Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

Juan G.
Sanchez
,
Francisco M.
Espinosa
,
Ruben
Miguez
and
Ricardo
Garcia
*

Instituto de Ciencia de Materiales de Madrid, CSIC, c/Sor Juana Inés de la Cruz 3, 28049 Madrid, Spain. E-mail: r.garcia@csic.es

Received
16th June 2021
, Accepted 1st September 2021

First published on 28th September 2021

AFM-based force–distance curves are commonly used to characterize the nanomechanical properties of live cells. The transformation of these curves into nanomechanical properties requires the development of contact mechanics models. Spatially-resolved force–distance curves involving 1 to 2 μm deformations were obtained on HeLa and NIH 3T3 (fibroblast) cells. An elastic and two viscoelastic models were used to describe the experimental force–distance curves. The best agreement was obtained by applying a contact mechanics model that accounts for the geometry of the contact and the finite-thickness of the cell and assumes a single power-law dependence with time. Our findings show the shortcomings of elastic and semi-infinite viscoelastic models to characterize the mechanical response of a mammalian cell under micrometer-scale deformations. The parameters of the 3D power-law viscoelastic model, compressive modulus and fluidity exponent showed local variations within a single cell and across the two cell lines. The corresponding nanomechanical maps revealed structures that were not visible in the AFM topographic maps.

The last 15 years have seen the consolidation of a live cell as a system that responds to mechanical interactions and biochemical processes. This realization emphasises the close relationship among mechanical forces, cell shape and physiology.

The intrinsic heterogeneity of a cell or the lack of a generally accepted contact mechanics model to fit the data and the changes of a cell (shape) during a measurement complicate the interpretation and/or standardization of AFM measurements on cells. Commonly, Sneddon expressions^{17} are used to fit AFM-based force–distance curves (FDCs). However, the finite-thickness of a mammalian cell might preclude the use of Sneddon expressions to determine the elastic modulus of a cell (see below).^{16} The above issue has led to a paradoxical situation. At a qualitative level, AFM measurements show a high degree of reproducibility. The observation that a cancer cell is softer than a non-malignant cell has been reproduced by many groups.^{18–22} Similarly, AFM data have firmly established that drug inhibitors of actin polymerization reduce the Young's modulus.^{23,24,54} On the other hand, the values of the Young's modulus measured on cells of the same line might differ by 5-fold.^{25} Even the methodology applied to determine the Young's modulus of a cell might be questionable. After all, a large body of experimental data has shown conclusive evidence about the viscoelastic response of a mammalian cell.^{7,16,24,26–31}

Here we aim to clarify the complex nanomechanical response of a mammalian cell as determined from AFM force–distance curves on three aspects. First, by showing that the intrinsic viscoelastic response of a cell is preserved in a force–distance curve measurement. Second, by implementing an analytical force reconstruction model that provides a faithful representation of the mechanical state of a live cell. This model which is called 3D power-law viscoelasticity (3D-PLR) hereafter expresses the dependence of the force exerted on a cell by a conical tip in terms of the indentation, the cell's thickness, the compressive modulus at an arbitrary time t_{0} and the fluidity exponent. Third, by showing that the compressive modulus and the fluidity exponent characterize the spatially-dependent nanomechanical response of a cell within the same cell line and across different cell lines. Specifically, we show that cytoplasmatic regions have more liquid-like properties than nuclear regions. By performing experiments on fibroblasts (NIH 3T3) and HeLa cells, we demonstrate that the finite-thickness of a cell must be accounted for to determine mechanical parameters from force–distance curves. In summary, our results underline the robustness of the compressive modulus and the fluidity exponent to characterize the mechanical state of a mammalian cell under micrometer deformations performed at low frequencies.

(1) |

Starting from the above expression, Garcia and Garcia developed an analytical expression to determine the force exerted by a conical tip on a cell of thickness h that rests on a rigid solid support,^{34}

(2) |

Eqn (2) enables to determine the Young's modulus of a cell without the influence of the rigid support. In this context, a polynomial expansion in terms of (I/h)^{n} is commonly called bottom-effect correction.^{8,16}

(3) |

(4) |

(5) |

E
_{0} is defined as the compressive modulus of the material at time t_{0} and γ is called the fluidity exponent. In a previous publication^{32} we defined E_{0} as the Young's modulus at a time of 1 s. However, we prefer to use the term compressive modulus to avoid the identification with an elastic modulus. A value of the fluidity exponent γ = 0 indicates an elastic solid while γ = 1 indicates a viscous liquid. Based on the above power-law equation, Garcia, Guerrero and Garcia^{31} deduced an analytical expression to determine the force as a function of indentation. The model accounts for the geometry of the contact, the finite-thickness of a cell and the history of the deformation. The analytical force expression was deduced in several steps. First, the force was deduced for a semi-infinite, incompressible (ν = 0.5) power-law viscoelastic material,

(6) |

(7) |

Third, we applied Ting's procedure^{44} to determine the contact area during the tip's withdrawal. A force–distance curve measurement involves the acquisition of the force as the tip approaches the cell and withdraws from it. The viscoelastic response implies the existence of a time shift between the maximum force and the maximum indentation. To determine the force applied to a viscous material during the withdrawal we applied the following transformations

(8) |

(9) |

The AFM data were used to plot the force as a function of time or indentation. The relevant AFM observations were the cantilever deflection, the z-piezo displacement and the displacement at which the tip contacts the cell surface. The distance was considered negative when the tip is indenting the cell. The indentation was stopped when the force reached a threshold level of 3 nN.

