Rheology of rounded mammalian cells over continuous high-frequencies

Understanding the viscoelastic properties of living cells and their relation to cell state and morphology remains challenging. Low-frequency mechanical perturbations have contributed considerably to the understanding, yet higher frequencies promise to elucidate the link between cellular and molecular properties, such as polymer relaxation and monomer reaction kinetics. Here, we introduce an assay, that uses an actuated microcantilever to confine a single, rounded cell on a second microcantilever, which measures the cell mechanical response across a continuous frequency range ≈ 1–40 kHz. Cell mass measurements and optical microscopy are co-implemented. The fast, high-frequency measurements are applied to rheologically monitor cellular stiffening. We find that the rheology of rounded HeLa cells obeys a cytoskeleton-dependent power-law, similar to spread cells. Cell size and viscoelasticity are uncorrelated, which contrasts an assumption based on the Laplace law. Together with the presented theory of mechanical de-embedding, our assay is generally applicable to other rheological experiments.


Supplementary Figure 3 | Lock-in amplification allows for low noise detection of the measured signal.
The power spectral noise density (PSD) of a single cantilever was recorded with a lock-in amplifier under three actuation conditions: i) thermal noise (no photothermal actuation, black data), ii) photothermal actuation at a frequency of 1 kHz (red data), and iii) photothermal actuation at a frequency of 16 kHz (blue data). The average laser power of 3.5 mW was the same for both frequencies. The PSD at the frequencies of actuation is 200 times higher compared to the thermal PSD, therefore rendering the contribution of the thermal noise negligible for our rheological measurements and allowing for a high data acquisition speed, as averaging multiple measurements is not necessary. The Fourier components at multiples of the excitation frequency of the blue laser, arise due to a slight asymmetry of the sinusoidal movement of the photoactuated cantilever, which is caused by thermal effects. oscillator (red line) having an effective mass m*, a quality factor , and a natural resonance frequency 4567 , and being actuated by a constant force $ . Dividing the experimental amplitude response by !"# , results in the scaling factor ℎ( ) (green dashed line). b, Multiplication of $ with the scaling factor function ℎ( ) gives the frequency-dependent actuation force ( ) at which !"# matches the experimental amplitude response (red line). As predicted by the theory of photothermal actuation 1 , ( ) increases at lower frequencies.
Supplementary Figure 5 | Shifting the position of the photothermally actuating laser on the master microcantilever changes its mechanical response. Shown are finite elements-simulations (Methods) of the phase of a microcantilever, which is either photothermally actuated at the base (continuous curve) or the free end (dashed curve). At actuation frequencies below the resonance frequency of the microcantilever 4567 ≈ 18.5 kHz, the difference in phase is negligible. However, deviations occur above the resonance frequency of the microcantilever. To apply Eq. 1, which is an implicit assumption of the lumped-mass model (Fig. 2b), such deviations have to be corrected. Accordingly, to account that the blue photothermally actuating laser and the red readout laser are not at the same positions, the position of the actuating blue laser is mathematically shifted (Supplementary Fig. 6).
Supplementary Figure 6 | Phase and amplitude correction curves to account for the blue laser position on the photothermally actuated microcantilever. a, Correction curves to transform the phase response of rectangular (red) and triangular (blue) microcantilevers. Addition of the phase correction offset to the experimental phase mathematically shifts the position of the photothermally actuating force, which is exerted by the blue laser (Methods). b, Correction curves to transform the amplitude response of rectangular and triangular microcantilevers. Eq. 1 does not account for the blue actuation laser and the red readout laser being at different positions on the microcantilevers. Transforming the measured data using the correction curves does this. The correction curves are calculated from the difference (phase) or the ratio (amplitude) of the curves simulated in Fig. 2c and Supplementary Fig. 5. Applying the corrections to the experimentally measured data, virtually shifts the position of the oscillating force applied by the photothermally actuating blue laser. In the example shown the position of the photothermally actuating laser shifts from the free end ( = 110 µm) to the base ( = 20 µm). Calculations to mathematically shift other positions of the photothermally actuating laser are applied accordingly (not shown).
Supplementary Figure 7 | The mechanical de-embedding framework functions independently of the cell mechanical properties. a, b, Amplitude and phase response curves of the slave cantilever extracted from finite elements (FEM) simulations in the MS configuration (Fig. 2). To generate the data, the same geometries of a rounded cell sandwiched between two triangular cantilevers, and of the cantilever driving force were used, however, the dependencies of the complex modulus 4897 * on the cantilever oscillation frequency differed as following: 1) Double power-law model: (black data points) between 1 kHz and 10 kHz as it has been observed for crosslinked gels 2 , to which alginate belongs to in the presence of 10 mM calcium ions (CaCl2) in the buffer solution at 37 °C. Figure 9 | HeLa cells confined between two microcantilevers stabilize within 3 minutes. a, Subsequent frequency sweeps recorded of a single rounded HeLa cell compressed by 1 µm using the parallel microcantilever assay (Fig. 1a-c). Frequency sweeps were recorded every 3 min. b, Amplitude changes among the frequency sweeps. The first sweep recorded at 0 min was set as reference (black). In the second sweep recorded at 3 min, the amplitude reduced by ≈ 2%. The sweeps subsequently recorded at 6, 9 and 12 min showed much smaller amplitude variations compared to the sweep recorded at 3 min (< 1%). Based on this result, we confined single HeLa cells in our parallel microcantilever assay and allowed each cell to relax for 5 min before starting the rheology measurements. the maximum driving amplitude of the actuated master cantilever from 10.6 nm to 4.6 nm (≈ 45% reduction), which results from reducing the power of the actuating laser of 3.5 mW by 50%, the (a) amplitude and (b) phase response of the slave microcantilever was measured in configuration MS (Fig. 2a). The amplitude response of the slave microcantilever shows peak amplitudes of 6.9 nm (3.5 mW laser power) and 3.0 nm (50% of 3.5 mW laser power). c, The linear ratio of both amplitude responses measured in (a) shows an equal reduction of ≈ 45% for the response amplitude over all frequencies. d, Relative phase difference of the slave microcantilever showing a peak amplitude of 4.5 nm (50% laser power) compared to the slave cantilever showing a peak amplitude of 10 nm. The phase change fluctuates around zero over all frequencies. The data scattering at higher frequencies (≈ 30 -38 kHz) is due to lower amplitudes and, therefore, lower signal-to-noise ratio. Both amplitude ratio and phase change curves show that the mechanical response of the living HeLa cell is linearly proportional to the actuated master cantilever amplitude (Supplementary Note 1). It can be reasoned, that the experimentally found linear response of the cell stems from the actuating amplitude of 10 nm, which is a small perturbation compared to the thickness of the cell cortex (≈ 200 nm, Fig. 4e,f).

