Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Mechanics of the cytoskeleton
J.F. Joanny1
1 Physico-Chimie
Curie
Institut Curie
ICTP Trieste
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Outline
1
Actin filaments and gels
Actin filaments
Actin based propulsion, Listeria
Dynamical regulation of the size of stereocilia
2
Molecular motors
Molecular motors in cells
Two level system description of molecular motors
Hair cells and motility assays
3
Active gel description of the cytoskeleton
Keratocytes and other active systems
Constitutive equations of active polar gels
Active gel flow
4
Active gels and mechanical properties of cells
Lamellipodium Motion
Cortical actin
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Outline
1
Actin filaments and gels
Actin filaments
Actin based propulsion, Listeria
Dynamical regulation of the size of stereocilia
2
Molecular motors
Molecular motors in cells
Two level system description of molecular motors
Hair cells and motility assays
3
Active gel description of the cytoskeleton
Keratocytes and other active systems
Constitutive equations of active polar gels
Active gel flow
4
Active gels and mechanical properties of cells
Lamellipodium Motion
Cortical actin
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Actin polymerization
Actin monomers
Actin polymers
molecular weight 45kDa
size δ = 5.5nm
2 protofilaments
ATP binding pocket
right-handed helix, 72nm
pitch, 24 monomers per
turn
polar monomer
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Actin Treadmilling
Polymerization Kinetics
Polymerization kinetics
dn
dt = kon ca − koff
Treadmilling
Critical concentration
Kc = koff /kon ∼
0.12µm
Non-equilibrium
polymerization due to
ATP hydrolysis after
polymerization
Different reaction rates
at + and − end
Polymerization velocity
vp = kon ca δ/2 ∼ 1µm
In vitro treadmilling concentration
∼ 0.13µm
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Polymerization and Depolymerization under force
Analogy with a particle in a double well potential
Kramers rate theory rate kd0 = τ −1 exp −E/kT
Application of a force: Barrier reduction E − fb
fb
Depolymerization rate kd = kd0 exp kT
Microscopic models for polymerization and
depolymerization Oster et al.
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Actin rigidity
Actin gel elasticity
Persistence length
Actin gel
Bending free
R energy
1
F = 2 kT `p ds ρ12
Persistence length
`p ∼ Eδ 4 /kT ∼ 15µm
Scaling law for Young
modulus E = kT
f [L/`p ]
L3
E local modulus ∼ `p
Gittes et al.
E ∼ kT `p /L4 ∼ 104 Pa
ignores entanglements
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Actin in vivo
Actin interacting proteins
Revenu et al.
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Outline
1
Actin filaments and gels
Actin filaments
Actin based propulsion, Listeria
Dynamical regulation of the size of stereocilia
2
Molecular motors
Molecular motors in cells
Two level system description of molecular motors
Hair cells and motility assays
3
Active gel description of the cytoskeleton
Keratocytes and other active systems
Constitutive equations of active polar gels
Active gel flow
4
Active gels and mechanical properties of cells
Lamellipodium Motion
Cortical actin
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Listeria motion
Listeria in a cell
Actin comet
Kocks et al.
Steals actin of the host cell
Polymerisation at the surface of the bacterium
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Biomimetic systems for Listeria
Vesicles and drops
Plastic beads
Bead radius 1-5 µ
Same advancing velocity
1µm/min.
Spontaneous adsorption of
Actin polymerization
promoters
Similar velocity
Saltatory motion Plastino
et al.
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Gel growth around spherical objects Sykes et al.
Maxwell viscoelasticity
Gel deformation field
Incompressible velocity
field v = vp Rr
Constitutive equations
dσrr
σrr
+
dr
τ
dσθθ
σθθ
v
+
dr
τ
v
Visoelastic gel
σαβ
dσαβ
+
= 2Evαβ
dt
τ
Grows from the surface at
a velocity vp
dv
dr
v
= 2E .
r
= 2E
Convected derivative and
d
∂
treadmilling dt
= ∂t
+v·∇
Force balance
dσrr
σrr − σθθ
dP
+
=
dr
r
dr
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Gel Thickness and symmetry breaking
Gel thickness
Gel conservation law
dh
= v (R + h) − vd ' vp − vd .
dt
Stress dependent depolymerization vp : Kramers rate
theory vd = vd0 exp {[σθθ − P](R + h)/σ0 }
Gel thickness
σ0 gR
h = −vp τ log 1 −
4Eτ vp
Dimensionless polymerization free energy g = log vp /vd0
Elastic limit h/R = σ0 g/4E
Instability of the growing gel, Sekimoto et al.
