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The authors have declared that no competing interests exist.

3D traction analysis: JPB. Discussed and interpreted the results and commented on the manuscript: DTT UC XT CYP JHK EM JPB JJF. Conceived and designed the experiments: DTT JPB JJF. Performed the experiments: DTT UC. Analyzed the data: DTT JPB. Wrote the paper: DTT JPB JJF.

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.

The human body is a cooperative of about 80 trillion cells, each one of which receives constantly from its neighbors both chemical and physical signals

In collective migratory processes, cells ordinarily move not as individual entities but as collective sheets, ducts, strands, or clusters

For the particular case of the cellular monolayer

Implementation of MSM begins with recovery of local tractions exerted by the monolayer upon its substrate. As originally described

Whether or not two-dimensional conditions might obtain, still other assumptions are required. The distribution of internal stresses within the monolayer is computed from the recovered in-plane tractions based on the two-dimensional balance of forces in the cell plane as demanded by Newton's laws

Here we reexamine these assumptions quantitatively in detail. We begin by formulating the problem in two dimensions and highlighting the necessary assumptions. We go on to consider boundary conditions and associated ambiguities, and then describe our analytical, numerical, and experimental approaches. Taken together, these approaches show quantitatively that errors attributable to out-of-plane tractions and displacements, and to cell material properties, are slight, and that errors attributable to unknown boundary conditions, if suitably handled, are manageable.

We consider a monolayer comprising a collection of contiguous cells which forms a sheet that is flat and thin (

(

If inertial effects are negligible, then according to Newton's second law, the local traction must be balanced by local gradients in monolayer stresses. This is simply a force balance argument, which implies

Since the monolayer always remains in mechanical equilibrium, Newton's laws demand that this force balance must be independent of material properties of the monolayer. The recovered stresses, however, are not independent of the specific material properties. Tambe et al.

In MSM, stresses are determined with standard finite element procedures which use

If the monolayer is neither isotropic, homogeneous, nor elastic, then the Beltrami-Michell compatibility equation becomes more complex, In

In most experimental monolayer systems, two types of boundaries require consideration (

To overcome the difficulty posed by the boundary conditions at the optical edge, Tambe et al.

Recognizing that this assumption is

At the optical edge, Tambe et al.

In the measurement of monolayers bounded by free edges alone (Fig. S2c in

Based on geometric scaling arguments, we first estimate the magnitude of out-of-plane tractions in comparison with in-plane tractions; based on analytic arguments, we then quantify the errors in recovered in-plane tractions when out-of-plane displacements are neglected. This is followed by an analysis of the errors in monolayer stresses attributable to the assumed simplified material properties of the cells. We conclude by considering the distinction between monolayers whose entire extent falls within the microscopic field-of-view versus those that extend outside the field-of-view.

Concerning traction recovery, we first consider the role of out-of-plane tractions,

To assess the accuracy of recovered monolayer stresses experimentally, we start by considering the monolayer bounded by free edges alone; we call such a monolayer a cell island (Fig. S2c, for experimental protocol see Supporting Information S1, in

In this analysis, we compute local principal stress components (

For

To assess the effect of heterogeneity of

To quantify boundary artifacts, as in the approach used by Tambe et al.

To analyze further how boundary perturbations and associated artifacts propagate into the region of interest, we imposed sinusoidal perturbations at optical edges. For example, along the right optical edge in

Two other monolayer geometries are of interest. The first is the monolayer bounded by two optical edges separated by two free edges (Fig. S2a in

There are at least three important length scales in the problem. (1) The lateral extent of in-plane traction fluctuations

The remaining dimensionless number is

In the case of extended monolayers, by contrast, geometry alone gives bounds on the errors. Suppose there are true traction vectors of unit magnitude acting at two remote points separated by

If, for example, a typical lateral scale is 100

If

In Supporting Information S2 (in

We now consider a unit displacement that has three-dimensional components

When

For example, taking an extreme value for polyacrylamide gels

Finally, we remark that for substrates of finite thickness, when the tractions are smoothly distributed over distances larger than gel thickness, the substrate deformation approximates pure uniform shear, and in this case Poisson's ratio in the substrate is irrelevant.

In an island of rat pulmonary microvascular endothelial (RPME) cells, tractions demonstrated extreme spatial fluctuations (

(a) Island of RPME cells. (

(

When we reduced the assumed value of

When

Across all instances examined, the correlation coefficient

Compared to the gold standard (

Maps of (

When the entire stress map was compared, the agreement between the gold standard and stresses from case 2 was weak, as expected (

Across all instances examined, in the region away from the optical edge, the correlation coefficient

First, we considered an optical edge adjacent to the free edge, and imposed boundary perturbation comprising sinusoidal normal displacements

(a) A thin sheet subjected to sinusoidal perturbations in normal displacements

Next, we considered the boundary perturbations comprising sinusoidal shear stress

Analysis of case 3 and 4 produced results that were qualitatively similar to those of the analysis of case 2 (Supporting Information S4; Figs. S3–S7 and Fig. S14; in

Our principal findings are these. For in-plane traction recovery and for in-plane monolayer stress recovery, the contributions of out-of-plane components of tractions and displacements are negligible. Second, optical edges cause artifacts that are mostly confined to regions near the optical edge but are otherwise homogeneous and small. Third, assumptions of homogeneity and incompressibility of the monolayer material cause artifacts that are also small. Finally, characteristic features of the recovered stress maps that underlie plithotaxis are insensitive to these artifacts, thus demonstrating that the observation of plithotaxis is robust. Below we discuss in greater detail monolayer stress recovery, associated findings, and their implications.

Recent reports have asserted that three dimensional methods are necessary because the out-of-plane tractions or displacements can be sufficiently large to invalidate tractions and monolayer stresses recovered from two-dimensional methods

First, to what degree do out-of-plane displacements, which are unmeasured in MSM, impact recovery of in-plane tractions? When the substrate is incompressible (

Second, are the recovered in-plane tractions insensitive to the difference between cells adhering tightly versus loosely to one another, as asserted by

Third, to what degree do nonzero out-of-plane displacements and out-of-plane tractions affect the recovered in-plane monolayer stresses? If out-of-plane tractions

Moreover, it has been suggested in several reports that the problem of intercellular stress recovery, given the traction field, is mathematically intractable

Subsequent to Tambe et al.

On the one hand, a recent study

If monolayer material properties are homogeneous, then stress recovery using MSM is unaffected. If the properties are heterogeneous (

Although movement of the epithelial sheet has been shown to respond to the viscosity of the substrate

According to the principle of plithotaxis, the local orientation of cell migration and the local orientation of maximum principal stress are strongly associated

The association between the stress and the motion is related to the local stress anisotropy; the higher is the anisotropy, the tighter is the association

Based upon this evidence and these arguments, we agree that out-of-plane tractions may be present and are of some interest. However, out-of-plane tractions are ordinarily uncoupled from, and therefore have no appreciable effects upon, recovered in-plane stresses.

For the cellular monolayer

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