one-and two-particle microrheology in solutions of actin
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Microrheology of Biopolymers (ITP Complex Fluids Program 3/05/02)
Dr. Christoph Schmidt, Vrije University, Amsterdam 1
One- and two-particle microrheology in solutions of actin, fd-virus and inorganic rods
Christoph SchmidtVrije Universiteit Amsterdam
Collaborators:
Frederick MacKintosh, Frederick Gittes, Peter Olmstedt, BernhardSchnurr , all over the place
Jay Tang, Karim Addas, Indiana University
Alex Levine, UC Santa Barbara
Gijsje Koenderink, Albert Philipse, Universiteit Utrecht
Cytoskeleton of cells
Complex dynamic machinery,based on polymer networks
and membranes
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are needed to see this picture.
Microrheology of Biopolymers (ITP Complex Fluids Program 3/05/02)
Dr. Christoph Schmidt, Vrije University, Amsterdam 2
Single filament dynamics
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semiflexible: lp >> alP = persistence length,a = monomer radius
lp =EI
kbTE = Young’s modulus,I = area moment of inertia
(for solid rod) I =π4
a4
� aspect ratio:
lp
a∝
a4
a= a3
with E - 1 Gpa, a - 20 – 50 Å,
lp ≈ 10 − 2500µm
Couette
parallel plate
plate and cone
Macrorheology
floating plate
Microrheology of Biopolymers (ITP Complex Fluids Program 3/05/02)
Dr. Christoph Schmidt, Vrije University, Amsterdam 3
Shear deformation of a viscoelastic object
F
F
stress = G x strain
complex shear modulus: G*(ω) = G’(ω) + i G”(ω)
storage modulus loss modulus
1µm
Microrheology
Advantages:— study inhomogeneities— study small samples, biological cells— reach high frequencies (-> MHz) w/o inertial effects— probe scale dependent material properties
by varying probe size— active vs. passive, single bead vs multiple bead (Weitz &Co.)
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Microrheology of Biopolymers (ITP Complex Fluids Program 3/05/02)
Dr. Christoph Schmidt, Vrije University, Amsterdam 4
Objective
Trap
Condenser
Back-focalplane
Quadrantdetector
∆p
pin
outp
θ
Interferometric Position and Force Detection
c
I
f
x
c
IF = = sinθ =
dt
dp
Gittes, Schmidt, (1998), Opt. Lett. 23: 7-9.
Laser
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Interferometric position detection
Microrheology of Biopolymers (ITP Complex Fluids Program 3/05/02)
Dr. Christoph Schmidt, Vrije University, Amsterdam 5
Experiments with semidilute semiflexible polymer networks
Parameters:
Bead diameter >> • • • •� • • • • Entanglement length > mesh sizePersistence length > mesh size0.1 Hz < ω/2π < 20 kHz
For larger beads and higher frequencies:
generalized Stokes-Einstein:
α * (ω ) = 16πG* (ω)a
Data processing: time-series data, x(t)
power spectral density, <xω2>
fluctuation-dissipation: xω2 = 4kBTα" (ω)
ω
Kramers-Kronig: α' (ω ) = 2π
Pζα"(ζ )ζ 2 − ω 2
0
∞
∫
G' (ω ),G"(ω)
xω =α * (ω) f (ω)
Power spectra of thermal bead motion in actin gel
Microrheology of Biopolymers (ITP Complex Fluids Program 3/05/02)
Dr. Christoph Schmidt, Vrije University, Amsterdam 6
Single bead microrheology with actin solutions
G’(ω)
G’’(ω)
Log ω
Log
G
scaling regime
rubberplateau
rep-tation
Gittes, F., B. Schnurr, P. D. Olmsted, F. C. MacKintosh and C. F. Schmidt (1997). “Microscopic viscoelasticity: shear moduli of soft materials determined from thermal fluctuations.” Physical Review Letters 79(17): 3286-9.Schnurr, B., F. Gittes, F. C. MacKintosh and C. F. Schmidt (1997). “Determining microscopic viscoelasticity in flexible and semiflexible polymer networks from thermal fluctuations.” Macromolecules 30: 7781-7792.
