Open Access Article

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

Pablo D.
Garcia‡
,
Carlos R.
Guerrero‡
and
Ricardo
Garcia
*

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

Received
14th May 2017
, Accepted 20th July 2017

First published on 10th August 2017

Single cell stiffness measurements consider cells as passive and elastic materials which react instantaneously to an external force. This approximation is at odds with the complex structure of the cell which includes solid and liquid components. Here we develop a force microscopy method to measure the time and frequency dependencies of the elastic modulus, the viscosity coefficient, the loss modulus and the relaxation time of a single live cell. These parameters have different time and frequency dependencies. At low modulation frequencies (0.2–4 Hz), the elastic modulus remains unchanged; the loss modulus increases while the viscosity and the relaxation time decrease. We have followed the evolution of a fibroblast cell subjected to the depolymerization of its F-actin cytoskeleton. The elastic modulus, the loss modulus and the viscous coefficient decrease with the exposure time to the depolymerization drug while the relaxation time increases. The latter effect reflects that the changes in the elastic response happen at a higher rate than those affecting the viscous flow. The observed behavior is compatible with a cell mechanical response described by the poroelastic model.

The cytoplasm consists of solid elements such as cytoskeleton, organelles and ribosomes in a liquid fluid (cytosol).^{25,26} Several experiments have demonstrated the existence of viscous processes in the cytoskeleton remodeling and in the inner cell dynamics.^{26–28} In fact, stiffness measurements performed on cancer cells of different pathological degrees^{17,29,30} have given similar elastic (Young) moduli values. These findings could be related to the limitations of stiffness measurements to describe the mechanical state of a cell.

Several factors explain the extensive use of AFM stiffness studies on cells. (1) The existence of an established protocol to determine the elastic modulus from force–distance curves.^{31,32} (2) Stiffness measurements have provided insightful information about the mechanics and physiology of cells.^{1,3} (3) On the other hand, there is not an established AFM protocol to perform time or frequency-dependent experiments.^{33–38} In addition, there are several competing theoretical models to interpret the data.^{40–43}

Let's summarize the AFM studies devoted to measure the viscoelastic properties of cells. These contributions can be broadly classified into two main categories, force oscillation and time response methods. In a force oscillation experiment, the tip is indented on the cell to a given depth and a subsequent sinusoidal perturbation is applied.^{28,33,38} Multi-harmonic AFM experiments^{8,36,39} could also be included in this category although in multi-harmonic experiments there is not a distinction between the indentation and the oscillation stages. Force oscillation approaches require the use of an elaborated theoretical framework.^{36} In many cases, it is not always possible to deduce parameters that could be compared with the elastic modulus and viscous coefficient obtained from force–distance curves. As an alternative, the loss tangent of a cell has been measured.^{29,36} This parameter represents the ratio between the energy dissipated and the energy stored in one cycle of the oscillation.^{44} The loss tangent avoids the estimation of the tip radius. However, this parameter has not been a straightforward link to the mechanical properties of a cell.

Time response methods include two different approaches, load relaxation and creep compliance experiments. In a load relaxation experiment,^{26,35,45} the tip is moved towards the cell until a certain indentation is reached, then the tip is held still and the dependence of the force with time is recorded. Creep compliance experiments^{33,34,37} keep the force applied on the cell at a fixed value. Due to the cell internal reorganization processes, keeping a constant force requires a continuous change in the z-position of the cell support. Time response methods require theoretical models to transform the data into rheological parameters. These models have to deal with the determination of the contact area for a viscoelastic contact. In general, time response methods are not compatible with high resolution imaging.

Force microscopy offers other schemes to measure inelastic interactions. The force–distance curves obtained during an approach–retraction cycle usually show the presence of a hysteresis loop. The area enclosed by the loop determines the energy dissipated by the tip–sample interaction^{46–48} (energy hysteresis). The measurement of the energy hysteresis could provide a method to determine the viscoelastic properties within the cell.^{49} The transformation of energy hysteresis values into nanomechanical properties implies two stages. First, it requires the existence of well-defined models in the energy dissipation processes.^{50} Second, these models require the determination of the contact area during the approach–retraction cycle. Currently, there are no analytical expressions to determine the contact area in the presence of viscoelastic processes.

Here we develop a force microscopy method to determine the time and frequency responses of a single live cell under the influence of an external compressive force. The method determines the Young's (elastic) modulus, the loss modulus, the viscosity coefficient and the relaxation time of a cell as a function of the frequency of the external force. This method can be divided in two main steps, the acquisition of force–distance curves and the application of a model to extract the cell rheological response from these curves. The elastic and dissipative components of the force are determined by considering the approach and retraction sections of the force–distance curve around the maximum indentation. Finite element simulation shows that near the maximum indentation the contact area is well described by conventional contact mechanics models. The method has been applied to follow the evolution of a single fibroblast cell exposed to the action of an actin polymerization inhibitor drug. The elastic modulus, the viscous coefficient and the loss modulus decrease while the relaxation time increases with the exposure time to the drug.

