The roles of

Momentum flux in

active multi-cellular media

Antoine Fruleux (ESPCI/P7)

Fruleux, Dresden 04/09/14

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Phys. Rev. Lett. 108, 160601 (2012)Phys. Scr. 86, 058508 (2012)

Acta Physica Polonica B 44, 847 (2013)

adiabatic piston

My thesis: Roles of Momentum Fluxes at Different Scales

Part I: non-equilibrium steady state of adiabatic piston

Part II: Roles of Momentum Fluxes in active multicellular media

— Today’s presentation

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Dictyostelium Discoideum : life stages

Migration stage

Vegetative stage

Aggregation stage

Culmination stage

Introduction

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Dictyostelium Discoideum : life stagesIntroduction

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Dictyostelium Discoideum : life stages

Migration stage

Vegetative stage

Aggregation state

Culmination stage

Introduction

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Dictyostelium Discoideum : Chemotaxis and motility

[D. Dormann et al, Biophysical chemistry 1998 ]

single cell with cortical flow

polarization of cells:

polymerization / depolymerization of cortex

, amplitude of cortical flow

aggregate ("slug") with propagating cAMP wave

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Cell undergoing cortical flow — neither "pusher" nor "puller"

puller pusher

pullerpusher

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Known facts: setups

Centrigual force

Pressure difference[K. Inouye et al, J. Cell Science 1980 ]

[K. Inouye et al, Protoplasma 1984 ]

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Known facts: results

Volume[K. Inouye et al, J. Cell Science 1980 ]

[K. Inouye et al, Protoplasma 1984 ]

Pressure differenceCentrifugal force

Total active force (E-3 N)

volume of slug (E-5 cc)

Velocity (mm/h)

Force Force

Velocity (mm/h)

Ø= 108 μm L = 528 μmØ= 167 μm L = 967 μm

Ø= 69 μm L = 432 μmØ= 122 μm L = 380 μm

Non CentrifugalCentrifugat

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Why surprising ?

cancellationof

cortical flow

Why no total acive force contact area ??

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Known facts: experimental results

[M.Kitami. J. Cell Science 1982 ]

confined in a tube on substrate

Force

Force

Ø= 108 μm L = 528 μmØ= 167 μm L = 967 μm

Velocity (mm/h)

Velocity (mm/h)

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1. Coarse-graining of conserved fluxes2. Finding constitutive law for conserved fluxes3. Applying to experimental setup

key word : momentum & angular momentum fluxes

Our Approach

To understand:

&

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1. Coarse graining of conserved fluxes2. Finding constitutive law for conserved fluxes3. Applying to experimental setup

key word : momentum & angular momentum fluxes

Our Approach

To understand:

&

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Conservetion Equations

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System of interest : Dense aggregation of cellular elements

Macroscopic conservation equationsfor momentum AND angular momentum

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Macroscopic conservation equationsfor momentum AND angular momentum

microscopic linear momentum flux

macroscopic linear and angular momentum fluxes

+ Expressions in terms of microscopic parameters

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Momentum fluxes at microscopic level:

Linear/Angular momentum conservation:

Linear momentum balance:

Angular momentum balance:

: microscopic momentum flux (= –[stress tensor])

, rank-3 pseudo-tensor. e.g.

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Macroscopic description of the momentum flux

At the cell-level:

: Too detailed

a few quantities associated to cell / cell-cell interface

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Macroscopic description of the momentum flux

At the cell-level:

The force at interface => force F + torque M

force: torque:

By cell on .by cell on .

= + + ...

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Macroscopic description of the momentum flux

At the cell-level:

The relative position

The force at interface => force F + torque M

force: torque:

By cell on .by cell on .

= + + ...

