# Determination by Relaxation Tests of the Mechanical Properties of Soft Polyacrylamide Gels Made for Mechanobiology Studies

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Polyacrylamide (PAM) Hydrogels Preparation

_{2}O) and degassed in a desiccator for at least 20 min to fabricate soft PAM hydrogels of expected elastic moduli of 1.10 ± 0.34 and 4.47 ± 1.19 kPa [5]. Then, mixtures were polymerized by adding 0.01% (w/v) of ammonium persulfate (APS) and 0.001% (v/v) of Tetramethylethylenediamine (TEMED). To achieve a thickness of ~233 µm, 75 µL of the solution were deposited onto 20 mm round glass coverslips which were treated previously with (3-aminopropyl)triethoxysilane (APTES)/glutaraldehyde following [34]; the polymerization reaction was carried out during 20 min at room temperature. Polymerized hydrogels were rinsed with ddH

_{2}O once and immersed in Dulbecco’s phosphate-buffered saline (DPBS) 1× at 4 °C overnight to swell and equilibrate.

_{2}O. The solution was incubated at 37 °C for 2 h to ensure a complete polymerization. Then, mixtures of 4% acrylamide and 2% bis-acrylamide stocks were prepared to fabricate soft viscoelastic PAM hydrogels of expected dynamic modulus of G′ = 1.6 kPa and G″ = 200 Pa respectively, in ddH

_{2}O [4]. TEMED was added at the acrylamide/bis-acrylamide and linear polyacrylamide mixture and APS was added just before the deposition; mixtures were incubated for 20 min at room temperature. Once polymerized, hydrogels were rinsed with ddH

_{2}O and immersed in DPBS 1X at 4 °C overnight to swell and equilibrate. It is possible to define the dynamic moduli that will be calculated here by considering that the relation $E=\left(1+2\nu \right){G}^{*}$ is true for soft PAM hydrogels [35], with ${G}^{*}=\frac{{G}^{\prime}+{G}^{\u2033}}{2}$ and a Poisson ratio of ν = 0.457 [36]. Here, the storage and loss shear moduli that were prepared, therefore, should lead to an expected Young’s modulus of the viscoelastic PAM hydrogels of E = 1.723 kPa. These gels will be called “soft V-PAM” in the following sections.

#### 2.2. Poly-Dimethylsiloxane (PDMS) Preparation

#### 2.3. Mechanobiology Test with Fibroblast Culture

^{5}cells were seeded on PAM hydrogel (HG) conjugated with [100 µg/mL] rat tail collagen type I (from Corning Inc, Corning, NY, USA), as described in [38] and after 48 h of culture were fixed with 4% paraformaldehyde in DPBS at 37 °C for 15 min. Cells were permeabilized with 0.1% Triton X-100 and blocked with 10% horse serum. For immunostaining, samples were incubated with a monoclonal antibody against Yes-associated protein (YAP) at a dilution of 1:200 (sc-101199, from Santa Cruz Biotechnology, Dallas, TX, USA). After that, Alexa594-coupled secondary antibody was used for immunodetection (Jackson ImmunoResearch, West Grove, PA, USA). Actin filaments and nuclei were detected by Alexa488-coupled phalloidin and 4′, 6-diamidino-2-phenylindole dihydrochloride (DAPI) staining (from Molecular probes, ThermoFisher Scientific, Waltham, Massachusetts, USA), respectively. Samples were mounted with Mowiol over a rectangular coverslip following [39]. Samples were visualized with an epifluorescence microscope Eclipse Ci-L coupled to a D750 FX digital SLR camera (from Nikon, Tokyo, Japan). Images were captured by using a Plan Fluor 40× objective. The images were quantified, edited and merged by using the open source image processing package Fiji. The cell spreading was measured as the area detected by phalloidin (actin filaments) in isolated cells only. Cell density was measured by counting the number of nuclei (DAPI) covering an area of 547.95 µm × 365.3 µm. Finally, localization of nuclear YAP protein was calculated by measuring fluorescence intensity, using the corrected total cell fluorescence [CTCF = Integrated Density − (Area of selected cell × Mean fluorescence of background readings] [40] in the area merged by DAPI.

