Atomic Force Microscopy: Imaging and Rheology of Living Cells



Fig. 15.1
Principles of AFM. The force between the cantilever tip and the sample is detected by deflection of the cantilever and an optical lever coupled with a position-sensitive photodetector (PSD). The PSD signal is used to regulate the force



In contact mode AFM operation, the force between the tip and the surface is kept constant by maintaining a static deflection of the cantilever (Fig. 15.2a). Thus, the z-position corresponds to the topography of the cell sample. In the case of soft samples such as cells, the contact force may cause local deformation (Figs. 15.2c and 15.3a). To minimize deformation, the contact force should be <1 nN [4], and a low-stiffness (∼0.1 N/m) cantilever should be used.

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Fig. 15.2
Imaging modes of AFM: (a) contact mode and (b) AM mode. The sample is a hard elastic material with a soft viscoelastic region. (c) Topography and (d) deflection images in contact mode. (e) Topography and (f) phase-shift images in AM mode


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Fig. 15.3
Contact mode height (a) and deflection (b) images of a living cell showing cytoskeletal structures (Reprinted with permission from [5])

The deflection of the cantilever would be unchanged if the feedback was perfectly regulated. During imaging, however, the deflection signal changes slightly with abrupt changes in the cell surface morphology. The corresponding error signal can be used to construct images in which changes in topography are sharply identified (Fig. 15.2d). In Fig. 15.3, a deflection error image reveals underlying cytoskeletal structures more clearly than the corresponding topography (z-piezo) image [5].

In contact mode, the scanning tip may generate lateral forces that could drag the cell surface. To reduce lateral forces, amplitude modulation (AM) mode or “tapping” mode AFM was developed. For intermittent contact, the cantilever is continuously oscillated in the z-direction (Fig. 15.2d). The small amplitude of the oscillation near the cantilever’s fundamental resonance frequency f 0 is used as the feedback signal for maintaining intermittent contact during topographic image acquisition. As in contact mode, the intermittent contact force and lateral forces could also deform a soft sample surface (Fig. 15.2e). The phase difference between the oscillation drive signal and the oscillating cantilever is sensitive to the change in the topography and reflects energy dissipation in the sample (Fig. 15.2f).

AM mode is used for high-speed AFM (HS-AFM) [6, 7], at video rates for high-resolution observation of dynamic processes. Recently, HS-AFM combined with a wide-area scanner was used to acquire video images of endocytosis on a living cell surface (Fig. 15.4) [8].

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Fig. 15.4
HS-AFM images showing the dynamics of endocytosis (dotted circles) in HeLa cells. Scan range and imaging rate are 5 × 5 μm and 5 s/frame over 200 × 200 pixels (Reprinted with permission from [8])

When the cantilever is oscillated at frequency f 0, the motion of the cantilever also includes higher harmonic modes (2f 0, 3f 0, etc.) because of nonlinear interactions between the AFM tip and the sample [9]. The interaction of the harmonic components with the cell surface can be employed to characterize local stiffness, stiffness gradients, and viscoelastic dissipation at high resolution [10].

In a technique called scanning near-field ultrasonic holography (SNFUH), the sample and the cantilever are simultaneously excited at different ultrasonic frequencies (MHz) f S and f C, respectively (Fig. 15.5). The ultrasonic vibration of the cantilever allows us to generate images of subsurface structures in cells because the mechanical waves propagate through the cell and are perturbed by internal structures [11, 12]. The amplitude and the phase shift reflect the local mechanical properties of the subsurface structures. In Fig. 15.5, SNFUH images of nanoparticles embedded in red blood cells are clearly shown [11, 12]. The imaging depth strongly depends on the material properties of the sample [13], but the mechanism of SNFUH imaging is not fully understood.