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Macroscopic description of the momentum flux

Further coarse-graining : i) No individual cells ii) Fluxes as averages of ``microsopic'' Fij , Mij , εij

Result :

Linear moment flux:

Angular moment flux:

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Macroscopic description of the momentum flux

Neighbor distribution function

Definition:

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Macroscopic description of the momentum flux

Redundancy

Law of action / reaction:i j

Avoid subtle cancellations of forces / torques

— key for 2nd coarse­graining

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Macroscopic description of the momentum flux

Macroscopic fluxes :

Linear moment flux: Angular moment flux:

, coarse-grained number of neighbors.

, cell density

, average of weighted by .

The linear/angular momentum conservation

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1. Coarse graining of conserved fluxes2. Finding constitutive law for conserved fluxes3. Applying to experimental setup

Our Approach

To understand:

&

F and M — Mechanics (dynamics)

ε — Geometry ( kinematics )

Task : express in terms of macro-state variables (order parameters)

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Geometry

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Reference state:

Assumptions :­ No cell­cell reconnection

Fast dynamics

reference state

virtualisolation

reference state— : cells' polar vector — : orientation around — : chemical gradient ( // in reference state)

Slow dynamics and Fast dynamics

Geometry,

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Three-neighbor distribution function :

In reference state :

In actual state :

Geometry,

(reference state) (actual state)

Aim :

, the mean deviation of the relative positions

Macroscopic parameters

Complicated ? Minimum necessary informations of correlations are in

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Symmetry / redundancy:

exchange symmetry:

redundancy of viewpoint :

Symmetry / redundancy constraints on the geometry, .

Geometry,

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The redundancy of and

Plausible assomptions : : constraints of dense packing fast varying with : statistically averaged movements slowly varying with

“K-theorem”

are linear combinations of

are uniquely determined

Geometry,

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K-theorem δε(1,0) :

: intrinsic part due to cell polarity

macroscopic deformation

, orientational gradient

: macro variables

Geometry,

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affine deformation non affine

term

Coarse-grained deformation of medium

Geometry,

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(intrinsic part) (affine part) (non-affine part)

Geometry,

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1. Coarse graining of conserved fluxes2. Finding constitutive law for conserved fluxes3. Applying to experimental setup

Our Approach

To understand:

&

● F and M — Mechanics (dynamics)

• ε — Geometry ( kinematics )

Task : express in terms of macro-state variables (order parameters)

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Mechanics

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( constitutive equations)

micro-environment

Passive force deviation :

Mechanics : in the bulk

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Mechanics : in the bulk

— “Molecular-field” model of the interactions

micro-environment collectively determines

Passive force deviation :

Passive torque deviation :

— determined similarly.

changes in micro-environement

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(reference state) (actual state)

Key ingredients: cells polarization : polar direction Chemotactic signal : gradient of cAMP

— chemotaxis of cells

disorientation of from  Flow of cells Cell-cell interaction

Active force deviation :

Mechanics : in the bulk

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passive

— “Molecular-field” model of the interactions

micro-environment

Active force deviation :

Mechanics : in the bulk

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wi , scalar order parameter of activity of cell — magnitude of cortical flow velocity.

active

passive

— “Molecular-field” model of the interactions

micro-environment

Active force deviation :

Mechanics : in the bulk

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From the mesoscopic parameters to the momentum fluxes:

Micro-environment: Internal cellular process:

Passive momentum fluxes: Active momentum fluxes:

Symmetry+ Weak deviation :Geometry :

(disorientation)²

Mechanics : in the bulk

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: upper-convected time derivative. : slow dynapics= (cell-cell reconnections) +(p's relaxation)

Actual state Reference state

Commutativity of mappings:Mechanics : in the bulk

virtual relaxation

time

evol

utio

n

: flow velocity

: « elastic » deformation

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Mechanics : in the bulkactive contribution

disoriented cell less effective cortical flow shear stress (= lateral momentum tansfer)

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Mechanics : in the bulkactive contribution

intrinsic term passive term

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Mechanics : in the bulkactive contribution

intrinsic term

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Mechanics : in the bulkactive contribution

intrinsic term passive term

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Mechanics : boundary conditionsGeometric condition :

Mechanic conditions :

Impenetrability :

Conservation of the momentum fluxes :

, rate of linear momentum passing boundary

, rate of angular momentum passing boundary

Free boundary :

Rigid boundary :

: boundary normal

and

ex.

cortex flow at  boundary ``anchoring''

~ "sticky" boundary condition (active fluid version).

