Post on 07-Feb-2018
transcript
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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'( )(Cells in 3D confinement in matrigel
• 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
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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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'( )(Cortical Model
• 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