#### 2.4. Microindentation and Relaxation Tests

#### 2.5. Force Curves Processing

#### 2.5.1. Determination of the Contact Point

#### 2.5.2. Indentation Data Analysis

_{r}and the separation distance at equilibrium z

_{0}(defined in [46] as the range of attraction of adhesive forces, close to atomic distance and ranging in 0.1–0.3 nm) must be determined [42,47]. Here, we used z

_{0}= 0.3 nm [48] and $\Delta \gamma $ was determined using the magnitude of the adhesion force F

_{adh}in the retraction section of the curve of Figure 2D in the following Equation (2):

_{r}can be calculated from Equation (3):

#### 2.5.3. Relaxation Data Analysis

_{max}was detected to extract its associated time t

_{Fmax}and displacement d

_{Fmax}values, which in turn allowed us to determine the indentation depth δ as d

_{Fmax}-d

_{c}where d

_{c}is the contact distance, as defined by the microindenter. Then, the GMM was used to determine the relaxation modulus E(t) as follows [49], Equation (4):

#### 2.6. Statistical Analysis

## 3. Results

_{l}is combined in parallel with a consecutive association of N parallel dashpots and springs (depicted as (η

_{i}; k

_{i}) in Figure 1B). The GMM is advantageous over the simpler viscoelastic linear models, such as the standard linear solid (SLS) model since it considers the non-homogeneous disorder at microscale: the material relaxation occurs according to a time distribution rather than at a single time. This is a definition that falls closer to the mechanics of a substrate as perceived by biological cells, which appear to be able to maximize their spreading if their dissipative timescales match that of the material [20]. Moreover, relaxation tests have been proven to allow the separate quantification of the viscous and poroelastic contributions occurring in soft hydrogels used in mechanobiology [51]. To validate our relaxation test method, we have tested the mechanical properties of soft polyacrylamide (PAM) hydrogels and compared the results with classical indentation tests, either static or using oscillations over a frequency range, as classically used in material characterization (see Appendix C, Figure A8).

#### 3.1. Force Curves and Correction for Tip Displacement

_{2}O + 0.1% Extran while an identical soaked sample was measured in air (Figure 2). The role of the detergent solution (Figure 2A,C) was to reduce the surface tension of the liquid and thus avoid the tip–hydrogel attraction that was observed in all the samples moistened with only ddH

_{2}O water (Figure 2B,D).

^{2}= 0.999). Therefore, the difference between the indentation curve of the samples and the indentation curve of the glass is the real indentation depth (as shown in panel C).

#### 3.2. Influence of Velocity and Depth of Indentation

^{3}suggesting the use of a JKR model in order to fit the experimental data. However, the JKR shown in Figure A3 panel B clearly shows that this is far from being perfect to accurately fit the data.

#### 3.3. Relaxation Tests to Characterize PAM Mechanical Properties

_{max}before relaxation started. We registered the indentation depths for all measurements and they were very similar and varying accordingly with the softest being indented the most: 23.94 ± 1.17 µm for 4 kPa PAM hydrogels, 25.12 ± 3.85µm for the 1 kPa PAM hydrogels and 28.89 ± 3.09 µm for the viscoelastic gels. As expected, the different soft materials presented distinct temporal relaxation responses and the two purely elastic PAM hydrogels relaxed more rapidly than the viscoelastic one. Also, the stiffest (4 kPa PAM) elastic gel stopped relaxing after t = 50 sec and settled to a fixed plateau while the other two materials kept relaxing further, although at different rates. The dissipation of force by an elastic material at a constant strain may seem abnormal, but it has been explained by the poroelastic nature of PAM hydrogels: they are made of a porous elastic matrix interpenetrated by an interstitial fluid that can flow and escape, similar to a sponge [33,51,55]. This behavior may explain the hysteresis found in loading–retracting curves of Figure 2 and the difference between the relaxation rates of the two elastic gels thus lies in their porosity difference [56].

