Vesicle micro-hydrodynamics
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1 Vesicle micro-hydrodynamics Petia M. Vlahovska Max-Planck Institute of Colloids and Interfaces, Theory Division CM06 workshop I Membrane Protein Science and Engineering IPAM, UCLA, 27 march 2006 future contact: from August 1 st at the Thayer School of Engineering, Dartmouth College permanent petia@aya.yale.edu 1
2 Outline Vesicles in equilibrium Vesicles in flow: deformation does matter! Motivation: cell hydrodynamics Examples: deformation in unbounded flow migration in wall-bounded flow parachutes in microchannels (rheology of suspensions) Theory: simulations: boundary integral method analytical: small deformation expansion Vesicle adhesion 2
3 Vesicles: equilibrium bilayer thickness d ~ 5 nm a ~ µm κ 10κ B T bending stretching free energy (Helfrich) area conservation volume conservation bending modulus mean curvature=1/r 1 +1/R 2 surface tension A area V volume 3 p pressure jump across the membrane
4 Vesicles: equilibrium shapes Reduced volume ( excess area) Seifert, Adv.Phys. 46 (1997) 4
5 Motivation: cell hydrodynamics Fahraeus effect decrease in blood apparent viscosity layer depleted of red blood cells near the wall (A. Viallat, Grenoble) Cell traffic between blood stream and tissues inflammatory response tumor metastasis formation of atherosclerotic plaques 5
6 Motivation: cell hydrodynamics inflammatory response (healing an injury) blood flow (white blood cell) 6
7 Vesicle dynamics in flow: unbounded shear 1 shear + = φ The shape is not given a priori! low shear rates: tank-treading vesicle deformation? orientation? stationary shapes? flow-induced shape transitions? Kantsler and Steinberg, PRL 95 (2005) Seifert, Eur. Phys. J. B 8 (1999) Misbah, PRL 96 (2006) 7
8 Vesicle dynamics in flow: unbounded shear 2 φ high shear rate: tumbling theory (Misbah) a=0.5 Kantsler and Steinberg, PRL 96 (2006) transition to tumbling? high shear rate, high viscosity contrast λ breathing vesicle? 8
9 Vesicle dynamics in flow: near a wall vesicle detachment? cell adhesion Abkarian et al. PRL 88 (2002), Biophys. J. 89 (2005) side view microscope lift force? 9
10 Vesicle dynamics in flow: microchannel flow vesicle parachutes? RBC in capillaries Vitkova et al. Europhys. Lett. (2004) discocyte parachute transformation: shapes? shape transitions? Noguchi and Gompper, PNAS 102(40) (2005) (see supplemental info) 10
11 Deformable objects in flow a free-surface boundary problem Drops: compressible interface, surface tension rules Vesicles: incompressible interface, bending stresses rule 11
12 Drop dynamics in flow Equilibrium: Drop shape is spherical Surfactant distribution is uniform Laplace s equation Flow: Drop deforms nonuniform curvature capillary stresses Surfactant is redistributed gradients in surface tension Marangoni stresses area changes (compressible interface) Vesicles what s new? 12
13 Vesicle dynamics in flow Equilibrium: shape need not to be spherical Flow: vesicle is rough fluctuations shape deforms nonuniform curvature capillary stresses bending stresses Laplace s equation area is conserved incompressible interface gradients in effective tension Marangoni stresses 13
14 vesicle dynamics: time scales shear extension rotation distorting convection by the extensional component of the flow capillary relaxation restoring bending relaxation relaxation driven by interfacial tension gradients rotation (gets to be important at high λ ) 14 viscosity ratio
15 Dimensionless parameters Capillary number relaxation distortion Bending parameter Marangoni parameter rotation parameter interplay of different time scales complex dynamics 15
16 Dimensionless parameters Capillary number =flow strength bending number elastic vesicle regime (area not conserved) 16
17 Problem formulation: governing equations Stokes flow λ i=in, out viscosity ratio Boundary conditions: continuity of velocity across the membrane shape evolution 17
18 Problem formulation: stress boundary condition & more hydrodynamics stress balanced by interfacial stresses capillary bending Marangoni incompressible surface: area conservation compressible surface: lipid conservation 18
19 Problem solution: general unbounded flow, single particle point force solution (stokeslet) Green s function (Oseen tensor) stokes equation is linear flow at infinity (integration along the particle surface) integral representation for the velocity 19
20 Problem solution Small deformations: Analytical solutions perturbation expansions for small deviation from spherical shape (quasi-spherical vesicle) r θ, φ spherical coordinates strong bending/strong tension/weak flow: 20
21 Problem solution Large distortions: Numerical simulations boundary integral method viscosity ratio λ=1 adaptive remeshing bending stresses Cristini et al. Phys. Fluids 10 (1998) ellipsoid red color signifies the magnitude21 of the bending stresses
22 Bending effects on deformation Be=0 (no bending) strong flow Ca=0.5 endtime=10 same drop! Be=0.1 Be=10 22
23 Bending effects on deformation Deformation parameter (ellipsoid) 0.4 No bending stiffness D Be=0.1 = flow strength symbols: BIM simulations lines: small deformation theory bending tension 23
24 Effects of the wall: particle migration Why do red blood cells go away from the wall? spherical neutrally-buoyant particle does not drift! Stokes flow equations linear reversing the flow 24
25 Particle migration away from a wall in shear flows deformed drop surfactant-covered, spherical drop far-away from the wall nonlinear effect close to the wall lubrication type analysis, numerical simulations. 25
26 Problem solution wall-bounded flow singularities formalism f image system -f fluid (shear free surface) stokeslet -2hf stokeslet doublet 4h 2 f source doublet rigid wall 26
27 Vesicle detachment flow reflection image drop BIM simulation (shear free surface) experiment: side view microscope vesicle on a glass substrate 27
28 Vesicle migration velocity BIM simulations: preliminary results h Be=0 Be= t, h Bending stiffness decreases migration velocity time? 28
29 Vesicle adhesion A non-destructive method to obtain adhesion strength and bending rigidity. idea: compare vesicle shapes for different reduced volumes Gruhn and Lipowsky PRE 71 (2005) Materials: DOPC, DOPC+DOPG, DOPC+cholesterol Experiment 1: Side-view observation with the phase-contrast microscope reflection 29
30 Vesicle adhesion Experiment 2: side-view observation with the confocal microscope DOPC:DOPG 9:1 DOPC:Cholesterol 8:2 D=70 microns Data for DOPC:DOPG on a glass substrate substrate (approximately) increase in bending rigidity less deformation (weakly adhering) 30
31 Summary vesicle = deformable drop with bending stiffness BIM simulations and small deformation analytical theory quantify: * deformation in unbounded flow * migration in wall-bounded flow Future work include membrane incompressibility study: flow-induced shape transformations budding lift of adherent vesicles near-contact motion 31
32 Acknowledgements Theory group at MPIKG: Prof. Reinhard Lipowsky, Dr. Rumiana Dimova, Dr. Ruben Gracia, Said Aranda Espinoza, Natalya Bezlyepkina Prof. Thomas Powers (Brown) Prof. Michael Loewenberg and Prof. Jerzy Blawzdziewicz (Yale) Prof. Vittorio Crisitini (UC Irvine) Dr. Manouk Abkarian (Montpellier) Thank you! Dimova s group 32
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