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Fig. 15.5
(Left) Schematic of scanning near-field ultrasonic holography (SNFUH). The cantilever and the sample are vibrated at fc and fs ultrasonic frequencies, respectively. The AFM cantilever is locked in at the frequency difference |fc-fs|, providing information about local intracellular nano-mechanical structures. (Right) Images of nanoparticles in red blood cells (Reprinted with permission from [11])



15.1.2 AFM Probe


The shape of the AFM probe in the region that interacts directly with the sample surface affects imaging resolution as well as mechanical properties. Sharp tips required for a high-resolution imaging are more likely to damage soft cells with fragile structures. Moreover, the exact profile of a sharp tip is hard to determine precisely, precluding quantitative mechanical measurements. For those reasons, a silica or polystyrene colloidal bead [14] with a well-defined spherical shape is widely used for force measurements on cells. It can be attached to a cantilever in various ways (Fig. 15.6). When the bead of the colloidal probe cantilever contacts the cell surface in a liquid environment, the liquid between the cantilever and the cell surface is highly confined and squeezed. This enhances viscous damping of the cantilever and affects the rheological observations. Because the squeezing effect can be reduced by increasing the distance between the cantilever and the surface [17], attaching the bead to the apex of the probe (Fig. 15.6c, d) is suitable for single-cell rheology.

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Fig. 15.6
Different mounting geometries for a colloidal bead probe on a cantilever. (a) Attached to a tip-less cantilever. (b) Attached beside a sharp tip. (c) Attached at the apex of a sharp tip. Reprinted with permission from [15]. (d) Electron microscope image of a probe attached as in (c) (Reprinted with permission from [16])

Adhesion between the colloidal bead and the cell surface should be minimized for single-cell rheology because the Hertz model [2, 3, 18, 19], which is the standard model for estimating contact mechanics, assumes that there is no adhesion between contacting materials. Hydrophobic perfluorodecyltrichlorosilane-coated colloidal beads [20] work well to prevent adhesion to the cell surface during force measurements.


15.1.3 Force Measurements


Elastic properties, i.e., reversible deformation of cells, are estimated from the relationship between the loading force F and the normal displacement Z of the AFM probe as it deforms the cell surface (Fig. 15.7a). The indentation δ in the cell surface is determined by subtracting the cantilever deflection d from the displacement Z. The loading force is estimated from Hook’s law (F = kd), where the spring constant k of the cantilever can be determined by thermal fluctuations [21].

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Fig. 15.7
(a) Force curve measurements. (i) The cantilever is separated from the sample surface and no deflection occurs. (ii) The AFM cantilever probe contacts the cell surface at the position Z = 0. (iii) The AFM probe indents the cell sample a Z from the position in (ii). The deflection of the cantilever is d, while the indentation is δ, where δ = Z – d (Reprinted with permission from [15]). (b) Characteristic features of force–distance curves measured in viscoelastic materials

F(δ) depends on the shape of the AFM probe. According to the Hertz model for a spherical probe with radius R [2, 3, 18, 19], F is given by


$$ F=\frac{4}{3}\frac{E{R}^{1/2}}{1-{\nu}^2}{\delta}^{3/2}, $$

(15.1)
where ν is Poisson’s ratio, which is assumed to be 0.3–0.5 for cells [2, 3], and E is Young’s modulus. E has been measured in various animal cells [2, 3], and its spatial heterogeneity has been resolved [5, 22, 23]. In one case, local values of E were attributed to actin filaments rather than microtubules and intermediate filaments [5].

Cells are not completely elastic, but behave more like a compliant viscoelastic material. Therefore, because of energy dissipation in the cells, force–distance curves tend to exhibit hysteresis between approach and retraction (Fig. 15.7b) [24]. Thus, E estimated by force–distance curve measurements may depend on the speed of approach or retraction. Because it increases with increasing speed (Fig. 15.7b), E from force curve measurements is an “apparent” Young’s modulus. Thus, frequency and/or time domain AFM measurements are indispensable for quantifying intrinsic mechanical properties of cells.


15.1.4 Frequency Domain AFM


In the force modulation mode [25–27], the dynamic response due to an external periodic strain is measured (Fig. 15.8a). The strain is due to a cantilever that is sinusoidally oscillated with fixed amplitude (usually 10–50 nm) at several frequencies during indentation. The amplitude and phase shift of the cantilever displacement are measured with a lock-in amplifier.