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1. Coarse graining of conserved fluxes2. Finding constitutive law for conserved fluxes3. Applying to experimental setup

Our Approach

To understand:

&

F and M — Mechanics (dynamics)

ε — Geometry ( kinematics )

Task : express in terms of macro-state variables (order parameters)

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Applications

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Confined geometry:Setup

Open geometry:

Small parameters:

rigid BC

rigid BC

rigid BC

free BC

2h

steady state

x

y

(assumption)

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Bulk equations and Boundary conditionsBulk equations:

Boundary conditions:Rigid BC: Mixed BC:

Momentum balance

Angular momentum balance

: coefficients

rigid BC

Mixed BC

rigid BC

rigid BC

Confined geometry: Open geometry:

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Solution

(Active force)

(Torque)

Angular momentum balance :

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Solution

Velocity :

Polar deviation :

"Riccati equation"

non-linearactive term characteristic length is state-dependent

Basic scales :

from the literature

comparison with the "target" data

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Solution

Spacial scales in the solutions :

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Solution

Spacial scales in the solutions :

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Solution

Spacial scales in the solutions :

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Velocity &polar vector

pre

sssu

re d

iffer

enc

e

thicknessConfined geometry

appearance ofboundary layers

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velocity of slug

force/volume force/volume

velocity of slug

total active force

Confined geometry

[K. Inouye et al, J. Cell Science 1980 ][K. Inouye et al, Protoplasma 1984 ]

Ø= 108 μm L = 528 μmØ= 167 μm L = 967 μm

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total active forcetotal active force

total volume total volume

Confined geometry

Non centrifugalCentrifugal

[K. Inouye et al, J. Cell Science 1980 ]

[K. Inouye et al, Protoplasma 1984 ]

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total active force

total volume

Confined geometry

at fixed length : Saturation of total force with thickness boundary layer effect

thin sample angular momentum play a role active force volume

thick sample active force only from the boundary

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Open geometry

[M.Kitami. J. Cell Science 1982 ]

rigid BC

free BC

Force

Velocity (mm/h)

Force

Velocity (mm/h)

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Open geometry

[M.Kitami. J. Cell Science 1982 ]

rigid BC

free BC

Force

Velocity (mm/h)

Force

Velocity (mm/h)

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→ Angular momentum flux is important

→ Bulk force is in fact from boundary layers

→ Active Nonlinearity modifies boundary layer thickness

CONCLUSION

PERSPECTIVE – Future projects•Numerical scheme for mesoscopic dense active medium — based on the momentum+angular momentum fluxes

•Analysis of topological defects of cell polarity (development)

•Dynamic coupling with chemical waves (cAMP)

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Acknowledgements:

Ken Sekimoto – Univ. Paris7, ESPCI (Paris, supervisor)For collaboration, Ryoichi Kawai — Alabama Univ. (USA)For helpful discussions : Vincent Fleury, Annemiek Cornelisson, Yves Couder — Univ. Paris 7 (MSC) Bernard Derrida — ENS (LPS) Jean François Joanny (ESPCI)Reporters & Jury members : Karsten Kruse, Andrea Parmeggiani, François Graner Daniel RivelineColleagues of Gulliver (ESPCI)

Thank you for your attention.

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The Roles of Momentum Fluexes in Adiabatic Piston

Other subject : Momentum Flux in NESS

Phys. Rev. Lett. 108, 160601 (2012)Phys. Scr. 86, 058508 (2012)

Acta Physica Polonica B 44, 847 (2013)

69

Solution

(Active force)(Torque)

Angular momentum balance :

Caused by : Enchoring Pressure difference

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Confined geometry

Open geometry

experiments ?

Miror symmetric/

free surface