Supplementary Figure 12 | Geometry-of rounded HeLa cells confined between two parallel surfaces
agree with the finite elements simulation of the cell shape within the experimental error. a, Sideview of a SiR-actin stained, living unperturbed HeLa cell before (left) and during parallel-plate compression (right). The fluorescence images were recorded using super resolution (stimulated emission depletion, STED) microscopy. The unperturbed cell is outline yellow. The red outline of the compressed cell has been generated from finite elements simulations, which took the experimentally approximated circumference of the unperturbed HeLa cell (yellow) and the amount of compression (≈ 10% of cell height) as input. Due to an optical distortion arising from the gold-coated cantilever reflecting and scattering the lasers used for the STED microscopy, the SiR-actin signal is broadened at the cantilever-cell interface (top). The scale bar is 6 µm. b, Two HeLa cells of different size being compressed by our parallel-plate assay. Red lines show outlines from the finite elements simulation for the respective experimental compression of ≈ 1 µm. For parallel-plate compression a wedged microcantilever was pressed onto (top of image) a rounded HeLa cell plated on a Petri dish (bottom of image) under cell culture conditions. c, Finite element simulated spring constant of cell geometries with different eccentricities . The spring constant depends linearly on for a compression ≈ 1 µm, which is used in the experiments. An = 1.1 (10% away from an ideal sphere) leads to a spring constant that is ≈ 10% higher than for = 1.0 and to an overestimation of the storage and loss moduli by 10%, if the geometry was not correctly accounted (i.e., the cell is assumed spherical). In our experiments, however, only small eccentricities (≈ 1 -4%) were observed for the cells. However, using STED microscopy on a different setup, we found that the cellular cortex thickness varies between 150 nm and 250 nm (Fig. 4) and does not correlate with the cell size (Supplementary Fig. 15). To investigate, how this variation affects the extraction of the complex modulus 4897 * , and likewise 4897 F and 4897 FF , we used the solid-shell-liquid-core model (Supplementary Note 2, Methods) to simulate a rounded cell with diameter 16 µm, a cortical storage modulus A897 F of 100 kPa, and cortex thicknesses of 250 nm. Under the purposefully wrong assumption of a cortex thickness of 150 nm we then extracted storage moduli. The ratio of the extracted moduli and the real modulus at 150 nm cortex thickness and 100 kPa is shown in the graph against the compression distance . The same dependency holds true for the loss modulus A897 FF . As can be seen, assuming a wrong cortex thickness affects the extraction of moduli differently, which depends on . For experimental conditions = 1 µm, we found a ratio of 1.4, which means that extracting the storage and loss moduli of the cell having a 250 nm thick cortex, but assuming it would be 150 nm thick, leads to overestimate the storage and loss moduli by 40%. In the manuscript, we assume a cortex thickness of 200 nm, which would leads under-and overestimate the cortex moduli by ≈ 20% for cells having a 150 nm and 250 nm thick cortex, respectively. Importantly, the ratio does not change a lot around the experimental conditions used in our work (other than it would change more at smaller compression distances).

