One- and two-particle microrheology in solutions of actin, fd-virus and inorganic rods

Transcription

1 Microrheology of Biopolymers (ITP Complex Fluids Program 3/05/02) One- and two-particle microrheology in solutions of actin, fd-virus and inorganic rods Christoph Schmidt Vrije Universiteit Amsterdam Collaborators: Frederick MacKintosh, Frederick Gittes, Peter Olmstedt, Bernhard Schnurr, all over the place Jay Tang, Karim Addas, Indiana University Alex Levine, UC Santa Barbara Gijsje Koenderink, Albert Philipse, Universiteit Utrecht Cytoskeleton of cells QuickTime and a Grafik decompressor are needed to see this picture. Complex dynamic machinery, based on polymer networks and membranes Dr. Christoph Schmidt, Vrije University, Amsterdam

2 Single filament dynamics QuickTime and a Sorenson Video decompressor are needed to see this picture. semiflexible: l p >> a l P = persistence length, a = monomer radius l p = EI k b T E = Young s modulus, I = area moment of inertia (for solid rod) I = π 4 a4 QuickTime and a Sorenson Video decompressor are needed to see this picture. aspect ratio: l p a a4 a = a3 with E - Gpa, a Å, l p µm Macrorheology plate and cone Couette floating plate parallel plate Dr. Christoph Schmidt, Vrije University, Amsterdam 2

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 Microrheology QuickTime and a Sorenson Video decompressor are needed to see this picture. Advantages: µm 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.) Dr. Christoph Schmidt, Vrije University, Amsterdam 3

4 Interferometric position detection QuickTime and a Interferometric are Sorenson needed Video to see decompressor this picture. Position and Force Detection Quadrant detector dp I I F = = sinθ = dt c c x f Back-focal plane Condenser p Trap p out θ p in Objective Laser Gittes, Schmidt, (998), Opt. Lett. 23: 7-9. Dr. Christoph Schmidt, Vrije University, Amsterdam 4

5 Experiments with semidilute semiflexible polymer networks Parameters: Bead diameter >> Entanglement length > mesh size Persistence length > mesh size 0. Hz < ω/2π < 20 khz For larger beads and higher frequencies: generalized Stokes-Einstein: x ω =α * (ω) f (ω) α * (ω) = 6πG * (ω)a Data processing: time-series data, x(t) power spectral density, <x ω2 > G' (ω),g"(ω) fluctuation-dissipation: x ω 2 = 4k BTα"(ω) ω Kramers-Kronig: α' (ω) = 2 π P ζα"(ζ ) 0 ζ 2 ω 2 Power spectra of thermal bead motion in actin gel Dr. Christoph Schmidt, Vrije University, Amsterdam 5

6 Single bead microrheology with actin solutions G (ω) Log G reptation rubber plateau G (ω) scaling regime Log ω Gittes, F., B. Schnurr, P. D. Olmsted, F. C. MacKintosh and C. F. Schmidt (997). Microscopic viscoelasticity: shear moduli of soft materials determined from thermal fluctuations. Physical Review Letters 79(7): Schnurr, B., F. Gittes, F. C. MacKintosh and C. F. Schmidt (997). Determining microscopic viscoelasticity in flexible and semiflexible polymer networks from thermal fluctuations. Macromolecules 30: Apparent shear elastic moduli of trapped bead in water Apparent shear elastic moduli [Pa] G ~ trap stiffness G = iωη (viscosity) Frequency ω [Hz] Dr. Christoph Schmidt, Vrije University, Amsterdam 6

7 Effects of cross-linking, bundeling etc. shear modulus G(0) = κ 2 k B Tl e 3 network single filament ω κ γl e 4 log ω Storage modulus for actin gels of different concentrations Dr. Christoph Schmidt, Vrije University, Amsterdam 7

8 Artifacts with -bead microrheology/2-bead microrheology r r QuickTime and a Sorenson Video decompressor are needed to see this picture. G d, G d Go, G o Depletion layer ~ r, l p 2-bead microrheology: Mutual compliance: x ω m,i = α ij nm (ω) f ω n, j n,m: particle index, i,j = x,y,z See: A. J. Levine, T.C. Lubensky, PRL, 85: 774 (2000) Relation to Lamé-coefficients α (,2) (ω) = 4πrµ 0 (ω ), µ 0 (ω) = G(ω) λ α (,2) (ω) = 0 (ω ) + 3µ 0 (ω ) 8πrµ 0 (ω ) λ 0 (ω ) + 2µ 0 (ω ) In incompressible limit: α (,2) (ω) α (,2) (ω) = 2 -bead/2-bead microrheology quadrant photodiodes X-Y time series sample Optical traps laser Dr. Christoph Schmidt, Vrije University, Amsterdam 8

9 Data evaluation for 2-bead microrheology Time series position data: r i (t), r i 2 (t), i = x,y Power spectral density of position cross-correlation: S xij 2(ω) = R i (ω)r 2 j (ω) Fluctuation-dissipation: R i (ω)r 2 j (ω) = 4kT ω α 2 " ij (ω) Kramers-Kronig: 2 α ' ij (ω ) Elastic coefficients: µ(ω ) = G(ω),λ(ω) fd virus Uniform length 0.9µ m Persistence length 2.2µ m Diameter 7nm Molecular mass kda Linear mass density 8,200 µ m 6 Da Collaboration with Jay Tang, Karim Addas Dr. Christoph Schmidt, Vrije University, Amsterdam 9