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Fig. 15.8
Schematics of AFM rheology measurements: (a) force modulation mode, (b) stress relaxation, and (c) creep relaxation (Reprinted with permission from [15])

Using the Hertz model from Eq. 15.1, the complex loading force F* with a small complex amplitude indentation oscillation $$ {\delta}_1^{\ast } $$ around an operating indentation δ 0 is approximately expressed [25–30] by a first-order Taylor expansion:


$$ {F}^{\ast }=\frac{4{R}^{1/2}}{3\left(1-{\nu}^2\right)}\left({E}_0{\delta}_0^{3/2}+\frac{3}{2}{E}_1^{\ast }{\delta}_0^{1/2}{\delta}_1^{\ast}\right), $$

(15.2)
where E 0 is Young’s modulus at zero frequency and $$ {E}_1^{\ast } $$ is the frequency-dependent Young’s modulus, given by $$ 2\left(1+\nu \right){G}^{*} $$ [19]. Since the oscillating probe experiences hydrodynamic drag forces $$ {F}_{\mathrm{d}}^{\ast } $$ [17], G * is given by


$$ {G}^{\ast }=G^{\prime }+iG^{{\prime\prime} }=\frac{1-\nu }{4{\left(R{\delta}_0\right)}^{1/2}}\left[\frac{F_1^{\ast }}{\delta_1^{\ast }}-ib(0)f\right], $$

(15.3)
where $$ {F}_1^{\ast }=2{\left(R{\delta}_0\right)}^{1/2}{E}_1^{\ast }{\delta}_1^{\ast }/\left(1-{\nu}^2\right) $$ and b is the drag factor [17]. F d* at a separation distance h between the sample surface and the probe with $$ {\delta}_1^{\ast } $$ is defined as F d*/δ 1* = ib(h)f. The value b(0) can be determined by the extrapolation of b(h) measured at an oscillating frequency [17]. The phase shift and amplitude of the AFM instrument at different frequencies can be calibrated with a stiff cantilever in contact with a clean glass substrate in air [17, 28]. By using this method, G * for several cell types has been measured in detail.


15.1.5 Time Domain AFM


In AFM stress relaxation [16, 20, 24, 31–33], Z is kept constant at the position where the initial force is applied, and F is measured as a function of time t (Fig. 15.8b). The cantilever defection d and the corresponding indentation δ both change during stress relaxation. This situation is not like conventional stress relaxation measurements where the strain is kept a constant value and the stress is measured as a function of t. In cell experiments, the change in d is typically about 1 % (10 nm) of that for δ (1 μm). Therefore, it is assumed that d is approximately constant relative to δ. According to the Hertz model, in which the contact radius a is dependent only on δ with a fixed probe radius R, the average stress is F/(πa 2).

Since F, δ, and E are time dependent, and based on the Hertz model of Eq. 15.1, F is given by


$$ F(t)=\frac{4{R}^{1/2}E(t)}{3\left(1-{\nu}^2\right)}{\delta}^{3/2}(t), $$

(15.4)
where E(t) is the relaxation modulus at t.

In the case of stress relaxation with a constant indentation δ 0, F(t) is proportional to E(t)H(t) by Eq. 15.4, using the Heaviside step function H(t) [32, 34]. As shown below, G(f) for cells follows a single power law of frequency f  α at low frequencies. Since the relaxation modulus in the Laplace domain E(f) is proportional to f  α from the relation $$ E(f)=2\left(1+\nu \right)G(f) $$ (in the case where ν is independent of t) [19], F(f) is proportional to $$ {f}^{\alpha -1} $$ . Therefore, the inverse Laplace transform of F(f) yields the functional form of the loading force for stress relaxation: $$ F(t)\propto {t}^{-\alpha } $$ . Since cells are generally soft, F decreases significantly over long time periods and may approach zero. Therefore, stress relaxation, when compared to creep relaxation, is insensitive for long-term measurements of cell rheology. Furthermore, large initial loading forces required to enhance the signal-to-noise ratio for stress relaxation curves over long times cause large deformations in cells. This may also induce the cells to actively escape from the stress.

