Lecture Note October 1, 29 Nanostructure characterization techniques UT-Austin PHYS 392 T, unique # 5977 ME 397 unique # 1979 CHE 384, unique # 151 Instructor: Professor C.K. Shih Subjects: Applications of AFM in Bio-physics-I: Viscoelastic properties of Cells
Ph.D. Thesis of See also: Scanning probe-based frequency-dependent microrheology of polymer gels and biological cells, Mahaffy RE et al., Phys.Rev. Lett. 85, 88 (2). Quantitative analysis of the viscoelastic properties in thin regions of fibroblasts using AFM, R. E. Mahaffy et al., Biophysical Journal 86:1777-1793 (24).
Goals and Overview Interest Previous studies Improved AFM technique to obtain the quantitative viscoelastic properties Descriptions of models that apply to cells Conclusions and future directions
Interest Better understanding of the cell cytoskeleton dynamic polymer network essential to cell structure and motility Applications cell motility cancer metastasis
Measurements involving the entire cell or a cluster of cells Rheology Eichinger et al., Biophys. J., 7, 154 (1996) Microplate manipulation Thoumine et al., J. Cell Sci., 11, 219 (1997) Micropipette aspiration Evans et al., Biophys. J., 56, 151 (1989) Optical tweezers Henon et al.,biophys. J., 76, 1145 (1999) Optical stretcher Guck et al., Phys. Rev. Lett., 84, 5451 (2)
Local measurements Cell poking Petersen et al., Proc. Natl. Acad. Sci., 79, 5327 (1982) Driven attached magnetic beads Bausch et al., Biophys. J., 75, 238 (2) Driven embedded magnetic beads Bausch et al., Biophys. J., 76, 573 (1999) Brownian motions of embedded polystyrene beads Yamada et al., Biophys. J., 78, 1736 (2)
Atomic force microscopy Scanning probe technique used for imaging and nanoindentation experiments Advantages Works in air and liquid local measurements of forces and indentations Limitations Sharp Commercial probes with poorly defined contact areas Quantitative for only the elastic constant Radmacher et al, Biophys. J., 7, 556 (1996) Rotsch et al, Proc. Natl. Acad. Sci., 96, 921 (1999)
Atomic Force Microscope (AFM) Photodiode detector Cantilever (k =.3-.5 N/m) F = kz c Feedback electronics Sample z s Scanner photodiode detectors Piezo scanner xy range = 1 µm z range = 6 µm
Requirements for quantitative viscoelastic information with AFM Calibrated cantilever Cantilever with known k Controlled probe shape F Cantilever with unknown k δ Accurate model for the sample system R Time dependence? Substrate effect? h δ
Improvements introduced in our AFM studies Well-defined probe shape Modifications to the microscope to obtain viscoelastic data A modified Hertz model for viscoelastic analysis Application of two additional models for thin regions of the cell
Modified probe shape advantages Probe Sharp tip Topographic image of a probe (R = 3 µm) Provides a well-defined surface area radius ~ 1.5-6 µm reverse imaging provides exact radius Lowers the stresses on the sample
Hertz model and limitations F = 4 R 3π E 2 1 ν δ 3 2 E - the Young s modulus ν - Poisson ratio (.5 for incompressible medium) Describes spherical body in contact with a elastic sample R - radius of probe F - total force on the probe δ - indentation into the surface of the sample Reliability: height (h) >>δ (depending on sample adherence to the substrate) R > δ
Applying the Hertz model to the polyacrylamide gel (h = 5 µm) h =.5 mm z s (nm) Force (nn) Contact point Force Curve notes z scanner displacement (z s ) cantilever deflection (z c ) F = kz c contact point at change in slope Force (nn) F = 4 R E δ 2 3π 1 ν 3 2 Hertz curve notes force vs. indentation (δ) δ =z s -z c Hertz model fits for δ < R δ (nm)
Quantitative Young s modulus for thick samples 3.5 3 R = 3 µm For ν =.5, K (kpa) 2.5 2 1.5 1.5 E K = = 3π F 2 3 1 ν 4 R 2 δ δ/r.1.2.3.4.5.6.7 AFM data: E = 1.9 ±.2 kpa Standard rheometry data (converted from the shear modulus): E = 2.6 ±.2 kpa Range of the Hertz model Agreement between AFM and rheometer
