Simple yet effective methods to probe hydrogel stiffness for mechanobiology

We focused on the development of two simple methods to measure the elastic modulus of hydrogels: static macrosphere indentation and micropipette aspiration. With these tools, hydrogels are deformed under the application of a pressure, which derives from the sphere weight or the aspiration pressure, and the deformation of the gel is quantified and related to the elastic modulus of the gel as described below. Protocols for their experimental realization, set-up details and all technical components are supplied in the methods. In this study we have applied these tools on classic Polyacrylamide (PAA) hydrogels, synthesized with variable rigidities, and the resulting values of stiffnesses have been compared with those obtained using more sophisticated techniques. Six PAA hydrogels formulations named PAA1-6 were prepared as previously reported^10,11, with increasing crosslinking densities (see Fig. 1a for the scheme of a soft and a stiff hydrogel structure, and Table 1 for Acrylamide (AA) and Bis-acrylamide (BA) copolymerization), in order to cover a range of stiffnesses from < 0.7 to 40 kPa (as measured by AFM in¹¹). To prepare gels for macroindentation (Fig. 1b left), the precursor solutions were polymerized in a 4 mm height 48-well polystyrene plate (diameter 10 mm). For micropipette aspiration (Fig. 1b right), the precursor solution was pipetted inside PDMS rings (thickness of 1–2 mm, area of about 1cm²) placed between a Kapton film (non-PAA adhesive) and a functionalized (PAA adhesive) coverslip (see “Methods” ). Once polymerized, all gels were left overnight to reach the swelling equilibrium before performing the mechanical tests.

Static macrosphere indentation

Multiple efforts over the last two decades have been dedicated to developing accessible ways of measuring gel mechanics, including methods based on material indentation. The latter can be applied at different scales, from macro to microindentation, in static or dynamic strain conditions^{9,12,13,14,15,16}. Original work from the Wang laboratory carried out elasticity measurements by hanging weights from PAA gels and measuring gel deformation. That said, a drawback of microindentation is the need of complex imaging procedures^{10,17,18,19,20}. To overcome this limitation, here we have applied a macroindentation procedure that can be easily applied using simple equipment. In principle, static macroindentation, measuring the penetration of the gel by a sphere with diameter of the order of millimeters, can be easily performed using a common digital camera and a microscope. In practice, however, this requires the availability of suitable mathematical tools to obtain the Young modulus (E) from imaging data, that are not currently available. Indeed, an experimentally validated theoretical model is essential for a correct determination of E, considering that indentation causes a complex stress state inside the sample. Unfortunately, few studies have addressed this problem on hydrogels and, when done, refer to a limited range of stiffnesses and sample geometries^15,16.

To advance in this direction, we carried out macroindentations and implemented the theoretical models underlying interpretation of macroindentation data, including the finite geometrical size effects of the experimental samples, all in all allowing to perform straightforward static indentation at the macro-scale and readily derive the E value of gels. Figure 2 shows a workflow of the static macroindentation process. The hydrogels were prepared in multiwell plate as described above, resulting in cylindrical shaped gels of height h and diameter D, and indented using a rigid sphere of radius R placed on top. The image is captured immediately so avoiding drying of the hydrogel. Figure S1 shows the set-up of the static macroindentation process and Fig. S2 an example of pictures for each of the 6 PAA hydrogels, from which the image analysis is made.

The experimental data (indentation depth δ, gel height h and gel diameter D) are measured from the picture taken using any common digital camera and repeating the measure 10 times. Images were analysed with the MATLAB software, to obtain δ, h and D. From each set of data, the calculation of the elastic modulus of the hydrogel is made using Eq. (1). The Young’s modulus is given as a function of the experimental data: the indentation force F (i.e., the weight of the sphere) and the sphere radius R (details on the derivation of Eq. (1) are given below).

Table 1 reports, for the 6 PAA gels, the achieved maximum and minimum values of δ/h ratio and the mean value of the 10 elastic moduli calculated for each gel, indicated as E^NH-fdh, using Eq. (1). As indicated in Table 1, a different indenter is used for each gel. In fact, we proceeded with macroindentation tests on each PAA hydrogels using different spheres, i.e. spheres of different weight and size as reported in Table S1, and obtaining different values of elastic modulus, as reported in Table S2. The most appropriate indenter for each PAA gel was empirically selected from Table S2 to minimize reading errors.

