Viscoelastic relaxation of fibroblasts over stiff polyacrylamide gels by atomic force microscopy

Cell viscoelasticity provides mechanistic insights into fundamental biological functions and may be used in many applications. Using atomic force microscopy in time and frequency domains, we find a peculiar behavior in the viscoelastic relaxation of L929 mouse fibroblasts that may help understand how cells perceive and adapt to distinct extracellular environments. They are stiffer when cultured over polyacrylamide gels (20-350 kPa) than over glass-bottom Petri dishes. The stiffness enhancement of cells over gels is attributed to a significant increase in the low-frequency storage shear moduli compared to the loss moduli, indicating that gels induce a remodeling of cytoskeleton components that store elastic energy. Morphological alterations are then expressed by the fractal dimension measured on confocal images of the f-actin cytoskeleton. We show a direct scaling between the fractal dimension and the substrate’s rigidity.


Introduction
Living cells hold complex mechanosensitive machinery composed of many biochemical gears that propagate information from the extracellular matrix (ECM) to the interior of the cells and vice versa. These mechanisms allow cells to sense and adapt to the topography, mechanics, and biochemistry of the environment to execute biological functions like migration, proliferation, morphogenesis, tissue differentiation, and regeneration [1][2][3][4][5][6][7][8][9][10][11][12]. To perform such sophisticated functions, cells sense forces through conformational changes in focal adhesion proteins (e.g., integrins, talin, paxillin, vinculin) that bind to the cytoskeleton inside the cell to the ECM [13][14][15]. Although it is not entirely clear how those proteins work together to perform mechanosensing and mechanotransduction functions, a recent work showed that the stoichiometry of those proteins in the cellmatrix adhesion depends on substrate rigidity [16].
Among the physical cues perceived by cells, the stiffness of the ECM is crucial. It is well known that cells grown on rigid substrates exhibit larger areas and longer actin stress fibers than those grown on soft substrates [17][18][19][20][21]. The elastic moduli of tissues in the human body range within many orders of magnitude: 10 2 Pa for the soft tissues, 10 4 Pa for muscles, and 10 9 Pa for cartilage and bones [18]. Abnormal changes in the ECM stiffness may trigger disease appearance and progression. Besides rigidity, cells can also sense viscous dissipation in viscoelastic substrates so that cell spreading is maximized for intermediate viscosities on soft substrates, and independent of viscosity on stiff substrates [3,6].
It is widely accepted that cells modulate their stiffness in response to substrate rigidity [18,[22][23][24][25][26][27][28]. Many reports pointed out a direct proportionality between cell and substrate rigidity. However, some recent works reported new behaviors. For instance, Rheinlaender et al showed that the rigidity of the cell cortex is invariant concerning substrate stiffness [29]. Contrary to the belief that cancer cells are softer than healthy ones [30,31], Rianna et al showed that this trend seems valid only for cells cultured on stiff substrates. When cultured on soft gels, cancer cells become stiffer [8]. These reports reveal the gap of information about how cells interpret and adapt to the mechanical properties of the extracellular matrix.
This work investigates the viscoelastic properties of L929 mouse fibroblasts cultured on stiff polyacrylamide gels using force measurements in both time and frequency domains with Atomic Force Microscopy (AFM). The gel stiffnesses studied in this work range between 20 kPa and 350 kPa, thereby compatible with the stiffness of the ECM in muscles and bones [15]. We show that the stiffness of the substrate induces alterations in the internal viscoelasticity of those cells in such a way that they are stiffer over collagen-treated polyacrylamide gels than over glass substrates treated or not with collagen. In contrast, the fluidity level of the cells increases for increasing substrate stiffness.

Methods
Cell culture. L929 mouse fibroblasts were cultured on bare glass (GLS), collagen-treated glass (CLG), and polyacrylamide (PAA) gels with different concentrations of bisacrylamide to modulate gel stiffnesses. Cells were grown in high-glucose Dulbecco's Modified Eagle's Medium (GIBCO, USA) supplemented with 10% fetal bovine serum (GIBCO, USA) and 1% penicillin-streptomycin and incubated at 37°C in 5% CO 2 . Before AFM measurements, one-third of the medium was replaced by PBS solution to keep pH stable out of the incubator. All measurements were performed up to 2 hours after cells were moved out of the incubator.