0.001
0.01
0.1
1
10
100
1 10 100 1000 104 105
App
aren
t she
ar e
last
ic m
odul
i [P
a]
Frequency [Hz]
G’~ trap stiffness
G”= iωη (viscosity)
ω
Apparent shear elastic moduli of trapped bead in water
Microrheology of Biopolymers (ITP Complex Fluids Program 3/05/02)
Dr. Christoph Schmidt, Vrije University, Amsterdam 7
Effects of cross-linking, bundeling etc.
log ω
shea
r m
odul
us G(0) = κ 2
kBTle3
ω ∝ κγle
4
single filament
network
Storage modulus for actin gels of different concentrations
Microrheology of Biopolymers (ITP Complex Fluids Program 3/05/02)
Dr. Christoph Schmidt, Vrije University, Amsterdam 8
Artifacts with 1-bead microrheology/2-bead microrheology
Depletion layer ~ r, lp
See: A. J. Levine, T.C. Lubensky, PRL, 85: 1774 (2000)
Gd’, Gd” Go’, Go”
r⊥r||
xωm,i = α ij
nm(ω) fωn, j
2-bead microrheology:
Mutual compliance:
n,m: particle index,i,j = x,y,z
Relation to Lamé-coefficients
α ||(1,2) (ω) = 1
4πrµ0 (ω ) , µ0(ω ) = G (ω)
α⊥(1,2) (ω) = 1
8πrµ0 (ω )λ0(ω ) + 3µ0 (ω )λ0(ω ) + 2µ0 (ω )
In incompressible limit:α ||
(1,2) (ω)α⊥
(1,2) (ω)= 2
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1-bead/2-bead microrheology
laser
sample
quadrantphotodiodes
X-Y time series
Optical traps
Microrheology of Biopolymers (ITP Complex Fluids Program 3/05/02)
Dr. Christoph Schmidt, Vrije University, Amsterdam 9
Data evaluation for 2-bead microrheology
Time series position data: ri1(t) , ri
2(t) , i = x,y
Power spectral density of position cross-correlation: Sxij
12(ω) = Ri1(ω)R j
2(ω)†
Fluctuation-dissipation: Ri1(ω)R j
2(ω)† = 4kTω
αij12"
(ω)
Kramers-Kronig: α ij12'
(ω )
Elastic coefficients: µ(ω ) = G(ω),λ(ω)
fd virus
m
kDa 18,200 density massLinear
104.16 massMolecular
7 Diameter
2.2 length ePersistenc
9.0 length Uniform
6
µ
µµ
≈
×≈≈
≈≈
Da
nm
m
m
Collaboration with Jay Tang, Karim Addas
Microrheology of Biopolymers (ITP Complex Fluids Program 3/05/02)
Dr. Christoph Schmidt, Vrije University, Amsterdam 10
10 -20
10 -19
10 -18
10 -17
0.1 1 10 100 1000 104
21 µm
14 µm
11 µm
9.2 µm
7.9 µm
7.0 µm
para
llel m
utua
l com
plia
nce
[a.u
.](im
agin
ary
part
)
Frequency [Hz]
10-20
10-19
10-18
400 600 8001000 3000
10-19
10-18
10-17
400 600 8001000 3000
Mutual compliance as function of bead distance d,α||”, fd solution 18 mg/ml
α||”
α||”x d
α ||(1,2) (ω ) = 1
4πrµ 0 (ω ) , µ0 (ω ) = G (ω )
α ⊥(1,2) (ω ) = 1
8πrµ 0 (ω )
λ0 (ω ) + 3µ 0 (ω )
λ0 (ω ) + 2µ 0 (ω )
1
10
0.1 1 10 100
21 µm
14 µm
11 µm
7.9 µm
7.9 µm
7 µm
Sto
rage
mod
ulus
G' [
Pa]
Frequency [Hz]
Shear elastic storage modulus for fd solution 18 mg/ml,measured by two-bead microrheology
Microrheology of Biopolymers (ITP Complex Fluids Program 3/05/02)
Dr. Christoph Schmidt, Vrije University, Amsterdam 11
1
10
100
0.1 1 10 100 1000
G' G"sh
ear
elas
tic m
odul
i [P
a]
frequency [Hz]
Shear elastic moduli for 12.5 mg/ml fd virus solution
slope 1
3/4
1/2
c = 300 x c*
1
10
100
1000
0.1 1 10 100 1000
5 mg/ml
7.5 mg/ml
10 mg/ml
12.5 mg/ml
15 mg/ml
She
ar e
last
ic lo
ss m
odul
us, G
" [P
a]
Frequency [Hz]
Concentration dependence of loss modulus
fd concentrationslope 1
3/4
1/2
c* = mfd/l3 � 0.04 mg/ml
Microrheology of Biopolymers (ITP Complex Fluids Program 3/05/02)
Dr. Christoph Schmidt, Vrije University, Amsterdam 12
1
10
100
0.1 1 10 100 1000
5 mg/ml
7.5 mg/ml
10 mg/ml
12.5 mg/ml
15 mg/ml
She
ar e
last
ic s
tora
ge m
odul
us, G
" [P
a]
Frequency [Hz]