In Eq. (1), f and g represent correction factors to the Hertz contact theory, the easiest approach to evaluate the elastic modulus of the specimens from indentation data, which provides a simpler relationship between load F, indentation depth δ and radius of the sphere R (Eq. 2) but valid for a semi-infinite sample:

The analytical expressions of correction factors f and g are given in Eqs. (3) and (4) respectively, and their relevance is discussed in the next section

The results show that, through this simple procedure and application of Eq. (1)—i.e. something readily applicable by any cell biology laboratory—we could measure a range of Young moduli from fractions of kPa to > 50 kPa, as such covering most of the physiological rigidity values of natural tissues. As shown below in Fig. 7, these values are in line with those obtained with other and more complex methods.

Correction factors to the Hertz theory

In this paragraph we detail how Eq. (1), used to calculate the elastic modulus of hydrogels, is achieved. One must notice that Hertz theory on how to calculate rigidities from indentation data is based on some assumptions: one is that the specimen is assumed to be a linear-elastic incompressible half-space. Indeed, it can be safely assumed that the hydrogel is an incompressible material, as macroindentation experiments are typically carried out in a much shorter timescale than diffusion of the small solvent molecules in and out the gels network, and thus characterized by a Poisson’s ratio ν = 0.5. Another assumption of the Hertz theory is that the deformation is within the limits of small-slope and small-strain conditions is however unlikely satisfied in a macroindentation experiment where the specimen has finite and rather small dimensions with respect to the indenter. Indeed, the small slope conditions (defined by the relation a² = R δ), are satisfied only for specific geometries of the contact problem. This is shown in Fig. S3, indicating for three different specimen dimensions expressed by D/h (namely D/h = 5, 2, and 1), the values of R/D and δ /D below which the small slope conditions are fulfilled. However, since our experimental data had D/h = 2 or higher (Fig. S3) they violate the small-slope assumption. Additionally, the measured strain δ/h in our experiments was always greater than 0.07 (refer to Table 1 for the experimental values). This indicates that the assumption of small strain, for which a linear dependence between stress and strain should be observed, is also not satisfied as also shown by uniform compression tests performed on our specimen (Fig. S4).

Thus, the Hertz equation cannot be used as it is in macroindentation, and a new model is necessary to calculate the elastic modulus of our specimens. To this end, it is important to first identify the material behavior that best describes the deformation response. While the literature often reports a linear elastic behavior for the hydrogels at low loading, non-linear elastic behaviour has also been reported and described with different models, such as neo-Hookean, Mooney-Rivlin, etc.^21,22. To ascertain which of these non-linear models better resembles the hydrogel specimen response, we used the uniform compression tests of Fig. S4 and performed a fit of the data using different non-linear models. Two examples of the stress–strain response are reported in Fig. 3a,b together with the linear-elastic and the neo-Hookean fits to the data. From this figure, one can see that the non-linear neo-Hookean model well describes the material behaviour.

Therefore, we adopted the expression for the indentation proposed in the work by Long et al.^15,23, in which hydrogels were, in fact, treated as neo-Hookean material. This expression, indicated as E^NH-fh, accounts also for the finite thickness of the specimens, deformed by a spherical indenter:

Notice that the equation reads like the Hertzian equation albeit for a correcting term, f(R/h,w) defined in Eq. (3) and valid for a slip condition, without adhesion between the gel and the indenter.

We then asked ourselves if the elastic response of our hydrogels is also affected by their lateral dimension, because hydrogel specimens do not only have a non-linear elastic behaviour and a finite height but also have a finite diameter, not considered in the Eq. (5). In this work, the effect of the finite diameter was assessed by means of Finite Element simulations performed using the commercial software ABAQUS (Fig. 4a-b). In the simulations, the indenter is modelled as a rigid sphere and the substrate is modelled as an incompressible neo-Hookean cylindrical solid, with the experimentally measured dimensions. Then, among the experimental data, the more severe values of indentation depths were chosen for each gel to simulate the indentation and derive the elastic modulus. In Fig. 4a,b, the finite element mesh is shown along with the normal displacement field of the deformed hydrogel. Data used for the simulation are reported in Table S3. Through this finite element simulations for different geometries and indentation depth, we can identify the conditions for which a correction is necessary and provide the correction in the form of a polynomial equation. The need of a correction factor for the diameter is found to depend on the δ/h ratio and the geometric factor Rh/D², the dimensionless strain and dimensionless diameter respectively, as shown in Fig. 4c. The correction factor g(δ/h,Rh/D²) previously reported in Eq. (4) is so obtained, through the method of least square regression.