_{tip}is the radius of the probe tip, δ is the indentation depth and D is the diffusivity of the gel (typically ranging in the 10

^{−10}m

^{2}/s) [58]. It can be inferred from the data that the nature of each gel (different porosity and free volume) may explain the variations of the relaxation times found here. This also suggests that the viscosity of the medium will probably influence the mechanics of the gels bathed in it, as found in [59,60].

_{∞}of the 4 kPa PAM sample. When compared with the same graph constructed in Figure 3D for data obtained from the Hertz model, it is clear that the variations are smaller for the GMM-derived modulus, around a value of 6 kPa (which has to be compared to the higher ~10 kPa results obtained by indentation and the Hertz model represents at least 50% less variation than that obtained with microindentation). Interestingly, for indentation velocities of ~140 µm/s the calculated long-term stiffness E

_{∞}is almost independent of the indentation depth. This is consistent with previously reported results showing that a higher indentation velocity is preferable, for such experiments [55].

#### 3.4. Comparison between Microindentation and Relaxation Mechanical Tests

_{H}) while the JKR was preferred for the soaked samples (E

_{JKR}). For relaxation tests, two elastic moduli obtained from the relaxation data and 3rd-order GMM fitting are shown: the long-term elastic modulus E

_{∞}and the storage elastic modulus at 1 Hz E′(ω = 1 Hz). A significant difference between the value of the elastic modulus was obtained with the Hertz model between the data of the microindentation performed in detergent and the results computed from relaxation tests. This may be explained by the fact that relaxation tests results are more robust when adhesion effects have to be taken into account. Indeed, Figure 5A shows that there is no significant difference between the instantaneous modulus calculated from measurements of 4 kPa PAM samples either immersed or soaked. Moreover, the use of the relaxation spectra obtained from the GMM fitting enabled a further analysis of differences between the two conditions that helped explain the impact of the indentation velocity as depicted in Figure 3D. Although the relaxation spectra of the 4 kPa PAM samples measured in the two different conditions looked very similar, the samples characterized in detergent solution presented an additional relaxation time ${\tau}_{i}$ = 3.43 ± 0.24 s which is absent from the soaked samples measured in air (see Appendix A Figure A4A). The GMM viscosity associated with this relaxation time was calculated to be η

_{i}= 5.73 ± 0.67 kPa.s, a value very similar to what was reported for other soft hydrogels under compression [65,66], and could explain the increment of the apparent Young’s modulus when indentation takes place at higher velocities. Indeed, we recall that the instantaneous modulus sums up all the moduli and it only appears in the immersed samples and not in the soaked ones, suggesting that the material viscosity (resistance to flow) may be playing a role in the excess found here. Therefore, when aiming at describing the intrinsic mechanical properties of a material under test, the long-term modulus E

_{∞}is clearly an appropriate characteristic property to use as it appears to be mostly independent of the measurement conditions and is almost not impacted by indentation depth for high indentation velocities (as seen in Figure 4D).

**Table 1.**Comparison of values of elastic moduli obtained for different methods, materials and conditions: E = calculated Young’s modulus, E

_{∞}= long-term stiffness, E′ = storage modulus evaluated at 1 Hz and E

_{ref}= reported Young’s modulus. Detergent and soaked are the conditions depicted in Figure 2 and dry is a condition for which the sample is not moistened and measured in air at room temperature. The JKR model was used to fit experimental data for soaked and dry samples, and the Hertz model was used for samples immersed detergent solution. All reported values are mean ± standard deviation for each condition (two samples were characterized in at least 6 different locations).

