Relationship between velocities, tractions, and intercellular stresses in the migrating epithelial monolayer

The relationship between velocities, tractions, and intercellular stresses in the migrating epithelial monolayer are currently unknown. Ten years ago, a method known as monolayer stress microscopy (MSM) was suggested from which intercellular stresses could be computed for a given traction field. The core assumption of MSM is that intercellular stresses within the monolayer obey a linear and passive constitutive law. Examples of these include a Hookean solid (an elastic sheet) or a Newtonian fluid (thin fluid film), which imply a specific relation between the displacements or velocities and the tractions. Due to the lack of independently measured intercellular stresses, a direct validation of the 2D stresses predicted by a linear passive MSM model is presently not possible. An alternative approach, which we give here and denote as the Stokes method, is based on simultaneous measurements of the monolayer velocity field and the cell-substrate tractions. Using the same assumptions as those underlying MSM, namely, a linear and passive constitutive law, the velocity field suffices to compute tractions, from which we can then compare with those measured by traction force microscopy. We find that the calculated tractions and measured tractions are uncorrelated. Since the classical MSM and the Stokes approach both depend on the linear and passive constitutive law, it follows that some serious modification of the underling rheology is needed. One possible modification is the inclusion of an active force. In the special case where this is additive to the linear passive rheology, we have a new relationship between the active force density and the measured velocity (or displacement) field and tractions, which by Newton's laws, must be obeyed.

Figure 1

(a) 3D schematic of a thin film of fluid (or epithelial monolayer) on a rigid substrate. Not drawn to scale—the length and width of the film are substantially larger than the height. (b) $x-z$ cross section. The top surface is stress free while at the bottom, tractions are balanced by viscous shear stresses. In this work, tractions (${\mathit{T}}_x$ and ${\mathit{T}}_y$) refer to the measured tractions by the cells on the substrate, ${\mathit{T}}_{\mathrm{cell}on\mathrm{substrate}}$. The dashed lines refer to boundary conditions at the edges, discussed in Sec.2e.

Figure 2

Schematic of a cell island approaching an obstacle. The cell island is given by a solid black line. Different scenarios, where the cell island is imaged fully or partially, are color coded. See text for case 4 edge conditions.

Figure 3

(a) Phase image of the migrating cells encompassing the obstacle. The field of view is 725 × 725 $\mu {\mathrm{m}}^2.$ Measured velocities (b) $v_x$ and (c) $v_y$ (color bar is in units of ${10}^{}\left[\mathrm{nm}/\mathrm{s}\right]$). Raw data from Ref. [16].

Figure 4

(a) Measured tractions ${T_x}^{\mathrm{meas}}\left[\mathrm{Pa}\right]$ used in Ref. [16]. (b) Calculated tractions $\mu h{t_x}^{\mathrm{calc}}\left[\mathrm{Pa}\right]$ with $h=5\mu \mathrm{m}$ and $\mu =23\mathrm{kPa}\mathrm{s}$. (c) The active force density, $h{F}_x^{\mathrm{active}}={T_x}^{\mathrm{meas}}-\mu h{t_x}^{\mathrm{meas}}$ [see Eq. (28) below]. (d)–(f) A zoomed view of the marked green box from panel (c), showing (d) $|{\mathit{T}}^{\mathrm{meas}}|\left[\mathrm{Pa}\right]$, (e) $|\mu h{\mathit{t}}^{\mathrm{calc}}|\left[\mathrm{Pa}\right]$, and (f)$|h{\mathit{F}}^{\mathrm{active}}|\left[\mathrm{Pa}\right]$. The green arrows show the direction of each respective vector of (d)–(f) where the lengths of the arrows are proportional to the magnitude.

Figure 5

Scatter plot of ${{\mathit{T}}_x}^{\mathrm{meas}}\left[\mathrm{Pa}\right]$ vs $\mu h{t_x}^{\mathrm{calc}}\left[\mathrm{Pa}\right]$. The color bar is proportional to the count density.