Hertz model for fibroblast data Fibroblast A 2 1.5 Fibroblast A(1) Fibroblast A(2) 1 µm 1 2 K (kpa) 1.5 Topography AFM image Error mode AFM image.1.2.3.4.5.6 δ /h Range of Hertz model in terms of total height of the sample (h) h (1) = 6.2 µm h (2) = 4.7 µm Elastic constant includes the Poisson ratio (K = E/(1 - ν 2 ) Not effective for all regions of the cell
Obtaining viscoelastic data Elastic force ~ x Viscous force (drag) ~ dx/dt If x ~ cos ωt then dx/dt ~ sin ωt or dx/dt ~ cos(ωt+π/2) 9º phase shift in the viscous component Cantilever/ detector system Lock-in amplifier Sample Scanner Phase and amplitude data
Viscoelastic Hertz model Complex viscoelastic modulus K K * = K + ik Expanded Hertz model 4 3 f δ 2 bead K R + 2K 3 * R ~ δ δ f osc Effective range same as Hertz model Complex frequency-dependent term (f osc )
Complication of a frequencydependent measurement Cantilever drag Viscoelastic force on bead Drag exists on whole cantilever f drag ~ ω (cantilever size) (viscosity of medium) Measured forces include this drag kz c =f bead +f drag
Evaluating the cantilever drag contribution f α = drag ~ ωδ 2 15 1 5 ~ δ (nm) i δ e δ f i δ -2-1 1 2 δ (nm) f ~ drag & δ ω t drag = ωαδ~ ω -frequency α -cantilever size and medium viscosity
Viscoelastic data from a cell z drag (nm) 3 µ m.4 -.4 -.8 -.12 Fibroblast B z c (nm) 1..5 -.5 c. p. in-phase 9 out-of-phase c. p..5 1. z s ( µ m).5 1. in-phase z s ( µ m) 9 out-of-phase c. p. Oscillatory cantilever response Cantilever drag
The Hertz model used to evaluate the viscoelastic constants 4 3 K Fibroblast B Polyacrylamide gel K* (kpa) 2 1-1 -2-3 2 4 6 8 1 K δ (nm) K * = fosc ~ 2δ Rδ -4 Produces a constant for some regions of the cell
Remaining limitations and other models R h δ Hertz model inaccurate in thin and rigidly adhered regions Numerical models Rigidly adhered sample model (Chen et al.,int. J. Solids & Structures, 8, 1257 (1972)) Frictionless substrate model (Tu et al., J. of Appl. Mech., 659, Dec., 1964)
Rigidly adhered sample model Comparison with Hertz 2 1.8 1.6 1.4 1.2 1.8.6.4.2 a Chen /a Hertz K Chen /K Hertz ν =.5 ν =.1.1.1 1 1 1 1 δr/h 2 a - contact radius a Adhered solution for a thin sample on a hard substrate Depends on the Poisson ratio
Data supporting the rigidly adhered sample model Fibroblast C 3 µm Strong substrate effect even far from substrate Poisson ratio effect Normalized K = K(δ)/K(δ max ) K ( kpa) 3 2.5 2 1.5 1.5 Hertz Chen.4 Chen Normalized K 1.5 1.5 Hertz Chen.4 Chen.1.2.3.4 δr/h 2.1.2.3.4 δr/h 2
Frictionless substrate model 2 Comparison with Hertz 1.8 1.6 1.4 1.2 1.8.6.4.2 K Chen /K Hertz a Chen /a Hertz frictionless substrate Chen.5.1.1.1 1 1 1 1 δr/h 2 Hertz model approximation holds longer for the frictionless substrate.
Data that supports the frictionless substrate model Normalized K 2 1.8 1.6 1.4 1.2 1.8.6.4.2 Fibroblast D Hertz Chen Chen.5 frictionless.2.4.6.8 1 δr/h 2 K (kpa) 2 1.8 1.6 1.4 1.2 1.8.6.4.2.2.4.6.8 1 δr/h 2 Hertz Chen Chen.5 frictionless Effective in poorly adhered regions of the cell
Review of models Useful models and additional information from each Hertz model elastic constant Rigidly adhered sample model elastic constant Poisson ratio Frictionless substrate model elastic constant adherence information
Extensions for viscoelastic information Force on bead some function of the indentation, radius of the probe and total height of the sample * R f bead = Kg( x) + K1 2 h dg ~ δ dx δr x = 2 h Expansion in the indentation Small oscillations on δ
Viscoelastic data from thin areas 3 2 K* (kpa) 1-1 -2-3.2.4.6.8.1 Hertz Chen.5 Chen frictionless δr/h 2 Constants are produced by one of the models in the thin region K' K"
Frequency dependent viscoelastic data K* (kpa) Polyacrylamide frequency dependent data frequency (Hz) K* (kpa) 1 1.1 3 µ m K K Fibroblast C point 1 Fibroblast E.1 1 1 1 frequency (Hz) Cell frequency dependent data
Summary and conclusions Modified probe shape provides well-defined contact area Quantitative viscoelastic properties for gels and thick regions of cells determined using an extended Hertz model Viscoelastic constants found for thin areas of the fibroblast Future directions Modified cell and normal cell comparisons Active response