Rheological Measurements: Viscoelastic Properties

Transcription

1 Rheological Measurements: Viscoelastic Properties Cone-plate rheometer oscillated over a range of frequencies () Imposed: sinusoidal torque () Measured: sinusoidal strain () 1 RADA16-II 1% Gel Self-assembling peptide. Modulus (Pa) Frequency (rad/s) G' Pa G'' Pa Gel: G >> G Little frequency dependence 1 RADA16-II 1% Solution Viscous solution: Lower values of moduli Modulus (Pa) 1.1 G' Pa G'' Pa Frequency dependent Frequency (rad/s) G. Kim, 23 1

2 Interpretation of displacement (x) of a microsphere (radius a) inside the cytoplasm. Assumes elastic and viscous contributions are additive as in a Voigt model. (G = µ) dx F(t) = 6a G x + µ dt Sphere in an infinite elastic medium G = F cos 6ax G = Sphere in an infinite Newtonian fluid F sin 6ax Smooth muscle cells -- with and without activation. Obtained using magnetic twisting cytometry. (a) 1 4 G''(Pa) G'(Pa) (b) G' (Pa) G * = G 1 5 (c) (G, φ /2π) f(hz) (d) 1 4 G' G'' f(hz) x1 f(hz) (1 + i )(2 x)cos 2 (x 1) + iµ Image removed due to copyright restrictions. Storage (a) and loss (b) moduli plotted over 5 decades of frequency for smooth muscle cells under control conditions (solid squares), after treatment for 1 minutes with histamine to produce smooth muscle activation (open squares), an agent to eliminate baseline tone, DBcAMP (solid triangles), or cytochalasin D, to disrupt actin filaments (open triangles). (c) shows the extrapolation of the data for illustrating the intersection at high frequency, and (d) directly compares the data from (a) and (b) under control conditions. Solid lines are the fit to the data by eqn. (68) with G =53.6 kpa and =2.5x1 8 rad/s. (Reproduced from Fabry, et al., 21) 2

3 Oscillatory shear tests. Storage and loss moduli (in Pa). 1 G' Storage modules (Pa) Viscosity (Pa.s) Loss modulus (Pa) G" ω(rad/s) Shear oscillation tests of pig kidney at strain.2%, which specifies the components of the dynamic modulus in terms of frequency. The standard deviations are also shown as error bars Experimental measurements made by cell poking (lymphocyte) (Zahalak, et al., 199).45 A F.35 FORCE (NEWTONS) a DISTANCE IN ( METERS ) 3

4 Terminology for poroelasticity H = 2G + = confined _ compression _modulus v f = local _ fluid _ velocity u v s = local _ solid _ velocity = t U = mean _ fluid _ speed _(relative_ to _ solid _ phase) A f = fluid _ area A s = solid _ area A f = porosity = Af + A s k = hydraulic _ permeability Poroelastic materials 11 Governing equations: 1. Constitutive law x 1 ij 3. Conservation of mass u U = (v f v s ) = v rel v s = t 4. Conservation of momentum ( tot = 2G ij ) ij p ij 2. Fluid-solid viscous interactions (Darcy s Law) U = kp u 1 1D forms tot 11 = (2G + ) 11 p U 1 = k p x 1 U 1 = u 1 + U t tot 11 = x 1 tot = 2 u 1 u 1 U = Hk 2 t x 1 4

5 Poroelasticity -- confined compression Impose displacements at boundaries: u 1 (x 1, t=) = 11 u 1 (x 1 =L, t>) = u 1 u 1 (x 1 =, t>) = u U = 2 u 1 u = Hk 1 t x 1 u 1 x 1 Characteristic time ~ L 2 /Hk x 1 Solution (Fourier series) t u 1 (x 1,t) = u 1 x 1 A n sin n x 1 exp L n L n L 2 n = 2 n 2 Hk Stress relaxation results for 4 tumor types Confined compression experiments 11 Stress, mmhg u 1 x Time, s 5

