Cell motility due to active gel flows

Rhoda J. Hawkins

Newton Institute 28th June 2013

Acknowledgements

Paris Raphaël Voituriez Jean-François Joanny Matthieu Piel

Edinburgh Davide Marenduzzo

Sheffield Carl Whitfield (PhD student)

Outline

•  Introduction & motivation: cell motility •  Active nematic droplet model •  Results & discussion of force moments •  Extension to 3D •  Comparison to cortical model •  Conclusions

Cell motility

Poincloux et al PNAS 2010

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•  MDA-MB-231 cell (breast cancer) •  In 3D Matrigel •  Cell expressing mCherry-Lifeact - labels F-actin •  Renaud Poincloux, Philippe Chavrier

Cytoskeleton: out of equilibrium soft matter theory of active gels

Cytoskeleton polymers: microtubules + actin

Molecular motors: myosin + actin   contractility

Modelling Cytoskeleton dynamics

Lattice Boltzmann simulations

From Davide Marenduzzo (with Elsen Tjhung & Mike Cates)

Model

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active polar fluid polarisation p

velocity v

•  long time limit – viscous fluid •  we have polar order, p, but essentially nematic

•  distortion free energy

•  Lattice Boltzmann simulation passive polar droplet:

Polarisation field p

radial anchoring

Steady state polarisation field from LB simulations of active droplet

adapted from Tjhung, Marenduzzo & Cates, PNAS, 2012

Imposed polarisation field p

Splay parameter

Active polar fluid equations

viscous stress

nematic distortion stress

active stress

•  Constitutive equation (Kruse et al)

•  Actomyosin contractile •  Fixed polarisation •  Incompressibility •  Low Re steady state force balance (Cauchy)

Active polar fluid equations

•  Up to second order in splay

Boundary conditions

•  No fluid across boundary: at

•  Effective viscous friction: at

•  If external medium is solid, determines slip •  If external medium is fluid of viscosity and we assume non-slip, is related to

R

Velocity Polarisation

•  Polarisation imposed, velocity & pressure field calculated analytically by assuming power series solutions

•  Non-slip boundary condition

Results non-slip

•  4 vortices, asymmetric pairs •  Symmetric vortices in no splay limit

Results - finite friction

Results – varying splay and friction

no splay

small splay large friction

large splay large friction

large splay small friction

Pressure field implies deformation

Deformation from LB simulations

Force moments •  Net force zero

•  Dipole deformation

Force moments •  Quadrupole

motion

Force moments

3D

3D

3D

3D

3D

3D

3D

3D

3D

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•  Spherical cell with thin shell “cortex” •  Actin 2D compressible (≅ variable thickness) isotropic fluid •  Myosin active stress local myosin concentration •  Myosin (de)attaches to cortex, diffuses in cell bulk & cortex & advected

MDA-MB-231 cell in 3D Matrigel Renaud Poincloux, Philippe Chavrier

µ(θ, φ, t)

Dynamical equations

friction active stress

•  force balance (low Reynolds number):

•  diffusion of myosin in cytoplasm:

•  conservation of myosin at cortex/cytoplasm interface:

diffusion in cortex •  conservation of myosin in the cortex:

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•  mass conservation for actin gel of density :

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Results – actin velocity 0

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actin (num.)

actin (exp.)

myosin (num.)myosin (exp.)

•  MB-231 tumor cells seeded in 3D matrigel •  Intensity mCherry-Lifeact labeled actin •  Kymographs give velocity of actin

Conclusion

Imposed splay polarisation of an active droplet   Internal flow in an active nematic droplet   Force quadrupole   Motion