A realistic description of a three-dimensional system should consider that the vertical and lateral deformations are coupled. For this system, the dependence of the force with the deformation in the presence of viscoelastic processes is obtained by combining the relaxation function Ψ(t) with the known force–indentation relationship obtained for an elastic material of the same geometry.^{53} This step involves the replacement of the elastic (Young) modulus E in the elastic equation for an integral operator that is expressed in terms of the relaxation function Ψ(t). The process for a three-dimensional axisymmetric indenter starts with the expression that relates the force with the indentation for an elastic material

F = αEI^{β} | (1) |

(2) |

To express Ψ(t) in terms of mechanical properties requires the use of a viscoelastic model. This step needs to balance numerical accuracy and the capability to provide an analytical expression in a closed form. The standard linear solid model provides an accurate approach to determine creep and relaxation processes.^{52} However, this model does not allow deducing expressions in a closed form when the contact area depends on the indentation. The above dependence should always be considered because the force–distance curves on cells involve indentations of several micrometers.

On the other hand, the Kelvin–Voigt model^{52} enables deducing analytical expressions to link the parameters with the observables without restricting the variation of the contact area with the deformation. For a given frequency, we demonstrate that the results provided by the Kelvin–Voigt model are equivalent to those given by any other linear viscoelastic model, in particular, the standard linear solid model (see the ESI†). Then,

ψ(t) = E + η_{com}δ(t) | (3) |

In many contributions the viscosity is described in terms of the one-dimensional laminar flow model. In this model the relevant coefficient is the shear viscous coefficient η_{sh}. For an incompressible material (ν = 0.5), the compressive and shear viscous coefficients are related by

η_{com} = 3η_{sh} | (4) |

In the following we express our results in terms of the commonly used shear viscous coefficient η_{sh} = η. With the above assumptions the force is calculated using the following equation:

F(t) = αI(t)^{β−1}[3βηİ(t) + EI(t)] | (5) |

The above expression can be simplified for Hertz and Sneddon contact mechanics models. For the latter (conical geometry),

(6) |

The interaction force includes conservative (elastic) F_{c} and dissipative (inelastic) F_{dis} processes. In general the force can be expressed as

F(t) = F_{c} + F_{dis} | (7) |

(8) |

(9) |

In eqn (9) the force is proportional to the indentation. This result is different from a previous expression deduced without the above considerations.^{50}

The loss modulus is calculated from the expression (see the ESI†)

E_{loss} = −iωη | (10) |

To deduce an expression for the relaxation time, we assume the Kelvin–Voigt approximation, where an initial deformation I_{0} decays exponentially with time,

(11) |

(12) |

The hydrodynamic drag due to the interaction of the cantilever body with the liquid causes an additional hysteresis in the force curve.^{55,56} The hysteresis is characterized by the separation between the baselines of the approach and withdrawing sections of a force curve. We have developed an algorithm to correct the data from the drag effect (see the ESI†).

Force–distance curves.
The tip–sample distance has been modulated by applying a triangular waveform. The data acquisition rate has been calculated to be 4000 Hz. Bottom effect corrections for a conical tip have been applied to correct the cell finite thickness.^{7} The force–distance curves have been measured on a region above the nucleus.

Chemical treatment for disrupting the cell cytoskeleton.
Cells were treated with 5 μM cytochalasin D (Sigma-Aldrich) to depolymerize the actin filaments. This step was performed in vivo by adding Cyt-D to the cell medium during the AFM measurements.

Fig. 1c shows the z-piezo displacement, the indentation and the force as a function of time during the acquisition of a force curve above the nucleus of a MEF. The raw data represent the variation of the cantilever deflection Δz with respect to the z-piezo displacement z(t). To obtain the force as a function of the tip–cell distance, the cantilever deflection is multiplied by the force constant k and the z-piezo displacement is converted into indentation values I. The latter is achieved using the following relationship,

S(t) = z(t) + Δz(t) − z_{0} | (13) |

I(t) = −S(t) for S ≤ 0 | (14) |

Fig. 1d shows that the approach and retraction sections of the force curve do not coincide. This observation reveals the existence of dissipative interactions between the tip and the cell. The area enclosed by the approach and retraction sections of the force curve coincides with the energy dissipated on the sample.^{46,47,50}

(15) |

(16) |

If the approach and retraction sections of the force–distance curve overlap, F_{a}(I) = F_{r}(I) and F_{dis} = 0.

For a conical tip, we obtain

(17) |

(18) |

We also assume that for a given indentation I(t), dI_{a}/dt ≈ −dI_{r}/dt.