6 Equilibrium stress-strain curves for 4 tumor types 3 Stress (mmhg) Strain Used to calculate confined compression modulus, H. Assumed linear for strains < Hydraulic permeability as a function of tissue deformation 1-5 K, cm 2 /mmhg sec Four different tumor tissues. Obtained from stress relaxation experiments in confined compression Strain 1-7 cm 2 / (mmhg. s) = 1.4x1-13 m 4 / (N. s).3 6

7 Typical Length Scales in Biology DNA width microtubule width typical animal cell length of DNA in a chromosome human meters histone chromatin Proteins width Width of lipid bilayer nucleus length of DNA contained in a typical human cell Similar spectra exist in time scales or energy scales. Typical Eukaryotic Cell 1 µm = 1-6 m 1 nm = 1-9 m 1 Å = 1-1 m 7

8 Plasma Membrane Plasma Membrane 2-D Elastic Plate A Typical Epithelial Cell Microvillus Apical Surface Tight Junctions Adherens Junction Spot Desmosome Lateral Surface Gap Junction Intermediate Filament Basal Surface Basal Lamina Hemidesmosome 8

9 Cell-cell junctions Images removed due to copyright restrictions. Fig Schematic showing the different types of cell junctions present in an epithelial cell as found in the small intestine. Tight junctions near the apical surface essentially prevent the passage of all molecules. The spot desmosomes and adherens junctions provide for cell-cell anchoring, and the hemidesmosomes for anchoring to the basal lamina. Gap junctions provide a means for communication between neighboring cells. [Reproduced from Lodish et al., Molecular Cell Biology, 2.] Detailed structure of a focal adhesion Actin Fim ERM Nex VASP Ten Zyx CRP Abl Parv Pl3K Csk ASAP1 Vnx Vin Tal Calp II Fil α Act PAK FAK Grb-7 α Vin β SHP-2 SHPS-1 α Pall β PIX Vin PKL Pax Src FK PINCH ILK β α β β Cav α β CAS SHIP-2 α β DRAL β α α Vin Pon β α PKC Lay Synd Synt β Syn 4 upar Extracellular Matrix Transmission of Forces from the Extracellular Matrix to The Cytoskeleton 9

10 Staining of actin and nuclei in fibroblast cells (J. Lammerding) Image removed due to copyright restrictions. The cytoskeleton as a homogeneous, isotropic, elastic material. Image removed due to copyright restrictions. The cytoskeleton of a macrophage lamellipodium as seen by electron microscopy. The fibrous structure is mainly comprised of actin filaments. (John Hartiwick, 1

11 Forces and deformations are transmitted throughout the cell by the cytoskeleton Figure removed due to copyright restrictions. Fibroblast with fluorescent mitochondria forced by a magnetic bead D. Ingber, P. LeDuc A variety of methods have been used to probe cell mechanics 1 2 AFM Magnetic Bead 3 Micropipette Aspiration 4 Optical trap Silica bead Red blood cell Optical trap Silica bead 5 Shear Flow 6 Stretching Focal adhesion complex Soft Membrane Bao & Suresh, 23 11

12 Are cells linear elastic materials? One example: Indentation Experiments FORCE (NEWTONS) X Neutrophils (Zahalak et al., 199) 2a Micropipette aspiration 2 Linear relationship 2a L/a 1 L p (dynes/cm 2 ) Experimental result obtained on a from an endothelial cell micropipette experiment. Data from Theret et al.,

13 Measurements Cell type Measurement method Shear modulus (Pa) Reference lymphocyte poking 3 Zahalak et al. of cell shear lymphocyte (activated) poking 7-11 Zahalak et al. modulus found neutrophil (activated) poking Zahalak et al. poking 11 Zahalak et al. in the literature. neutrophil NIH 3T3 fibrobalst magnetic tweezers 2,-4, Bausch et al. NIH 3T3 fibroblast AFM 4,-1, Haga et al. J774 mouse magnetic tweezers 343 Bausch et al. macrophage Values range over 3T3 and NRK AFM 1,-1, Rotsch & fibroblast 3-4 orders of magnitude! al. Rademacher mouse fibroblast poking 16 (E) Peterson et endothelial cell aspiration 4-5 Theret et al. bovine endothelial cell indentation 4-6 Sato et al. porcine endothelial cell aspiration 75 (E) Sato et al. endothelial cell magnetic twisting 2.2 (round) (E) Wang et al. cytometry (MTC) 4.5 (spread) (E) bovine endothelial cell MTC Wang & Stamenovic human chondrocytes aspiration 33 Trickey et al. smooth muscle cell MTC 11.5 (E) Stamenovic & Coughlin COS7 (kidney laser tracking ( G * ) Yamada, et epithelial) microrheology al. Elastic or viscoelastic?? Cells are viscoelastic FORCE (NEWTONS) x Indentation depth (-1.6µm) Images removed due to copyright restrictions. Micropipette Aspiration DISTANCE ( METERS ) x 1-5 Indentation (Zahalak et al., 199) 13

