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Copyright © 2008 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 95, Issue 1, 155-165, 1 July 2008

doi:10.1529/biophysj.107.114678

Biophysical Theory and Modeling

Modeling the Influence of the E-Cadherin-β-Catenin Pathway in Cancer Cell Invasion: A Multiscale Approach

Ignacio Ramis-Conde^*, †, ,  , Dirk Drasdo^*, ‡, ,  , Alexander R.A. Anderson^† and Mark A.J. Chaplain^†

^* French National Institute for Research in Computer Science and Control (INRIA), Domaine de Voluceau-Rocquencourt, Le Chesnay, France
^† Division of Mathematics, The University of Dundee, Dundee, Scotland
^‡ Interdisciplinary Centre for Bioinformatics (IZBI), University of Leipzig, Leipzig, Germany

Address reprint requests to Dirk Drasdo, Tel.: 1-39-63-5036.

Address reprint requests to Ignacio Ramis-Conde.

Cancer is characterized by multiple mutations in a single cell leading to a loss of control in cell replication accompanied by an uncontrolled growth of the total cell mass, eventually leading to the formation of an in situ solid tumor. After the tumor reaches a certain size, genetic instability in the cancer cells may lead to further dedifferentiation within the malignant cell mass. These secondary mutations are relevant for the tumor to gain advantage over neighboring cells and to invade further the local tissue and organs. In the transition from a normal cell to a malignant cell, the modification of intracellular pathways related to cell-cell adhesion and cell-matrix adhesion are important and determine the compactness of the tumor surface and the invasiveness of the tumor 1. Cadherins are cell-cell adhesion proteins which form part of multiprotein complexes at the cell membrane to bind neighboring cells and determine the tissue architecture. The different types of cadherins are named from the type of tissue where they originate from, e.g., E-cadherin in epithelia, N-cadherin in the nervous system, etc. Of particular interest is E-cadherin, sometimes considered as a tumor suppressor protein due to its functionality in maintaining the compactness of the epithelium. The role of E-cadherin in the malfunction of cell-cell adhesion observed in colorectal cancer, and in the β-catenin degradation system after mutations that affect the wnt-pathway, belong to the most studied examples 2,3,4. Greater than 80% of colorectal tumors show malfunctions in APC, a key protein in the wnt-pathway related also to intracellular interactions where E-cadherin plays a main role. These mutations are correlated with higher cancer invasion and therefore poorer prognosis.

Mathematical modeling of cell adhesion has been approached by different strategies including continuum models 5,6 (for a general review of mathematical continuum models of cancer see 7), individual-based lattice-free models 8,10,11,12 and lattice-based models where each lattice site can at most be occupied by one single cell 13,14,15 (for general reviews of individual-based models, see 16,17,18,19). Although all these different approaches have shown the importance of cell adhesion to keep tumor compactness and prevent invasion, there is still a wide field to explore linking the intracellular dynamics of signaling pathways to the adhesion molecules at the cell surface and the extracellular consequences in invasive tumors. In this article, we approach this problem using a multiscale individual-based lattice-free model which accounts for the intracellular dynamics of the E-cadherin-β-catenin interactions and the physical forces on the cells.

When a cell adheres to adjacent neighbors, the E-cadherin molecules are situated in an intermembrane position, forming bonds with local neighbors at the intercellular space. The cytoplasmatic tail of the E-cadherin molecule binds to the proteins of the catenin family: p120-catenin, α-catenin, and β-catenin. The α-catenin and β-catenin then form a complex to link the actin filaments of the cytoskeleton and the E-cadherins. When bonds are released, caused by intracellular signaling or the effect of mechanical stress, the multiprotein complex is broken and the E-cadherins are internalized, i.e., transported into the cytoplasm by the endocytosis apparatus within cadherin-coated vesicles. It is not well known yet if, after being endocytosed, E-cadherins are degraded or if they are kept by the vesicles for later recycling 20.

