F.Jülicher, K.Kruse, J.Prost, K.Sekimoto Active polar gels: actin-myosin complexes Hydrodynamic theory Lamellipodia Keratocyte motion Physical Review Letters 92, 078101 (2004), European Physical Journal E in press

Keratocyte cells Verkhovsky

Lamellipodium spreading propagating waves: period 27s

Giannone et al.

Actin filaments

C.Revenu, D.Louvard et al. Treadmilling: actin flow Polar filaments: local orientation, polarization vector p Gel-like structure: physical gel

Actin polarization

Polarization vector Local unitary vector n: polarization p= Nematic order Ferromagnetic order Conjugate field Torque

dF=−h d p 2

=K ∇

K =Frank constant

parallel field hll fixes the degree of orientation p

Myosin motors

R.Vale

Myosin motor proteins ● ● ● ●

Form small aggregates Move along actin polar filaments towards + end Consume energy (ATP) Provoke contractions (muscles) and actin flow

Maxwell viscoelasticity Maxwell model Elastic at short time, viscous at long time single relaxation time τ ∂ ij ij =2 E u ij viscosity η=Eτ ∂t

Reactive and Dissipative stress Elastic stress reactive, viscous stress dissipative d

r

ij =− 2

∂ ij ∂t

d 1− ∂ 2 ij =2 u ij ∂t 2

velocity gradient

Onsager hydrodynamic theory of actin-myosin gels

Fluxes and forces fluxes forces

ij P=dp/dt

uij

h

r

molecular fluxes chem.pot gradients

Onsager relations time inversion translational and rotational invariance

1−2 P di =

D2 Dt hi 1

d ij =2 u ij 2

1 pi

r d = 1 p i h i U p i ∂i m

Reactive and dissipative fluxes reactive stress r ij

=−

[ ] D dij Dt

active stress d

i u− p i p j− ' ij

convected derivative

reactive polarization rate

1 pi h j−p j h i 2

1 2

p i h jp j h i 1 'p k h k ij coupling to polarization

antisymmetric stress

P ri =−ij p j−1 u ij p j−1 'u kk p i vorticity

reactive ATP consumption rate Energy dissipation

˙ ∫dxr T S=

r

r = pi p j u ij 'u kk

Motion of a thin gel layer

dv d Gel constitutive equation 2 =v−v c dx dx

gel reference frame Maxwell model contraction due to active stress ∂ v = ∂x h

Viscous friction on substrate Boundary conditions dL f dt

=v L f v p

dL r dt

=v L r v d

active stress

Liquid-like motion Retrograde flow Friction length velocity profile

=− /2 E≪1 2h = h sinh x / v= cosh L /2 2

Stability of movement even if the friction force is a nonmonotonous function of velocity Gel velocity vc = (vp +vd)/2 Critical polymerization velocity Density profile Contraction at the back

2 h v =v d − c p

Polymerization kinetics

Actin polymerization promoter Concentrated at the contact line wa x =0 exp −x / ', Forces local polarization orientation Polymerization velocity

v p=k p wa x

Lamellipodium thickness h=0 k p '/ v d

' =Dwa / v

Polarization defects Nematic point defects in two dimensions topological charge 1 singular solutions of the director equilibrium equations ∇ 2 =0

aster

=

vortex

spiral

Active orientational defects Rotating spiral nematodynamics short distances

1 cos 2 = 1 v =0 r log r / r 0 ,

stable if ν1>1 0 =

2 1 sin 2 4 1 2 1 sin2 2

Keratocyte motion Two coupled vortices advancing velocity 1μm/s adhesion not treated

Other active gel problems Propagating waves 1d travelling waves

= c q ,

c =

v 0

c c 0

1T c k off

c c 0

Cortical actin K.Storm Finite thickness if -ζΔμ large enough Unstable C.Sykes, E.Paluch Bacterial « Turbulence » R.Voituriez Compressible gel unstable towards lattice of rotating vortices