

Collision
Dynamics

Alignment EffectCollective Motion of Microtubules on Kinesin Bed
Short Microtubules (~ 1 um)

Long Microtubules (~ 5 -7um)

Bundle formation with nematic ordering

Loop formation, Oscillating Bundles

Cell Migration
Actin polymerization

force generation by molecular motor
membrane deformation
polarity (symmetry breaking),
sensing direction

cell-cell/substrate contact(focal adhesion),

Cell Motility

How cells coordinate deformation and movement?
What is the relationship between Force, Shape, and Motion?
Sensing & Feedback
Dipole

Shape

Force

Deformation
Cell Rheology

Active dipole

MotionCell migration might be understood as acollective non-equilibrium dynamics of matter.

Adhesion &Cell Substrate Interaction
Detachment
Look at dynamics of slow variables:
Shape, Force, Centroid Motion

Various Modes in Migration
1) Steady motion:ballistic
2) Turning(Oscillation, Rotation) :short term behavior
3) Random migration:long term behavior

Spontaneous Symmetry Breaking of Cell Shape
Hiraiwa, Matsuo, Ohkuma, Ohta, Sano, EPL (2010)

Sano, Ohta, Matsuo, (World Scientific, 2011)

Ordered Patterns of Cell Shape & Cell Migration
TailHead

Front is more
noisy than thetailTime (sec)

Maeda, Inose, Matsuo, Iwaya, MS ,Plos One, 3, e3734 (2008)

Angle θAuto Correlation Function:

CA( , t ) A(0,0) 0,0A( , t ) 2

 ,tBranches:

Lamellipodia adhere tothe substrate and fixed
while cell body movedforward.

Ordered Patterns in Cell Shape Dynamics:
Maeda, Inose, Matsuo, Iwaya, MS ,Starved Cells
Plos One, 3, e3734 (2008)
ElongatedRotation

Oscillation

Amplitude

Time (min)AutoCorrelation
FunctionAngle θAbout 70% of time series are classified into 3 ordered patterns.

Transition of patterns occurs in a single cell along the time course.

Time series and ACF of asingle cell for time duration
30min.

Liang, Cox, PLoS one (2008)

Cells show zig-zag motion.

Goal and ideas for measuring traction force
 GoalFind out a force-motion relationship of migrating cells

 ChallengeCharacterize the spatial properties of the traction stress IdeasForce spot tracking: Fine-grained approach Multi-pole analysis: Coarse-grained approachNet Traction Force Vanishes for Small Cells∝ 𝐿3∝ 𝐿2=0～1 Pa～100 Pa

Newton’s Third Law:≪=0

Sum of traction force (torque) is zero.

 f  0,  M  0ii

Measurement of Traction Force in Single Cell

Ayari et al. Cell Motility andthe Cytoskeleton (2008)

Traction Force Microscopy

Transmission

Fluorescence (beads)

We quantified the traction stress of migrating Dictyostelium cell (simple uni-cellularamoeba) using flexible poly-acrylamide gel embedded with fluorescence beads.
Substrate: poly-acrylamide gel (E=800Pa)
Bead: red (diameter 200nm, excitation 543nm)

Traction Force Microscopy 3

Improving measurement protocol, we measured the traction stress of migrating Dicty
more than one hour with a better spatio-temporal resolution.

Highly localized traction stress 2

Traction stress amplitude

Lab-frame integration

The force spots are evident in the integrated plot.
- ‘foot prints’ of migrating cell.

Multi-pole expansion
𝑀𝑖 =

𝑇𝑖 𝑑𝑆

0th moment: net force

𝑀𝑖𝑗 =𝑥𝑖 𝑇𝑗 𝑑𝑆

1st moment: force dipole

Anti-mirror symmetric

Mirror symmetricMulti-pole expansion
𝑀𝑖 =

𝑇𝑖 𝑑𝑆

0th moment: net force

𝑀𝑖𝑗 =𝑥𝑖 𝑇𝑗 𝑑𝑆

1st moment: force dipole

Force Dipole → Deformation

Anti-mirror symmetric

Mirror symmetricForce dipole𝑥𝑇𝑥
𝑀=𝑦𝑇𝑥

𝑥𝑇𝑦
𝑦𝑇𝑦

The force dipole is a 2*2 symmetric matrix with two orthogonal eigenvectors.

Force dipole
Relationship with motion

𝛿𝜃The dipole axis and cellular velocity orientation are agreed well.
- Cell migrates along with its force dipole axis.
Left: Snapshot (white arrow: velocity vector)
Right: PDF of angle formed by dipole and velocity axes (N=5; ~3000frames)

Multi-pole expansion
𝑀𝑖 =

𝑇𝑖 𝑑𝑆

0th moment: net force

𝑀𝑖𝑗 =𝑥𝑖 𝑇𝑗 𝑑𝑆

1st moment: force dipole

Anti-mirror symmetric

Mirror symmetricMulti-pole expansion
𝑀𝑖 =

𝑇𝑖 𝑑𝑆

0th moment: net force

𝑀𝑖𝑗 =𝑥𝑖 𝑇𝑗 𝑑𝑆

1st moment: force dipole

𝑀𝑖𝑗𝑘 =

𝑥𝑖 𝑥𝑗 𝑇𝑘 𝑑𝑆

2nd moment: force quadrupole

Anti-mirror symmetric

Mirror symmetricAnti-mirror symmetricForce quadrupole

𝑀𝑖𝑗𝑘 =

𝑥𝑖 𝑥𝑗 𝑇𝑘 𝑑𝑆

The force quadrupole is a 3rd order tensor with 6 independent components.