4kPa PAM (kPa) | 4kPa PAM (kPa) | Soft V-PAM (kPa) | 1kPa PAM (kPa) | Stiff PAM (kPa) | PDMS 10:1 (MPa) | PDMS 15:1 (MPa) | ||
---|---|---|---|---|---|---|---|---|

measurement conditions | soaked | detergent | detergent | detergent | soaked | dry | dry | |

reported values [reference] | E_{ref} | 4.47 ± 1.19 [5] | - | 1.723 ^{1} [22] | 1.10 ± 0.34 [5] | 34.88 [5] | 1.35–2.01 [67] | 0.9–1.2 [37] |

micro indentation | E | 8.92 ± 0.55 | 12.60 ± 0.42 | 3.73 ± 0.27 | 3.37 ± 0.17 | 38.72 ± 8.71 | 0.91 ± 0.09 | 0.63 ± 0.01 |

relaxation tests | E_{∞} | 6.00 ± 0.57 | 6.41 ± 0.44 | 1.55 ± 0.31 | 1.63 ± 0.39 | 36.01 ± 2.67 | 1.61 ± 0.48 | 0.61 ± 0.17 |

E′ | 8.25 ± 1.47 | 8.78 ± 0.80 | 2.09 ± 0.40 | 2.57 ± 0.50 | 39.93 ± 2.54 | 1.68 ± 0.47 | 0.65 ± 0.16 |

^{1}expected E′, see Section 2.1.

#### 3.5. Relevance for Cellular Mechanobiology: Cell Response to PAM Hydrogels of Different Mechanical Properties

_{∞}= 1.6 kPa and E

_{∞}= 6.4 kPa instead, less than a 5 kPa difference. This much smaller stiffness difference between the two gels indicates the large impact on cells of small variations of substrate rigidities, which may occur from one preparation to the other if no particular care is taken in preparing the samples and conserving most stringently the components of gel preparation. These results hence offer a better precision and resolution of the range of the mechanical properties of a material to which the cells respond.

## 4. Discussion

_{∞}(Pearson correlation of 0.92, see Appendix C Table A1 and Figure A9) but the absolute values differ by less than expected. Biological cells seem very sensitive to slight changes in absolute gel stiffness, calling for very precise measurements of the mechanics sensed by cells, especially when designing biomimetic materials aiming at recapitulating native tissue properties. In addition to the determination of elastic moduli, PAM hydrogels are known to present dissipation and, here, it was possible to quantify it, as shown in the hysteresis curves in Figure 2. This needs to be accounted for in the characterization of materials as recent findings have shown that the relaxation times of culture substrates are critical in mechanobiology, controlling cell spreading dynamics in culture [20].

_{∞}, which is clearly an appropriate characteristic, independent of the measurement conditions and almost not impacted by indentation depth for high indentation velocities. It could then be useful to report long-term modulus E

_{∞}, in decellularized matrices or tissue slides to correctly mimic this mechanical cue in biomimetic polymers. Finally, the relaxation test method, combined with a correct GMM fitting, provides useful information on the dynamic mechanical behavior in a range of frequencies that is relevant in mechanobiology.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Notes on FD Curve Processing and Analysis

_{2}O + 0.1% extran solution (to avoid undesired attraction of the tip) the contact point coincides with the position at which indentation begins. However, in the soaked-only samples, it has been reported that the contact point is not necessarily there [14]. We then considered the change of the slope in the FD curve (its first derivative) as a useful way to identify the position of a switch in the force regime. Indeed, when performing indentation tests, the origin of the major contribution of the measured forces is changed from an attraction-led regime in which the force is proportional to the indentation depth δ (F

_{attraction}~ δ) to an indentation-driven regime in which F

_{indentation}~ δ

^{3/2}. If these two types of forces are the only ones present during the physical characterization, then the maximum of the second derivative precisely and objectively marks the beginning of indentation (see Figure A1). The variations of the slope in the FD curves are also observed in the FT curves. If ${F}_{indentation}\sim {\delta}^{3/2}$ and we suppose that ${F}_{indentation}=F\left(t\right)$ therefore $F\left(t\right)\sim \delta {\left(t\right)}^{3/2}\Rightarrow dF\left(t\right)\sim \frac{3}{2}\delta {(t)}^{1/2}dt$and so$dF/dt\sim \frac{3}{2}{\delta}^{1/2}$. On the other hand, we have $dF/d\delta \sim \frac{3}{2}{\delta}^{1/2}$ leading to $dF/d\delta =dF/dt$. This implies that the procedure to find the contact point in the FD curves for microindentation tests is equivalent for the FT curves obtained in relaxation tests. Figure A1 shows the determination of the contact point associated with the maximum of the 2nd derivative with respect to time computed for a relaxation curve for a 4 kPa PAM hydrogel immersed in detergent solution.