PAA gels. PAA gels were obtained by mixing a stock solution of acrylamide 40% and bisacrylamide with concentrations of 0.01%, 0.1%, and 1.0% in volume to obtain gels with increasing stiffnesses. After polymerization, the gels were rinsed with PBS and attached to glass Petri dishes with 2.5% glutaraldehyde for 30 minutes at room temperature (RT) and exhaustively rinsed with PBS to remove non-reagent material. Then, the gels were treated with a 500 μg/ml collagen type I solution and refrigerated overnight. The gels were soaked for 6 h in a culture medium at 37°C. Finally, 3 ml of medium with 5 × 10 5 cells/ml were seeded and left to attach overnight. Before measurements, two-thirds of the medium was replaced by PBS to maintain pH out of the CO 2 incubator. The resulting average elasticity moduli of the gels are 23.3 kPa (0.01%), 49.6 kPa (0.1%), and 337.6 kPa (1.0%).
AFM measurements workflow. Cells were probed with an Asylum Research MFP3D-BIO coupled to a Nikon IX51 inverted optical microscope. Measurements were performed at RT in nearly identical conditions. We used regular AFM cantilevers (pyramidal tip) with nominal spring constants of 0.08 N/m. Determining the viscoelastic properties of soft materials with pyramidal indenters is much less susceptible to finite thickness effects than with spherical and flat indenters [32,33]. A maximum force F 0 = 3 nN and a piezo extension frequency of f z = 1 Hz (cantilever speed of 6 μm/s) were employed in all measurements. Figure 1 shows the measurements and data analysis workflow adopted in this work. We combine dwell force curves (DFCs) with a small amplitude (30 nm) frequency-sweep z modulation 1 s after the start of the dwell portion with frequency components varying from 1Hz up to 100 Hz to obtain the so-called frequency-sweep dwell force curves (FS-DFC), depicted in figure 1(b) [34]. The approach portion of the force curve is analyzed to estimate the fast and slow relaxation exponents and the crossover timescale t c between relaxation regimes, shown in figure 1(c) [34,35]. The oscillatory portion of the FS-DFC is analyzed in the frequency domain to obtain the complex dynamic shear modulus *( ) According to linear deformation theory, the relaxation function E(t) and G * (ω) are connected by *( ) , and the viscoelastic regimes of the cells must appear in both E(t) and G * (ω). The theory to determine the viscoelastic properties of samples from FS-DFCs is briefly described in the Supplementary Material (SM). All measurements were performed in liquid, and adhesion effects between tip and sample are disregarded. All measured cells exhibited at least h = 5 μm of height in their thickest part, and the deepest indentation achieved in our measurements ranged between 1 μm and 2 μm, depending on the type of substrate. According to the nanoindentation theory of thin viscoelastic materials of reference [33], this represents a maximum of 20% of indentation depths compared to the sample thicknesses, which exhibits nearly negligible errors in the determination of viscoelastic parameters when using DFC's.
The measured data is pooled as follows. For each substrate, a total of n = 15 cells were measured. In each cell, we acquired 16 force curves equally distributed over an area of 4 μm × 4 μm to probe different cell regions, and calculated the median values of the viscoelastic properties per cell. The histograms shown in this work present the distribution of median values of the cells, which also carry important information about intercellular variability.
Confocal imaging and fractal analysis. Cells were fixed with paraformaldehyde (4% in PBS for 15 min), permeabilized with Triton X-100 (0.5% in PBS for 30 min), and treated with BSA (3% in PBS for 60 min). F-actin filaments were stained with phalloidin (5 μg/ml in PBS). Cell nuclei were stained with 4',6-diamidine-2phenylindole hydrochloride (DAPI) (100 ng/ml in PBS). Confocal images were obtained at RT with a laser scanning confocal microscopy system LSM 710 (Zeiss, Jena, Germany) with excitation at 405 nm and 488 nm and peak emission at 457 nm and 518 nm for the nuclei and F-actin, respectively.