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Force Balance J.Prost
Elastic force, thin shell theory
Elastic stress σ ∼ Eh/R
Elastic energy
σ2
F ∼ 2E
2πLRh
Elastic propulsion force
3
fel ∼ F /L ∼ E hR
Friction force
Hydrodynamic force
negligible
Attachment-detachment
kinetics: friction constant
ξ = cl κ k (kkon+kon )
off
off
Friction force ff = 2πLRξV
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Advancing velocity
Force velocity relation Y.Marcy
Advancing velocity
Force balance fel − ff = fext
Steady state condition
h/vp = L/V
Absence of external force
3/4
V = ( RL )1/2 ( Eξ )1/4 vp
Ignores actin monomer
diffusion through the gel
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Soft Listeria
Actin propulsion of oil droplets O.Campas
2γ0
+ δP(h) = γ
R
cos θ(h) d cos θ(h)
+
h
dh
− σnn (h)
Non-uniform interfacial tension γ = γ0 + δγ(h)
Volume conservation vp (h) = V sin θ(h)
Stress dependent polymerization
vp (h) = vp0 exp[σnn (h)/σ0 ],
σ0 ≡ kT /a2 δ
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Experiments on oil drops
Steady shape motion H.
Boukellal, L.Trichet
Soft Listeria H.Boukkellal,
L.Trichet
Velocity V = 0.15µm/s
Interfacial tension
γ = 4mN/m
Strong dependence on cell
extract
vp0 = 1.4nm/s,σ0 = 32nN/µm2
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Saltatory motion A.Bernheim
Non linear friction force
Stick-slip motion of beads
Non linear elasticity of
binding proteins
ξ = cl
2 k k
κδm
on off
V 2 kon + koff
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Outline
1
Actin filaments and gels
Actin filaments
Actin based propulsion, Listeria
Dynamical regulation of the size of stereocilia
2
Molecular motors
Molecular motors in cells
Two level system description of molecular motors
Hair cells and motility assays
3
Active gel description of the cytoskeleton
Keratocytes and other active systems
Constitutive equations of active polar gels
Active gel flow
4
Active gels and mechanical properties of cells
Lamellipodium Motion
Cortical actin
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Hair cells
Stereocilia and tiplinks
Ciliated cells of the inner ear
Bundle of actin cilia
Graded sizes in a bundle
Connections via tip-links
Hudspeth, Martin
Bundles of varying sizes
Shape and size regulation
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Other ciliated cells
Microvilli
Actin ciliated structures
Intestinal epithelial brush
border cells
Microvilli (actin
structures)
Filopodia, lamellipodia
spines and boutons in
neurons
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Stereocilia shape in hair bundle
Myosin XV
Penetration in cutical layer
Radzinska et al.
Fettiplace and hackney
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Actin and espin in stereocilia
Structure of stereocilia
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Actin and espin in stereocilia
Actin and espin incorporation
Radzinska et al.
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Actin and espin in stereocilia
Kachar law
Organ of Corti and Vestibule
Treadmilling velocity proportional to stereocilium length
Constant treadmilling time 48h
Thicker cilia have larger treadmilling velocity
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Membrane dynamics and polymerization front
Polymerization front motion
Penetration in
cutical layer
Free energy
Z
Z
1
Fm =
dsm σ + κH 2 − Pdv
2
Z
1
Fi =
dsm k (δ − δ0 )2
2
Linear dynamic equation
δF
δrn
δF
= −λa
+ vp0
δh
vn = −λm
∂h
+ vT
∂t
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Steady state treadmilling
Membrane shape equation
vp0 − vT
1 H(vT − vp0 )2
δFm
=
+
δrn
λa
2
k λ2a
vp0 −vT
λa
(vT −v 0 )2
+ 12 k λ2p
a
effective tension Peff = P +
effective pressure σeff = σ
Peff = 3 103 Pa, total force 100pN, force per filament
3 10−1 pN
treadmilling velocity vT = vp0 − λa
2σeff
r0
+
κ
r03
Thicker cilia have a larger treamilling velocity Gale et al.
