Monolayer stress microscopy: limitations, artifacts, and accuracy of recovered intercellular stresses

Abstract

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.

Figure 1. Balance of forces considered in MSM.

(a) Cell monolayer is considered as a thin sheet of cells. Each cell in the monolayer exerts traction, , on the substrate. According to the Newton's second law, the tractions are balanced by local monolayer stress, , such that, in the one dimensional force balance, . (b) The force balance is ensured only in the plane. Variation of stresses across the thickness is assumed to be negligible. (c) In classical wound healing assay, also referred to as case 2, the optical field-of-view has three optical edges and a free edge. For boundary conditions, all edges have shear stress to be zero. In addition, the free edge has normal stress to be zero and the optical edge has normal displacement to be zero.

Figure 2. Accuracy of in-plane tractions as a function of Poisson's ratio when out-of-plane components of displacements are neglected.

denotes the angle of the displacement vector relative to the plane. (a) Ratio of recovered in-plane traction to true in-plane traction. (b) Error in the phase of the recovered in-plane traction (degrees).

Figure 3. Substrate tractions for RPME cell island.

(a) Island of RPME cells. (b,c) Two components of tractions, and respectively, applied by these cells on the substrate. Size of the cell island: 4.2 mm×2.6 mm.

Figure 4. Sensitivity of the recovered monolayer stresses to change in the assumed elastic properties of the monolayer.

(a) Map of maximum principal orientation (for enlarged version of this image, see Supporting Information S8; Figs. S11–13; in File S1), (b) map of average normal stress, and (c) map of maximum shear stress obtained by assuming monolayer elastic properties to be homogeneous with . (d–f) The stress maps with instead. (g–i) The stress maps when is heterogeneous with , here was assumed to be proportional to the map of the average normal stress (b). (j) Scatter plots for maximum principal orientation where, red points quantify effect of on the maps (a) and (d), and blue points quantify effect of heterogeneity of on the (a) and (g). (k) Scatter plots for average normal stress, (l) scatter plots for maximum shear stress. Regression parameters for a straight line fit, in (j–l): blue points, (j) , (k) , and (l) ; red points, (j) , (k) , and (l) .

Figure 5. Influence of the optical edges on the monolayer stresses recovered for case 2.

Maps of (a) maximum principal orientation, (b) average normal stress, and (c) maximum shear stress extracted from the region of interest (Fig. 4a–c, yellow rectangle). (d–f) Stress map obtained by limiting the solution of equilibrium equations to the region of interest. (g) Map of difference between (a) and (d). (h) Map of (b) minus (e). (i) Map of (c) minus (f). The grey band in (a–i) represents cropped region; width of this region was 20% of the length of the optical edge. (j) Scatter plots of maximum principal orientations for quantitative comparison between (a) and (d). For the blue points cropped region was included, for the red points cropped region was exclude. (k) Scatter plots for average normal stress, (l) scatter plots for maximum shear stress. Regression parameters for a straight line fit, in (j–l): blue points, (j) , (k) , and (l) ; red points, (j) , (k) , and (l) . Size of the region of interest is 830 m830 m.

Figure 6. Propagation of boundary artifacts away from the optical edge.

(a) A thin sheet subjected to sinusoidal perturbations in normal displacements at one edge, and at two other edges. (b) Map of average normal stress, and (c) map of maximum shear stress when . (d–e) The stress maps when . (f) Decay of dominant Fourier mode in the stresses induced by the boundary conditions shown in the inset. Blue curves correspond to , and red curves correspond to . The curves marked with circle represent the induced average normal stress, and the curves marked with cross represent the induced maximum shear stress. (g–i) Decay curves of the stresses induced by boundary conditions shown in the inset. At all the boundaries along appropriate axis the natural boundary conditions, i.e. boundary stress = 0 are not mentioned but they are implied. The stresses in (b–f) are normalized with the amplitude of induced normal stress at the perturbed edge, the stresses in (g,i) are normalized with amplitude of imposed shear stress , and the stresses in (h) are normalized with the amplitude of induced normal stress at the perturbed edge.