For the fractal analysis, confocal images of the f-actin network of individual cells were binarized using Otsu's threshold method [36]. The box-counting method was applied to the binarized images to calculate the fractal dimension D f of the cells. To avoid interference of the image orientation, the cell images were rotated 360 o by increments of 15 o , and the final D f of each cell is the average of such rotations [37]. The cell area is also determined from the analysis of the binarized images. The number of cells in this analysis were n 0.01% = 35, n 0.1% = 26, n 1.0% = 32, n CLG = 30 and n GLS = 23.
Force model. The viscoelasticity of living cells has been modeled with either one or two power-law (PL) regimes [35, 38,39]. It is then instructive to describe the single power-law (SPL) force curve model and expand it to the double power-law (DPL) model. The viscoelastic relaxation function of a single PL material in the time domain may be written as is a reference value of the elasticity modulus at an arbitrary time t = t ref , and β is the relaxation exponent that lies in the range 0 β 1. The corresponding dynamic complex shear modulus in the frequency domain * = ¢ +  G G iG , where ¢ G is the storage modulus and G″ is the loss modulus, is given by where f 0 is an arbitrary scaling frequency (not necessarily =f t ref . In the SPL model, the phase angle between ¢ G and G″ is constant . As cells are known to exhibit frequency-dependent rheological properties [40][41][42][43], the SPL model allows to define a dynamic fluidity exponent γ(ω) directly from the oscillatory portion of the FS-DFCs using that must vary between γ(ω) = 0 (perfectly elastic solid) and γ(ω) = 1 (viscous fluid). For γ(ω) = 0.5, the material is equally solid and fluid. The dwell force curve (DFC) for materials obeying a single PL is given by In these equations, t is measured from the contact point, t l is the loading time, δ 0 is the maximum indentation depth, and λ and Ω λ are parameters that depend on the indenter geometry. In the particular case of conical/pyramidal tips, one has λ = 2 and An extension of the SPL force model for samples exhibiting a double PL viscoelastic relaxation behavior is straightforwardly obtained (see SM). The DPL model provides four parameters: the elasticity moduli E α (t l ) and E β (t l ), and relaxation exponents associated with the fast (α) and slow (β) relaxation regimes, respectively. Both moduli can be scaled for an arbitrary reference time t ref according to the single PL scaling rule of equation (1). However, the fast relaxation only dominates in the first few milliseconds of indentation, and E α (t) quickly vanishes before t ≈ t l . Therefore, our analysis focuses only on E β (t l ), and the label β will be dropped in the next sections.
The removal of the substrate effect from E β (t l ) is performed assuming that the system cell + substrate can be well approximated by an association of elastic springs in series [29]. The detailed derivation is provided in the SM. We anticipate that the actual elasticity moduli differ from the measured ones (due to the substrate effect) by approximately 0.7%, 0.2%, and 0.04% for the 0.01%, 0.1%, and 1.0% gels, respectively, and 0% for the CLG and GLS substrates. Therefore, for the range of substrate stiffnesses used in this work, the measured E β (t l ) values are approximately equal to the actual cell's elasticity moduli. Elastic analysis. Although FS-DFC curves allow obtaining the viscoelastic properties of samples in a wide range of timescales, many hints of the substrate effect on the cell can be determined from the approach curves alone. Figures 3(a)-(b) show the maximum indentation depth δ 0 and loading times t l grouped by substrate type. The indentation depth of cells on gels is approximately 1 μm, lower than the maximum indentation of cells on glass (approximately 1.5 μm). The effective spring constant, defined as d ¢ = k F 0 0 and shown in figure 3(c), is 25% larger for cells over gels than cells over the glass, with no clear relationship between ¢ k and the elasticity moduli of the gels (E s ).

Results
Although these findings were obtained for indentations of the order of 1 μm (deep in the cytoplasmic crowd), the same analysis focusing only on the indentation of the cells cortex (up to 400 nm) shows that the average force to produce such an indentation in cells over gels is ≈500 pN, while in cells over glass is only 300 pN. Remarkably, the loading times to reach such depths are identical for all cells, indicating that the cortex's timedependent (viscous) response is similar in all substrates despite the stiffness enhancement the cortex of cells on gels compared to that of cells on glass. From the simple perspective of cells described as elastic springs, it is clear that cells are sensing the substrate stiffness and adapting their internal and cortical structures to the external environment so that cells become stiffer over the gels.