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Shape of stereocilium tip
Deformation by tip-link force
Tip-link force 10 pN
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Shape of the stem depolymerization
Capping
proteins
Treadmilling
Polymerization at the barbed end (at the
tip)
rate kp
Capping proteins at the pointed end (espin)
prevents depolymerization
unbinds at a rate ku
Depolymerization at the pointed end
only uncapped protein
rate kd
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Depolymerization kinetics
Rate equations
∂pc
= −ku pc (n) + kp (pc (n − 1) − pc (n))
∂t
∂pu
= ku pc (n) + kp (pu (n − 1) − pu (n))
∂t
+kd (pu (n + 1) − pu (n))
Stereocilium shape
Constant inplane density, exponential decay over λ = vT ku−1
"
z
z #1/2
a
kd − kp
kp
kp a
r (z)
ku
=
−
r0
kd − ku − kp kp + ku
kd − ku − kp kd
k −k
p
d
)1/2
Size of stereocilium L = 2vT ku−1 log ra0 ( kd −k
p −ku
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Shape of Stereocilia
Calculated shapes
Membrane shape
Shape of Myosin XV
mutants
Surface
depolymerization?
Local radius of curvature ρ ∼ (κ/σ)1/2
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Emerging part of stereocilium
Force balance
Membrane force 2πrb σ
Viscous force αLi vT
Immersed length
Li
aσ log(Li /r0 )
Li
exp −( ) =
=
2λ
2λ
(α/π)λvT η
> 1 Totally immersed stereocilium
< 1 Finite immersed lengh smaller than 2vT ku−1
Maximum immersed length Li < LT /3
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Outline
1
Actin filaments and gels
Actin filaments
Actin based propulsion, Listeria
Dynamical regulation of the size of stereocilia
2
Molecular motors
Molecular motors in cells
Two level system description of molecular motors
Hair cells and motility assays
3
Active gel description of the cytoskeleton
Keratocytes and other active systems
Constitutive equations of active polar gels
Active gel flow
4
Active gels and mechanical properties of cells
Lamellipodium Motion
Cortical actin
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Molecular motor functions
Motor structure
Motor proteins
Muscle contraction
(myosin II)
Cilia and axonemes
(Dynein)
Mitosis
Intracellular transport
(kinesin, myosinV)
Inner ear hair cells (Myosin
1c)
Rotating motors
Vale
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Single molecule experiments
Motor steps
Processive motors, Bead
assays
Cappello
Processivity length ∼ 1µm
Stall force 6pNBlock
Velocity 1µm/s
Steps at the period of
microtubule
Existence of backward
steps
Processivity 200 steps
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Processivity and motility assays
Motility assay
Processivity
−1
−1
On and off rates ton = koff
, toff = kon
Duty ratio r =
ton
ton +toff
=
kon
kon +koff
Fraction of bound motors r , 1/r motors required on the
filament
Myosin are non-processive r = 0.02. Myosin filaments
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Thermodynamics of molecular motors
Linear flux-force relation
External forces on the
motor
Mechanical force
(optical trap) f
Chemical force
∆µ = µATP − µADP − µPi
Flux-force relation
v
= λ11 f + λ12 ∆µ
r
= λ21 f + λ22 ∆µ
Works as a motor if
f < 0, ∆µ > 0
Onsager symmetry
relations λ21 = λ12
Stall force
fs = −λ12 ∆µ/λ11
Yield η = −fv /r ∆µ, does
not satisfy Carnot’s law
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Outline
1
Actin filaments and gels
Actin filaments
