AFM-based microrheology

Averaging of force-indentation curves of one force map was performed by interpolating the indentation depth values of each preselected force-distance curve in the contact region by searching for the nearest indentation depth value at a given force (for pre-selection criteria see chapter 3.9.6). An example of an averaged force-indentation curve is shown in Figure 3.9. The resulting curves were subject to fitting with the parameters of the liquid droplet model as detailed previously. Experimental details can be found in chapter 3.9.5 and 3.9.6.

3.9.4 AFM-based microrheology

Although force-indentation curves can be well described by the liquid droplet model, a hysteresis between indentation of the cantilever tip into a cell and its retraction indicates, that cells cannot be regarded as purely elastic solids or shells as energy is dissipated during indentation. Instead, cells are viscoelastic bodies, which exhibit fascinating rheological features. A possibility to capture cellular rheology using the AFM has been developed by Shroff et al. and was refined by Alcaraz and coworkers.

(Alcaraz et al., 2003; Shroff et al., 1995) A detailed description of the method can be found in the publication of Alcaraz et al.. Briefly, upon indentation of the cantilever tip into the sample, it is excited to oscillate with small amplitudes and the force response is measured (see Figure 3.10).

Figure 3.10: A Schematic drawing of the experiment: The cantilever oscillates around the indentation depth δ0 with an amplitude δ at the frequency ω. B Time course of force during the

measurement of a force-distance curve. When the cantilever gets into contact with the sample, the force increases rapidly until the present trigger point is reached. During dwell, the cantilever

is excited to sinusoidal oscillations. Afterwards, the cantilever is retracted and the procedure repeated at a different position. C Indentation oscillation δ(ω) with frequencies from 5 to 200

Hz around the indentation depth δ0 and corresponding force signal F(ω ) after detrending.

A quantitative description can be achieved employing contact mechanics. Starting with Hertzian contact mechanical model for a four-sided pyramidal indenter (Bilodeau, 1992) one obtains, after linearization and by use of the relation G=2E/(1+ν), the frequency-dependent complex shear modulus G*:(Alcaraz et al., 2003)

0 0 be described by the ratio of the amplitudes of the force oscillation (AF) and indentation depth (Aδ) and the difference of phases (ϕF-ϕδ). The complex shear modulus consists of a real part G´ called storage modulus, which accounts for the energy stored in the system and an imaginary part G´´ called loss modulus, which accounts for energy dissipated in the sample. The ratio between loss modulus and storage modulus is called loss tangent η and is given by the tangent of the phase shift (ϕF-ϕδ) between the two sinusoidal signals of F(ω) and δ(ω):

''/ ' tan( _F )

G G δ

η= = ϕ −ϕ . (3.12)

The loss tangent is a model-independent parameter, which does not rely on geometrical factors. For an elastic solid η is zero, while it approaches infinity for a viscous fluid.

Furthermore, the force response of the cantilever during oscillation in an aqueous medium is influenced by hydrodynamics. To correct for these influences, a method introduced by Alcaraz et al. was used.(Alcaraz et al., 2002) Thereby, the force response of the cantilever at different frequencies and distances from a glass cover slide is measured (Figure 3.11 A and B).

Figure. 3.11: A Scheme of the correction of force data for hydrodynamic drag. The triangular cantilever is kept at a height h above the surface and oscillates with a z-displacement of z(ω_{) at}

its basis. The force response F(ω) is determined over the deflection of the cantilever and its spring constant. B Piezo movement z(ω) and the measured force signal F(ω) as a function of time. Dots represent the measured data; the dashed line shows the fit of both signals using the

sum of a sine and a cosine function with a pre-set frequency.

Before calculation of the drag coefficient, the phase shift between excitation and force response was set to 90°, as the experimentally obtained phase shifts differed from the theoretical 90° in a Newtonian viscous fluid. This was achieved by fitting the experimental mean deviation ϕlag from 90° using a linear fit (see Figure 3.12).