Fig. 2c shows the radius a of the contact area of a linear viscoelastic body that interacts with a spherical tip. The contact area is defined as the area of the contact projected on a plane perpendicular to the tip indentation (πa^{2}). The radius is presented as a function of the time. However, it could be easily transformed into a function of the indentation using eqn (13) and (14). During the approach, the contact is characterized by a contact radius that follows the shape and the numerical values given by Hertz contact mechanics. During the tip retraction, the contact radius of the viscoelastic body decreases more rapidly than that of an elastic material. This effect is somehow reduced by the presence of an adhesion force in the retraction section (red dots). The mechanical response of the body has been simulated with an elastic modulus of 4 kPa and a viscous coefficient of 300 Pa s. These values are similar to those obtained for a MEF cell (see below).

The above comparison shows that during the tips approach and near the maximum indentation, the contact area of the deformation of a linear viscoelastic body is well approximated by the contact area obtained from an elastic contact mechanics model.

Two relevant conclusions are derived from the data shown in Fig. 3. First, there is not a general trend in the dependence with respect to the frequency. The above observation implies that to understand and characterize the nanorheology of a single cell it is required to measure several parameters. Second, live cells show significant changes in the mechanical response in a relatively small frequency range.

The data shown in Fig. 3 report the measurements performed on a single cell. We have repeated the measurements on 20 different MEF cells. All the cells reproduced the frequency dependencies described in Fig. 3.

To simplify the discussion we limit the measurements to a single modulation frequency (1 Hz). Fig. 4a shows some optical microscopy images that illustrate the shape changes of a MEF under the action of Cyt-D (see the ESI:† time evolution of the MEFs affected by the Cyt-D). The cell increases its volume and becomes more rounded in the presence of Cyt-D. The evolution of the elastic modulus, the viscosity coefficient, the loss modulus and the relaxation time as a function of the exposure time to Cyt-D is shown in Fig. 4b–e. The elastic modulus of the cell decreases in the presence of Cyt-D. These results are in agreement with previous studies.^{4,53} However, the method allows us to observe three time domains. First, there is a sharp decrease from 6.2 kPa to 3.6 kPa that happens within the first 5 minutes. This decrease is characterized by a slope of −10 Pa s^{−1}. Between 5 and 30 minutes, the elastic modulus changes from 3.6 to 1.6 kPa. The rate is reduced to −1 Pa s^{−1}. After 30 minutes, the elastic modulus decays very slowly towards 1 kPa with a slope close to −0.3 Pa s^{−1}. The above results are in contrast with the trend observed for the viscosity coefficient and the loss modulus. The viscosity coefficient decreases monotonically with time from 165 Pa s to 90 Pa s. The loss modulus decreases from 3.2 Pa to 1.7 kPa. The rate is −0.4 Pa s^{−1}. The relaxation time increases with the exposure time from 0.07 s to 0.28 s. The relaxation time reflects a competition between elastic and dissipative processes within the cell. The treatment with Cyt-D softens the cytoskeleton structure and reduces the viscous flow. However, the softening of the cell happens at a faster rate that the reduction of the viscosity, thus, the increase of τ as the exposure time increases.

The above results are consistent with the behavior described by the poroelastic model.^{26,54} In this model, the cytoplasm consists of a porous, elastic solid (cytoskeleton, organelles, ribosomes) filled with an interstitial fluid (cytosol) that moves through the pores in response to pressure gradients. The disruption of the F-actin polymerization weakens the cytoskeleton architecture, and as a consequence the elastic modulus decreases. On the other hand, the fragmentation of the acting filaments is equivalent to the increased average pore size. If the size of the pores is increased it becomes easier to displace the cytosol through the cytoplasm, and then the viscosity decreases.

The method has been applied to follow the evolution of a fibroblast cell subjected to the depolymerization of its F-actin cytoskeleton with a time resolution of 1 minute. The loss modulus and the viscous coefficient monotonically decrease with the exposure time to the cytochalasin D. The relaxation time increases with the exposure time from 0.07 s to 0.28 s. The elastic modulus has three different time domains. First, there is a sharp decrease from 6.2 kPa to 3.6 kPa that happens within the first 5 minutes. Between 5 and 30 minutes, the elastic modulus changes from 3.6 to 1.6 kPa. After 30 minutes, the elastic modulus decays very slowly towards 1 kPa with a slope close to −0.3 Pa s^{−1}. The proposed method is general. It can be applied to study different types of cells and different types of cell–drug interactions.

The above behavior is compatible with a cell mechanical response described by the poroelastic model. The fragmentation of the actin filaments reduces the elastic modulus. At the same time, the depolymerization of the actin filaments leads to an increase of the average size of the cytoskeleton pores, which increases the cytosol flow. This process is equivalent to a reduction of the viscosity coefficient. The increase of the relaxation time implies that the weakening of the cytoskeleton structure as measured by the elastic modulus is a factor that dominates over the reduction of the viscosity.

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## Footnotes |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7nr03419a |

‡ These authors contributed equally to this work. |

This journal is © The Royal Society of Chemistry 2017 |