Viscoelastic analysis in time domain. The comparison of ¢ k among samples must be performed carefully because time is implicitly involved. Indeed, the SPL model shows that which is a time-dependent quantity. Although the piezo extension rate roughly controls t l (1 Hz in all measurements), non-negligible variations in t l are expected due to local inhomogeneities of the cell properties [38]. The values of t l for all substrates are shown in figure 3(b), where the role of the substrate appears again. For cells over gels, t l is shorter than in cells over glass, indicating that the force in the cantilever reaches the trigger value of 3 nN faster in the gels. Figure 3(d) compared the distributions of ¢ E scaled to t ref = 1 s, which is a relevant timescale for biological functions at the cellular level. The average values of ¢ E seem nearly identical among gels and do not exhibit significant differences. A small variance is also observed by grouping cells over the glass. However, cells over gels are significantly stiffer than cells over the glass. Figure 4(a) shows the distribution of relaxation exponents of the L929 cells in different substrates. The fast relaxation exponents (green boxes) lie around 〈α〉 ≈ 0.6 and do not exhibit significant statistical differences among substrates. However, the distribution of slow relaxation exponents (yellow boxes) presents median values 〈β〉 within the 0.15 − 0.20 range. The statistical comparison of all groups reveals that they are in the lower limit to be considered statistically independent (p < 0.05). It is also observed that the β exponents grow slightly with the gel stiffness. The broader distribution of α compared to β is explained because α is estimated from the portion of the force curves comprising the first 400 nm of indentation after the contact point, which is noisier and has fewer data points than the portion of the force curves within slow relaxation time-span.
The increase in the substrate stiffness also increases the crossover timescale t c between fast and slow regimes, as shown in figure 4(b). In recent work, it has been shown an inverse relationship between t c and the motility of living cells in wound-healing experiments [35]. Although we have not performed motility assays in the present work, those results suggest that the rigidity of the substrates reduces the motility of the L929 fibroblasts, as has been previously found for 3T3 mouse fibroblasts in substrates with increasing stiffnesses [44,45].
The low-frequency storage ( ¢ G ) and loss (G″) moduli determined by the DPL model are shown in figures 4(c)-(d). The direct comparison of ¢ G and G″ in gels and glass shows that the effect of the gels is more substantial in the storage moduli ( ( ) The effective viscosity, defined as η(ω) = G ″ (ω)/ω, is also larger for cells over gels than over glass at 1 Hz. It seems contradictory that an increase in fluidity is accompanied by a reduction in viscosity as the substrate stiffness increase. However, cells are active viscoelastic materials where the complex interaction with substrate changes both ¢ G and G ″ moduli. The degree of fluidity is estimated from the ratio  ¢ G G , while η is determined only by G ″ . This apparent conflicting behavior is only possible because the impact of the substrate in ¢ G is more significant than in G ″ . It is clear from figure 4 that the gels are inducing two alterations in the cells: (i) increasing their stiffness in such a way that most significant changes occur in the internal components responsible for their elastic response and (ii) increasing their fluid character. As the f-actin cytoskeleton is an essential load-bearing structure in cells, it is expected to be directly influenced by substrate rigidity. Later in this section, we will discuss how the substrates modify the organization of the f-actin in the cells.
Viscoelastic analysis in frequency domain. figures 5(a)-(b) show the dynamic storage and loss moduli for cells in different substrates at 1 Hz and 100 Hz. At 1 Hz, we observe that ¢ >  G G for all groups indicating that cells exhibit a predominantly solid behavior, and this trend holds for frequencies up to 80 Hz (not shown). The groups of cells over gels exhibit more significant values of ¢ G compared to those over glass. However, there are no significant differences in G″ among groups. These findings agree with the analyses in the time domain where we found that cells over gels are stiffer than cells over glass, and the changes induced by the gels are predominant in ¢ G . At 100 Hz, both ¢ G and G″ strongly increase compared to 1 Hz, but exhibiting ¢ <  G G for all substrates. The inversion between ¢ G and G″ occurs because the viscous response dominates at high frequencies. Such a behavior has been previously observed in crosslinked actin networks and living cells [41,43,46,47].