10 Mutual compliance as function of bead distance d, α, fd solution 8 mg/ml α (,2) (ω) = 4πrµ 0 (ω ), µ 0 (ω ) = G (ω) λ α (,2) (ω) = 0 (ω ) + 3µ 0 (ω ) 8πrµ 0 (ω ) λ 0 (ω ) + 2µ 0 (ω ) parallel mutual compliance [a.u.] (imaginary part) µm 4 µm µm 9.2 µm 7.9 µm 7.0 µm α α x d Frequency [Hz] Shear elastic storage modulus for fd solution 8 mg/ml, measured by two-bead microrheology Storage modulus G' [Pa] 0 2 µm 4 µm µm 7.9 µm 7.9 µm 7 µm Frequency [Hz] Dr. Christoph Schmidt, Vrije University, Amsterdam 0

11 Shear elastic moduli for 2.5 mg/ml fd virus solution G' G" slope shear elastic moduli [Pa] 00 0 /2 c = 300 x c* 3/ frequency [Hz] Shear elastic loss modulus, G" [Pa] Concentration dependence of loss modulus c* = m fd /l mg/ml fd concentration 5 mg/ml slope 7.5 mg/ml 0 mg/ml 2.5 mg/ml 5 mg/ml /2 3/ Frequency [Hz] Dr. Christoph Schmidt, Vrije University, Amsterdam

12 Concentration dependence of storage modulus Shear elastic storage modulus, G" [Pa] 00 0 /2 fd concentration 5 mg/ml 7.5 mg/ml 0 mg/ml 2.5 mg/ml 5 mg/ml Frequency [Hz] concentration dependence of elastic moduli at 00 Hz shear elastic moduli [Pa] 00 y =.6238 * x^(.2035) R= y = * x^(0.9742) R= G'[Pa] G"[Pa] fd concentration [mg/ml] Dr. Christoph Schmidt, Vrije University, Amsterdam 2

13 Gel (in)compressibility measured by 2-bead microrheology fd solution 8 mg/ml Frequency [Hz] Re(a / a ) Im(a / a ) α (,2) (ω) = 4πrµ 0 (ω ), µ 0 (ω) = G(ω) (,2) λ α (ω) = 0 (ω ) + 3µ 0 (ω ) 8πrµ 0 (ω ) λ 0 (ω ) + 2µ 0 (ω ) In incompressible limit: α (,2) (ω) α (,2) (ω) = 2 Translational Brownian motion: optical tweezers Host = rods (L/D = ) Host = spheres (R = 5 nm) 244 nm Dr. Christoph Schmidt, Vrije University, Amsterdam 3

14 Translational Brownian motion: optical tweezers Host = rods (L/D = 0) L = nm D = 7.8 nm L/D =.4 D probe /L = 6 D probe /D = 70 ρ =.7845 g/ml Host = spheres (R = 5 nm) D 5 nm (DLS, SLS) D probe /L = 83 ρ = 2 g/ml Material = silica Negative charge from Si-OH Core = AlOOH Shell = silica (4 nm) Negative charge from Si-OH Solvent = DMF (N,N-dimethylformamide Salt = LiCl (κ - = 2-22 nm) H 3 C H 3 C N H O ε r = 36.7 n =.4305 η 0 = mpa.s pk auto ~ 29 Depletion attraction Solvent Rod dispersion Koenderink et al., Depletion induced crystallization in colloidal rod-sphere mixtures, Langmuir 5(97)4693 Dr. Christoph Schmidt, Vrije University, Amsterdam 4

15 Optical tweezers: host rods in DMF, 0 mm LiCl 0. Solvent: -2 S(f) [V 2 /Hz] E-3 E-5 S 0 γ f 0 /γ E-7 E-9 0% rods 2% rods 3.5% rods κ - 22 nm f [Hz] Optical tweezers: host rods in DMF mm LiCl mm LiCl 0 mm LiCl -Slope φ rod Dr. Christoph Schmidt, Vrije University, Amsterdam 5

16 Dr. Christoph Schmidt, Vrije University, Amsterdam 6

17 6% rods Conclusions sampling of µm volumes is possible with microrheology frequency range can be extended to 0s of KHz and even MHz (DWS) wide frequency range makes it possible to probe different dynamic regimes especially interesting for semiflexible polymers, complex transitions, ω 3/4 scaling 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 measures compressibility directly open questions: plateau values,subtleties of transitions regimes, cross-linking, non-linearities, single-filament dynamics Dr. Christoph Schmidt, Vrije University, Amsterdam 7

18 shear elastic modulus [Pa] 000 Comparison: Munich data (F. G. Schmidt et al. (2000) PRE 62, 5509) with our data G'[Pa] - Munich G"[pa] - Munich G' - Amsterdam G" - Amsterdam ~3 mm salt (monovalent) 00 mm NaCl frequency [Hz] Dr. Christoph Schmidt, Vrije University, Amsterdam 8