Concentration dependence of storage modulus
fd concentration
1/2
10
100
3 4 5 6 7 8 9 10
G'[Pa]G"[Pa]
y = 1.6238 * x^(1.2035) R= 0.91199
y = 6.332 * x^(0.97142) R= 0.80853
shea
r el
astic
mod
uli [
Pa]
fd concentration [mg/ml]
concentration dependence of elastic moduli at 100 Hz
Microrheology of Biopolymers (ITP Complex Fluids Program 3/05/02)
Dr. Christoph Schmidt, Vrije University, Amsterdam 13
-2
-1
0
1
2
3
0.1 1 10 100 1000 104
Frequency [Hz]
Re(a|| / a⊥ )Im(a|| / a⊥ )
Gel (in)compressibility measured by 2-bead microrheologyfd solution 18 mg/ml
α ||(1,2) (ω) = 1
4πrµ0 (ω )
, µ0 (ω ) = G (ω)
α⊥(1,2) (ω) = 1
8πrµ0 (ω )
λ0 (ω ) + 3µ0 (ω )λ
0 (ω ) + 2µ0 (ω )
In incompressible limit:
α ||(1,2 ) (ω)
α⊥(1,2 ) (ω)
= 2
Translational Brownian motion: optical tweezers
Host = rods (L/D = 11) Host = spheres (R = 15 nm)
1244 nm
Microrheology of Biopolymers (ITP Complex Fluids Program 3/05/02)
Dr. Christoph Schmidt, Vrije University, Amsterdam 14
Translational Brownian motion: optical tweezers
Host = rods (L/D = 10) Host = spheres (R = 15 nm)L = 202.7 nmD = 17.8 nmL/D = 11.4Dprobe/L = 6Dprobe/D = 70ρ = 1.7845 g/mL
Core = AlOOHShell = silica (4 nm)Negative charge from Si-OH
D �
15 nm (DLS, SLS)Dprobe/L = 83ρ = 2 g/mL
Material = silicaNegative charge from Si-OH
Solvent = DMF (N,N-dimethylformamideSalt = LiCl (κ-1 = 2 - 22 nm)
N
H3C H3C
OH
εr = 36.7n = 1.4305η0 = 0.796 mPa.spKauto ~ 29
Depletion attraction
Solvent Rod dispersion
Koenderink et al., Depletion induced crystallization in colloidal rod-sphere mixtures, Langmuir 15(97)4693
Microrheology of Biopolymers (ITP Complex Fluids Program 3/05/02)
Dr. Christoph Schmidt, Vrije University, Amsterdam 15
0.1 1 10 100 1000 100001E-9
1E-7
1E-5
1E-3
0.1S
(f)
[V2 /H
z]
f [Hz]
0% rods 2% rods 3.5% rods
Optical tweezers: host rods in DMF, 0 mM LiCl
S0∝γ f0∝1/γ
Solvent: -2
κ-1 ≥ 22 nm
Optical tweezers: host rods in DMF
0 1 2 3 4 5 6 7 8 91.75
1.80
1.85
1.90
1.95
2.00
-Slo
pe
102 φrod
0 mM LiCl1 mM LiCl10 mM LiCl
Microrheology of Biopolymers (ITP Complex Fluids Program 3/05/02)
Dr. Christoph Schmidt, Vrije University, Amsterdam 16
Microrheology of Biopolymers (ITP Complex Fluids Program 3/05/02)
Dr. Christoph Schmidt, Vrije University, Amsterdam 17
6% rods
Conclusions
— sampling of µm volumes is possible with microrheology
— frequency range can be extended to 10s of KHz and even MHz (DWS)
— wide frequency range makes it possible to probe different dynamic regimes
— especially interesting for semiflexible polymers, complex transitions, ω3/4scaling
— effect of electrostatics, not understood.
— data interpretation has to be done with caution: depletion layers, different dynamics at low frequencies etc.
— two-bead microrheology avoids depletion layer effects and measurescompressibility directly
— open questions: plateau values,subtleties of transitions regimes,cross-linking, non-linearities, single-filament dynamics
Microrheology of Biopolymers (ITP Complex Fluids Program 3/05/02)
Dr. Christoph Schmidt, Vrije University, Amsterdam 18
0.001
0.01
0.1
1
10
100
1000
0.01 0.1 1 10 100 1000
G'[Pa] - MunichG"[pa] - MunichG' - AmsterdamG" - Amsterdam
shea
r el
astic
mod
ulus
[Pa]
frequency [Hz]
Comparison: Munich data (F. G. Schmidt et al. (2000) PRE 62, 5509)
with our data
100 mM NaCl
~3 mM salt(monovalent)
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