At 1 Hz, we observe that γ(ω) slightly increases with the substrate stiffness. This trend was also observed for the slow relaxation exponent β of the time domain analysis. The exponents γ and β also agree quantitatively, showing that cells are predominantly solid at low frequencies, with a differential enhancement of fluidity as the substrate stiffness increases. At 100 Hz, the fluidity of cells in all substrates is raised. The softest gels (0.01% and 0.1%) exhibit 〈γ〉 ≈ 0.7, while the stiffer substrates exhibit 〈γ〉 ≈ 0.52. The fluidity exponent at high frequencies is equivalent to the fast relaxation exponent α of the DPL model. In the time domain analysis, we found 〈α〉 ≈ 0.6 for all substrates, which is in qualitative agreement with γ at 100 Hz.
Cytoskeleton morphology. The f-actin cytoskeleton morphology is well known to be sensitive to substrate stiffness [20,21,48]. We perform a quantitative analysis of the cell's cytoskeleton to determine whether there is any correlation between cell rigidity and cytoskeleton morphology. Investigating changes in the cytoskeleton is difficult due to its self-similarity, i.e., a property where structural patterns replicate in different scales. A recent work quantified cytoskeleton morphology in terms of the orientation of the f-actin fibers [20]. Instead, we determine the fractal dimension D f of confocal images of individual cells, as shown in figure 6. In a rough sense, the fractal dimension indicates how efficiently a structure fills the space it is inserted in. Cells over glass exhibit longer f-actin stress fibers, while cells over gel exhibit shorter fibers with a higher concentration of f-actin in the cell periphery and protrusions. The average D f of individual cells grows slightly with the substrate stiffness. The comparison among groups shows that these distributions are statistically different, but there are no significant differences between the groups of stiffer substrates (1.0%, CLG, GLS). Moreover, D f scales inversely with the complex shear modulus * | | G c and directly with the fluidity exponent γ (at 1 Hz) with significant correlation coefficients. Therefore, it is clear that the cells are adapting their cytoskeleton to changes in the substrate stiffness. In addition, the fractal dimension shows to be a convenient method to quantify differential changes in the cytoskeleton morphology.

Discussion
Using measurements in both time and frequency domains, we found that L929 mouse fibroblasts seeded over polyacrylamide gels are stiffer than cells seeded over glass. Both methods revealed a viscoelastic relaxation compatible with a double PL relaxation regime, where the fast relaxation regime shows an exponent 〈α〉 ≈ 0.6 independent of the type of substrate, and the slow relaxation regime exhibits an exponent 〈β〉 ≈ 0.2 that increases as the substrate stiffness increases. Since the fast relaxation regime is attributed to the entropic response of the individual f-actin filaments [41], it is reasonable to expect that substrate rigidity does not affect the exponent α. In fact, our results shown in figure 3(b) demonstrate that the cortical layer indentation loading times are substrate-independent. On the other hand, the slow relaxation regime is attributed to the response of the cytoplasmic crowd, thereby related to the architecture and deformability of the whole cytoskeleton. The storage shear moduli of cells over gels are enhanced at low frequencies compared to cells over glass, implying that the cells are remodeling their f-actin cytoskeleton. This is in agreement with Fusco et al who shown that the Brownian dynamics of beads embedded in the cytoplasm of 3T3 fibroblasts were highly dependent on the substrate stiffness as a consequence of their cytoskeleton remodeling [49]. Here, we quantify the remodeling of the f-actin network in terms of its fractal dimension, which exhibits a positive correlation with the substrate stiffness and fluidity exponent, and a negative correlation with |G * |.