Force quadrupole
Relationship with motion
f3 Different Modes
in Crawling Cells
Inchworm modev

Squeeze-Stretch mode
Breaststroke mode

One strongly correlated pair: (1,1,1) and Vx- The sign of (1,1,1) determines the direction along the dipole axis.

(1,1,1;1) correlation

Symmetric stress pattern

Asymmetric stress pattern

Force quadrupole determines direction of migrationPredicting amoeboid motion of the cell
Multi-pole analysis reveals
a simple force-motion relationship in cell migration

Normal equations for multi-pole
traction force and shape is required.



The lowest two moments are coupled with cell migration
Consistent with the force spot tracking

H. Tanimoto and MS, Biophys. J. 106, 16-25 (2014).
H. Tanimoto and MS, Phys. Rev. Lett. 109, 248110 (2012).

Correlation between Fouriermodes and centroid velocity
M. Sano, unpublishedr ( , t )   an (t )e inFourier Expansion of Shape:

nShape and Centroid Motion400

Hitogram of Directional Cos: predictability=0.66346
250
120

350

100200300

Y200150

Histogram

Histogram

80250

1001505010050100

150200250X

300

3500-260

40
20-10

1

Directional Cos cos2v  a12q0-2-1

0Directional Cos cos3v  a3

12q

Deformable self-propelled particle
Ohta, Ohkuma PRL (2009)Introducing 3rd order rank tensors

z1  v1  iv2 ,1z 2  ( S11  iS12 ),21z3  (U111  iU222)
2

Normal Form Equations

Collective Behavior is much more rich !

Normal Form Equations Invariant under a transformation;

Hiraiwa, Matsuo, Ohkuma, Ohta, Sano, EPL (2010).

Straight motion
Ohta-Ohkuma modelZig-zag motion
Ohta-Ohkuma modelChaotic motion
Ohta-Ohkuma modelNon-polarized cell (Vegetative Cell)

Traction force of migrating cellDicyostelium cell
In two states:
1. Starved state
2. Vegetative statedForce Distribution

Traction force of migrating cell
Polarized cell (Starved Cell)

Force Distribution

Single Cell motility to Collective Motion

Wound healing

C.W. Wolgemuth, Physics 4, 4 (2011)

Short Summary
1. Multi-pole analysis revealsa simple force-motion
relationship in cell migration

2. Collective motion of NS
cells can be analyzed bynematicaly interacting moving(active nematic) material

The lowest two moments are coupled with cell migration
Consistent with the force spot tracking

What are active colloids?
Self-propelled colloid:
Self-propelled Janus particles moving their own directions
 By a local temperature gradientSelf-Thermophoresis

 By electric field:

Jiang, Yoshinaga, Sano, PRL (2010) APS Physics
Viewpoint(2010)

Induced Charge Electro-Osmosis (ICEO)

Gangwal et al. PRL (2008)

Self-propulsion of Janus particle II :by Electric field
ICEO (Induced Charge Electro Osmosis)
Gangwal et al. PRL (2008)
𝑉𝐼𝐶𝐸𝑂 ~𝜖𝜀𝑅𝐸 2𝛾∆𝑟 2 ≈ 4𝐷𝑡 + 𝑉 2 𝑡 2

Motion of Janus Particle: Top View

Surface slip flow drives particle

Bazant, PRL (2008)

Visualization: Tracer particle 200nm fluorescent beads
Janus Particle: 6 μm

Formation of Chains at frequency higher than fc

Doublet

TripletLinear Chain

Oscillation, Wave

Deformable self-propelled chain

Waving motion

Trajectory of the tail particle

Spiraling motion

Problem of self-propelled flexible chain
Self-propellingforce-Self-propelledstringContinuum
approximationP≠ u

Self-propelled(swimming ) string

P 〃uMotion of active filaments

K. Sekimoto, PRL (1995)

Motion of actin filaments on myosin bed.
Bourdieu, Leibler, Libchaber, PRL (1995)

Emergent dynamics of self-propelled stringwith cargo

53

Acknowledgements
 H. Tanimoto (Institut.

 Y. Maeda (Kyoto Univ.)

Jacques Monod, France) K. Kawaguchi (U. Tokyo)
 T. Ohta (U. Tokyo)
 Sakurako Tanida (U. Tokyo)
 Taiki Nisgiguchi (U. Tokyo)

 Miki Matsuo (JAMSTEC) T. Hiraiwa (Univ. Berlin)