Locomotive Force & Angular Momentum in 3D Cell Aggregates

Transcription

1 Locomotive Force & Angular Momentum in 3D Cell Aggregates Antoine Fruleux (UoS, Sheffield) Fruleux, Aberdeen 23/06/15

2 Dictyostelium Discoideum : social ameba

3 Dictyostelium Discoideum : Chemotaxis and motility aggregate ("slug") with propagating camp wave [D. Dormann et al, Biophysical chemistry 1998 ] polarization of cells: single cell with cortical flow, amplitude of cortical flow polymerization / depolymerization of cortex

4 Known facts: setups Centrigual force [K. Inouye et al, J. Cell Science 1980 ] [K. Inouye et al, Protoplasma 1984 ] Pressure difference 4

5 Known facts: results Centrifugal force Pressure difference Velocity (mm/h) Velocity (mm/h) Ø= 108 μm L = 528 μm Ø= 167 μm L = 967 μm Ø= 69 μm L = 432 μm Ø= 122 μm L = 380 μm Force Force Total active force (E-3 N) [K. Inouye et al, J. Cell Science 1980 ] Volume Non Centrifugal Centrifugat [K. Inouye et al, Protoplasma 1984 ] 5volume of slug (E-5 cc)

6 Why surprising? cancellation of cortical flow Why no total acive force contact area?? 6

7 Known facts: experimental results confined in a tube on substrate Velocity (mm/h) Velocity (mm/h) Ø= 108 μm L = 528 μm Ø= 167 μm L = 967 μm Force Force 7 J. Cell Science 1982 ] [M.Kitami.

8 To understand: Our Approach & 1. Coarse-graining of conserved fluxes 2. Finding constitutive law for conserved fluxes 3. Applying to experimental setup key word : momentum & angular momentum fluxes 8

9 To understand: Our Approach & 1. Coarse graining of conserved fluxes 2. Finding constitutive law for conserved fluxes 3. Applying to experimental setup key word : momentum & angular momentum fluxes 9

10 Conservation Equations 10

11 Macroscopic conservation equations for momentum AND angular momentum System of interest : Dense aggregation of cellular elements 11

12 Macroscopic description of the momentum flux At the cell-level: The force at interface => force F + torque M = force: By cell on torque: by. 12 cell on.

13 Macroscopic description of the momentum flux At the cell-level: The force at interface => force F + torque M = force: The relative position By cell on torque: by. 13 cell on.

14 Macroscopic description of the momentum flux Neighbor distribution function Definition: 14

15 Macroscopic description of the momentum flux Law of action / reaction: i j Redundancy key for 2nd coarse-graining Avoid subtle cancellations of forces / torques 15

16 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 16

17 To understand: Our Approach & 1. Coarse graining of conserved fluxes 2. Finding constitutive law for conserved fluxes 3. Applying to experimental setup Task : express in terms of macro-state variables (order parameters) ε Geometry ( kinematics ) F and M Mechanics (dynamics) 17

18 Geometry 18

19 Geometry, Three-neighbor distribution function : In reference state : (reference state) (actual state) In actual state : Aim :, the mean deviation of the relative positions Macroscopic parameters 19 Complicated? Minimum necessary informations of correlations are in

20 Geometry, Symmetry / redundancy: exchange symmetry: redundancy of viewpoint : Symmetry / redundancy constraints on the geometry, 20.

21 Geometry, Plausible assomptions : : constraints of dense packing fastly varying with Ɛ : statistically averaged movements slowly varying with Ɛ (intrinsic part) (affine part) (non-affine part), orientational gradient, polarization, 2nd 21 polarization

22 To understand: Our Approach & 1. Coarse graining of conserved fluxes 2. Finding constitutive law for conserved fluxes 3. Applying to experimental setup Task : express in terms of macro-state variables (order parameters) ε Geometry ( kinematics ) F and M Mechanics (dynamics) 22

23 Mechanics 23

24 Mechanics : in the bulk ( constitutive equations) Passive force deviation : micro-environment 24

25 Mechanics : in the bulk Passive force deviation : micro-environment collectively determines changes in micro-environement Molecular-field model of the interactions Passive torque deviation : determined similarly. 25

26 Mechanics : in the bulk Active force deviation : chemotaxis of cells Key ingredients: cells polarization Chemotactic signal (reference state) : polar direction : gradient of camp (actual state) disorientation of from Flow of cells Cell-cell interaction 26

27 Mechanics : in the bulk Active force deviation : Molecular-field model of the interactions micro-environment passive 27

28 Mechanics : in the bulk Active force deviation : Molecular-field model of the interactions micro-environment passive active wi, scalar order parameter of activity of cell magnitude of cortical flow velocity. 28

29 active contribution Mechanics : in the bulk disoriented cell less effective cortical flow shear stress (= lateral momentum tansfer) 29

30 active contribution intrinsic term Mechanics : in the bulk passive term 30

31 active contribution Mechanics : in the bulk intrinsic term 31

32 active contribution Mechanics : in the bulk intrinsic term passive term 32

33 To understand: Our Approach & 1. Coarse graining of conserved fluxes 2. Finding constitutive law for conserved fluxes 3. Applying to experimental setup Task : express in terms of macro-state variables (order parameters) ε Geometry ( kinematics ) F and M Mechanics (dynamics) 33

34 Applications 34

35 Setup Confined geometry: rigid BC 2h rigid BC Open geometry: y free BC x rigid BC (assumption) 35

36 Confined geometry thickness Velocity & presssure difference polar vector appearance of boundary layers 36

37 Confined geometry velocity of slug velocity of slug Ø= 108 μm L = 528 μm Ø= 167 μm L = 967 μm force/volume force/volume [K. Inouye et al, J. Cell Science 1980 ] [K. Inouye et al, Protoplasma 1984 ] total active force 37

38 Confined geometry total active force total active force Non centrifugal Centrifugal total volume total volume [K. Inouye et al, J. Cell Science 1980 ] [K. Inouye et al, Protoplasma 1984 ] 38

39 Confined geometry total active force 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 total volume 39

40 Open geometry free BC rigid BC Velocity (mm/h) Velocity (mm/h) Force Force [M.Kitami. J. Cell Science 1982 ] 40

41 Open geometry free BC rigid BC Velocity (mm/h) Velocity (mm/h) Force Force [M.Kitami. J. Cell Science 1982 ] 41

42 CONCLUSION Angular momentum flux is important Bulk force is in fact from boundary layers Active Nonlinearity modifies boundary layer thickness 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) 42

43 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) Colleagues in Sheffield Rhoda Hawkins Carl Whitfield Ian Estabrook Jamie Cumiskey Thank you for your attention. 43