Active behavior of the actin cytoskeleton
Hydrodynamic theory of active gels
Active gel flow
Cortical Actin
Active gels: toward a generic approach of cell
mechanics
J.F. Joanny1
F. Jülicher2
K. Kruse3
1 Physico-Chimie
Curie
Institut Curie
2 Max
Planck Institut für Komplexer Systemen
Dresden
3 Universität
Saarbrücken
University of Chicago
J. Prost1
Summary
Active behavior of the actin cytoskeleton
Hydrodynamic theory of active gels
Active gel flow
Outline
1
Active behavior of the actin cytoskeleton
Keratocytes
Actin-Myosin complexes
2
Hydrodynamic theory of active gels
Onsager description
3
Active gel flow
Spontaneous flow of active gels
Lamellipodium Motion
4
Cortical Actin
Cell oscillations
Cortical Actin
Summary
Active behavior of the actin cytoskeleton
Hydrodynamic theory of active gels
Active gel flow
Outline
1
Active behavior of the actin cytoskeleton
Keratocytes
Actin-Myosin complexes
2
Hydrodynamic theory of active gels
Onsager description
3
Active gel flow
Spontaneous flow of active gels
Lamellipodium Motion
4
Cortical Actin
Cell oscillations
Cortical Actin
Summary
Active behavior of the actin cytoskeleton
Hydrodynamic theory of active gels
Active gel flow
Cortical Actin
Motion of Keratocyte cells, Verkhovsky
Lamellipodium motion
Cell fragments
Fast motion: 10µm/min.
Flat lamellipodium
Keratocyte fragments: actin
+myosin II
Summary
Active behavior of the actin cytoskeleton
Hydrodynamic theory of active gels
Active gel flow
Outline
1
Active behavior of the actin cytoskeleton
Keratocytes
Actin-Myosin complexes
2
Hydrodynamic theory of active gels
Onsager description
3
Active gel flow
Spontaneous flow of active gels
Lamellipodium Motion
4
Cortical Actin
Cell oscillations
Cortical Actin
Summary
Active behavior of the actin cytoskeleton
Hydrodynamic theory of active gels
Active gel flow
Cortical Actin
Actin gel
Treadmilling, actin flow
Polar filaments
Gel like structure
Revenu et al.
Summary
Active behavior of the actin cytoskeleton
Hydrodynamic theory of active gels
Active gel flow
Cortical Actin
Summary
Actin gel properties
Actin polarization
Polarization vector
Local unitary vector n
Unitary vector p =< n >
Nematic or polar
ordering
Conjugate field
Free energy change
dF = −hdp
Torque aligning the
director h⊥ = K ∇2 φ
Longitudinal field
Maxwell viscoelasticity
Elastic at short time,
viscous at long time
Single relaxation time τ
η = Eτ
Constitutive equation
τ
∂σαβ
+ σαβ = 2ηvαβ
∂t
Elastic and viscous stress
Active behavior of the actin cytoskeleton
Hydrodynamic theory of active gels
Active gel flow
Cortical Actin
Summary
Myosin motors
MyosinII
Myosin move along actin
filaments towards + end
Consume energy ATP
Form small aggregates
(minifilaments)
Non processive motors
Provoke muscle
contraction
R.Vale
Active behavior of the actin cytoskeleton
Hydrodynamic theory of active gels
Active gel flow
Outline
1
Active behavior of the actin cytoskeleton
Keratocytes
Actin-Myosin complexes
2
Hydrodynamic theory of active gels
Onsager description
3
Active gel flow
Spontaneous flow of active gels
Lamellipodium Motion
4
Cortical Actin
Cell oscillations
Cortical Actin
Summary
Active behavior of the actin cytoskeleton
Hydrodynamic theory of active gels
Active gel flow
Cortical Actin
Hydrodynamic Theory
One component effective gel, incompressible
Linear relations between fluxes and forces
Description based only on symmetries
Polar symmetry: vector p, tensor qαβ = pα pβ − 13 p2 δαβ
Time reversal symmetry
reactive and dissipative components
Active effects (motors) described in terms of ATP
consumption
Summary
Active behavior of the actin cytoskeleton
Hydrodynamic theory of active gels
Active gel flow
Cortical Actin
Mechanical stress
Stress equation
Symmetric stress
σαβ + ζ∆µqαβ + τ Aαβ
2ηvαβ =
1+τ
D
Dt
2
ν1
− (pα hβ + pβ hα − hγ pγ δαβ )
2
3
Antisymmetric stress
a
σαβ
=
1
(pα hβ − pβ hα )
2
Convected Maxwell model
Coupling between stress and polarization
Active stress
myosin coupling to filaments
normal stress difference
activity coefficient ζ < 0 ∆µ = µATP − µADP − µP
Summary
Active behavior of the actin cytoskeleton
Hydrodynamic theory of active gels
Active gel flow
Cortical Actin
Polarization and energy dissipation
Rate of change of polarization
D
D hα
pα = 1 + τ
+ λ1 pα ∆µ − ν1 vαβ pβ
Dt
Dt γ1
Rotational viscoelasticity
Active field
ATP consumption
r = Λ∆µ + ζqαβ vαβ + λ1 pα hα
Onsager relation
Energy dissipation W =
R
dr r ∆µ
.