Actin based propulsion, Listeria
Dynamical regulation of the size of stereocilia
2
Molecular motors
Molecular motors in cells
Two level system description of molecular motors
Hair cells and motility assays
3
Active gel description of the cytoskeleton
Keratocytes and other active systems
Constitutive equations of active polar gels
Active gel flow
4
Active gels and mechanical properties of cells
Lamellipodium Motion
Cortical actin
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Thermal ratchet motion
Periodic potential
Qualitative study
Efficient motor a ∼ (2D/ω2 )1/2
Gliding velocity vg = 1ζ ( Ub − f )
Average velocity v = p
a+b
ω2−1 +b/vg
Stall force U = fb or p ∼ 0
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Fokker Planck equation
Probability conservation
∂P1 ∂j1
+
∂t
∂x
∂P2 ∂j2
+
∂t
∂x
Probability fluxes ji = −D
= −ω1 (x)P1 + ω2 P2
= −ω1 (x)P2 + ω2 P1
∂Pi
∂x
+
Pi ∂Wi
kT ∂x
−
Pi
kT f
No motion at thermal equilibrium
Non equilibrium transitions due to ATP consumption
R
−1 R
`
`
Advancing velocity v = 0 (P1 + P2 )dx
0 (j1 + j2 )dx
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Outline
1
Actin filaments and gels
Actin filaments
Actin based propulsion, Listeria
Dynamical regulation of the size of stereocilia
2
Molecular motors
Molecular motors in cells
Two level system description of molecular motors
Hair cells and motility assays
3
Active gel description of the cytoskeleton
Keratocytes and other active systems
Constitutive equations of active polar gels
Active gel flow
4
Active gels and mechanical properties of cells
Lamellipodium Motion
Cortical actin
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Force displacement measurements in Hair cells Martin
Micromanipulation of hair cells
Gating spring model
Role of myosin1c motors
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Force displacement measurements in motility assays
Balland et al.
Biomimetic system
Oscillations and force displacement curves
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Collective action of molecular motors Julicher et al.
Rigidly bound molecular motors
Trap position X0
bead position X
Equations of motion
Force balance
R`
∂W1
∂W2
ζf dX
=
−k
(X
−
X
)
+
N
0
dt
0 dx(P1 ∂x + P2 ∂x )
Fokker Planck Equation (no thermal noise)
∂P1 dX ∂P1
−
∂t
dt ∂x
∂P2 dX ∂P2
−
∂t
dt ∂x
= −ω1 (x)P1 + ω2 (x)P2
= −ω2 (x)P2 + ω1 (x)P1
random distribution of motors P1 + P2 = 1/`
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Force-displacement curves T.Guerin
Force-velocity curve
Spontaneous symmetry
breaking
Oscillations above activity
threshold ω = (Ω/τ )1/2
Ω = ω1 + ω2 , τ = ζf /k
Force-displacement curve
Large non-linear zone when Ωτ >> 1
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Outline
1
Actin filaments and gels
Actin filaments
Actin based propulsion, Listeria
Dynamical regulation of the size of stereocilia
2
Molecular motors
Molecular motors in cells
Two level system description of molecular motors
Hair cells and motility assays
3
Active gel description of the cytoskeleton
Keratocytes and other active systems
Constitutive equations of active polar gels
Active gel flow
4
Active gels and mechanical properties of cells
Lamellipodium Motion
Cortical actin
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Motion of Keratocyte cells, Verkhovsky
Lamellipodium motion
Cell fragments
Fast motion: 10µm/min.