**Figure A1.**Example of the determination of the contact point, start of relaxation (maximum force) and indentation depth for a relaxation test performed on a 4 kPa PAM hydrogel immersed in ddH2O + 0.1% extran. The first and second derivatives of the force are calculated as a function of time. The superior graph shows the displacement of the probe tip as a function of time. The middle graph shows the time behavior of the measured force. The 1st and 2nd derivatives are calculated from this graph and shown in the inferior graph. Experimental data were interpolated and it was considered that the maximum of the 2nd derivative implied a concavity change associated with the change in force behavior (a linear behavior for the attractive forces domain and a F

^{2/3}for indentation forces domain) Thus the position, representative of the contact point is thus precisely determined by the maximum of the second derivative. Here, raw data are shown and time is therefore equivalent to displacement.

**Figure A2.**Correction for displacement of the indentation tip. (

**A**) FD curve representative of the indentation of glass (indentation of 100 µm/s and a frequency of data acquisition of 100 Hz). (

**B**) Hertz model and Hook model fittings of glass indentation data. (

**C**) comparison between the FD curve of the indentation of a PAM hydrogel measured immersed in ddH

_{2}O + 0.1% Extran (blue curve) and the FD curve of the indentation of stiff glass (red), associated to the displacement of the probe tip mechanism. The difference between them is the true indentation depth.

**Figure A3.**Representative curves of FD fitting processing. (

**A**) Hertz model fitting of the indentation of a 4 kPa PAM hydrogel in ddH

_{2}O + 0.1% Extran. The viscous drag was calculated using the slope of the approach curve (Figure 2C in the main text) obtaining 0.006 ± 0.005 µN/µm at an indentation speed of 100 µm/s, which is negligible compared to the ~0.3 µN change due to relaxation with the same indentation speed. (

**B**) JKR model fitting of the indentation of the same hydrogel PAM measured while only soaked. There is an underestimation of the slope (therefore the elastic modulus) when modelling data with distances lower than ~10 µm. Also, for indentation depths greater than ~11 µm the Young’s modulus reaches a plateau value which is similar to the behavior presented in Figure 3 in the main text. (

**C**) JKR model fitting of a stiffer hydrogel (~35 kPa, see Table 1) when only soaked in deionized water. There is a clear attraction of ~10 µN, twice that of the softer PAM hydrogel (~5 µN), suggesting a possible dependency of the tip attraction with the stiffness of the sample (acrylamide/bisacrylamide proportions).

**Figure A4.**To report on the experimental variability, we plotted a distribution of characteristic relaxation times (represented by Dirac deltas in Figure 4B of the main text), grouping the results of several measurements of different samples. (

**A**) Comparison between the relaxation spectra of a 4 kPa PAM hydrogel when measured either immersed in detergent solution or soaked but in air. While the material is identical, the experimental conditions differ and a relaxation time appears at 3.43 ± 0.24 s when the sample is immersed in detergent. Also, an additional relaxation peak appears at τ = 0.05 ± 0.05 s in the soaked condition. These differences are probably caused by relaxation effects due to poroelasticity when diffusion happens at different times (see main text). (

**B**) Comparison between the different relaxation spectra of samples measured in detergent solution. Each peak represents a normal distribution centered in τ

_{i}with a standard deviation σ, obtained by gathering values of τ

_{i}from 2 independent samples for each condition (n = 11). Note: A normal distribution was supposed, with the standard deviation of the data being the deviation of the distribution and the weight is represented by ${k}_{i}/{k}_{\infty}$. Because the distributions are gaussian distributions, a smaller peak represents a greater standard deviation, hence a greater dispersion (variability) of the data.

## Appendix B. Notes on the Criterion of Cross Validation to Determine a Proper Fitting

**Figure A5.**Third order GMM fitting (N = 3) to a relaxation curve of a 4 kPa PAM hydrogel immersed in ddH

_{2}O + 0.1% Extran. The superior graph shows the residues of the fitting. Experimental data is normalized before adjusting with a GMM fitting.

