1. Large Amplitude Oscillatory Shear Rheology of Living Fibroblasts: Path-Dependent Steady States 
Mathias Sander, Heike Dobicki, Albrecht Ott 
Biophysical Journal 
Volume 113, Issue 7, Pages (October 2017)
DOI: /j.bpj
Copyright © 2017 Biophysical Society Terms and Conditions

2. Figure 1
(a) Schematic representation of the experimental setup. We use a commercial rotational rheometer to probe the mechanical properties of 106 cells in a single experiment. The cells adhere to a fixed glass plate and a glass ring, connected to the measurement head of the rheometer. The diameter of the glass plate is 50 mm. The outer (inner) diameter of the ring is 40 (35) mm. Rotation of the ring shears all the adherent cells. The rheometer applies a defined torque and records the angle of rotation. During measurements, cells can be observed through a microscope. The inset shows a schematic representation of the shearing procedure, as well as actual pictures taken during a measurement. (b) (Top) An experiment consists of time sweeps with a duration of 120 s for each amplitude, separated by periods during which cells are kept at zero shear, resting for 120 s. The events are repeated, each time with increased amplitude, until the cells detach from the surfaces. (For an example of other types of excitation, see Fig. S1.) (Bottom) Shown here are details of the time sweep (blue box) for DF-controlled excitation (DF in c) as an example. Each sweep consists of the initial cycles (red box, here DF) until steady-state oscillations (green box) are achieved, then followed by a pause (magenta box). The initial cycles (red box) occur again with each new sweep. (c) Shown here are examples for initiating deformation-controlled and stress-controlled excitations. (Top left) Shown here is DS-controlled (DS) excitation. The deformation amplitude is slowly increasing until steady state is reached after 45 s. (Bottom left) Shown here is fast deformation-controlled (DF) excitation. The initial deformation increases steeply, but there is a pronounced overshoot. Steady state is reached within 10 s. (Top right) Shown here is sinusoidal shear (SS) stress excitation. (Bottom right) Shown here is cosine shear (SC) stress excitation. Whereas the SS does not provide initial deformation rates faster than during steady-state loading, the SC stress excitation initially increases the load in a stepwise manner, entailing very high deformation rates. To see this figure in color, go online.
Biophysical Journal  , DOI: ( /j.bpj )
Copyright © 2017 Biophysical Society Terms and Conditions

3. Figure 2
Scheme of one filament inside the simulation box at an angle θ (left) and the stretched filament resulting from shearing the simulation box (right). To see this figure in color, go online.
Biophysical Journal  , DOI: ( /j.bpj )
Copyright © 2017 Biophysical Society Terms and Conditions

4. Figure 3
Maximum shear stress τ0 as a function of maximum shear deformation γ0 for experiments on 13 monolayers with different excitation types. Different colors denote the four different types of excitation (legend and Fig. 1 c); different symbols of the same color represent different monolayers with the same type of excitation. The parameters τ0 and γ0 were obtained from all oscillation cycles, including initial loading. For γ0 < 0.01, τ0 increases roughly linearly with γ0. Variation among different monolayers is very small in this regime. However, for larger γ0, these variations increase and τ0 deviates from the linear dependence on γ0. This indicates the onset of nonlinearity. To see this figure in color, go online.
Biophysical Journal  , DOI: ( /j.bpj )
Copyright © 2017 Biophysical Society Terms and Conditions

5. Figure 4
Average steady-state shear modulus 〈G〉 as a function of maximum shear deformation. Data corresponds to Fig. 3. Different colors denote the four different types of excitation (legend and Fig. 1 c); different symbols of the same color represent different monolayers with the same type of excitation. The modulus decreases as a function of maximum shear deformation, indicating cell softening for all excitation types. However, the amount of softening substantially varies between different monolayers. To see this figure in color, go online.
Biophysical Journal  , DOI: ( /j.bpj )
Copyright © 2017 Biophysical Society Terms and Conditions

6. Figure 5
The amount of softening ΔG as a function of linear steady-state modulus G0 for experiments on 31 different monolayers, regardless of the type of excitation. ΔG increases almost linearly with increasing linear modulus G0 = G(γ → 0).
Biophysical Journal  , DOI: ( /j.bpj )
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7. Figure 6
Shear stress as a function of shear deformation in steady-state oscillatory excitations for an experiment with slow deformation (DS), fast deformation (DF), and cosine shear stress (SC) control. The maximum shear deformation increases from bottom to top. For small excitations (bottom) (γ0 = 0.02), cells exhibit an ellipsoidal trace in the Lissajous plot. This corresponds to a linear response. It occurs independently of the type of excitation. A crossover from linear to nonlinear response occurs at γ0 = 0.05–0.1. For the three types of transitory, initial strain paths (DS, DF, and SC), the cells exhibit a different nonlinear response to identical steady-state excitations. Here a sinusoidal shear stress excitation (SS) cannot be distinguished by eye from the slow deformation (DS) and is not pictured. Only computation of the differential modulus g reveals less pronounced intracycle stiffening in SS as compared to DS.
Biophysical Journal  , DOI: ( /j.bpj )
Copyright © 2017 Biophysical Society Terms and Conditions

