Atomic force microscopy

Invented by Binning, Gerber and Quate in 1985, the atomic force microscope allows the characterization of various specimen including in vivo samples.^[109]

As it provides a high resolution and allows to measure small forces relevant for biological applications it is widely used for the analysis of biophysical processes such as ligand-receptor interactions and protein unfolding.^{[110, 111]}

Atomic force microscopes measure the deflection of a cantilever interacting with a sample. Light of a diode is focused on the cantilever and reflected onto a four quadrant photo diode. Forces between sample and tip deflect the cantilever and changes in the sub-nanometer range are detectable. When approaching the sample, attractive van-der-Waals interactions between cantilever and sample occur with decreasing distance. When further decreasing the distance, repulsive interactions rapidly increase the force affecting the cantilever. As a first approximation, the interactions between cantilever and sample can be described by a Lennard-Jones potential (equation 3.49).

V = 4

σ_L−J

h 12

−σ_L−J

h 6

(3.49) The potential V is a function of the distance of the cantilever to the sample h, the magnitude of the potential well and the distance σ_L−J at which repulsive and attractive interactions compensate each other. The different operation modes of in atomic force microscopy (AFM) require different interaction potentials and therewith different distances between cantilever and sample (Figure 3.13). The contact mode (C) requires a permanent contact of the tip to the sample applying a constant force using a feedback loop. The intermediate-contact mode (IC) is characterized by only short contacts between cantilever and sample and allows sensitive probing. The non-contact mode (NC) can provide a very high resolution but is only suitable for very flat samples as the cantilever deflection is measured at a constant distance.

Figure 3.13: Schematic illustration of the interaction potential between cantilever and sample and the regime of the different operation modes of an atomic force microscope. C:

contact mode, IC: intermediate-contact mode, NC: non-contact mode.

In force spectroscopy the deflection of the cantilever is measured as a function of the cantilever movement. Force-distance measurements allow to investigate processes at a single molecule level such as protein unfolding and rupture of covalent bonds.^[112] Atomic force microscopy was also used for local probing and ensemble measurements to analyze the mechanics of cellular and artificial membranes.[35, 113–115]Information about the mechanical properties of the sample can be extracted from force-distance curves. A typical force-distance curve is shown in Figure 3.14. The cantilever approaches the substrate (Figure 3.14 A)

and at a certain distance attractive electrostatic or van-der-Waals interactions can lead to a snap on of the cantilever which is then in contact with the specimen (Figure 3.14 B). By moving the cantilever further towards the sample, a force is applied and the response of the system is measured (Figure 3.14 C). After reaching a predefined force, the cantilever is retracted. Adhesion can result in a bending of the cantilever maintaining contact to the surface. When being further retracted the retraction force exceeds the adhesion force and the cantilever looses contact to the surface (snap off, Figure 3.14 D). The cantilever is then further retracted until the predefined tip-sample separation is reached (Figure 3.14 E). This procedure can be repeated to acquire a force-volume image containing mechanical information of each probed location.

Figure 3.14:Schematic illustration of the measurement of a force-distance curve.

As the atomic force microscope measures the deflection z_c of a cantilever, a conversion is required to obtain the applied force. According to Hooke’s law the forceFof an ideal, linear elastic spring is given by:

F=−k_c·z_c, (3.50)

withk_c being the spring constant andz_cthe deflection of the spring. The deflection is then converted into the tip-sample separationhby subtracting the cantilever deflectionz_cand sample indentationz_sfrom the piezo movementz_p:

h=z_p−z_c−z_s. (3.51)

For incompressible materials isz_sz_cand can be neglected. The spring constant of the cantilever was determined by the method of Hutter and Bechhofer measuring the intensity of the thermal noise of the cantilever.^[116] Assuming a harmonian oscillator, introducing two correction parameters accounting for deviations from the ideal behavior allows the calculation of the thermal noise as a function of the mean square displacement of the cantilever

z_c² to:

k_c= βkBT χ²

z_c². (3.52)

Hereby χ corrects for the cantilever not oscillating entirely free but being endloaded.^[117] As only a single mode of a real cantilever is measured for the determination of the thermal noise, the parameterβwas introduced as a second correction factor.^[118]

When investigating the mechanical properties of pore-spanning membranes, contributions from membrane bending, stretching and the lateral membrane tension have to be considered. However, for small indentations of circular pore-spanning membranes it was shown that the forceF increases linearly as a function of the indentation depth h and that membrane bending and stretching can be neglected.^{[113, 119]}The lateral membrane tensionσ is governed by the adhesion of the lipids to the pore rim and can then be extracted from:

h= F

withr_tipandr_porebeing the radius of the cantilever tip and the pore radius.^[120]

Experimental procedure

Pore-spanning membranes on porous glass substrates were generated as de-scribed in Section 3.5.2. Fluorescence microscopy was used for localization of the membrane-covered pores. Force volume images of 16×16 to 64×64 force-distance curves were measured by a Nanowizard 2 (JPK Instruments AG, Berlin, Germany) using a cantilever with a spring constant ofk_c= 0.01 N/m (MLCT, Bruker AFM Probes, Calle Tecate, USA). Areas ranging from 8×8 to 30×30 µm were probed with velocities ranging from 2 to 8 µm/s each probing a distance of 1 to

5 µm. Indentation was stopped after applying a force of 200 to 900 nN/m. The linear regime of the force-distance curves was fitted for small indentation depths (<300 nm) and the lateral membrane tension was then extracted by numerically solving equation 3.53 for the measured dependence of force to the indentation depth.