^{train}variables are obtained. Then, using these parameters, the validation data (the values that were not used) are fitted and the MSE

_{i}

^{test}of the validation set is calculated. This is iterated for all the N data points to average the cross validation MSE as a measure of the quality of the fitting.

**Figure A6.**Calculation of MSE as a criterion of cross validation to evaluate the quality of the GMM model fitting.

**Figure A7.**Example of the method of determination of best GMM fitting order for stiff PAM hydrogel samples. The left column shows the experimental data of 4 curves (yellow) together with their GMM fittings of different orders (corresponding to a different number of Maxwell elements, or arms). Each dot presented in the graphs at the center and right columns are the MSE criterion (center) and Akaike information criterion (right) for N arms in the GMM fitting, showing which order is the best.

## Appendix C. Validation and Comparison with Other Methods

**Figure A8.**GMM analysis of published experimental data (FT curve) obtained by nanoindentation-relaxation of a biological cell using AFM [61] and comparison of computed frequency behavior with experimental data of dynamic moduli. The data were recovered manually from the published article. This figure corroborates the usefulness of the proposed method to characterize soft materials used in mechanobiology by only using a simple FT curve with relaxation. The raw stress-relaxation data were collected from Figure 1 of reference [61]: a cell was indented and relaxed using an ARROW-TL1 cantilever with an attached 4.7 μm silica bead, and with a spring constant 0.05 N m

^{−1}. (

**A**) Graph reproducing the experimental relaxation data obtained from [61] together with the corresponding GMM 3rd order fit. (

**B**) Relaxation spectra associated with the hydrogel sample. We can observe that the relaxation times are of the same order as the ones found in Figure 4 and Figure A4, of our experiments. (

**C**) Representation of the dynamic moduli G′ and G″ obtained from our GMM model (calculated from E* using the relation in Section 2) and compared with the actual experimental data from rheology [61]. From frequencies of ~10

^{−1}s

^{−1}, the tendencies are very similar. It is striking that only with the data of the single relaxation curve of panel A the full frequency characterization is possible. Interestingly, in [61] the sample was indented 0.4 µm at a velocity of 30 µm/s; the indentation lasted 0.13 s corresponding to the highest frequency (10 Hz) and the relaxation lasted 30 s (~10

^{−1}Hz). It was obtained that G′(1 Hz) = 432.19 Pa and G″(1 Hz) = 98.75 Pa, values that are recommended to be used to compare the results obtained with other characterization methods. (

**D**) Losses in elastic energy are defined as tan(δ) = G″/G′ and it can be observed that the major loss is at frequencies around 10

^{−1}Hz and longer times (>10 s) dissipate less elastic energy.

**Figure A9.**Comparison between the different modules obtained with microindentation (E

_{H}) and relaxation (E

_{inst}and E

_{∞}) for a 4 kPa PAM hydrogel immersed in a ddH

_{2}O + 0.1% Extran solution.

**Table A1.**Pearson’s correlation between the Young’s modulus measured by microindentation (E

_{H}) and relaxation (E

_{inst}and E

_{∞}); in both cases a positive correlation was found indicating that if there is a change in the elastic behavior, both the Young’s modulus E and the long-term stiffness E

_{∞}will be affected in the same proportion since their correlation level is >0.9. On the other hand, with the instantaneous modulus E

_{inst}a correlation ~0.5 was obtained, which indicates a similar trend but not precisely in the same proportion. This effect is noted in Table 1 of the main text, where both the values of Young’s modulus and long-term stiffness of the 1 and 4 kPa samples measured in detergent maintained a 4-fold identical proportion between the expected values measured with a different technique (AFM). r (E

_{H}, E

_{i}) = Pearson correlation with E

_{H}.

