A hybrid model of intercellular tension and cell-matrix mechanical interactions in a multicellular geometry

Abstract

Epithelial cells form continuous sheets of cells that exist in tensional homeostasis. Homeostasis is maintained through cell-to-cell junctions that distribute tension and balance forces between cells and their underlying matrix. Disruption of tensional homeostasis can lead to epithelial-mesenchymal transition (EMT), a transdifferentiation process in which epithelial cells adopt a mesenchymal phenotype, losing cell-cell adhesion and enhancing cellular motility. This process is critical during embryogenesis and wound healing, but is also dysregulated in many disease states. To further understand the role of intercellular tension in spatial patterning of epithelial cell monolayers, we developed a multicellular computational model of cell-cell and cell-substrate forces. This work builds on a hybrid cellular Potts model (CPM)-finite element model to evaluate cell-matrix mechanical feedback of an adherent multicellular cluster. Cellular movement is governed by thermodynamic constraints from cell volume, cell-cell and cell-matrix contacts, and durotaxis, which arises from cell-generated traction forces on a finite element substrate. Junction forces at cell-cell contacts balance these traction forces, thereby producing a mechanically stable epithelial monolayer. Simulations were compared to in vitro experiments using fluorescence-based junction force sensors in clusters of cells undergoing EMT. Results indicate that the multicellular CPM model can reproduce many aspects of EMT, including epithelial monolayer formation dynamics, changes in cell geometry, and spatial patterning of cell-cell forces in an epithelial tissue.

Keywords: Cell mechanics; Cellular Potts model; Cell–cell junction forces; Epithelial–mesenchymal transition; Spatial patterning; Traction forces.

Fig. 1

Simulated cells (red pixels) migrate on a finite element substrate that responds to cell-generated traction forces. Traction forces are calculated based on either (A) individual cell geometries or (B) multicellular clusters. (C, left) Representation of traction forces with resulting strain for multicellular geometries, and (C, right) inset of time points from panel B. Time in units of Monte Carlo steps (MCS). Proliferation rate p_divide = 0 for this figure only.

Fig. 2

Spatiotemporal dynamics of simulated and in vitro tissue patterning. Visual comparison of time points from initial seeding to confluence illustrates parallels between (A) in vitro immunolfuorescence images of actin (red) and (B) simulated spatial patterns. Time in panel A in hours, and in panel in Monte Carlo steps (MCS). (C) Confluence, defined as the fraction of total cell area to total substrate area, is shown as a function of time, for in vitro and in silico experiments, for different conditions: In silico measurements of confluence are shown for different values of the cell contact inhibition to substrate inhibition (J_cc/J_cm; green, red, blue lines). In vitro mean confluence measurements ± standard error are shown for control (black) and 4 ng/mL TGF-β1 treatment (magenta). Time scale relating in vitro to in silico measurements: 4.8 min/1 MCS, J_cm = 2.5.

Fig. 3

Parameter sweep of interaction energies. (A–D) Single cell first moment of area (FMA) model and (E–H) multicellular FMA simulated confluence, cell area, cell count, and net traction force, shown as a function of the ratio of cell-cell contact inhibition to cell-matrix inhibition (J_cc/J_cm), varying J_cm values.

Fig. 4

Morphological characterization of the epithelial tissues with altered contact inhibition. (A) Representative immunofluorescence images of actin (red) illustrate a confluent MCF10A monolayer bounded by the 250 × 250 μm microfabricated square; scale bar = 50 μm. In vitro (B, D) and in silico (C, E) average cell count and cell area for the confined geometry are shown for each TGF-β1 dosage and ratio of contact interaction energies (J_cc/J_cm), respectively. Sample sizes: n=3 experiments, 7–10 monolayers per experiment per dose (in vitro); n = 5 simulations per parameter set (in silico). * denotes significance by one-way ANOVA test between each TGF-β1 dosage (B, D) or each contact energy ratio (C, E).

Fig. 5

Simulated cell-cell junctional force spatial patterns reflects TGF-β1 effects in vitro. (A) In vitro FRET intensities in MDCK II cells. (B) Corresponding heatmaps for average FRET intensities are binned into a 5 × 5 grid, and (C) their associated mean for corner, edge, and interior bins for 0, 2, and 4 ng/mL TGF-β1 dosages. Note the y-axis lower limit in panel C corresponds with a FRET ratio of 0.3. Schematic (right) illustrates bin positions. (D) Simulated cell-cell junctional force is depicted as the net magnitude for high, medium, and low interaction energy (J_cc/J_cm) ratios. (E) Cell-cell junctional force magnitudes are shown as a 5 × 5 grid with (F) their associated mean for corner, edge, and interior bins. Sample sizes: n=3 experiments, 7–10 monolayers per experiment per dose (in vitro); n = 5 simulations per parameter set (in silico). Binned and position values shown in panels B, C and E, F represent averages over all samples. In panel C, p-value denotes near significance for Student’s t-test comparing Corner and Interior spatial locations. * denotes significance by one-way ANOVA test between spatial location (F). Force in panels D, E, and F in arbitrary units (a.u.).

Fig. 6

Individual cell geometry spatial patterns (A) In vitro heatmaps for binned cell area treated with 0, 2, and 4 ng/mL TGF-β1 and (B) their associated bar graphs for average corner, edge, and interior. (C) In silico heatmaps for binned cell area at high, medium, and low contact inhibition and (D) their associated bar graphs. Sample sizes: n=3 experiments, 7–10 monolayers per experiment per dose (in vitro); n = 5 simulations per parameter set (in silico). Binned and position values represent averages over all samples. * denotes significance by one-way ANOVA test between each spatial location (B, D).

Fig. 7

Multicellular forces at mechanical equilibrium. (A) Representative snapshot of the traction and junction forces in the multicellular CPM model. (B) Plots of the traction and junction forces (in arbitrary units, a.u.) from the CPM simulations shows that traction force (blue lines, circles) scales linearly with distance from monolayer centroid and cell-cell junctional forces (red line, circles) drops off quadratically from the centroid. (C) One-dimensional tissue simplification illustrating the balance of traction and cell-cell junctional forces. See text for further description.