Fig. 1a shows a sequence of 5 FDCs obtained on the same spot of a HeLa cell. The thick line is the mean value. The triangular waveform and the dependence of the z-tip deflection with time are shown in Fig. 1b. Fig. 1c shows a compact representation of the FDC as a function of indentation.

The acquisition of a FDC implies the selection of a velocity or a modulation frequency. The velocity is easily determined for a triangular waveform; however, the contact time or its inverse (f_{m} = 1/t_{c}) provides more information on the cell's nanomechanical response. Indeed, the nanomechanical response of a live cell does depend on the tip's modulation frequency or velocity.^{10,24} To simplify the discussion and interpretation of the experimental data, we have chosen a velocity of 10 μm s^{−1} (t_{c} = 0.27 s, f_{m} = 3.7 Hz). A velocity of 10 μm s^{−1} provides a good compromise between moderately quick data acquisition speed and yet the measured parameters are representative of cell's response in the 0.5–5 Hz range.

Fig. 2 shows some FDCs obtained on HeLa and NIH 3T3 cells. The 3D elastic model leads to two Young's modulus values depending on the section of the FDC used for the fitting (approach or retraction). For both cell lines, the modulus obtained during the retraction is about 3 times higher than the one obtained during the approach. The existence of two values should question the use of elastic models. The AFM community has resorted to present the data obtained during the approach without providing a proper justification.

Fig. 3 compares the experimental FDCs to the force expressions of the viscoelastic models. The 3D linear viscoelastic model (3D-KV) provides an excellent fitting to the approach section for both cell lines. However, it shows a step discontinuity at the turning point which prevents a good fitting of the retraction section of the FDC. This artefact is associated with the change in the sign of the velocity. Therefore, we do not recommend the use of the 3D-KV model if the FDC was generated using a triangular waveform. On the other hand, the 3D-PLR model does provide a good fitting for the both sections (approach and retraction). This agreement cannot be considered fortuitous because the model has just two parameters. For this reason, in the next sections, the 3D-PLR model will be used to describe the nanomechanical response of live cells.

Fig. 3 Comparison of experiment and analytical viscoelastic models. (a) 3D power-law viscoelastic model. (b) 3D-Kelvin–Voigt. Parameters of the FDCs as in Fig. 2. |

From the above analysis, we conclude that the nanomechanical response of a cell cannot and should not be described by an elastic model. This conclusion comes from three reasons. First, the values E_{a} and E_{r} are very different. Second, the values of the Young's modulus do not coincide with the effective modulus provided by a viscoelastic model. Third, the FDCs show hysteresis and the existence of energy dissipative processes.

It is not straightforward to understand 3D power-law rheology in terms of physical concepts. However, we consider that different effects such as poroelasticity^{50} and the hydrodynamic drag of cell's organelles^{13} contribute to 3D power law rheology.

To illustrate the type of numerical errors associated with the Young's modulus of the rigid support, we have compared the compressive modulus and the fluidity coefficient measured on several HeLa and NIH 3T3 cells. For HeLa cells (Fig. 4a), the box plots show mean values, respectively, 2348 Pa (semi-infinite) and 2219 Pa (finite). For NIH 3T3 cells (Fig. 4b), we obtained 574 Pa (semi-infinite) and 533 Pa (finite). This is a semi-infinite viscoelastic model overestimates the compressive modulus by about 6–10%. On the other hand, the fluidity exponent (0.29) seems to be unaffected by the stiffness of the solid support. A similar trend is obtained on NIH 3T3 cells (Fig. 4c). The same trend, semi-infinite models overestimate the modulus of cells, was observed when elastic expressions were used. Table 1 shows a summary of all the measurements.

