Force Spectroscopy Methods to Study the Mechanics of Living Cells

mapping of the topology of biological surfaces, over single cell or molecule applications to quantitative force manipulations in a relevant force regime to measure cell mechanics (nN forces).

Here we use the AFM for two different purposes: (i) for force manipulation on cell monolayers, especially for microrheological measurements (for description see chapter 2.2.2.1.1) and (ii) to map the topography of the apical side of the cell monolayer before starting the measurement.

4.2.7.1. Active Microrheology Measurements by Atomic Force Microscopy 4.2.7.1.1. Theoretical Background

AFM oscillatory experiments to probe cellular viscoelasticity go back to Shroff et al. and were refined by Alcaraz et al.^76,244,275 A detailed description of the method can be found in the work of Alcaraz and coworkers. The most important steps are revised here. In force-indentation experiments with the AFM the contact geometry between the indenting probe and the sample has to be considered in the linear relation between the force and the Young's modulus. Hertzian and related contact mechanics yield:^266,276

𝐹 = 𝐶 ∙ 𝐸 ∙ 𝛿^𝑛 for a pyramidal tip geometry 𝐹 =^{3∙𝐸∙tan(𝜃half)}

4(1−²) ∙ 𝛿² (59)

where, 𝐹 is the force acting on the cantilever, 𝐶 is a pre-factor and 𝑛 an exponent depending on the tip geometry, 𝐸 is the Young modulus, 𝛿 the indentation depth, 𝜃_half denotes the half opening angle of a pyramidal cantilever tip and  the Poission's ratio of the viscoelastic medium. After linearization for small amplitudes after Mahaffy et al., transformation in frequency space and using equation (3), the following expression was obtained:²⁷⁷

𝐺^∗(𝜔) =_3δ ^(1−)

0∙tan (𝜃_half )∙^F(ω)_δ(ω), (60)

where 𝜔 is the angular frequency, 𝛿₀ denotes the indentation depth at which the oscillation was excited and F(ω)/δ(ω) is the term for amplitude damping and phase shift in the Fourier space after equation (10):

𝐹(𝜔)

𝛿(𝜔)=_𝐷(𝜔)^𝐴 ∙ exp (𝑖(𝜑(𝜔))), (61)

with 𝐴 the amplitude of the excitation amplitude, 𝐷(𝜔) the amplitude of the response amplitude and 𝜑(𝜔) the phase shift. Additionally, Alcaraz et al. corrected the response of the viscoelastic medium for hydrodynamic drag force 𝑖𝜔 ∙ 𝑏(ℎ₀) acting on the cantilever from the surrounding medium at the cell surface:²⁷⁸

𝐺^∗(𝜔) = 𝐺^′(𝜔) + 𝑖 ∙ 𝐺^′′(𝜔) =_3∙𝛿 ^1−

0∙tan(𝜃_half)∙ [^𝐹(𝜔)_𝛿(𝜔)− 𝑖𝜔 ∙ 𝑏(ℎ₀)] (62)

Methods and Experimental Procedure

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4.2.7.1.2. Experimental Procedure

Atomic force microscopy (AFM) experiments were carried out using a MFP-3D (Asylum Research, Santa Barbara, CA, USA) set-up equipped with a BioHeater mounted on an inverted Olympus IX 51 microscope (Olympus, Tokyo, Japan). MLCT cantilevers (C-lever, nominal spring constant 10 pN·nm^-1, length 200 mm, tip height 8 mm, Bruker, Camarillo, CA, USA) with a pyramidal tip (𝜃_half ≈ 17.5°) were used for imaging and force spectroscopic experiments. All experiments were performed at 37 °C in buffered cell culture medium. The spring constant and the hydrodynamic drag force acting on the cantilever in different distances from the surface were determined prior to each experiment on a stiff substrate such as glass. For the calibration of the spring constant the thermal noise method was used.^265,279 After the calibration the sample was mounted to the measuring cell and thermally equilibrated. Surface topography images (60 x 60 µm², 256 x 256 px²) were collected using the contact mode (sampling frequency=0.3 Hz). Following the topography images Force Maps of 32 x 32 force distance curves were collected at the same sample position. The force spectroscopy experiments were performed with a cantilever velocity towards the cell surface of 3 µm·s^-1. As soon as the cantilever reached a deflection corresponding to a force of 500 pN the cantilever was held in this indented position for 0.5 s before starting an oscillation protocol with oscillation frequencies ranging from 5 to 100 Hz at small amplitudes (2 ∙ 𝐴 = 40 nm, peak to peak) after another 0.5 s without oscillation the cantilever was retracted from the cell.