14 Comparison between experiments and Maxwell fluid model: Ramped force application.6 Bead Displacement, µm Numerical simulation Experimental data.4.6 kpa G =.4 kpa.2 1. kpa Time, s 2 5 pn Force Time 2s Experiments suggest a shear modulus of about 1. kpa for NIH T3T fibroblasts Sinusoidal Forcing.4 Numerical simulation (G =.4-1. kpa) Experimental data Bead Displacement, µm G =.4 kpa 2 3 Time, s 4 1. kpa Cells appear to behave as a Maxwell viscoelastic material with characteristic time constant of ~1s Force, pn Time, s

15 Magnetic Twisting Cytometry (a) G''(Pa) G'(Pa) (b) G' (Pa) 1 5 (c) (G, φ /2π) f(hz) (d) 1 4 G' G'' f(hz) f(hz) Testing over a wider range of frequencies illustrates a more complex, but still viscoelastic, behavior. T G * = G + ig = (t) (t) is a geometry-dependent prefactor determined from finite element analysis G * = G x 1 ( 1 + i ) (2 x)cos (x 1) 2 + i µ is the Gamma-function; G, and x are adjustable parameters Consistent with models for soft, glassy materials Viscoelastic or Poroelastic?? Microfilaments Intermediate Filaments Microtubules Cells have a porous, fluidfilled matrix. Images removed due to copyright restrictions. 15

16 Are thermal (Brownian) effects important? Lipid vesicles exhibit a fluctuationdominated regime and an elastic regime when inflated from zero tension. (Evans and Rawicz, PRL, 199) ln (N) (dyn/cm) N (dyn/cm) SOPC:CHOL Fluctuation-dominated A α = A SOPC:CHOL DGDG A α = A Elastic stretching DGDG Measurements of membrane tension and fractional area dilation for two vesicles with different lipid compositions. (a) A fluctuation-dominated regime appears at low tensions: a slope of 8πkc/kT. (b) Crossover to direct expansion at high tension: a slope of K. (open symbols, ascending pressure; solid symbols, descending pressure.) Homogeneous, isotropic?? Images removed due to copyright restrictions. Mapping cell surface elasticity using AFM Elasticity mapping in a fibroblast (NIH3T3) cell using atomic force microscopy. The elasticity map (A) shows gross differences with the lowest values corresponding to the nucleus (N), and a small pocket (arrow) low in actin content. The height of the cell from the substrate is shown in (B). The two lower images are stained for actin (C) and microtubules (D) from the same cell. (Reproduced from Haga, et al., 2) Cells are inhomogeneous and anisotropic. 16

17 What else needs to be considered?? Cells are dynamic. Properties are constantly changing. Cell Motility Images removed due to copyright restrictions. Fluorescently tagged actin Actin is a polymer The cytoskeleton is active Coordinated processes: adhesion, (de-) polymerization Active Cell Contraction Image removed due to copyright restrictions. Cardiac myocyte (Jan Lammerding) 17

18 Cells can sense and respond to mechanical stimuli Mechanotransduction: Hair cell stimulation tip link tension in tip link increases stereocilium Images removed due to copyright restrictions. SEM of the stereocilia on the surface of a single hair cell (Hudspeth) Tension in the tip link activates a stretch-activated ion channel, leading to intracellular calcium ion fluctuations. 18