Cell | Thickness | Elastic model | Single power law model | ||||
---|---|---|---|---|---|---|---|

E (Pa) |
E
_{
0
} (Pa) |
γ | |||||

NIH3T3 | Semi-infinite | 2254 | 2225 | 574 | 484 | 0.38 | 0.42 |

Finite | 2092 | 1977 | 533 | 431 | 0.38 | 0.41 | |

HeLa | Semi-infinite | 7320 | 5769 | 2348 | 1539 | 0.29 | 0.33 |

Finite | 6836 | 5293 | 2219 | 1416 | 0.29 | 0.32 | |

Nucleus | Cytoplasm | Nucleus | Cytoplasm | Nucleus | Cytoplasm |

Fig. 5 compares the parameters of the 3D power-law rheology model measured on NIH 3T3 fibroblasts and HeLa cells. The compressive modulus of the HeLa cells is significantly higher (about five-fold). For example, the mean values over the nuclear region are, respectively, 2219 and 533 Pa. Similar differences are observed for the cytoplasmatic regions (1416 versus 431 Pa). The fluidity exponent also reflects the differences between HeLa and NIH 3T3 cells. Higher fluidity exponents are found on NIH 3T3 cells. The mean values over the nuclear regions are, respectively, of 0.38 (NIH 3T3) and 0.29 (HeLa) Interestingly, the differences observed between HeLa and NIH 3T3 cells are larger than the differences measured among the regions of the same cell lines. Overall, HeLa cells are stiffer and less viscous than NIH 3T3 cells.

Fig. 5 also shows the differences between the nuclear and cytoplasmatic regions. Those differences were reported previously by measuring variations of Young's modulus between the nuclear and cytoplasmic regions. The Young's modulus values measured over the nucleus are higher than those measured on the cytoplasm.^{13,49} However, we showed above the limitations of elastic models to describe the deformation of a mammalian cell, for that reason, we have revisited this problem by applying the 3D power-law model.

The results for HeLa cells show that the compressive modulus obtained over the nucleus (2219 Pa) is 36% higher than the one measured on the cytoplasm (1416 Pa). The fluidity exponent is higher over the cytoplasmic regions (0.32) than over the nucleus (0.29). These findings indicate that, on one hand, the cytosol has a more liquid-like behaviour. On the other hand, the cytosol offers a certain resistance to the displacement of the nucleus. This conclusion is supported by the findings obtained by Efremov et al. on nucleoplasts and enucleated cells.^{51}

The analytical force–distance curve generated using a 3D linear viscoelastic model presented a step discontinuity. This effect was associated with the use of triangular waveforms to generate the tip's displacement. It is associated with a discontinuity of the velocity at the turning point. On the other hand, the analytical force expression deduced for the 3D power-law model reproduced the experimental force–distance curves for HeLa and NIH 3T3 cell lines. These cell lines are very different in terms of mechanical properties and biological functionalities. Based on this result, we hypothesise that the same viscoelastic model might be applicable to other mammalian cells.

We showed that HeLa and fibroblast cells are harder to deform when the force is applied in a region of the plasma membrane located above the nucleus. A cell has higher compressive modulus values and lower fluidity exponents above the nucleus. The differences observed in the mechanical response of nuclear and cytoplasmic regions support the use of nanomechanical maps to characterize the spatial heterogeneity of a cell. The nanomechanical maps revealed structures that are hidden in the topographic images. Finally, we underlined the relevance of using analytical expressions that include bottom-effect corrections to compare experimental and theoretical force–distance curves.

To generate the nanomechanical maps of Fig. 6, NIH3T3 fibroblasts and HeLa cells were fixed with 4% formaldehyde (ThermoFisher Scientific, USA) in phosphate-buffered saline-PBS 0.01 M (Sigma-Aldrich, UK) for 10–15 minutes and rinsed with PBS.

We used BL-AC40TS cantilevers. Those cantilevers have pyramidal tips with an aperture semi angle of 18° (7 μm in height and a tip radius of 8 nm). The cantilever force constant k was calibrated using the thermal noise method.^{52,53} Force–distance curves on NIH 3T3 cells were recorded with k in the 0.10–0.13 N m^{−1} range (f_{0} = 38.15 kHz in liquid) while on HeLa cells the force constant was in the 0.044–0.087 N m^{−1} range (f_{0} = 30.68 kHz in liquid). The cantilever sensitivities were between 4.8–6.7 nm V^{−1} (NIH 3T3) and 6.9–9.5 nm V^{−1} (HeLa).

For each cell line, the data involved 8 single cells. For each cell, the FDCs were obtained on 12 different regions of the cell, 6 above the nucleus and 6 in cytoplasmatic region; 10 FDCs were obtained on each region of the cell.

The FDCs were obtained using a triangular waveform at 10 μm s^{−1}. The indentation was stopped when the force reached a value of 3 nN. The z-displacement was performed with a closed-loop feedback and involved a z-displacement of 5 μm (sampling rate of 500 data points per μm). Hence the number of points was 2500 for the approach and 2500 for the retraction. The fittings were performed by applying a correlation coefficient R above 0.95.

The nanomechanical maps consisted of an ordered mesh of FDCs measured over the cell with a grid of 512 × 512 pixels. The velocity was 150 μm s^{−1} with a setpoint of 3 nN and a z-displacement of 1–1.5 μm. The sampling rate in the FDCs was 500 points per μm. The force constant and resonant frequency were, respectively, k = 0.13 N m^{−1} and f_{0} = 33.3 kHz (HeLa) and k = 0.16 N m^{−1}f_{0} = 30.22 kHz (NIH 3T3).

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