Figure 35: Measuring principle of the active microrheological measurement with AFM. A: Schematic representation of the measurement. A Cantilever with a pyramidal tip was indented approximately 𝛿₀ ≈ 1µm into a cell before an oscillation protocol was performed to probe the viscoelastic properties of the cell. B:

Measuring signal. Above: Amplitude diminution and phase shift between excitation and response signal (black and red curve respectively). Below: Deflection signal of the cantilever vs. time. Different regimes can be seen.

I: Approach; II: Indentation; III: Creep; IV: Oscillation; V: Retraction.

4.2.7.1.3. Data Processing

From the overall 1024 force-indentation curves only those obtained from the center of the cell were chosen for further mechanical analysis to avoid artefacts from the cell boundaries. Force-distance curves showing mechanical instabilities or artefacts were disregarded as well.^244,280 Contact point was identified by a sudden change in force. The hydrodynamic coefficient 𝑏(ℎ₀) was extrapolated from a plot of the hydrodynamic drag force versus the distance to the sample²⁷⁸. The shear modulus was evaluated after equation (64) using an in house code 'ShearFM' (version 14.07.2015) written by Jan Rother including subroutines by Ingo Mey and Andreas Janshoff in Matlab (Version 2014a, MathWorks Inc., Massachusetts, United States).

4.2.7.2. AFM Force Spectroscopy on Living Cells by Force Indentation Curves

Force indentation experiments on living cells measured with AFM allow not only to probe the viscoelastic properties of a cell but also to access other mechanical parameters. Mechanical parameters notably the sum of the cortical and the membrane tension 𝑇₀ and the apparent area compressibility 𝜅̃_A in this context were determined by fitting a mechanical model to the contact

59 area of a force distance curve, called the liquid droplet model introduced by Discher and co-workers²⁸¹, which describes the contact of a cone-like indenter with a spherically modeled cap of a cell. A suitable model for confluent monolayers of cells was developed by Pietuch et al.¹⁶⁸ Prior to mechanical analysis the cap of the cell needs to be modeled in order to obtain the restoring force as a function of the indentation depth. In order to determine the shape of the cell caps, contact images were collected at the sample position the measurements was taken. From these measurements the averaged radius of the contact zone and the wetting angle could be obtained from which the cell cap was modeled. It is important to note that the model assumes constant volume and curvature during the indentation process. The model assumes the tension to dominate the force on the cantilever in the first few hundred nanometers of indentation. Upon further indentation the force does no longer scale linearly but quadratic, which is attributed to area dilatation. The isotropic tension 𝑇 is given by the sum of the cortical tension and the membrane tension 𝑇₀ as well as a term describing the area dilatation of the membrane:

𝑇 = 𝑇₀+ 𝜅_A∙^∆𝐴

𝐴₀ (63)

where ∆𝐴 marks the change in surface area and 𝐴₀ the initial area of the computed cell. 𝜅_A denotes the area compressibility modulus which is dominated by the incompressible lipid bilayer rather than the underlying actin mesh. The real surface area in an epithelial cell is much larger than the geometrical area 𝐴₀ due to protrusions and invaginations. Thus, the area compressibility modulus 𝜅_A needs to be corrected for this excess area 𝐴_ex. The apparent area compressibility modulus 𝜅̃_A replaces 𝜅_A.

𝜅̃_A= 𝜅_A∙ ^𝐴0

𝐴₀+𝐴_ex (64)

The overall tension can be itemized in two parts the cortical 𝑇_c and the membrane tension 𝑇_t

𝑇₀= 𝑇_c+ 𝑇_t (65)

The former originates e.g. in myosin II contractility. The membrane tension is dominated by attachment sites (about 80 %) and in plane tension (about 20 %)¹⁴³ and can be determined experimentally from the tether force 𝐹_t (Figure 36), neglecting minor viscous contributions:

𝑇_t= ¹

2∙𝜅B∙ (^𝐹t

2𝜋)² (66)

𝜅_B is the bending module of a lipid bilayer, which we chose to be 𝜅_B = 2.7 ∙ 10⁻¹⁹J.²⁸²

Figure 36: Schematic overview of mechanical parameters, which can be probed by AFM force indentation experiments. Left: Relation of the cellular cortex components and the mechanical parameters probed by AFM indentation measurements. Microrheology measurements probe the viscoelasticity of the cellular cortex hence the F-actin cytoskeleton and its cross-links. Apparent area compressibility modulus dominated by the incompressibility of the lipid bilayer and the tension rising from either active contractility of the actin cytoskeleton or the interconnection of the membrane and the cytoskeleton. Right: Example of force distance curve measured on with an AFM. Regions are dominated by different mechanical properties of the cell. The trace of the contact regime is dominated by the overall tension and the area compressibility. The hysteresis between the trace and the retrace curve is a measure of the viscoelasticity of the cell. Sudden jumps in the retrace following a long plateau in force originate in the pulling of so called tethers.

Results and Discussion

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