In essence, our Eq. (1) represents a modification of Long et al.¹⁵, that accounts for a finite height neo-Hookean material, augmented by a function of the two dimensionless arguments δ/h and Rh/D², g(δ/h,Rh/D²).

To verify the extent of correction introduced by our model, that considers both finite diameter and non-elastic behavior, the elastic moduli obtained by our FEM simulations E^NH-fdh were compared with those calculated through different models, using the experimental data of Table S3. In particular, the calculation of the elastic modulus achieved by the Hertz model (E^H, Eq. (2)), the model proposed by Long et al.¹⁵ (E^NH-fh, Eq. (5)) and a model proposed by Dimitriadis²⁴, E^H-fh, that takes into account the finite height of the gel but in a linear elastic material (instead of neo-Hookean), were made and compared in Table 2. From the table it can be seen that the finite height correction is the most relevant one, about 20–30%, as it results from the comparison between the Hertz linear model and the other three calculated values, E^NH-fh, E^H-fh and E^NH-fdh, while the correction for a non-linear behavior, i.e., E^NH-fh and E^NH-fdh, partially counteract the finite height correction. Besides, comparison of E^NH-fdh (our results) and E^NH-fh (calculated without taking into account a finite diameter) indicates that a correction lower than 7% is needed in our experimental conditions. In fact, in the experiments we conducted here, we used a sufficiently small indentation depth and large diameter (Rh/D² = 0.05) that the results are not so far from Long’s equation. This would not be the case for experiments on smaller specimens or with a larger indentation depth, as shown by curves for Rh/D² = 0.075 and Rh/D² = 0.15 in Fig. 4c. In that case the final values would be more affected by the correction function g(δ/h,Rh/D²).

Micropipette aspiration

Micropipette aspiration²⁵ is also a simple and cheap technique. The main advantage of this method is its versatility, as it can be used to measure stiffness of cells and soft natural tissues^22,23,26, ex vivo or in vivo^27,28. In literature, few examples of its use on siloxane polymers or stiff gels are reported^29,30, but theoretical models have been exhaustively described to correlate tissue deformation, aspiration pressures and geometrical parameters, and can be extended to soft hydrogels. Here we advance by providing set-up protocols to build micropipette aspiration, so far poorly described in literature, and offer technical details and instruction to perform reliable measurements.

Figures 5 and S5 show a workflow and a picture of the micropipette aspiration apparatus respectively. To perform these analyses, we designed and fabricated a device consisting of a testing head that holds a capillary (external radius A = 0.5 mm and internal radius a = 0.375 mm) and moves the sample in the xy plane, a circuit enclosure containing the motors controller and the pressure sensors and a syringe pump to exert a controlled pressure on the sample (see methods for a detailed description of the system).

The mechanical characterization is made mounting the sample with the free surface perpendicular to the glass capillary and moving the sample until a full contact between the surface and the capillary section is achieved. Then, a given negative pressure is applied, through an aspiration pipette, that induces a small convexity in the gel surface, from which a certain aspiration length l can be observed. As a certain pressure P is reached, a picture is captured by a microscope imaging software, and the aspirated length l can be measured using an image analysis software. Figure S6 shows example of pictures for each of the 6 PAA hydrogels, from which the image analysis is made.

The local deformation of the hydrogel in the pipette is then related to the applied pressure p and the internal radius of the pipette a, using an explicit expression developed through FEM to calculate the hydrogel elastic modulus, as reported by Zhang et al.³⁰. In this model, friction between gel and the internal surface of the capillary should be avoided. For this, in our experiments, a fluorinated functionalization was made in the internal surface of the pipette.

