Modeling the influence of the E-cadherin-beta-catenin pathway in cancer cell invasion: a multiscale approach

Abstract

In this article, we show, using a mathematical multiscale model, how cell adhesion may be regulated by interactions between E-cadherin and beta-catenin and how the control of cell adhesion may be related to cell migration, to the epithelial-mesenchymal transition and to invasion in populations of eukaryotic cells. E-cadherin mediates cell-cell adhesion and plays a critical role in the formation and maintenance of junctional contacts between cells. Loss of E-cadherin-mediated adhesion is a key feature of the epithelial-mesenchymal transition. beta-catenin is an intracellular protein associated with the actin cytoskeleton of a cell. E-cadherins bind to beta-catenin to form a complex which can interact both with neighboring cells to form bonds, and with the cytoskeleton of the cell. When cells detach from one another, beta-catenin is released into the cytoplasm, targeted for degradation, and downregulated. In this process there are multiple protein-complexes involved which interact with beta-catenin and E-cadherin. Within a mathematical individual-based multiscale model, we are able to explain experimentally observed patterns solely by a variation of cell-cell adhesive interactions. Implications for cell migration and cancer invasion are also discussed.

FIGURE 1

The three states of E-cadherin considered in the model: free in the cytoplasm, just arrived to the cell membrane and forming bonds in a multiprotein complex which includes β-catenin. When two cells become in contact, the cadherin travels to the membrane determined by the function c_i(t), it binds, between other molecules, to β-catenin and forms a bond with the neighbor cell. When detachment occurs, the complex β-catenin-E-cadherin is broken at a rate governed by the function d_i(t).

FIGURE 2

The left plot shows the force function between two cells, variables are distance between the centers of the cells (in μm) and adhesion energy per unit of area in contact (in μN/m). The right plot shows the vertical view of the same graph, where it can be better observed the adhesive interaction between cells depending on the E-cadherin concentration forming bonds. The gridded part determines the zone where adhesive forces act.

FIGURE 3

Plot showing the concentrations over time of the intracellular variables of a cell that attaches to a group of cells at t ≈ 0.4 min. The β-catenin is rapidly sequestered by the cadherins that travel to the cell surface to form bonds.

FIGURE 4

Plots showing the concentrations over time of the intracellular variables under two different scenarios. On the left plot, a cell loses its contact with its neighbors at t ≈ 0.4 min. The β-catenin concentration increases dramatically and it enhances mechanisms which promote invasion. On the right plot only a few of the neighbors are detached, soluble β-catenin is maintained under the threshold levels (c_T = 0.5) that enhances migration.

FIGURE 5

Plots of the spatio-temporal dynamics of [β] in a layer of cells where a single cell with upregulated soluble catenin (white) is situated on a layer with defective proteasome system. As can be seen from the plots, it produces a wave of upregulated β-catenin (white) caused by the induced decision of detachment in other cells. After the wave has passed, strong adhesion is recovered (black). Unit of time is measure in minutes.

FIGURE 6

Plots showing the spatio-temporal dynamics of a scenario where two cells with upregulated soluble β-catenin are situated on a layer with defective proteasome system (white), inducing two detachment waves. The detachment waves collide and vanish. This outcome prevents the cell layer to become disorganized due to an excess of detachment signal. Unit of time is measured in minutes.

FIGURE 7

Plots showing how malfunctions in the proteasome system can alter the layer configuration producing the epithelial-mesenchymal transition. In this figure, the cells migrate toward a source of attractants escaping from the initial epithelial configuration. Migration can occur only when the catenin levels are over a determined threshold (yellow). Unit of time is measured in minutes.

FIGURE 8

Plot showing a transversal section of a tumor shows how the catenin spatial distribution depends on the tumor geometry. Cells in gray have fewer binding neighbors and the catenin concentration is not attached to the cadherins and free to go into the nucleus. Cells in the center of the tumor show how catenin is better downregulated by a larger number of binding neighbors (black).

FIGURE 9

Plot showing a scenario where cells aggregate and grow in a two-dimensional configuration in a petri dish where they show similar patterns of β-catenin distribution as those found by Brabletz et al. (4). Clearer cells denote a higher β-catenin nuclear concentration.

FIGURE 10

Plot showing how a small tumor invades further tissue stimulated by a source of morphogen located at the right-hand side of the tumor. Cells decide to detach gradually when the intracellular concentration of β-catenin is upregulated (light gray). Unit of time is measure in minutes.

FIGURE 11

Plot showing the simulation results of a cell invasion assay. The plot shows the number of cells that achieve a migration distance of 150 μm away from the principal tumor over time. It can be observed that as the β-catenin degradation rate is decreased (k₂ = 10, 1, and 0 min⁻¹), the malignant cells become more invasive.