Figure 3.12: Example of deviation from 90° phase shift between z(ω_{) and F(}ω₎ (squares) as a function of the oscillation frequency measured in viscous medium. Data were fitted using a

linear fit applied to all data above an oscillation frequency of 10 Hz.

The results of the fit were then subtracted from the phase shift between excitation and force response (ϕF-ϕz) for hydrodynamic correction as well as from (ϕF-ϕδ) for calculation of G*(ω_).

After correction of the phase shift, the complex transfer function

* ( )

( )^d ( )

d F

H z x

ω

ω ω

= − ^(3.13)

is determined, where z(ω) denotes the oscillation of the piezo, Fd(ω) is the drag force and x(ω) is the cantilever deflection, as a function of the frequency at a distinct distance h from the glass cover slide. A linear increase of the imaginary part of the transfer function Hd´´ is obtained, while the real part shows a constant value around 0 mN/m (see Figure 3.13 A). Thereby, the slope is dependent on the drag coefficient b(h).

Linear fitting of Hd´´ provides the drag coefficient b(h) as the slope m of the fit is given by m=2πb(h).(Alcaraz et al., 2002)

Figure 3.13: A Real part of the transfer function HD´ (squares) and imaginary part HD´´

(triangles) as a function of the oscillation frequency f at a fixed tip surface distance h. The dashed line shows linear fit of HD´´ with the slope 2πb(h). B Drag coefficient b(h) (squares) as a

function of the tip-sample separation h. The data were fitted using a scaled spherical model described by Alcaraz and coworkers (eq. 3.14, dashed line). Extrapolation of the fit to h = 0 nm

delivers the drag coefficient b(h0) used for the hydrodynamic drag correction.

The drag coefficient is determined at tip-sample separations ranging from 200 nm to 3500 nm. The dependency of the drag coefficient b(h) from the tip-sample separation is shown in Figure 3.13 B. At low tip-substrate distances, a higher drag coefficient is observed. The drag coefficient as a function of the tip-sample separation is fitted with the scaled spherical model of the cantilever to obtain the drag coefficient at an extrapolated tip-sample separation of 0 nm (b(h0)):(Alcaraz et al., 2002)

6 2

( ) ^eff

eff

b h a

h h

=

πη

+ ^{, (3.14)}

where η denotes the viscosity of the medium and aeff and heff are fitting parameters describing the cantilever geometry. Finally, b(h0) is subtracted from the imaginary part of F(ω_)/δ₍ω) leading to the following expression for the complex shear modulus:

0

AFM-experiments were carried out using a MFP-3D™ (Asylum Research, Santa Barbara, CA, USA) setup equipped with a BioHeater™ mounted on an inverted Olympus IX 51 microscope (Olympus, Tokio, Japan). MLCT cantilevers (C-lever, nominal spring constant 10 pN/nm, length 200 µm, tip height 8 µm, Bruker, Camarillo, CA, USA) with a pyramidal tip were used for imaging and force spectroscopic experiments. Prior to each experiment, the spring constant of the cantilever and the hydrodynamic drag force acting on the cantilever were determined on a flat glass slide.

After calibration of the spring constant and hydrodynamic drag coefficient, the glass slide was replaced by the substrates used for cell culturing. A homemade holder of spring steel fixed the substrates during the measurement inside the BioHeater™. The temperature was set to 37°C throughout the measurement. Before each force spectroscopic measurement, the area of interest was imaged in contact mode.

Force spectroscopy and frequency dependent rheological data were acquired by approaching the cantilever towards the surface with a velocity of 3 µm/s. When the pre-set trigger point was reached, the cantilever movement was stopped for 0.5 s before it was excited to oscillate with frequencies between 5 Hz and 100 Hz (5 Hz to 150 Hz (A549 cells)) at small oscillation amplitudes (Aδ = 20 - 50 nm,peak to peak). After an additional quiescent period of 0.5 sec it was retracted from the surface. Per area of interest, usually 1024 of these force-distance curves where recorded in a 32 point × 32 point grid (force map). Per force map 2 to 15 cells were probed.