Monolayer Stress Microscopy: Limitations, Artifacts, and Accuracy of Recovered Intercellular Stresses

In wound healing, tissue growth, and certain cancers, the epithelial or the endothelial monolayer sheet expands. Within the expanding monolayer sheet, migration of the individual cell is strongly guided by physical forces imposed by adjacent cells. This process is called plithotaxis and was discovered using Monolayer Stress Microscopy (MSM). MSM rests upon certain simplifying assumptions, however, concerning boundary conditions, cell material properties and system dimensionality. To assess the validity of these assumptions and to quantify associated errors, here we report new analytical, numerical, and experimental investigations. For several commonly used experimental monolayer systems, the simplifying assumptions used previously lead to errors that are shown to be quite small. Out-of-plane components of displacement and traction fields can be safely neglected, and characteristic features of intercellular stresses that underlie plithotaxis remain largely unaffected. Taken together, these findings validate Monolayer Stress Microscopy within broad but well-defined limits of applicability.

 m away from the monolayer), we again chose a square region approximately 96  96 pixels. We then shifted the after-trypsin mosaic relative to the before-trypsin mosaic by a displacement that maximized the cross correlation between them.

Calculation algorithms.
To calculate the gel deformation from the fluorescent bead mosaics, we used the particle image velocimetry (PIV) procedure of Trepat et al. [3]. To calculate tractions, we used the Fourier-transform traction algorithm described by Butler et al. [4] and Trepat et al. [3]. To calculate monolayer stresses, we used the finite element procedure described by Tambe et al. [2].

Boundary conditions for the monolayer.
For the monolayer bounded by free edges on all sides (Fig. 3a), the entire boundary was subjected to the stress-free condition  (Fig. 1c), we used the boundary conditions described by Tambe et al. [2].
Supporting Information S2. Fourier representation of the three dimensional Boussinesq solution.
Butler et al. [4] solved the two dimensional inverse problem recovering tractions from surface displacements restricted to in-plane tractions and displacements. Here we extend that analysis to the three dimensional relationship between i u , i T and  in Fourier space. We adopt the notation of Butler et al. [4]; let the in-plane coordinate system be given by . The response function (or Green's function) of the 3 dimensional surface displacements u  , given a point traction source of unit force at the origin on the 0  z plane is given by the Boussinesq solution,  is a (3 dimensional) unit vector, and ) (  is the Dirac delta function.
The matrix K is given by, We now seek the 2 dimensional Fourier transform of this kernel, with the transform defined by . The upper 2×2 block is the two dimensional relationship of in-plane and out-of-plane tractions and displacements; its transform is given in Butler et al. [4] (typographical sign errors in the off diagonals are here corrected). The transform of zz K requires k r FT / 2 ) / 1 ( 2   (also given in [4]   The transform of the full three dimensional relationship between tractions and displacements is therefore given by,

Supporting Information S3. Alignment between maximum principal orientation and cell orientation.
For a monolayer of elongated cells (e.g. RPME cells), Tambe et al. [2] found that monolayer stresses align with local cell orientation. Below we examine whether this alignment might be affected by the artifacts attributable to cell material properties.
In order to quantify the alignment, we define an angle  between the local maximum principal stress orientation and the local cell orientation (as these are non-directed orientations, we take   Recent studies of collective cell migration used a monolayer subsystem where the region-of-interest was bounded by two optical edges separated by two free edges (Fig. S2a) [5,6]. For this subsystem, stresses can be computed by solving the equations of equilibrium within the region of interest alone; we call this approach as case 3. For this case, boundary conditions were similar to those in case 2 (i.e.

Experimental results for case 3: Artifacts attributable to the boundary conditions:
The results were qualitatively similar to those obtained for case 2 (Fig. 5).

Numerical results for case 3:
Artifacts attributable to the boundary conditions: In case 2, the relative location of optical edges and free edges is asymmetric, and as such, it has two optical edges with slightly different forms of boundary artifacts (inset in Figs. 6f and 6h). By contrast, in case 3, the relative location of optical edges and free edges is symmetric, and as such, each optical edge has a similar form of boundary artifact (inset in Fig. S4a). The decay of stresses was similar to that in case 2 where the perturbed edge was located between an optical edge on one side and free edge on other (Figs. 6f,g).

Experimental results for case 4:
Artifacts attributable to the boundary conditions: Compared with cases 2 and 3, stresses obtained from case 4 were more closely correlated with stresses obtained from the gold standard; after data cropping, 66 . 0 2  r for case 2, 62 . 0 2  r for case 3, and 84 . 0 2  r for case 4. By contrast, the average normal stresses obtained from case 4 had a bigger offset with the gold standard than the offsets in cases 2 and 3; after data cropping, Pa for case 2, Pa for case 3, and Pa for case 4. In Supporting Information S5, we propose the source of better correlations in case 4.

Numerical results for case 4:
Artifacts attributable to the boundary conditions: Case 4 has optical edges on all sides, and hence every optical edge has similar form of boundary artifact (inset in Fig. S6a). The decay of stresses was similar to that in case 2 where the perturbed edge was located between two optical edges (Figs. 6h,i).

Experimental results for cases 2-4:
Artifacts attributable to the material properties: For cases 2-4, we quantified the artifacts attributable to material properties by using the same approach as for case 1 (Fig. 4) and generated the associated scatter plots (Fig. S7). as opposed to a free edge (Fig. 6g vs. Fig. 6i; re-plotted on a log scale in Fig. S8). This sort of rapid decay of the boundary artifacts is the source of superior correlations between the stresses from case 4 and the stresses from gold standard. , are plotted as a function of distance from the perturbed edge. Away from the perturbed edge, the stresses decay. The rate of stress decay is, however, faster if the perturbed edge is opposite to the free edge (red curves) instead of being adjacent to the free edge (blue curves).

Supporting Information S6. Mapping the decay of boundary artifacts.
In our numerical analysis, we found that the greater the spatial frequency of errors at the boundary the faster is the decay of boundary artifacts (Figs. 6, S4, S6). This result was quantified by plotting the amplitude of dominant mode of a one dimensional Fourier transform (along the axis parallel to the perturbed edge) of the stresses as a function of distance from the boundary.

Supporting Information S7. Substrate tractions: effect of substrate thickness.
When the lateral scale of traction fluctuations ( l ) is greater than or equal to substrate thickness ( H ), then the effective stiffness sensed by the cells is larger than the actual stiffness of the substrate [8]. Below we examine whether the RPME cells in our monolayer are sensing substrate stiffness that is larger than actual.
In our experiments, we used substrates of thickness  100  H m. From the spectral distribution of tractions, we found that the dominant contribution was from wavelengths close to 100 m (i.e. ,   Fig. S9). However, compared to the tractions obtained from the algorithm of Butler et al. [4], the tractions obtained from the algorithm of Trepat et al. [3] were identical (Fig. S10). As such, the effect of substrate thickness on the distribution of tractions was minimal. Taken together, for RPME cell monolayers, when

Supporting Information S8. Enlarged image of selected results
Principal stress orientations shown in Fig. 4.