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.