Supplementary Figure 17 | Master and slave microcantilevers show identical deflection sensitivities.
The deflection sensitivity of the master microcantilever was determined by touching the Petri dish with the free end and measuring the microcantilever deflection. Thereafter, the laser was shifted to the slave microcantilever. As the slave microcantilever was fixed on the wedge its deflection sensitivity was measured by bringing the support of the master microcantilever into contact with the slave microcantilever. The slave microcantilever shows very similar deflection sensitivity, which is not surprising because both microcantilevers were of the same type and exposed to identical conditions (microcantilever length, width and thickness, laser spot location on the microcantilever, light intensity at the photodiode).

Unperturbed
HeLa cells  Fig. 7 and Manuscript, Eq. 2). The fit quality was assessed via calculating in two ways how far off the fit is from the experimental data using the distance measures ℓ % * and ℓ E * . The measures differ in how they react to outliers (Methods). For each rheological model (row) and experimental condition (column) the two distance measures (top) and the relative distance to the double power-law distance (bottom) are given. The relative distance describes how much better (in this case the relative distance is < 1) or worse (the relative distance is > 1) the respective fit is compared to the double-power law fit. The double powerlaw fits the experimental data best since the relative distances of all other models are greater than 1. FF using a two-dimensional optimization (Methods). n is the number of biologically different cells. and are scaling factors, and are low and high frequency exponents, and $ is a normalization constant, which has been set to 1 kHz. Values represent the mean and standard deviation. Source data are provided as a Source Data file.

