Active gels and cell motility

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 jp 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 2h =    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

1T 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