Creep relaxation of single cells can also be performed by AFM [20, 35]. In this case, the probe contacts the cell surface at a constant F under feedback, and Z is monitored as a function of t (Fig. 15.8c). Because the contact radius a changes during creep relaxation, the stress applied to the cell is not constant and decreases with t. The relationship between a(t) and the creep compliance J(t) becomes $$ {a}^3(t)\propto J(t) $$ [18]. If G(f) follows a single power law of the form f  α , J(t) is proportional to t  α . Moreover, by using $$ {a}^2(t)=R\delta $$ [18, 19], the indentation for creep relaxation is given by $$ \delta (t)\propto {t}^{2\alpha /3} $$ . For soft cells during creep measurements, δ significantly increases over long time periods, allowing us to easily monitor the relaxation. Conversely, the observed relaxation curve may reflect the highly heterogeneous cell structure with depth. Fluctuation and active movement of the cell also occur because of large δ.



15.2 Single-Cell Rheology



15.2.1 High-Throughput Measurements


Among single cells of the same source and type, rheological properties exhibit spatial, temporal, and intrinsic variations. High-throughput techniques, based on magnetic or optical trapping with micron-sized beads [36–42], micro-fluidic systems [43, 44], and AFM with micro-fabricated substrates [45–48], have been developed to characterize large numbers of cells.

Magnetic twisting cytometry (MTC) is one of the most common methods for investigating rheology statistics of adherent cells. A micron-sized magnetic bead is attached to a cell surface via binding proteins, and the cell modulus is estimated from the displacement of the beads under a periodic, external magnetic force (Fig. 15.9a). Lateral [36–42] or vertical [49] displacement of a large number of microbeads can be simultaneously monitored with optical microscopy. The disadvantages of MTC are that the contact geometry and the degree of binding between the microbeads and the cell surface are not well known, and the positions on the cell surfaces are not precisely controlled. Thus, it is difficult to assess cell-to-cell variations from the experimental data. Furthermore, focal adhesion complexes form at the microbead binding sites; thus, local reorganization of the cytoskeleton may alter the rheology [28].

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Fig. 15.9
Depiction of single-cell rheology for a large number of cells: magnetic twisting cytometry (MTC), micro-fluidic optical stretcher, and AFM on a micro-fabricated substrate

Micro-fluidic techniques can provide very high-throughput measurements of single-cell rheology. Suspended cells flowing in micro-channels can be deformed by optical pressure (Fig. 15.9b) [43] or hydrodynamic forces [44], and the deformability of whole cells floating in a micro-fluidic chamber is estimated. One disadvantage is that adherent cells have to be detached from their substrate, which may perturb intracellular structures that, in turn, affect cell mechanics.

With micro-fabricated substrates, one can use AFM to characterize the rheology of a large number of single cells rapidly (Fig. 15.9c). It has the advantage of measuring mechanical properties of single adherent cells at any region on the surface without cell surface modification [2, 3]. Thus, AFM is a less-invasive technique for measuring intrinsic mechanical features of single cells.


15.2.2 Power-Law Rheology Model


Because cells have internal organelles, their spatial–temporal rheological properties will vary from cell to cell. In spite of the structural complexities, the rheology of cells has been widely explained in terms of linear viscoelastic [34] or structural dampening models [36, 37, 50, 51].

In linear viscoelastic models, the cell is simulated with linear springs and linear viscous dashpots, and inertia effects are neglected. Therefore, creep and stress relaxations are sums of single-exponential functions in the time domain [34].

Power-law behaviors as a function of f have been observed for cell rheology with MTC and AFM. G′ exhibits one single-power-law behavior in the range of 100–102 Hz, whereas, for other frequency ranges, other power-law models have been proposed: single (Fig. 15.10a) [36, 37] and multiple (Fig. 15.10b) [38, 52].