E_{H} | E_{inst} | E_{∞} | |
---|---|---|---|

r (E_{H}, E_{i}) | 1.000 | 0.556 | 0.924 |

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**Figure 1.**Diagram (not to scale) of the method employed here to study with great precision the mechanical properties and dynamic behavior of polyacrylamide (PAM) hydrogels used in mechanobiology assays. (

**A**) Relaxation test performed with a microindenter; it collects all the information necessary to analyze the hydrogels mechanically. The bead is typically 50 µm in diameter and the gel ~200 µm thick. (

**B**) The data collected from the relaxation test is adjusted using a generalized Maxwell model (GMM) with the appropriate relaxation times τ

_{i}that then allows to obtain the corresponding k

_{i}and η

_{i}. The long term stiffness k

_{l}(or k

_{∞}) is also determined from the model fitting. Finally, the dynamic storage (E′) and loss (E″) moduli that describe the soft material were calculated, assuming a linear viscoelastic behavior. This complete characterization is more accurate than a microindentation and calculation of the elastic (Young’s) modulus.

**Figure 2.**Force curves of microindentation-relaxation assays. Graphs of (

**A**,

**B**) are force-time curves representative of the relaxation of a soft polyacrylamide hydrogel using a spherical tip with a 25 µm radius. Red arrows signal the different sections of the curve and the embedded insets show the constant position of the tip, equivalent to a constant indentation of the sample. (

**A**): relaxation curve obtained from a sample measured when submerged in a detergent solution of ddH

_{2}O and 0.1% Extran. (

**B**): relaxation curve from a measurement in air of a similar hydrogel sample soaked with water (ddH

_{2}O). Graphs of (

**C**,

**D**) are force-displacement curves, representative of the microindentation-retraction cycle of a soft hydrogel using the same tip. Red arrows signal the different sections of the curve and the magnitude of the attraction between the tip and the sample. Hysteresis can be observed between load and retraction in both cases. (

**C**,

**D**) represent the same measurement conditions as in panels (

**A**,

**B**), respectively.

**Figure 3.**Influence of velocity and depth of indentation on the computed Young’s modulus of a 4 kPa PAM hydrogel, measured in a detergent solution of ddH

_{2}O + 0.1% extran. Panels (

**A**)–(

**C**) present the results for velocities of 1, 50 and 100 µm/s respectively. The values presented correspond to calculations using a Hertz model fitting. By increasing the indentation depth, the fitted modulus decreases down to a plateau level of 14.14 ± 1.14 kPa, 12.74 ± 1.74 kPa and 10.94 ± 0.98 kPa, respectively. It was possible to fit those values to exponential decays as a function of the indentation depth, shown in the legend of each graph. In addition to the velocities shown in panels A–C, velocities of 80 and 10 µm/s were tested and shown in the density diagram of panel (

**D**) This graph is divided into different Voronoi regions (grey mesh) to visually present the influence of velocity and indentation depth. In panels A–C, error bars represent the mean standard deviation and when not visible, they are smaller than the data point visible dimension.

**Figure 4.**Mechanical characterization of 3 types of PAM hydrogels: 1 kPa elastic PAM, 4 kPa elastic PAM and 4 kPa viscoelastic PAM. (

**A**) Comparison of force-time (FT) curves (60 s relaxation indentation at a velocity of ~100 µm/s with a frequency of 100 Hz) with their respective 3rd-order GMM fitting. The curves were normalized to F

_{MAX}. (

**B**) Relaxation spectrum obtained from the 3rd-order GMM fitting with Dirac deltas. Distributions of 11 experiments are presented in Appendix A, Figure A4B. (

**C**) From the 3rd-order GMM fittings, the dynamic moduli E′(ω) and E″(ω) were calculated for each sample type. The experimental resolution is the range of frequencies corresponding to the actual measurement times (0–60 s at 100 Hz). (

**D**) Influence of the indentation depth and velocity on the long-time stiffness E

_{∞}for the 4 kPa PAM sample. For all graphs, 2 independent samples were measured on at least 5 different locations for each condition.