.
Summary
Active behavior of the actin cytoskeleton
Hydrodynamic theory of active gels
Active gel flow
Outline
1
Active behavior of the actin cytoskeleton
Keratocytes
Actin-Myosin complexes
2
Hydrodynamic theory of active gels
Onsager description
3
Active gel flow
Spontaneous flow of active gels
Lamellipodium Motion
4
Cortical Actin
Cell oscillations
Cortical Actin
Summary
Active behavior of the actin cytoskeleton
Hydrodynamic theory of active gels
Active gel flow
Cortical Actin
Flow induced by boundary conditions
Film on a solid substrate
Flow field
Boundary layer cos2θ0 = 1/ν1
ζ̃∆µ sin 2θ0
Constant velocity gradient u = γ [4(η/γ
2
1
1 )+ν1 −1]
RL
2 ζ̃∆µ sin 2θ
0
Macroscopic flux Q = 0 dz v (z) = 2γ L[4(η/γ
)+ν 2 −1]
1
Cellules dans des canaux A. Zumdieck
Undulated microfluidic channels
1
1
Summary
Active behavior of the actin cytoskeleton
Hydrodynamic theory of active gels
Active gel flow
Cortical Actin
Summary
Spontaneous Frederiks transition
Parallel anchoring conditions
Flow bifurcation R.Voituriez
Same anchoring
condition on both
surfaces
Active stress equivalent
to an external magnetic
field along x axis
Instability for a finite
thickness
Lc =
1/2
2 4η
π K ( γ +(ν1 +1)2 )
1
− 2ζ̃∆µ(ν +1)
1
Active behavior of the actin cytoskeleton
Hydrodynamic theory of active gels
Active gel flow
Outline
1
Active behavior of the actin cytoskeleton
Keratocytes
Actin-Myosin complexes
2
Hydrodynamic theory of active gels
Onsager description
3
Active gel flow
Spontaneous flow of active gels
Lamellipodium Motion
4
Cortical Actin
Cell oscillations
Cortical Actin
Summary
Active behavior of the actin cytoskeleton
Hydrodynamic theory of active gels
Active gel flow
Cortical Actin
Keratocyte motion
Retrograde flow in a keratocyte
One dimensional active gel
Advancing velocity
h0 1/2
u = vd − ζ∆µ( 4ξη
)
Retrograde flow
Valloton et al.
ζ∆µ ∼ 103 Pa
Summary
Active behavior of the actin cytoskeleton
Hydrodynamic theory of active gels
Active gel flow
Outline
1
Active behavior of the actin cytoskeleton
Keratocytes
Actin-Myosin complexes
2
Hydrodynamic theory of active gels
Onsager description
3
Active gel flow
Spontaneous flow of active gels
Lamellipodium Motion
4
Cortical Actin
Cell oscillations
Cortical Actin
Summary
Active behavior of the actin cytoskeleton
Hydrodynamic theory of active gels
Active gel flow
Cortical Actin
Summary
Oscillating cells
Depolymerized microtubules
E.Paluch et al.
Non-adhering cell P.Pullarkat
depolymerized
microtubules
actin and myosin labeling
cell fragments
Role of calcium
Active behavior of the actin cytoskeleton
Hydrodynamic theory of active gels
Active gel flow
Cortical Actin
Summary
Active gel theory G.Salbreux
Cortical actin layer on a flat surface
Actin polymerizes from the membrane, actin quasi-parallel
to membrane
Myosin create tensile stresses in the membrane
ζ∆µ = ζ̄∆µ(1 − exp −z/v τm )
Actin depolymerization is enhanced by stress
vd = vd0 exp σxx /σ0
Cortical layer thickness
h = −vp τm log(1 −
gσ0
).
ζ̄∆µ
Active behavior of the actin cytoskeleton
Hydrodynamic theory of active gels
Active gel flow
Cortical Actin
Summary
Cortical actin layer in a spherical geometry
Active tension
Tensile stress due to myosin activity and curvature
Rh
Active tension γ = γm + 0 dz(σxx − σzz ) = γm + hζ∆µ
Laplace law ∆Π =
2
R
(γm + hζ∆µ)
Instabilities of the cortical layer
Perturbation decomposed into spherical modes
Mode n = 0 stable (volume not conserved: requires
permeation)
Mode n = 1 (Paluch oscillations) and n = 2 (Mitosis
shapes) can be unstable
Instability with respect to membrane detachments from the
cytoskeleton (blebs)
Active behavior of the actin cytoskeleton
Hydrodynamic theory of active gels
Active gel flow
Cortical Actin
Summary
Calcium Oscillations
Oscillation model
Calcium increases myosin
activity
Gated calcium channels
Radius decreases if activity
decreases
stability diagram
y = ζ∆µ/3E
0<α<1
characterizes
channel opening
Active behavior of the actin cytoskeleton
Hydrodynamic theory of active gels
Active gel flow
Summary
Generic hydrodynamic description
Bacterial films description
Tissues
Other active systems S.Ramaswamy
Wave propagation
Outlook
Noise and fluctuations
Permeation effects and 2 fluid description
Cortical Actin
Summary