Fast, multi-frequency, and quantitative nanomechanical mapping of live cells using the atomic force microscope

A longstanding goal in cellular mechanobiology has been to link dynamic biomolecular processes underpinning disease or morphogenesis to spatio-temporal changes in nanoscale mechanical properties such as viscoelasticity, surface tension, and adhesion. This requires the development of quantitative mechanical microscopy methods with high spatio-temporal resolution within a single cell. The Atomic Force Microscope (AFM) can map the heterogeneous mechanical properties of cells with high spatial resolution, however, the image acquisition time is 1–2 orders of magnitude longer than that required to study dynamic cellular processes. We present a technique that allows commercial AFM systems to map quantitatively the dynamically changing viscoelastic properties of live eukaryotic cells at widely separated frequencies over large areas (several 10’s of microns) with spatial resolution equal to amplitude-modulation (AM-AFM) and with image acquisition times (tens of seconds) approaching those of speckle fluorescence methods. This represents a ~20 fold improvement in nanomechanical imaging throughput compared to AM-AFM and is fully compatible with emerging high speed AFM systems. This method is used to study the spatio-temporal mechanical response of MDA-MB-231 breast carcinoma cells to the inhibition of Syk protein tyrosine kinase giving insight into the signaling pathways by which Syk negatively regulates motility of highly invasive cancer cells.

photothermal (3)(4) actuation. Indirect excitations are acoustic (1) and sample (5) actuation. Below is a brief description of each excitation method:  Acoustic mode is the most widely used method and consists of a piezoelectric transducer or dither piezo attached to the cantilever holder and vibrated at high frequencies to excite the microcantilever. This excitation method not only drives the cantilever but also the chip, holder and surrounding liquid. This generates an effect called forest of peaks that masks out the real microcantilever vibration response. A second, more difficult effect to understand, is that of fluid borne excitation which has a major influence on cantilever dynamics even if the forest of peaks effect is resolved (6).  Sample excitation can be done with a piezo-electric transducer located underneath the sample and vibrated. The microcantilever is then brought into direct physical contact with the vibrating sample exciting the cantilever. An issue is that this excitation excites the sample and the liquid creating unwanted resonances that masks out the real microcantilever vibration response.  Photothermal excitation uses a high powered laser to excite AFM microcantilevers in liquids yielding clean/smooth resonant peaks without spurious peaks and with a wide frequency bandwidth. However, phototermal efficiency is low requiring large amounts of laser power to mechanically actuate the cantilever a few nanometers, resulting in local heat that can potentially damage sensitive samples and accelerate liquid evaporation.  For direct "magnetic" excitation, as the name implies, only the microcantilever is directly excited and a clean vibration response is obtained. There are two ways of doing this: (a) magnetic, which consists of a paramagnetic coating on the microcantilever backbone that will be excited by a solenoid such that applying a alternating current to it generates a magnetic field that interacts and excites the coated microcantilever. (b) iDrive, which is a technology that consists of triangular V-shaped cantilever that is gold coated. An alternating current is applied to the cantilever generating an electric field that will interact with a magnetic field generated by a permanent magnet. This is the so called Lorentz force excitation.
Choosing the optimal excitation method for the microcantilever is important for quantitative spectroscopy measurements. We put to test the 3 excitation schemes while in contact with a sample and determined the ideal for extracting quantitative information. Figure S1 shows the response spectra of a TR400PB cantilever acquired in contact with a glass slide in PBS for the 3 excitation methods. As shown clearly in Fig. S1, the excitation method that yields a smooth transfer function in which changes in amplitude and phase can be easily recorded and used to reconstruct the conservative and dissipative tip-sample interactions is the directly excited iDrive method. Thus, we chose this method over the other 2 conventional actuation methods for our quantitative experiments.

b. Linearized spring and dashpot of a two-element Kelvin-Voigt viscoelasticity model
Because we use special soft cantilever with short tips for cell imaging, the hydrodynamic loading changes both the natural frequency and the damping of the cantilever as it comes closer to the sample surface [7][8]. As a consequence the theories for nanomechanical properties mapping requires two important considerations: (a) they must account for the difference (often significant) due to viscous hydrodynamics in the resonant response of the cantilever when located far and near the sample surface, and (b) the dynamics of the harmonically oscillating cantilevers interacting with the sample surface. From linear vibration theory, for a point mass oscillator of natural frequency n  Using the above it can be easily shown that when the drive frequency is tuned to achieve maximum amplitude, the following hold:  We will use these relationships several times in the following derivation.

Extracting local cellular mechanical properties:
Far from the sample, the natural frequency and Q-factor of the cantilever are ,  far far Q . The drive excitation frequency is tuned to achieve maximum amplitude. Hence, Eq. S2 applies and we have the following relationships for the amplitude and phase far from the sample 11

Dynamics while interacting with the sample
so that the dynamic tip indentation into the sample is: Accordingly, Substituting (S9) and (11) into (S8): 11 0 1 1 2 These equations apply for both tapping mode observables and also for contact mode (with resonant excitation) observables.