Flat lamellipodium
Keratocyte fragments: actin
+myosin II
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Single cell elasticity, Asnacios
Creep measurement
Weak power law increase of creep compliance J(t) ∼ t 0.24
Complex elastic modulus G(ω) ∼ ω 0.24 . Large distribution
of relaxation times
Active effect
Cell regulation at times larger than 100s
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
In vitro active gels, G.Koenderink
Actin Myosin gel
Build-up of contractile stress
Actin-myosin gel in a 400µm diameter
capillary
Accelerated 180 times
ATP introduced at time t = 0. Tensile
stress increases with time
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Outline
1
Actin filaments and gels
Actin filaments
Actin based propulsion, Listeria
Dynamical regulation of the size of stereocilia
2
Molecular motors
Molecular motors in cells
Two level system description of molecular motors
Hair cells and motility assays
3
Active gel description of the cytoskeleton
Keratocytes and other active systems
Constitutive equations of active polar gels
Active gel flow
4
Active gels and mechanical properties of cells
Lamellipodium Motion
Cortical actin
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
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
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Motion of a thin active gel layer
Actin polarization
Surface friction σxz = ξv
Local force balance
d
dx hσ = ξv
Constitutive equation
2η
∂v
= σ + ζ∆µ
∂x
Polymerization Conditions
U = v (L) + vp = v (0) + vd
Velocity profile
Friction length d 2 = 2hη/ξ
Stress and velocity profiles
σ
=
−ζ∆µ
cosh((2x−L)/2d)
1−
cosh L/2d
vdξ
ζ∆µh
=
sinh((2x−L)/2d)
cosh(L/2d)
Advancing velocity
U = (vp + vd )/2
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
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
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Dissipative and reactive fluxes for polar active liquids
Dissipative fluxes
Onsager relations: crossed terms are equal
d
σαβ
= 2ηvαβ
hα
Pαd =
+ λ1 pα ∆µ
γ1
r d = Λ∆µ + λ1 pα hα
Reactive fluxes
Onsager relations: crossed terms are opposite
r
σαβ
= −ζ∆µpα pβ +
Pαr = −ν1 vαβ pβ
rr
= ζpα pβ vαβ
ν1
(pα hβ + pβ hα )
2
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
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
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
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
.
.
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Outline
1
Actin filaments and gels
Actin filaments
Actin based propulsion, Listeria
Dynamical regulation of the size of stereocilia
2
Molecular motors
Molecular motors in cells
Two level system description of molecular motors
Hair cells and motility assays
3
Active gel description of the cytoskeleton
Keratocytes and other active systems
Constitutive equations of active polar gels
Active gel flow
4
Active gels and mechanical properties of cells
Lamellipodium Motion
Cortical actin
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
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
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
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
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Multicomponent active gels
z
u
h
p
v
x
-h
-u
Constitutive equations for a two component active gel
σαβ = 2ηvαβ − (pα ∂β µ̄ + pβ ∂α µ̄) − ζ∆µpα pβ
2
jα = ρg vgα = −γ∂α µ̄ + κ∆µpα − pβ vαβ
Conservation laws
Force balance ∂α σαβ = ∂β P
Steady state diffusion ∂α jα
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Simple shear flow
Velocity and composition
Shear geometry
d µ̄
σxz = η dv
dz − 2 dz
jx = ρg v = κ∆µ
jz = −γ ddzµ̄ − 2 dv
dz
Boundary condition, gel
friction on the surfaces
σxz (h) = −ξ(vg (h) − u)
σxz (−h) = −ξ(vg (−h) + u)
Velocity field v (z) =
ξu
z − ρκg ∆µ
ξh+η+2 /(4γ)
Finite average velocity
due to active current
Composition field
µ̄ = µ0 + κ∆µ
2γρg −
ξu
2γ ξh+η+2 /(4γ) z
Finite composition
gradient due to polarity
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Outline
1
Actin filaments and gels
Actin filaments
Actin based propulsion, Listeria
Dynamical regulation of the size of stereocilia
2
Molecular motors
Molecular motors in cells
Two level system description of molecular motors
Hair cells and motility assays
3
Active gel description of the cytoskeleton
Keratocytes and other active systems
Constitutive equations of active polar gels
Active gel flow
4
Active gels and mechanical properties of cells
Lamellipodium Motion
Cortical actin
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Lamellipodium motionVerkhovsky
Actin velocity field
Stress distribution
Velocity field obtained by
speckle microscopy
Valloton et al.
Advancing velocity
u = 10µm/min.
Retrograde flow
v = 1µm/min.
Stress distribution on the
substrate
σxz = 4 102 N/m2 Oliver
et al.
Actin viscosity
η = 105 Pa.s Kaes et al.
