Vesicles and Red Blood Cells under Flow Gerhard Gompper Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Ju¨lich, Germany
Soft Matter Hydrodynamics • Red blood cells in microvessels:
– Flow behavior depends on cell elasticity. – Diseases such as diabetes reduce deformability of red blood cells!
Soft Matter Hydrodynamics Example: Flow behavior of malaria-infected red blood cells in microchannels Just after infection:
Late stage:
Diameter: 8 µm
6 µm
J.P. Shelby et al., Proc. Natl. Acad. Sci. 100 (2003)
4 µm
2 µm
Some Facts about Blood One micro-liter of blood contains • liquid called blood plasma, 55% by volume • blood cells suspended within the plasma, 45% by volume – 5 million red blood cells (RBCs) – 5,000 white blood cells (leucocytes) – 250,000 platelets (thrombocytes)
Red blood cells contain hemoglobin, iron-containing protein which facilitates transportation of oxygen by reversible binding.
Red Blood Cells • RBCs are produced in the red bone marrow • at rate of 2 million per second • production stimulated by hormone erythropoietin (EPO) • lifetime in human vascular system about 120 days • heart pumps volume of about 5 liters/min • RBCs travel through the human body about 180,000 times • corresponds to traveled distance of 180 km!
Mesoscale Flow Simulations Complex fluids: length- and time-scale gap between • atomistic scale of solvent • mesoscopic scale of dispersed particles (colloids, polymers, membranes) −→ Mesoscale Simulation Techniques Basic idea: • drastically simplify dynamics on molecular scale • respect conservation laws for mass, momentum, energy
Examples: • Lattice Boltzmann Method (LBM) • Dissipative
Particle
Dynamics
(DPD) • Multi-Particle-Collision Dynamics (MPC)
Alternative approach: Hydrodynamic interactions via Oseen tensor
Mesoscale Hydrodynamics Simulations Multi-Particle-Collision Dynamics (MPC)
• coarse grained fluid • point particles • off-lattice method • collisions inside “cells” • thermal fluctuations
A. Malevanets and R. Kapral, J. Chem. Phys. 110 (1999) A. Malevanets and R. Kapral, J. Chem. Phys. 112 (2000)
Multi-Particle Collision Dynamics (MPC) Flow dynamics: Two step process Streaming
• ballistic motion ri(t + h) = ri(t) + vi(t)h
Multi-Particle Collision Dynamics (MPC) Flow dynamics: Two step process Streaming
Collision
i
• ballistic motion ri(t + h) = ri(t) + vi(t)h
• mean velocity per cell Pni 1 ¯ i(t) = n v j∈Ci vj (t) i • rotation of relative velocity by angle α ¯ i + D(α)(vi − v ¯ i) vi′ = v
Mesoscale Flow Simulations: MPC • Lattice of collision cells: breakdown of Galilean invariance • Restore Galilean invariance exactly: random shifts of cell lattice
T. Ihle and D.M. Kroll, Phys. Rev. E 63 (2001)
Mesoscale Flow Simulations: MPC • Lattice of collision cells: breakdown of Galilean invariance • Restore Galilean invariance exactly: random shifts of cell lattice
T. Ihle and D.M. Kroll, Phys. Rev. E 63 (2001)
Low-Reynolds-Number Hydrodynamics • Reynolds number Re = vmaxL/ν ∼ inertia forces / friction forces For soft matter systems with characteristic length scales of µm: Re ≃ 10−3 • Schmidt number Sc = ν/D ∼ momentum transp. / mass transp. Gases: Sc ≃ 1, liquids: Sc ≃ 103 1000
α=130,ρ=30 α=130, ρ=5 α=90, ρ=5 α=45, ρ=5 α=15, ρ=5
Sc
100 10 1 0.1 0.01
0.1
h
1
10
M. Ripoll, K. Mussawisade, R.G. Winkler and G. Gompper, Europhys. Lett. 68 (2004)
Multi-Particle Collision Dynamics Advantages: • Unconditionally stable • Galilean invariance • Boundary conditions easily implemented ◮ Curved wall ◮ Deformable boundaries ◮ Slip / no-slip walls • Thermal fluctuations naturally included • High Schmidt numbers
Multi-Particle Collision Dynamics Some applications:
• Vesicles in flow • Red blood cells in flow • Microswimmers • Rod-like colloids in shear flow • Colloid sedimentation • Semi-dilute polymer solutions • Polymers in microchannels • Star polymers in flow
Do it Yourself — Colloidal Hydrodynamics Get a feeling for hydrodynamic interactions: Interactive MPC Simulations
See webpage: http://www.fz-juelich.de/ics/ics-2/EN/Forschung/HydrodynamicsSimulation/ node.html
Membranes Hydrodynamics of Membranes and Vesicles
Simulations of Membranes Dynamically triangulated surfaces
Hard-core diameter σ Tether length L: σ < L < √3 σ --> self-avoidance
Dynamic triangulation:
G. Gompper & D.M. Kroll (2004)
Capillary Flow: Red Blood Cells
• Spectrin network induces shear elasticity µ of composite membrane • Elastic parameters: κ/kB T = 50, µR02/kB T = 5000
Equilibrium Vesicle Shapes Minimize curvature energy for fixed area A = 4πR02 and reduced volume V ∗ = V /V0, where V0 = 4πR03/3:
stomatocyte
discoctyte
U. Seifert, K. Berndl, and R. Lipowsky, Phys. Rev. A 44 (1991)
prolate
Membrane Hydrodynamics Interaction between membrane and fluid: • Streaming step: bounce-back scattering of solvent particles on triangles • Collision step: membrane vertices are included in MPC collisions
implies impenetrable membrane with no-slip boundary conditions.
H. Noguchi and G. Gompper, Phys. Rev. Lett. 93 (2004); Phys. Rev. E 72 (2005)
Membrane Hydrodynamics Red Blood Cells in Shear Flow
Blood Viscosity (DPD) Blood viscosity at physiological conditions (hematocrit 45%)
Relative viscosity
a)
10
MS-RBC (no aggregation) MS-RBC (aggregation) LD-RBC (no aggregation) LD-RBC (aggregation)
2
x x x 10
10
Experiments: ◦ Chien et al., Science (1967) Skalak et al., JBE (1981)
x
1
x
no attraction: xx
x
without fibrinogen xx
aggregation: whole blood
0
10
-2
10
-1
10
0
10
1
10
2
10
3
Shear rate (s-1) D.A. Fedosov, W. Pan, B. Caswell, G. Gompper, G.E. Karniadakis, Proc. Natl. Acad. Sci. USA 108 (2011)
Blood Viscosity (DPD) • Two shear-thinning regimes: attraction / deformation dominated • Estimate attraction strength: shear stress ≃ 0.02P a; force ≃ 2pN • Blood cells: shape and energy (c)
(b) mean (no aggregation) mean (aggregation) +/- standard deviation +/- standard deviation
RBC asphericity distribution
25
0.045 s-1 (no aggregation) 45 s-1 (no aggregation) 1424 s-1 (no aggregation)
20
15
Equilibrium - 0.154
10
5
0
0
0.1
0.2
0.3
Asphericity
0.4
0.5
0.6
Average membrane bending energy (k BT)
30 5500
5000
4500
4000
3500
value in equilibrium
3000
10
-2
10
-1
10
0
10
1
Shear rate (s-1)
10
2
10
3
Membrane Hydrodynamics Vesicle and Cells in Capillary Flow
Capillary Flow: Fluid Vesicles
(fluid particles not shown)
• small flow velocities: vesicle axis perpendicular to capillary axis −→ no axial symmetry! • discocyte-to-prolate transition with increasing flow H. Noguchi and G. Gompper, Proc. Natl. Acad. Sci. USA 102 (2005)
Capillary Flow: Elastic Vesicles Elastic vesicle:
• curvature and shear elasticity (κ = 20 kB T , µ = 110 kB T /R02 ) • model for red blood cells
parachute shape
Capillary Flow: Elastic Vesicles Elastic vesicle:
• curvature and shear elasticity • model for red blood cells
Tsukada et al., Microvasc. Res. 61 (2001)
Capillary Flow: Red Blood Cells Shear elasticity suppresses prolate shapes (large deformations) Flow velocity at discocyte-to-parachute transition
bending rigidity Implies for RBCs: vtrans ≃ 0.2mm/s for Rcap = 4.6µm
shear modulus
RBC Clustering & Alignment in Flow Physiological conditions: Hematocrit (volume fraction of RBCs) H = 0.45 Lower in narrow capillaries HT = 0.1...0.2 Therefore: Hydrodynamic interactions between RBCs very important Note: No direct attractive interactions considered!