To phenomenologically explain the stiffening of the L929 cells over stiff PA gels, we consider the hypothesis of prestress [50,51]. Prestress is the internal tension in the cell cytoskeleton necessary to maintain its structure and function. The minimal cell model of Fischer-Friedrich shows that the effective elasticity modulus of cells in the presence of cytoskeleton prestress can be written as [52]: where E c (σ p , ω) is the prestress-and frequency-dependent elasticity modulus of the cell cytoskeleton and σ p is the active prestress in the cytoskeleton. The adimensional factor A can be incorporated in the prestress as Aσ p = σ p . There are two limiting cases: (i) an adherent cell over an infinitely stiff substrate (e.g., glass) with an effective modulus E c,stiff , and (ii) an adherent cell over a soft substrate with an effective modulus E c,soft , where the prestress and frequency dependencies were dropped. Those quantities are connected by where Δσ p is an additional prestress. E c,soft and Δσ p depend on the substrate stiffness, surface treatment, and even on the type of cell. For E s → ∞ , E c,soft → E c,stiff and Δσ p → 0. Moreover, for Δσ p < 0, the cells over soft substrates are stiffer than cells over stiff substrates and vice-versa. Rheinlaender et al [29] identified that the total indentation depth of an AFM tip in a cell over a soft substrate is given by δ = δ c + δ s , where δ c is the indentation of the AFM tip in the cell, and δ s is the indentation of the cell in the substrate. The total indentation is equivalent to the association of two hookean springs in series, where each element feels the same external force that is applied to the whole system. In this case, we can use Sneddon's model [53] in each element of the cell/substrate system to obtain where F is the force that the AFM tip exerts vertically to cell/substrate system, ¢ E is the measured elasticity modulus of the compound system, E c,soft is the elasticity modulus of the adhered cell, and E s is the elasticity modulus of the substrate. λ is the indenter exponent of the AFM tip, and λ s is the indenter exponent of the cell bottom indenting the substrate. These formulae allow us to obtain an analytical expression for the measured Some recent data [8,18,[22][23][24][25][26][27]29] on the mechanobiology of living cells over soft substrates are compiled in figure 7. Most of these works agree with our phenomenological model (solid blue line) with Δσ p = 0. Minor variations of the experimental data for Δσ p = 0 line are possible due to different experimental conditions (e.g., indenter geometry, surface treatment) adopted in each report. The model for Δσ p = 0 is in excellent qualitative agreement with the finite element modeling of the nanoindentation of half-elastic spheres over soft substrates with different indenter geometries of reference [29].
A few data sets, though, disagree with the theory (L929, A498, ACHN, and MDA-MB-231 cells), for which cells over gels are stiffer than over glass. This behavior cannot be solely described in terms of the stacking of materials with distinct elastic properties (Δσ p = 0) because this assumption ignores that cells actively exert  [29]. In contrast, ¢ E is the measured elastic modulus, E c is the elastic modulus measured in glass, and E s is the elastic modulus of the substrate. The normalization is performed with the data measured in the stiffest substrate for the works that did not probe cells over the glass. The open and closed symbols represent cancer and normal cells, respectively. The curves represent the theoretical model of equation (9)  forces and adapt to the external environment. Except for ours, the diverging cases are cancer cells, which are able to degrade the ECM and invade tissues. For instance, Rianna et al showed that A498 and ACHN cells indented the gels they were seeded, resulting in a roundish-like cell/substrate interface of a few micrometers of depth. Such ability requires the remodeling of the cytoskeleton to apply both traction (parallel to the cell/substrate interface) and protrusion (perpendicular to the interface) forces to the substrate. Recent optical interference measurements have shown that focal adhesion are able to exert traction forces of the order of a few nN in fibroblasts [54]. At the same time, mature invadopodia of highly invasive cancer cells produce protrusion forces of 10 pN [55]. Li et al showed that breast cancer cells exert traction forces of 15 nN, more than double the forces exerted by normal cells [56], and metastatic cells can exert even larger forces than their non-invasive counterparts [57,58]. Departing from the assumption that the remodeling of the cytoskeleton affects the internal prestress of cells, a constant value of Δσ p = − 0.8 kPa raises the plateau of the theoretical curve to the order of magnitude of the diverging cases. Such a value is comparable to the traction stress exerted by invasive cancer cells [57]. On the other hand, a value of Δσ p = + 0.2 kPa lowers the plateau and nearly matches the experimental data of Yousafsai et al (HBL-100 and MCF-7).
