Atomic force microscopy methodology and AFMech Suite software for nanomechanics on heterogeneous soft materials

Atomic force microscopy has proven to be a valuable technique to characterize the mechanical and morphological properties of heterogeneous soft materials such as biological specimens in liquid environment. Here we propose a 3-step method in order to investigate biological specimens where heterogeneity hinder a quantitative characterization: (1) precise AFM calibration, (2) nano-indentation in force volume mode, (3) array of finite element simulations built from AFM indentation events. We combine simulations to determine internal geometries, multi-layer material properties, and interfacial friction. In order to easily perform this analysis from raw AFM data to simulation comparison, we propose a standalone software, AFMech Suite comprising five interacting interfaces for simultaneous calibration, morphology, adhesion, mechanical, and simulation analysis. We test the methodology on soft hydrogels with hard spherical inclusions, as a soft-matter model system. Finally, we apply the method on E. coli bacteria supported on soft/hard hydrogels to prove usefulness in biological field.

with N2 gas at 5 °C for 30 min. After that, the N2 purged APS stock solution was added to the pre-gel mixture through a rubber septum by a degassed syringe, mixed immediately and allowed to react at 5 °C for 12 hrs.
The prepared gels were cut into small square type shapes (approximately 6x6 mm 2 and 3 mm thickness), and dipped into the de-ionized water for dialysis to remove the unreacted chemicals, and water were changed twice a day for 7 days. After the dialysis, the gels were dried under vacuum at 50 °C for 72 hours. Nearly 100% conversion was determined gravimetrically for all prepared samples. The procedure used the radius calibration method was fully documented by Indrieri et al. 4 All AFM operations involve detection of cantilever deflection through laser beam reflected on a segmented photodetector. Photodetector response around the center (zero voltage) is linear but laser spot near edges show non-linearity towards saturation at 12V (maximum photodetector voltage), therefore, restricting the usable voltage range on photodetector. Supplementary Fig. 3a shows typical force curve on rigid substrate with non-linear response for deflection signals from 5 to 8 V. The correction is performed by fitting the nonlinear part using a 4 th degree polynomial law and subtracting it from raw data. Correction is generated on force curves on rigid substrate where indentation is negligible. Finally, the correction is transported on experimental data with indentation where, generally, non-linearity causes a forward bending and underestimation of Young's moduli. Examples of correction on indentation curves on 30 kPa hydrogels are presented in Supplementary Fig. 3b showing improved performance of Hertz fit after correction all over the indentation range.
Standardized Nanomechanical procedure (SNAP) was recently introduced, 5 discussing problematics in nanomechanical measurements experienced in 11 different AFM labs across Europe and participating to COST Action TD1002, European network on applications of atomic force microscopy to nanomedicine and life sciences. The main issue was about the error/imprecision during optical lever sensitivity calibration, then propagating in thermal noise calculation. Through an external, independent measurement of elastic constant of cantilever, zsens can be corrected using the following relations: As external and independent measurements in place of vibrometer or interferometer we use 5 calibrated samples with rheometer. After the confirmation of quantitative characterization with colloidal probes, 6 we decided to prepare and array of 5 homogeneous gels with moduli equally distributed between 1 and 30 kPa in order to perform the calibration of the probe. So far, recent works from our group demonstrated how the series of well-defined and homogenous gels show Young's modulus at micro-nanoscale in excellent agreement with rheological macroscopic measurements, highlighting their importance for calibration purpose. [6][7][8] Supplementary Fig. 4 shows the comparison between Young's moduli measured by rheology and colloidal probe AFM. Due to the linear relationship between moduli and applied force, the slope coefficient can be used to correct elastic constant and optical lever sensitivity using Eq.2, as reported in SNAP. 5 The method is independent of intrinsic AFM optical alignment, still based on rheometer accuracy.

Supplementary Note 3 | Adhesion Analysis.
Regarding the samples under investigation (hydrogels and bacteria), especially working completely immersed in water solution, a negligible adhesion was detected.
'AFMech Suite' software allows focusing on adhesion, specifically, on single force spectroscopy data showing that adhesion is detectable but negligible in comparison with the indentation forces. This is a crucial point for the validity of Hertz model used for data analysis. Adhesion analysis was performed for all sample investigated, here, we present the analysis in correspondence to Figure 3 of main text. Adhesion map is shown in Supplementary Fig. 5a confirming low adhesion all over the surface sample. Quantitative analysis in Supplementary Fig. 5b shows an average adhesion force of FAD = 0.6 ± 0. The main scope of the work is evaluating the transition from homogeneous to heterogeneous samples, still a justification and validation in using Hertz model is necessary.
1. Hydrogel samples are isotropic. Changing direction during cutting did not influence moduli values either in rheology and AFM 2. Non-linear elasticity behavior is negligible for shallow indentation. Hydrogels and bacteria are generally treated as hyperleastic (stress and strain non-linear) following Neo-Hookean model. 9 In this framework, under spherical indentation, the relations between the force based on Neo-Hookean model and Hertz solution was provided by Zhang et al. 10 and confirmed in our recent work. 11 In current work, R = 2500 nm, and the maximum indentation during AFM experiments is around 800 nm, which leads to an error of 3% accordingly.
3. Hydrogels used here show purely elastic behavior and negligible viscosity. This is directly confirmed by rheology measurements. Moreover AFM measurements did not show frequency dependence (see Galluzzi et al. 6 ) and there is no relaxation hysteresis between approaching and retracting indentation curves (see Supplementary Fig. 5).
4. Adhesion of hydrogels samples in solution is negligible as shown in previous section.
5. Sample thickness is macroscopic (in millimeter range), while radius and indentation characteristic scale are in micrometer range. Even if in our study is not necessary, 'AFMech Suite' grants the possibility to apply Finite Thickness correction (see 'AFMech Suite' guide). 6. Indentation is small compared with probe radius. Sneddon spherical model is describing the exact relation of a sphere indenting pure elastic plane, still Hertz is a good approximation at shallow indentations. Following the work of Puricelli et al. 12 , for spheres with R = 2.5 µm in cases of E = 30 kPa and F = 40 nN, the relative discrepancy between the two models is well below 3%.
'AFMech Suite' allows choosing several geometries and models. Here, focusing on sphere geometry, Hertz, Sneddon (sphere) and adhesive DMT, JKR and non-linear hyperelastic Neo-Hookean models are compared in Supplementary Fig. 6 directly analyzing experimental data of FV measurement on homogeneous hydrogel (30 kPa).
Equations relative to this study are reported as follows: Hertz = Where F, E, R, ν, δ, a and FAD represent respectively force, Young's modulus, radius of spherical probe, Poisson's ratio, indentation, radius of contact area and adhesion force.
Besides DMT model, usually employed for systems showing strong capillary adhesion (our system is completely immersed in solution, thus making one of the key assumptions of this model invalid), all models show comparable Young's moduli values, well inside experimental error. Hertz approximation validity was confirmed directly, allowing to focus our discussion in solving heterogeneity problems.