Moreover, the model discriminates between linear or nonlinear elastic deformations. The proposed parameter to discriminate the two behaviors is l/a, set at values l/a < 0.3 or l/a > 0.3 for the linear or non-linear regimes respectively, and the relative equations used to calculate E, named E^PA, are the following:

Given that in our experimental conditions a reliable reading of the aspiration length is achieved when l/a > 0.3, Eq. (7) was used to calculate E^PA values that are reported in Table 3. In particular, to verify that measures achieved in a range of l/a > 0.3 are substantially constant with applied pressures, the six PAA compositions were measured at different pressures. The results are shown in Fig. 6 (graphs on the right) in which each dot represents a single measure taken at a specific value of pressure. Then the elastic moduli E^PA are calculated for each value of p and l/a and reported in Fig. 6 (graphs on the left). Mean values and standard deviations are reported in Table 3, together with the minimum and maximum values of pressure used (and the respective ratio l/a).

Comparison of methods to determine the elastic modulus

To test the reliability of our methods, we compared the E values measured on the same six PAA hydrogels by static macroindentation and micropipette aspiration with those measured with traditional methods, adopting more expensive equipment and operating at lower throughput. First, the values we obtained with either method were nicely matching those expected by the six PAA formulations, that were indeed chosen from the recipes of Engler et al.¹¹, who empirically derived such formulations using AFM measurements. We further experimentally supported the validity of our methods by comparing them with uniaxial compression (Fig. S4) and parallel plate rheometry (Fig. S7). Figure 7 provides a schematic representation of all the techniques involved in the comparison. A direct comparison between all these measurements is offered in Fig. 8 and Table 4. From the comparison, we found a consistent trend: the elastic modulus increases with the amount of crosslinking and a maximal 2/3-fold variation between elastic moduli calculated with the different techniques is observed over the same gradient of rigidities, except for the softest hydrogel.

Of note, these setups have proved to be suitable to perform a large set of measurements rather quickly (up to 1 measurement in less than 5 min and 10 min for micropipette aspiration and static macro-indentation respectively, for the less challenging samples), while a minimum of 15–20 min is generally required for AFM and rehometry. The rapid measurement is important also to avoid drying of the hydrogel.

As proof-of-concept of the applicability of our methods to a different material, we prepared an OH-functionalized PAA hydrogel, as previously reported³¹ (PAA-OH, Fig. 9, and Table 1). PAA-OH formulations are prepared replacing part of the AA of PAA formulations with a new monomer, N-hydroxyethyl acrylamide (HEA), copolymerized in different concentrations and molar ratio. The presence of highly polar group OH, as shown in Fig. 10 and Table 4, is sufficient to modify the mechanical properties of PAA-OH in respect to PAA, of twofold (e.g., from 2 to 4 kPa, that is within the range of biological significance), in spite of the fact crosslinking degree between the PAA and PAA-OH gels should be identical. This confirms the effectiveness and sensitivity of approach.

We next asked to if and to what extent measures obtained with macroindentation and micropipette aspiration really reflect biologically meaningful values of extracellular rigidity for mechanobiology applications. To address this issue, we measured YAP/TAZ nuclear vs. cytoplasmic localization in PAA-OH hydrogels for a gradient of PAA-OH rigidities. YAP/TAZ are transcription factors serving as universal readers of the cell mechanical state, and prime mediators of the mechanical challenges, turning them into gene-expression programs. YAP/TAZ are nuclear and transcriptionally active in cells experiencing high-rigidities and progressively relocalize to the cytoplasm in cells experiencing more and more compliant substrates^2,32. Each PAA-OH gel tuned over a gradient of rigidity values was biofunctionalized with the same concentration of the adhesive ECM protein Fibronectin (FN) and used as substrate for adhesion of MCF10A cells. By immunofluorescence, shown in Fig. 11, we found that YAP/TAZ was robustly nuclear in hydrogels of measured E ≥ 13 kPa, to become gradually evenly distributed between the nucleus and the cytoplasm at intermediate stiffness values (around 6–13 kPa), and completely cytoplasmic or excluded in softest hydrogels (< 1 kPa). Of note, this gradient of YAP/TAZ nuclear vs. cytoplasmic localization nicely parallels hydrogels’ rigidity values, allowing to appreciate differences in biological behavior which might arise from gels with the same crosslinking degree but different chemistries.