**Figure 5.**Comparison between the elastic modulus obtained from microindentation and the most relevant model for data fitting, and relaxation tests, using the GMM model. (

**A**) Comparison between values calculated for hydrogels immersed in a detergent solution of ddH

_{2}O + 0.1% Extran (cyan) and only soaked with ddH

_{2}O and measured in air (grey). In the case of the microindentation test, the Hertz model was used to determine E

_{H}in immersed samples while the JKR (Johnson, Kendall and Roberts) model was preferred to calculate E

_{JKR}for soaked gels. In the case of relaxation tests, the 3rd order GMM model allowed the calculation of long-term elastic modulus E

_{∞}as well as the determination of E′(ω) and E″(ω) for a given frequency range delimited by 10

^{−2}–10

^{1}Hz. In this graph, E′(ω = 1 Hz) is shown and noted E′. (

**B**) Same comparison for the soft viscoelastic PAM and 1 kPa elastic PAM. Their apparent elastic moduli when submerged in detergent are very similar when using microindentation (see Table 1), but are much larger than expected (see text). Boxes are the interquartile range (Q1–Q3) and bars extend to the maximum and minimum values. All the experimental data are presented as scatter plots in the Appendix C Figure A9, for more details. A one-way analysis of variance (ANOVA) analysis with Tukey correction was employed for multiple comparisons. It was considered significant statistically with p < 0.05, * stands for p < 0.01.

**Figure 6.**Mechanoresponse of human immortalized fibroblasts on elastic hydrogels with two different apparent stiffness. Human BJ fibroblasts were cultured for 48 h on 1 kPa (panel

**A**) and 4 kPa (Panel

**B**) polyacrylamide hydrogels functionalized with [100 µg/mL] commercial collagen type I. Subcellular localization of Yes-associated protein (YAP)/TAZ (WW-domain-containing transcription regulator) proteins (yellow) were analyzed by immunofluorescence and epifluorescence microscopy DAPI (4′, 6-diamidino-2-phenylindole dihydrochloride) (cyan) and Alexa488-coupled phalloidin (magenta) were used to stain nuclei and actin filaments (F-actin), respectively. Nuclear localization of YAP/TAZ proteins was highlighted in the zoom squares (gray). (

**C**) Cell spreading (cell area) was quantified Figure 1. kPa and 4 kPa substrates. (

**D**) Quantification of nuclear localization of YAP/TAZ proteins in fibroblasts cultured on 1 and 4 kPa substrates. Fluorescence intensity was presented in arbitrary units derived from the corrected total cell fluorescence (CTCF), see methods. (

**E**) Cell density of fibroblasts cultured on 1 and 4 kPa substrates for 48 h. Data shows that for cells cultured on the stiffer condition there are more cells adhered on the substrates per unit area. Data shown are representative of 3 independent experiments (n = 3). In order to prove statistical differences, an unpaired t-test with Welch’s correction was performed. Bars presented in plots are mean ± standard deviation. Scale bar = 50 µm.

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**MDPI and ACS Style**

Pérez-Calixto, D.; Amat-Shapiro, S.; Zamarrón-Hernández, D.; Vázquez-Victorio, G.; Puech, P.-H.; Hautefeuille, M. Determination by Relaxation Tests of the Mechanical Properties of Soft Polyacrylamide Gels Made for Mechanobiology Studies. *Polymers* **2021**, *13*, 629.
https://doi.org/10.3390/polym13040629

**AMA Style**

Pérez-Calixto D, Amat-Shapiro S, Zamarrón-Hernández D, Vázquez-Victorio G, Puech P-H, Hautefeuille M. Determination by Relaxation Tests of the Mechanical Properties of Soft Polyacrylamide Gels Made for Mechanobiology Studies. *Polymers*. 2021; 13(4):629.
https://doi.org/10.3390/polym13040629

**Chicago/Turabian Style**

Pérez-Calixto, Daniel, Samuel Amat-Shapiro, Diego Zamarrón-Hernández, Genaro Vázquez-Victorio, Pierre-Henri Puech, and Mathieu Hautefeuille. 2021. "Determination by Relaxation Tests of the Mechanical Properties of Soft Polyacrylamide Gels Made for Mechanobiology Studies" *Polymers* 13, no. 4: 629.
https://doi.org/10.3390/polym13040629