RBC Clustering & Alignment in Flow Low hematocrit HT :
• Single vesicles more deformed → move faster • Effective hydrodynamic attraction stabilizes clusters J.L. McWhirter, H. Noguchi, G. Gompper, Proc. Natl. Acad. Sci. 106 (2009)
RBC Clustering & Alignment in Flow Low hematocrit HT :
G(z*nb)
6
v*0=7.7 =10 =10
4
Positional correlation function a
2 0 0
1
2
3
4
5
z*nb
0.7
Probability for cluster size ncl
0.6 P(ncl)
0.5 0.4
b
0.3 0.2 0.1
Clustering tendency increases with
0 0
1
2
3
4 ncl
5
6
7
increasing flow velocity
RBC Clustering & Alignment in Flow Low hematocrit HT :
G(z*nb)
6
v*0=7.7 =10 =10
4
Positional correlation function a
2 0 0
1
2
3
4
5
z*nb
0.7
Probability for cluster size ncl
0.6 P(ncl)
0.5 0.4
b
0.3 0.2 0.1
Clustering tendency increases with
0 0
1
2
3
4 ncl
5
6
7
increasing flow velocity
RBC Clustering & Alignment in Flow High hematocrit HT :
disordered discocyte
aligned parachute
zig-zag slipper
J.L. McWhirter, H. Noguchi, G. Gompper, Proc. Natl. Acad. Sci. 106 (2009)
RBC Clustering & Alignment in Flow High hematocrit HT :
disordered discocyte
aligned parachute
zig-zag Skalak, Science (1969)
Clustering & Alignment in Flow Phase diagram (Rcap = 4.6µm):
Hematocrit HT = 0.28/L∗ves Transition to zig-zag phase despite higher flow resistance than alignedparachute phase! 2
4
6
8
10
flow velocity v0∗
12
RBC Clustering & Alignment in Flow Larger capillary radius Rcap = 5.2µm: Phase diagram: 1.6
D
0.2
Pc
New phase: Asymmtric lane (Al) HT
0.25
*
L RBC
1.4 S,Al
1.2
0.3
1
0.35
0.8 6
8
10
g*
12
14
16
18
J.L. McWhirter, H. Noguchi, and G. Gompper, New. J. Phys., submitted (2012)
White Blood Cell Margination White blood cells: • important for immune response • requires interaction with vessel wall
(a)
(b)
WBC margination for • low shear rate
• intermediate hematocrit
D.A. Fedosov, J. Fornleitner, G. Gompper, Phys. Rev. Lett. 108 (2012)
Vesicles and Cells with Viscosity Contrast near Wall
1.08 1.04
FL Rp /(kB T )
Vesicle in gravitational field near wall:
100
1.00 10
rcm FG
0.96 ycm
111111111111111 000000000000000 000000000000000 111111111111111 000000000000000 111111111111111 (a)
λ=1: λ=2: λ=3: λ=4: λ=7: λ = 10 : Oseen: −2 ycm :
(a)
1
0.5
0.92
2
(b) 180
Lift force FL balanced by gravitational
Lift force depends on viscosity
Simulation Oseen
160 140 2 /(kB T Rp ) FL ycm
force FG
contrast λ = ηin/ηout
5
ycm /Rp
120 100 80 60 40 20 0 0
2
4
6 λ
S. Messlinger, B. Schmidt, H. Noguchi and G. Gompper, Phys. Rev. E 80 (2009)
8
10
White Blood Cell Margination Phase diagram: hematocrit and shear rate ’hard’ WBC
(b)
1
20
y/W
0.8
15
0.6 0.4
H t = 0.45
γ*
0.2 0
10
1
P(x,y)
P(x,y) 0.0024
y/W
0.8 0.8 0.6 0.6
y/H
5