Supplementary Note 1: Derivation of Equation 1
Supplementary Figure 18 | The system of coupled microcantilevers is described as a lumped-mass model.
* and ? describe the spring constant and * and ? the damping of the master and slave microcantilevers that are actuated by the photothermal forces * and ? and have the effective masses * and ? , respectively. The coupling of both microcantilevers by the sandwiched cell is modelled with the spring constant A and the damping A of the cell. The effective mass of each cantilever comprises the mass of the cantilever, the mass of the water dragged by the cantilever, and the mass of the cell adhering to the cantilever.
Derived from the exchange model shown in Supplementary Fig. 18 We can describe the equation of motions with the matrix notation: Hereby, @ and = are the positions of the point masses @ and = , respectively, and the dot notation indicates the total derivatives in time. @ and = describe the spring constant and @ and = the damping of the master and slave microcantilevers that are actuated by the photothermal forces @ and = . Making use of the harmonic drive and the harmonic motion of the microcantilevers we can write Eq. S1 in Fourier space: where we define We invert and generalize , writing: where G PQ ( )J = ( ), and we made use of the symmetry of the matrix. It is important to note, that the desired coupling properties ( 4 , 4 ) are given as the complex spring constant %E ( ) which can be identified as the cellular transfer function 4 ( ), with 2 = . If the system can be described as linearlycoupled two masses (terms only linear in ), the off-diagonal elements of contain only the coupling between @ and = . Therefore, the cellular spring constant and damping, are stored in the diagonal elements of the matrix. This is independent on how the spring constant and damping of the cell are realized, e.g. For the application of Eq. S5 the actual values for @ and = are not of importance as they are contained in the three measured functions "" , "R , RR . Therefore, also the distribution of the cell mass across the cantilevers does not need to be known.

Supplementary Note 2: Extraction of F and FF from Eq. 1
Eq. 1 yields the frequency dependent, complex cellular transfer function A ( ), which is a complex spring constant stemming from the spatially averaged mechanical properties of the entire, single cell. In order to convert the geometry dependent A ( ) into a geometry independent, dynamic modulus 4897 * ( ) = 4897 F ( ) + 4897 FF ( ), we introduce a shape factor S: A ( ) = • 4897 * ( ). captures all geometry dependencies of A ( ) whereas 4897 * ( ) captures all geometry-independent, cortex-material dependencies. To find we employed finite-elements modelling. Using a solid shell liquid core model (Methods, Supplementary Fig. 19a), we found the relationship between the compression distance Δ and real part of the cellular transfer function A F for a fixed frequency, i.e. Re( 4897 * ( )) = 4897 F ( ) = 4897 F . Hereby, g 4 ' is in the units of a spring constant. Several of these spring constant versus compression distance curves are shown in Supplementary Fig. 19b. We then made a third order polynomial ansatz for S: where ℎ 4897 is the thickness of the cell cortex and R the radius of the uncompressed cell. To find % , E , C , we fitted Eq. S6 to a single spring constant versus compression distance curve that was generated with 4897 F = 125 kPa and = 8.4 µm. This radius corresponds to a cell with mass of 2.5 ng. We found % = 15 • 10 .%E , E = 9.7 • 10 .%E , C = 5 • 10 .%E . The three parameters were then used to extract 4897 F . from the spring constant versus compression distance curves (Supplementary Fig. 19b,c) according to 4897 F = A F /(ℎ 4897 ( % Δ/ + ( E Δ/ ) E + ( C Δ/ ) C ). We successfully extracted different 4897 F for a fixed , (Supplementary Fig. 2b). This means that that the parameters % , E , C found for 4897 F = 125 kPa are valid for a wide range of 4897 F , as encountered in the experiment. Also, we successfully extracted 4897 F for varying and a fixed 4897 F (Supplementary Fig. 19c). This means that that the parameters % , E , C found for = 8.4 µm are valid for a wide range of , as encountered in the experiment. It is important to note that the geometry of the cell, and hence the shape factor S of the cell, is determined by the initial cell radius and the height of the compressed cell. The cell radius is extracted via the cell mass measurement, whereas Δ can be accurately chosen via the microcantilever distance during the experiment. The contact area of the cell and a sandwiching cantilever (Supplementary Fig. 19a), is not necessary to be measured as it results from compressing a spherical cell of radius by the distance Δ and is therefore already captured in the model. The same is used in Eq. S6 to extract 4897 / Θ) E A where Θ is the meridian angle and the cell radius (Supplementary Fig. 20). With / Θ ≈ 2Δ/ we can approximate the integral to be ≈ l'm E T. Thus, the energy stored in the cell cortex is proportional to the cortex tension and quadratic to the compression distance Δ. The volume of the cell cortex having a thickness ℎ 4897 corresponds to 4897 = 4 E ℎ 4897 . Thus, the storage modulus is proportional to / E or if expressed in terms of pressure using the Laplace law 4897