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Fig. 15.10
Power-law models of G′ as a function of f. (a) Single-power-law model for single cells under different conditions and (b) multiple-power-law model

Fabry et al. reported that G′ followed a single-power-law function over 10−2–103 Hz, where the exponent α depended on the cytoskeletal architecture, regardless of modifications by chemical drugs, and appeared to cross at G′ = g 0 at a high frequency f = Φ0 [36, 37] (Fig. 15.10a). In this model, the complex shear modulus G * is given by the structural damping model [36, 37, 50, 51]:


$$ {G}^{\ast }={G}_0\left(1+i\eta \right)\;{\left(\frac{f}{f_0}\right)}^{\alpha}\Gamma \left(1-\alpha \right) \cos \left(\pi \alpha /2\right)+i\mu f, $$

(15.5)
where η is the hysteresivity, which is expressed by tan(πα/2), G 0 is a modulus scale factor at a frequency scale factor of f 0, and Γ denotes the gamma function. The α-value was 0.1–0.4 depending on cell type, where α = 0 is solid-like and α = 1 is fluidlike. The Newtonian viscous term μf is small, except at high frequencies. Single power laws have been discussed in detail in terms of soft glassy rheology (SGR) [50, 53, 54].

In contrast, two power-law exponents in the frequency domain have been observed; they cross over at around 100 Hz or 102 Hz (Fig. 15.10b). The exponent for the lower frequencies was 0.5, because of noncovalent protein–protein bond rupture during near-equilibrium loading [52]. Meanwhile, the exponent for the higher frequencies was about 0.75, because of entropic fluctuations of semi-flexible filaments and soft-glass-like dynamics [38].

Multiple-power-law cell rheology has also been observed in time domain experiments. Overby et al. reported that α = 0.18 for pulling a single cell in a creep experiment over several seconds and α = 0.5 for longer time scales [55]. Using magnetic microbeads, Stamenovic et al. reported that in creep experiments of single cells over a wide range of time scales, there were two power-law regimes with an intervening plateau over 10 s [56]. Desprat et al. employed a uniaxial stretching rheometer to observe that the creep function of pulling a whole cell follows a power-law exponent of 0.24 for periods <200 s, while for periods >200 s, the exponent increased to ≈ 0.5 [57]. These studies commonly showed that in the intermediate frequency range of 100–102 Hz, the single power law is an intrinsic feature of cell mechanics and is valid at size scales from a few tens of nanometers to the entire cell. However, it is not elucidated whether passive and active cell behaviors are involved in the mechanics over longer time scales.


15.2.3 Ensemble Averaged Single-Cell Rheology



15.2.3.1 Frequency Domain AFM


Mahaffy et al. used force modulation mode with a colloidal probe on an AFM cantilever to measure the mechanical properties of cells as a function of indentation depth δ and estimated the viscoelastic parameters and Poisson’s ratio quantitatively [29, 30]. Using the AFM model in Eq. 15.3, Alcaraz et al. revealed a characteristic feature of averaged G′ and G″ for single cells as a function of f [28]. The typical behavior of G* is shown in Fig. 15.11. G′ increased linearly in a log–log scale, exhibiting a weak power-law dependence on oscillation frequency. Conversely, G″ displayed a similar frequency dependence at values <10 Hz, and the frequency dependence was more pronounced at higher frequencies. The results fit the structural damping model shown in Eq. 15.5. This power-law frequency dependence has been observed with AFM in different cell types [45, 58, 59]. However, the absolute values of G′ and G″ were different between cell types.

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Fig. 15.11
Storage modulus G′ (left) and loss modulus G″ (right) of adherent mouse fibroblast cells


15.2.3.2 Time Domain AFM


Darling et al. measured the stress relaxation of single cells for ≈ 60 s with a colloidal probe cantilever [32, 33]. They observed that the stress relaxation was a single-exponential function, obeying a linear viscoelastic model. Moreno-Flores et al. reported that heterogeneities in single-cell rheology could be imaged with stress relaxation AFM [60].

Wu et al. investigated the relationship between viscoelastic properties and the cytoskeletal architecture of cells by using creep relaxation AFM. They demonstrated that creep relaxation for 60 s could be fit with a standard linear solid model consisting of two springs and one dashpot [35]. The creep relaxation of cells treated with various chemical drugs affecting the cytoskeleton was examined. Cytochalasin D (cytoD), which depolymerizes actin filaments, reduced both elasticity and viscosity, whereas nocodazole or colcemid, which depolymerizes microtubules, exhibited a marked increase in elasticity and a slight increase in viscosity. Thus, changes in cytoskeletal structure can be detected by using AFM in the time domain.