Although the hypothesis of a constant value of Δσ p is enough to qualitatively adjust the phenomenological model to match the experimental data in figure 7, it has two weaknesses: (i) it does not explain the large volume of experimental data demonstrating that the traction forces exerted by non-invasive cells increase with the substrate stiffness [18], and (ii) it does not obey the normalization condition for E s /E c,stiff → ∞ . Since our model requires that Δσ p → 0 for E s → ∞ , we adopted Δσ p = ae − bEs , where a E c,stiff and b is parameter that controls the slope of the curve. The resulting theoretical curve (dash-dotted curve in figure 7) is similar to the Δσ p = 0 case. Increasing values b makes both curves identical. This means that the simple model of the stacking of materials with distinct elastic properties can only explain the mechanical properties of wellbehaved living cells over soft substrates. By well-behaved, we mean cells whose traction forces and cell area increase with E s . However, the Δσ p profile of well behaved cells does not adjust the data of the diverging cases. Then we adopt , which is a superposition of the profile of well behaved cells with a peak reduction of additional stress around a target substrate stiffness E s,0 of width given by d. In the case of cancer cells, such a form is physically justified by a stiffness range that favors cell invasiveness [59]. For such profile, the diverging data are qualitatively adjusted to the theoretical model (red line).
Cells exhibiting E c,soft > E c,stiff may also be explained by nonlinear mechanical responses. Kollmannsberger et al showed that cells stiffen when subjected to increasing forces and that the cellʼs elastic moduli can nearly double by increasing successive force steps [60]. Since indenting cells can exert forces of the order of tens of nN on the substrate, the counterbalanced forces applied by the substrate on them have the same magnitude. Therefore, nonlinear stiffening may play an important role during substrate indentation [61].
Finally, most of our justification of the peculiar enhancement of cell stiffness over stiff polyacrylamide gels was based on the raised levels of prestress of invasive cancer cells. Although we have not measured whether the normal L929 fibroblasts are indenting the gels, it is well known that migration and infiltration are essential roles of fibroblasts in wound healing processes [62,63]. The work of Kollmannsberger et al, who showed that the larger the prestress, the lower is the relaxation exponent [60], strongly suggests that L929 cells over the 0.01% gels are subjected to larger prestress than cells in CLG and GLS substrates.
It is important to note that the gels used in this work are much stiffer (20-350 kPa) than the L929 cells, and most mechanotransduction mechanisms are expected to occur in soft substrates where E s /E c,stiff  1. For instance, the data of A498 and ACHN cells in figure 7 were measured in polyacrylamide gels with 3 kPa, 17 kPa, and 31 kPa of rigidity, while those cells measured roughly E c,stiff ≈ 1 kPa over stiff Petri dishes [8]. However, 18% and 42% stiffening are observed in the A498 cells over the 31 kPa and 17 kPa gels. For the ACHN cells, significant 26% stiffening and above was only observed for the gels with 17 kPa and below. Even more remarkable are the data of MDA-MB-231 cells from Yousafsai et al [27], whose PDMS substrates measured 17 kPa, 88 kPa, and 173 kPa, and their respective stiffness-enhancing are 39%, 81%, and 89%. Our results, as Riannaʼs and Yousafsaiʼs data, reveal that mechanotransduction processes also occur for substrates were E s /E c,stiff ? 1.

Conclusions
The study of the viscoelastic relaxation of L929 mouse fibroblasts cultured over stiff polyacrylamide gels and glass substrates in time and frequency domains revealed that these cells are stiffer over gels than over glass. The gels induced the remodeling of the f-actin network whose fractal dimension scales inversely with substrate rigidity and directly with the degree of fluidity of the cells. In particular, cells over gels exhibited larger storage ¢ G (at 1 Hz) compared to cells over glass, while the loss moduli G″ were weakly affected. The simplistic assumption of the stacking of two inert materials with different mechanical properties is not sufficient to explain the stiffening of the L929 cells because cells are able to exert forces and adapt their cytoskeleton to the ECM. By considering the hypothesis that cells are subjected to substrate stiffness-dependent prestresses, we were able to phenomenologically explain the stiffening of the L929 cells over stiff polyacrylamide gels. In conclusion, the study of the viscoelastic relaxation of cells on substrates with a wide range of stiffnesses is crucial to understand important mechanotransduction phenomena and, ultimately, may shed new light in the understanding of pathologies like fibrosis and cancer.