c. Viscoelastic Sneddon's contact mechanics model
The fact that at each point on the image we can solve for the local force and damping gradients allows the extraction of unknown constitutive material properties, which are a more fundamental physical properties of cells, by using a tip-sample contact mechanic model of interest [9][10]. The Sneddon's contact mechanics model, which is a modification of the standard Hertz contact mechanics model for axisymmetric tips was used with a linear viscoelastic expansion [11]. The viscoelastic Sneddon's model for a cone-shaped AFM tip used in this study;  0 , sample mean indentation [11][12].
Using the small oscillation assumption which means that the AFM probe oscillation amplitude is much smaller than the average indentation  0 on soft samples, the tip-sample interaction force as a Taylor series expression in     0 : Using Eq. S16 for small oscillations assumptions as above and neglecting the contribution of the higher order terms in ts F Taylor series expansion and in the multiplicative correction, we find that: Now, we present in further detail the method to quantify the local mechanical properties by combining the experimental multi-harmonic observables 0 th and 1 st data on live cells, let first write the expression of the dynamic and average tip indentation    t ,  0 into the sample as: where Z is the piezo movement, 0 A is the cantilever mean deflection, 1 A is the first harmonic amplitude, and  1 is the first harmonic phase lag.
Substituting Eqs. S18 into resulting Eqs. S16 and S17, we can solve for the unknown constitutive parameters:

d. Bottom effect cone correction (BECC) of the Sneddon's model with viscoelasticity extension
Sneddon's contact mechanics model requires small sample indentations <10% of sample height. However, a model that takes into account the artifact generated by moderate and large indentations of conical tips in AFM measurements on thin samples and adherent cells is required. In this case, we chose to use the BECC contact model [13], which is a multiplicative analytical correction done to the commonly used Sneddon's model. This is a non-artifactual contact mechanics model that takes into consideration topographical effects by large indentations in soft samples like live cells. For a cell thickness of ~4 µm as used in this work, an indentation of larger than 400-800 nm would be needed to violate the assumptions of the standard Sneddon's model. However all the measurements made here have been for indentations less than 400 nm. Because as shown before [9][10]  Using Eq. S16 for small oscillations assumptions as previously presented and neglecting the contribution of the higher order terms in ts F Taylor series expansion and in the multiplicative correction, we find that: Substituting Eqs. S18 into resulting Eqs. S16 and S22, and evaluating the Fourier coefficients of the tip-sample interaction force are:   harmonics, formulas Eqs. S24 and S20 [9][10]. It is important to keep in mind that these equations actually extract the effective properties of the live cell at a specific mean indentation  0 and excitation frequency  dr [9]. With the above briefly discussed theory and the maps of multi-harmonic amplitudes and phases ( 0 A , 1 A , and  1 ), that can be easily acquired on a live cancer cell in vitro, it's possible to map the mean indentation ( ).

e. Bimodal AFM imaging for viscoelastic property mapping
For bimodal experiments, because of the cell softness and the low Q-factor of the soft cantilever in liquids, the vibrational mode shapes of the cantilever are assumed to be unperturbed. Moreover, since the cantilever oscillations are much smaller than the net indentation into the cell it can be assumed that the equation-of-motion of the cantilever can be separated into two independent simple harmonic oscillators [14]. Therefore, we can model the governing dynamics of the soft cantilever in permanent contact on the cell surface in liquid as: (S26)

f. Additional images of nanomechanical properties of rat fibroblast cells
In this work we performed multiple fast AFM imaging of live rat fibroblasts and MDA-MB-231 human breast cancer cells using Lorentz-force microcantilever excitation with feedback on the cantilever mean deflection. After imaging we extracted their nanomechanical properties. In Fig S3 we provide additional images of a living fibroblast cell in culture media, showing that this novel technique can be easily implemented with repeatability and confidence yielding reasonable quantitative nanomechanical values. Figure S4 shows the viscoelastic tangent loss  tan maps obtained at two widely spaced high frequencies (7 kHz and 61 kHz) on a live rat fibroblast cell in culture media.
 tan maps, Figs. S4(a and b), clearly shows the classical viscoelastic frequency dependence. Figure S4c is the difference between the low and high frequency  tan maps showing a reduction by ~0.3-0.9 on the cell.

g. Adhesion experiments
The expression in MDA-MB-231 cells of Syk decreases cell motility and enhances adhesion. To confirm that this effect is an intrinsic property of the active kinase, we compared cells either lacking Syk or expressing Syk-EGFP (wild-type Syk with a green fluorescent protein tag) or Syk-AQL-EGFP, an analog-sensitive version of Syk. The treatment with 1-NM-PP1, an orthogonal inhibitor of Syk-AQL-EGFP, of cells expressing the engineered kinase, but not the wild-type enzyme, reduced adhesion to the level seen with Syk-deficient cells (Fig. S5). These experiments illustrate the ability of Syk to enhance cell adhesion in a manner dependent on its catalytic activity.

h. How well the BECC contact mechanics model fits the experimental data
We used a nonlinear least-squared fit algorithm to best fit the unknown physical properties to the experimental data dynamic AFM observables. In order to check the applicability of the contact model and the experimental data we extracted the residuals and resnorm of the fit. The residuals measure the differences between a data point and the corresponding mechanics models estimate, therefore the smaller the difference the better the fit. However, residuals can be positive or negative making it difficult sometimes to judge if the fit is good. The resnorm is a better estimate consisting in the sum of squared residuals. Figure S6 show the extracted values for the residuals and resnorm are very small confirming the goodness of the fit. The movies provide insights into the kinetics of cytoskeletal changes. Interestingly, rapid changes in the cytoskeletal architecture at the cell periphery could be visualized within 1.5 min including the formation and movement of lateral actin bands or transverse arcs characteristic of retrograde actin flow that preceded the release of focal adhesions. Thus, the rapid loss of Syk activity was correlated with dramatic rearrangements in the cortical actin cytoskeleton.