8. Figure 7
Differential shear moduli g = ∂τ/∂γ as a function of deformation γ for an experiment with slow deformation (DS), sinusoidal shear stress (SS), fast deformation (DF), and cosine shear stress (SC) excitation. Each type of excitation is studied on a separate monolayer. Strain softening is reflected by a decrease of the differential modulus with increasing maximum deformation (red arrow). Intracycle stiffening is reflected by an increasing value of the differential modulus within one oscillation cycle. Except for SC stress, the differential modulus increases for deformations beyond a certain threshold γc. Here, we only present the result from a single monolayer for each type of excitation. Results for different monolayers, using the same type of excitation, are qualitatively similar. However, quantitatively, they differ in a way comparable to the linear viscoelastic moduli in Fig. 4. To improve readability, for each type of stimulus only five curves with increasing maximum deformation (color code) are shown. The entire data set can be found Fig. S2. To see this figure in color, go online.
Biophysical Journal  , DOI: ( /j.bpj )
Copyright © 2017 Biophysical Society Terms and Conditions

9. Figure 8
(Top) Differential modulus as a function of deformation in DF control with a maximum deformation γ0 = 0.5. The critical deformation γc (red) marks the crossover from constant (gs, green line) to increasing differential moduli. The parameter gmax represents the maximum of the differential modulus. (Bottom) The value γc is given as a function of 1/gs. The scaling suggests that a well-defined critical stress level causes the stiffening, regardless of the amplitude of shear excitation. Data are shown for DS control. The increase is observed for experiments in DF control as well. To see this figure in color, go online.
Biophysical Journal  , DOI: ( /j.bpj )
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10. Figure 9
Comparison between experimental data (black) and model (red). Given here is shear stress as a function of shear deformation for increasing deformation amplitudes. The filament model captures the experimental results well, considering the simplifying assumptions. For small and very large excitation amplitudes, the model deviates from experimental results. However, the transition from linear response to intracycle stiffening along with the intercycle softening is reproduced. To see this figure in color, go online.
Biophysical Journal  , DOI: ( /j.bpj )
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11. Figure 10
Comparison between experimental data (black) and filament model (red). Given here is shear modulus as a function of the shear deformation for DS (left) and fast deformation (DF, center) and as a function of shear stress for SC stress (right). The softening response for all excitation types is reproduced quite well. In the case of SC stress excitation, the shear modulus for the first two lowest shear stress amplitudes is underestimated. To see this figure in color, go online.
Biophysical Journal  , DOI: ( /j.bpj )
Copyright © 2017 Biophysical Society Terms and Conditions

12. Figure 11
Comparison of DF control experimental data (black) with cell spanning filament modeling (DF-CS, red). Given here is shear stress as a function of shear deformation for increasing amplitudes of excitation. In the model, new filaments are only allowed to form as spanning between the top and bottom surfaces. The agreement between model and experiment is significantly improved compared to the standard filament model as discussed before (Fig. 9, DF). To see this figure in color, go online.
Biophysical Journal  , DOI: ( /j.bpj )
Copyright © 2017 Biophysical Society Terms and Conditions

13. Figure 12
Filament model. Given here is angular distribution of filaments as a function of maximum shear deformation γ0 for DS (upper left panel), DF (upper right), SC stress (lower left), and modified fast deformation (DF-CS, lower right). In all cases initially the filaments are mainly distributed around an angle of θ = 90° with respect to shear direction. For DS, DF, and SC, the filaments reorient toward an angle of θ = 0° (or θ = 180°). This diminishes filament stretch caused by shearing. For the modified fast deformation (DF-CS), filaments orient in a more pronounced way toward θ = 90° with increasing deformation amplitude. To see this figure in color, go online.
Biophysical Journal  , DOI: ( /j.bpj )
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14. Figure 13
Filament model. (Left) Given here is mean filament length in steady state as a function of maximum shear deformation γ0 for different response regimes. Only for the DF-CS does the mean filament length increase with increasing maximum shear deformation. In all cases, the initial mean filament length is ≈13.7 μm. (Right) Shown here is mean number of cross-links as a function of maximum shear deformation γ0 for different response regimes. The number of cross-links diminishes with maximum shear deformation almost independently of the particular regime considered. To see this figure in color, go online.
Biophysical Journal  , DOI: ( /j.bpj )
Copyright © 2017 Biophysical Society Terms and Conditions