0.4 0.4
0.1
0.2
0.3
H
0.4
0.5
H t = 0.5
0.2 0.2
0.6
t
0 1
Mechanism: push WBC towards wall
y/W
• non-spherical RBCs experience lift force,
0.8 0.6 0.4 0.2 0
• large hematocrit: RBCs invade space in front of WBC, act as stepping stone
H t = 0.55
0.0024 0.0018 0.0018 0.0012 0.0012 0.0006 0.0006 0 0
Vesicles in Structured Channels
Vesicle motion through zig-zag shaped channel (Lx = 100µm): time-dependent flow
Vesicles in Structured Channels
Vesicle motion through zig-zag shaped channel (Lx = 100µm): time-dependent flow
H. Noguchi, G. Gompper, L. Schmid, A. Wixforth, and T. Franke, EPL 89 (2010)
Vesicles in Structured Channels
s1
s2
s4 e4
e3 s3
e2 e1
∆l /
0.2
ly
e1
0.3
0.1
Large reduced volume V : ∗
e2
• Fast flows: Symmetric shape oscillations
0
s1
0.8
lx V*
0.9
1
s3
s4
• Slow flows: Orientational oscillations Smaller reduced volume V : ∗
s2
e3
• Symmetric double tail • Asymmetric single tail e4 e2 20µm
Synchronization and Bundling of Rotating Helical Flagella Bacteria swim by a bundle of rotating helical flagella. Synchronization and bundling due to hydrodynamic interactions: P5
5
4 /R
h
P4
3
P3
2
P1
1
P2
0
-1 0
50
100
150
200
t
• synchronization before bundling • end-distance initially increases • synchronization time ∼ d2
S.-Y. Reigh, R.G. Winkler, and G. Gompper, Soft Matter 8 (2012).
Summary • Mesoscale simulation techniques are powerful tool to bridge the length- and time-scale gap in complex fluids • Multi-particle-collision dynamics well suited for hydrodynamics of embedded particles: colloids, polymers, vesicles, RBCs, microswimmers • Red blood cells in shear flow: roleaux formation, shear thinning • Red blood cells in capillary flow: shear elasticity implies parachute shapes, hydrodynamic clustering and alignment • White blood cell margination: interaction with RBCs • Vesicles in structured channels: single- and double-tailed shapes Review Multi-Particle Collision Dynamics: G. Gompper, T. Ihle, D.M. Kroll, R.G. Winkler, Adv. Polym. Sci. 221, 1 (2009)
Acknowlegments Many thanks to: Daniel M. Kroll
Triangulated surfaces; membrane defects NDSU, Fargo (USA)
Tamotsu Kohyama
Membrane defects; scars
Shiga University
Gerrit A. Vliegenthart Virus buckling; crumpling
FZ Ju¨lich
Hiroshi Noguchi
Vesicles & RBCs in flow
ISSP, U Tokyo
Dmitry Fedosov
Blood flow
FZ Ju¨lich
Roland Winkler
Mesoscale hydrodynamics
FZ Ju¨lich
Liam McWhirter
RBCs in Microchannels
U Helsinki
Sebastian Messlinger
Vesicle in shear flow
PhD student
Benjamin Schmidt
Vesicle in shear flow
DAAD (U Toronto)
Thomas Franke
Vesicle experiments
U Augsburg
and all other members of Theoretical Soft Matter and Biophysics group at FZ Ju¨lich.
