Aggregates: solid or liquid?

Transcription

1 Aggregates: solid or liquid? François Graner Polarity, Division and Morphogenesis team dir. Yohanns Bellaïche Génétique et Biologie du Développement UMR 3215, CNRS & Institut Curie, Paris, France 2011

2 Outline Mechanics of cellular patterns 1 Solid & liquid 2

3 Cellular materials solid & liquid Foams for fire fighting

4 Neighbour exchange T1" = 1st topological process cells tile the space no gap nor overlap to move, cells need to displace their neighbours pass through a 4-fold vertex 1

5 Neighbour exchange T1" = 1st topological process cells tile the space no gap nor overlap to move, cells need to displace their neighbours pass through a 4-fold vertex 2

6 Neighbour exchange T1" = 1st topological process cells tile the space no gap nor overlap to move, cells need to displace their neighbours pass through a 4-fold vertex 3

7 Viscous, elastic, plastic (VEP) behaviour Marmottant 2007 local energy minimum Small deformation elastic solid reversibly comes back to its initial shape

8 Viscous, elastic, plastic (VEP) behaviour Marmottant 2007 local energy minimum T1: neighbour change Small deformation elastic solid reversibly comes back to its initial shape Large deformation plastic solid irreversibly sculpted, new shape

9 Viscous, elastic, plastic (VEP) behaviour Marmottant 2007 local energy minimum T1: neighbour change relaxation other minimum Small deformation elastic solid reversibly comes back to its initial shape Large deformation plastic solid irreversibly sculpted, new shape Quick deformation rate viscous liquid irreversibly flows, stress increases with rate

10 Numerical methods Cellular Potts Model Similar to experiments a cell = a set of pixels same size as in experiments a movement = a pixel changes to another cell V. Grieneisen

11 Numerical methods Cellular Potts Model Similar to experiments a cell = a set of pixels same size as in experiments a movement = a pixel changes to another cell Energy Energy cost at cell boundaries Size conservation Energy minimisation Surface minimisation Differential adhesion Membrane fluctuations... and much more V. Grieneisen

12 Compression set-up dissociate cells, then aggregate them middle cross section (2-photon microscopy) Vial & van der Sanden simulations Käfer side view Mgharbel, Delanoë-Ayari, Rieu

13 Stress relaxation cells deform and rearrange energy barrier no gap nor overlap: T1 rearrangement Mgharbel, Delanoë-Ayari, Rieu Two regimes high stress: stress-induced rearrangements low stress: fluctuation-induced rearrangements

14 Fluctuations Marmottant Rate at which barriers are passed f + = f exp{( E T 1 + τv δε)/ξ} τv = work gained δε = deformation

15 Fluctuations Marmottant Rate at which barriers are passed f + = f exp{( E T 1 + τv δε)/ξ} τv = work gained δε = deformation f + δε f δε = τ η sinh ( ) τ τ characteristic stress τ = ξ/v δε effective viscosity η = ξ exp( E T 1 /ξ)/2fv (δε) 2

16 Fluctuations Marmottant Rate at which barriers are passed f + = f exp{( E T 1 + τv δε)/ξ} τv = work gained δε = deformation f + δε f δε = τ η sinh ( ) τ τ characteristic stress τ = ξ/v δε effective viscosity η = ξ exp( E T 1 /ξ)/2fv (δε) 2 Stress relaxation τ = 2τ tanh 1 [ tanh ( τ 0 2τ ) exp ( t t c )] time over which stress disappears: t c exp ET 1 ξ Relaxation after compression 12 x 106 model vs simulations Force time (MCS) x 10 5

17 Fluctuations Marmottant Rate at which barriers are passed f + = f exp{( E T 1 + τv δε)/ξ} τv = work gained δε = deformation f + δε f δε = τ η sinh ( ) τ τ characteristic stress τ = ξ/v δε effective viscosity η = ξ exp( E T 1 /ξ)/2fv (δε) 2 Stress relaxation τ = 2τ tanh 1 [ tanh ( τ 0 2τ ) exp ( t t c )] time over which stress disappears: t c exp ET 1 ξ Relaxation after compression 12 x 106 model vs simulations Residual plasticity after 400 s model vs experiments Force time (MCS) x 10 5

18 Characteristic time t c relaxation time: 5h for F9 cell lines confirmed by fusion experiments

19 Take home message individual cell mechanics cells deform cells rearrange : T1 - energy barrier internal degrees of freedom non-newtonian liquid fluctuation-driven regime: high viscosity stress-driven regime: decreasing viscosity time scales relaxation time(s) experiment duration residual solid-like stresses P. Marmottant et al., PNAS (2009)