one-and two-particle microrheology in solutions of actin

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

QuickTime™ and aGrafik decompressor

are needed to see this picture.



Single filament dynamics

QuickTime™ and aSorenson Video decompressorare needed to see this picture.


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



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.)




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


Interferometric position detection



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



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



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



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


1-bead/2-bead microrheology

laser

sample

quadrantphotodiodes

X-Y time series

Optical traps



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



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



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



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



-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



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



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



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



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)

one-and two-particle microrheology in solutions of actin

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