solid & liquid cell aggregates - crans...génétique et biologie du développement umr 3215,...
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Solid & liquid Cell aggregates
Aggregates: solid or liquid?
François Graner
Polarity, Division and Morphogenesis teamdir. Yohanns Bellaïche
Génétique et Biologie du DéveloppementUMR 3215, CNRS & Institut Curie, Paris, France
2011
Solid & liquid Cell aggregates
Outline Mechanics of cellular patterns
1 Solid & liquid
2 Cell aggregates
Solid & liquid Cell aggregates
Cellular materials solid & liquid
Foams for fire fighting
Solid & liquid Cell aggregates
Neighbour exchange “T1" = 1st topological process
cells tile the spaceno gap nor overlap
1to move, cells need to displace their neighbours
pass through a 4-fold vertex
Solid & liquid Cell aggregates
Neighbour exchange “T1" = 1st topological process
cells tile the spaceno gap nor overlap
2to move, cells need to displace their neighbours
pass through a 4-fold vertex
Solid & liquid Cell aggregates
Neighbour exchange “T1" = 1st topological process
cells tile the spaceno gap nor overlap
3to move, cells need to displace their neighbours
pass through a 4-fold vertex
Solid & liquid Cell aggregates
Viscous, elastic, plastic (VEP) behaviour Marmottant 2007
local energy minimum
T1: neighbour change relaxation → other minimum
Small deformationelastic solid
reversibly comes backto its initial shape
Large deformation
plastic solid
irreversibly sculpted,new shape
Quick deformation rateviscous liquid
irreversibly flows,stress increases with rate
Solid & liquid Cell aggregates
Viscous, elastic, plastic (VEP) behaviour Marmottant 2007
local energy minimum T1: neighbour change
relaxation → other minimum
Small deformationelastic solid
reversibly comes backto its initial shape
Large deformation
plastic solid
irreversibly sculpted,new shape
Quick deformation rateviscous liquid
irreversibly flows,stress increases with rate
Solid & liquid Cell aggregates
Viscous, elastic, plastic (VEP) behaviour Marmottant 2007
local energy minimum T1: neighbour change relaxation → other minimum
Small deformationelastic solid
reversibly comes backto its initial shape
Large deformation
plastic solid
irreversibly sculpted,new shape
Quick deformation rateviscous liquid
irreversibly flows,stress increases with rate
Solid & liquid Cell aggregates
Numerical methods Cellular Potts Model
V. Grieneisen
Similar to experiments
• a cell = a set of pixelssame size as in experiments
• a movement = a pixel changesto another cell
Energy
• Energy cost at cell boundaries• Size conservation
Energy minimisation
• Surface minimisation• Differential adhesion• Membrane fluctuations• ... and much more
Solid & liquid Cell aggregates
Numerical methods Cellular Potts Model
V. Grieneisen
Similar to experiments
• a cell = a set of pixelssame size as in experiments
• a movement = a pixel changesto another cell
Energy
• Energy cost at cell boundaries• Size conservation
Energy minimisation
• Surface minimisation• Differential adhesion• Membrane fluctuations• ... and much more
Solid & liquid Cell aggregates
Cell aggregates dissociate cells, then aggregate them
Compression set-up
middle cross section
(2-photon microscopy)
Vial & van der Sanden
side view
Mgharbel,
Delanoë-Ayari, Rieu
simulations
Käfer
Solid & liquid Cell aggregates
Stress relaxation cells deform and rearrange
Mgharbel, Delanoë-Ayari, Rieu
energy barrierno gap nor overlap: T1 rearrangement
Two regimeshigh stress:stress-induced rearrangementslow stress:fluctuation-induced rearrangements
Solid & liquid Cell aggregates
Fluctuations Marmottant
Rate at which barriers are passed
f + = f exp{(−∆ET1 + τV δε)/ξ}τV = work gained δε = deformation
f +δε− f −δε = τ∗
η∗ sinh(
ττ∗
)characteristic stress τ∗ = ξ/Vδε
effective viscosity η∗ = ξ exp(∆ET1/ξ)/2fV (δε)2
Stress relaxation
τ = 2τ∗ tanh−1[tanh
(τ02τ∗
)exp
(− t
tc
)]time over which stress disappears: tc ∼ exp
“∆ET1
ξ
”
Relaxation after compression
model vs simulations
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
6
7
8
9
10
11
12x 10
6
time (MCS)
For
ce
Residual plasticity after 400 s
model vs experiments
Solid & liquid Cell aggregates
Fluctuations Marmottant
Rate at which barriers are passed
f + = f exp{(−∆ET1 + τV δε)/ξ}τV = work gained δε = deformation
f +δε− f −δε = τ∗
η∗ sinh(
ττ∗
)characteristic stress τ∗ = ξ/Vδε
effective viscosity η∗ = ξ exp(∆ET1/ξ)/2fV (δε)2
Stress relaxation
τ = 2τ∗ tanh−1[tanh
(τ02τ∗
)exp
(− t
tc
)]time over which stress disappears: tc ∼ exp
“∆ET1
ξ
”
Relaxation after compression
model vs simulations
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
6
7
8
9
10
11
12x 10
6
time (MCS)
For
ce
Residual plasticity after 400 s
model vs experiments
Solid & liquid Cell aggregates
Fluctuations Marmottant
Rate at which barriers are passed
f + = f exp{(−∆ET1 + τV δε)/ξ}τV = work gained δε = deformation
f +δε− f −δε = τ∗
η∗ sinh(
ττ∗
)characteristic stress τ∗ = ξ/Vδε
effective viscosity η∗ = ξ exp(∆ET1/ξ)/2fV (δε)2
Stress relaxation
τ = 2τ∗ tanh−1[tanh
(τ02τ∗
)exp
(− t
tc
)]time over which stress disappears: tc ∼ exp
“∆ET1
ξ
”
Relaxation after compression
model vs simulations
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
6
7
8
9
10
11
12x 10
6
time (MCS)
For
ce
Residual plasticity after 400 s
model vs experiments
Solid & liquid Cell aggregates
Fluctuations Marmottant
Rate at which barriers are passed
f + = f exp{(−∆ET1 + τV δε)/ξ}τV = work gained δε = deformation
f +δε− f −δε = τ∗
η∗ sinh(
ττ∗
)characteristic stress τ∗ = ξ/Vδε
effective viscosity η∗ = ξ exp(∆ET1/ξ)/2fV (δε)2
Stress relaxation
τ = 2τ∗ tanh−1[tanh
(τ02τ∗
)exp
(− t
tc
)]time over which stress disappears: tc ∼ exp
“∆ET1
ξ
”
Relaxation after compression
model vs simulations
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
6
7
8
9
10
11
12x 10
6
time (MCS)
For
ce
Residual plasticity after 400 s
model vs experiments
Solid & liquid Cell aggregates
Characteristic time tc
relaxation time: 5hfor F9 cell lines
confirmed by fusion experiments
Solid & liquid Cell aggregates
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)