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Active gel description
One dimensional lamellipodium
Liquid-like motion
Equation of motion
∂h(σ−P)
= ξv
∂x
Fully polarized p=1
Polymerization at the tip
mediated by Wasp
vp = nkp ρwa (x)
Depolymerization at the
back
Mass conservation
d
dx h(v + u) = kp ρwa
Surface friction
σxz = ξv
ξ ∼ 1010
SI
Active behavior
2η dv
dx = σ + ζ∆µ
Stress distribution on the
substrate
σxz = 4 102 N/m2
Friction length
d = (4ηh/ξ)1/2 ∼ 6µm
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Active gel description
Retrograde flow
Liquid-like motion
Boundary conditions
Force at the back due to
cell body
No force at the front
Interfacial tension
forces: Wetting
phenomena
Velocity field
No movement at the
center of lamellipodium
Retrograde flow at the
front
v = − dζ∆µ
4η exp −x/d
Anterograde flow at the
rear depending on the
applied force
Advancing velocity
h0 1/2
u = vd − ζ∆µ( 4ξη
)
Retrograde flow
ζ∆µ ∼ 103 Pa
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Solid-like motion of lamellipodium
Tip of size ` = (u + v (0))τ
Force balance on the solid region
vp0 −(u+v (0))
λa
solid tip F (0) = Fext + σ + σ 0 − ξv (0)`
membrane Peff h0 = σ + σ 0 + Fext
Effective polymerization pressure Peff =
Force balance on the
Force balance on the
Matching to the liquid region v (0), h0 , −F (0) = σh0
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Outline
1
Actin filaments and gels
Actin filaments
Actin based propulsion, Listeria
Dynamical regulation of the size of stereocilia
2
Molecular motors
Molecular motors in cells
Two level system description of molecular motors
Hair cells and motility assays
3
Active gel description of the cytoskeleton
Keratocytes and other active systems
Constitutive equations of active polar gels
Active gel flow
4
Active gels and mechanical properties of cells
Lamellipodium Motion
Cortical actin
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Cells with depolymerized microtubules, E. Paluch et al.
Actin labeling L929 fibroblast fragments
Myosin labeling
Cell with depolymerized microtubules, cell fragments
Oscillation period 10 min, 2 min for fragments
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Non-adhering cells, P. Pullarkat
Cell oscillations (T3 fibroblasts)
Oscillations depend on
actin contractility
Oscillations depend on
calcium (threshold density)
Period
Oscillation period ' 30s
Oscillation period
decreases with myosin
activity
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Blebs
Blebs on spreading cells
D.Cuvelier
Membrane detachment from
the cortical layer
Charras
Detachments of the membrane form the cortical layer
Bleb lifetime 30s
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Planar cortical layer G.Salbreux
Cortical actin layer on a flat membrane
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
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Cortical actin layer in a spherical cell
Spherical model of the cell
Active tension
Tensile stress due to
myosin activity and
curvature
Active tension
Rh
γ = γm + 0 dz(σxx −σzz ) =
γm + hζ∆µ
Osmotic pressure
difference between interior
and exterior
constant membrane
tension
Laplace law
∆Π = R2 (γm + hζ∆µ)
v
σ0 ln vp0 =
d
ζ∆µ +
e
− τv
6ηvp
p
R (e
− 1)
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Stability of the cortical layer
Instabilities of the cortical layer
Perturbation decomposed into
spherical modes
P
0
r (θ) = R + ∞
0 un Yn (θ)
Mode n = 0 volume not
conserved: requires water
permeation through the
membrane
Mode n > 0 , no volume
change
n = 1: No shape
deformation: local thinning
of the layer. Paluch
oscillations?
n = 2 quadrupolar shape:
mitosis?
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Oscillations of non adhering cells
Oscillation model
Calcium increases myosin activity
Gated calcium channels
Radius decreases if activity decreases
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Cell Stability
stability diagram
First unstable mode: n=1
y = ζ∆µ/3E, 0 < α < 1 characterizes channel opening
Three regions
stable cell
oscillating cell
unstable cell (Paluch oscillations)
Actin filaments and gels Molecular motors Active gel description of the cytoskeleton Active gels and mechanical properties of cells
Aknowledgements
Active gels
Frank Juelicher
Karsten Kruse
Jacques Prost
Theory
R.Voituriez
O.Campas
A.Zumdieck
G.Salbreux
Y.Kafri
A.Callan-Jones
P.Dewimille
K.Sekimoto
A.Basu
H.Boukkellal
E.Paluch
J.Y. Tinevez
P.Martin
M.Balland
P.Y.Placais
Experiments
C.Sykes
J.Plastino
L.Trichet