The results from stress and creep relaxation experiments are inconsistent with those obtained with force modulation mode, where power-law behavior in the frequency domain has been widely observed. In this context, the relaxation behavior of individual cells placed and cultured in microarray wells was characterized with AFM by averaging several relaxation curves (Fig. 15.12) [20]. Tails in both stress and creep relaxation curves at long times follow single power laws over 60 s. Also, α = 0.1–0.4, which varies between cells and has an average value in good agreement with that estimated from the force modulation mode [45].

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Fig. 15.12
Linear plots of the averaged AFM stress (a) and creep (b) relaxation curves for NIH3T3 cells on a microarray. The insets show the corresponding relaxations on logarithmic axes. Solid lines represent the fit of the power-law functions described in the main text (Reprinted with permission from [20])


15.2.4 Cell-to-Cell Variability



15.2.4.1 Statistics of Single-Cell Rheology


The statistics of single-cell rheology is crucial to reveal a universal behavior of single cells and to conduct single-cell diagnostics automatically. Hoffman et al. found that the distribution of G * measured by MTC followed a lognormal and that the amplitude of the rocking motion or mean-square displacement of beads in cells varied dramatically for different methods [40]. Moreover, Massiera et al. showed that by using MTC and laser tracking micro-rheology, the magnitude of G * at a low frequency exhibited a lognormal distribution, whereas the single-power-law exponent was a normal Gaussian [41]. Using optical trapping and uniaxial stretching of single cells, Balland et al. also showed that α is distributed normally over a cell population and that the prefactors of G* and J follow a lognormal distribution [42].

The statistical properties of single-cell rheology were investigated with AFM on a microarray of wells [45] (Fig. 15.13). Experimental variation is minimized because the cell shape is highly controlled in each well and the measurement position of cell is well defined. Force measurements are automatically performed at the centers of each well without confirming the cell positions. Figure 15.14 shows the distributions of single cells cultured in the wells [45]. We observed four characteristic features from the number distributions of mouse fibroblast cells [47]. First, $$ {G}^{\ast } $$ consistently exhibited a lognormal distribution. Second, the geometric mean of $$ {G}^{\ast } $$ ( $$ \overline{G}^{\prime } $$ and $$ \overline{G}^{{\prime\prime} } $$ ) shifted to higher values with increasing f. Third, the distribution of G′ became narrower with f, and the distributions of G″ were narrower than those of G′. Fourth, the distribution of $$ {G}^{\ast } $$ for the cytoD-treated cells was narrower than that of the untreated cells.

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Fig. 15.13
Schematic of AFM for microarrays of cells. Force modulation mode measurements are automatically examined at the centers of the wells after a specific cell is chosen with optical microscopy (Reprinted with permission from [47])


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Fig. 15.14
Distributions of the storage G′ (left) and loss G″ (right) moduli of untreated fibroblast cells in microarray wells at different frequencies: (a) 5, (b) 100, and (c) 200 Hz. The solid line represents the fitted result using a lognormal distribution function

$$ {\overline{G}}^{\prime } $$ and $$ {\overline{G}}^{{\prime\prime} } $$ increased with f and closely followed the structural damping in Eq. 15.5 (Fig. 15.15a, b). The depolymerization of actin filaments resulted in a decrease in $$ {\overline{G}}_0 $$ and an increase in the arithmetic mean of α, 〈α〉, which were similar to those characteristics measured with MTC [36, 37, 61, 62]. The standard deviation of the complex modulus $$ {\sigma}_{\ln {G}^{\ast }} $$ was reduced in the treated cells (Fig. 15.15c, d), indicating a strong coupling between cell-to-cell variation and the cytoskeleton, where σ X represents the standard deviation of X. The $$ {\sigma}_{\ln {G}^{\ast }} $$ in the untreated and treated cells crossed at the point where the extrapolated lines of $$ {\overline{G}}^{\prime } $$ for the treated and untreated cells intersect; this was defined as $$ {\overline{G}}^{\prime }={\overline{g}}_0 $$ at $$ f={\overline{\varPhi}}_0 $$ (Fig. 15.15) [36, 37].
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Mar 22, 2018 | Posted by in BIOCHEMISTRY | Comments Off on Atomic Force Microscopy: Imaging and Rheology of Living Cells

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