When the bond is broken, β-catenin is released in a phosphorylated state. In this form, it is ready to interact with other molecules and can be recognized and degraded in the proteasome systems. Intracellular control of β-catenin concentration is important in preserving the tissue architecture. Upregulation of free β-catenin (also known as soluble β-catenin) is related to cell migration and the epithelial-mesenchymal transition 22,23, a process where a well-ordered and polarized layer of cells changes into an unstructured configuration to facilitate collective cell migration. Sufficiently large concentrations of soluble β-catenin then move from the cytoplasm into the nucleus, where it interacts with transcription factors which modify cell behavior—for example, by promoting cell proliferation. Although the precise mechanisms which relate the β-catenin nuclear translocation to cell migration are not yet very well known, it has been observed that invasive cells show a higher nuclear accumulation of soluble β-catenin. Wong and Gumbiner 22 used matrigel chambers and different cell chimeras to show how upregulation of soluble β-catenin induces cell migration. They used cell clones with different malfunctions in the cytoplasmatic tail of E-cadherin and found that invasion was enhanced when β-catenin was not able to bind E-cadherin, despite cell-cell bonds being formed at the extracellular part where it was intact. Different biological mechanisms capable to trigger invasion after activating the β-catenin pathway have been proposed 24,25,26. In particular, Kemler et al. 27 proposed that β-catenin is translocated into the nucleus above a certain concentration threshold leading to downregulation of E-cadherin-mediated adhesion, partly referring on earlier work by Huber et al. 25 and Jamora et al. 28.

In this article, we study the possible effects of a nuclear upregulation of soluble β-catenin as a consequence of being released from the E-cadherin-β-catenin complex, and the biological consequences of the existence of a β-catenin threshold which may downregulate cell adhesion during the epithelial-mesenchymal transition. Instead of considering the pathways that involve β-catenin in every detail known, we focus on a simplified β-catenin pathway which captures the key features of the cell adhesion process. The main interactions of β-catenin in our model are with the E-cadherins at the cell membrane and a generic proteasome-related complex in the cytoplasm which accounts for the whole set of proteins involved in the process of β-catenin degradation. We include soluble β-catenin and the E-cadherin-β-catenin complex as the main variables of our model. Upregulation of soluble β-catenin is assumed to interact with transcription factors in the nucleus, and induce cell migration 26.

We first present our model of the intracellular β-catenin dynamics and show the importance of this regulation system on the E-cadherin system. This is then followed by our single-cell model and a subsection on how the intracellular dynamics couples to the cell-biophysical and cell-biological single-cell parameters.

It is not completely known how E-cadherin is transported from the cytosol to the intermembrane position to form bonds 29. However, there is some evidence that cytoplasmatic E-cadherin translocates to the membrane after binding to β-catenin at the endoplasmic reticulum 30. For simplicity, we assume that the complexes are formed in the cell membrane. This is translated into our model considering three possible different states of the E-cadherins: catenin free in the cytoplasm ([E_c]), catenin free at the cell membrane ([E_m]), and the complex E-cadherin-β-catenin forming bonds at the cell membrane ([E/β]). As one cell comes into contact with another cell, the cadherin in the cytoplasm moves to the cell surface. Here we make the assumption that the amount of cadherin stimulated to move is proportional to the contact area. At the cell membrane, the cadherin binds to β-catenin and the cell forms bonds with its neighboring cells. On the other hand, if cell detachment occurs, then the β-catenin-E-cadherin complex is ruptured, the β-catenin becomes soluble β-catenin, and E-cadherin is sequestered into the cytoplasm by endocytosis. Further, we assume that E-cadherin can be recruited to form bonds again. In this way we can consider the total E-cadherin concentration (E_T=100nM) to be constant:

(1)

(2)

(3)

(4)

Fig. 1 summarizes the intracellular interactions of the different forms of E-cadherin. (We also considered the backward reaction in Eq. (3) and found that it neither modifies the β-catenin steady-state concentrations within a cell, nor changes the multiscale qualitative dynamics as we validated by fixed point analysis and by computer simulations). Following Lee et al. 31, we assume that the production of β-catenin occurs at a constant rate (k_m=0.01nMmin⁻¹). The degradation process takes place after forming a complex with the proteasome. In the framework of our model, this proteasome variable should be understood as a complex of proteins which, after different biochemical interactions, degrade soluble β-catenin, i.e.,

(5)

Display large version of this figure

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).

In a similar way to the E-cadherin concentration, we assume that the proteasome total concentration remains constant (P_T=0.33514 nM), and therefore

(6)

(7)

(8)

(9)

(10)

The functions a(t)_c,j and a(t)_d,j determine the area stimulated to interchange the cadherin from the cytosol to the membrane and from the membrane to the cytosol, respectively. We define these functions as

Hence, both attachment and detachment of cells lead to an exchange of E-cadherin between the membranes in the contact zone of the interacting cells. We assume that Eqs. (8) determine the concentrations of β-catenin and β-catenin-E-cadherin only as long as the soluble β-catenin concentration is below a threshold c_T. In the case [β]>c_T, we consider the soluble β-catenin in cytoplasm to be large enough and as a consequence it is free to enter the nucleus and interact with transcription factors causing the cell to migrate. As a necessary step of migration, cell detachment occurs. To model the active detachment process we assume that for [β]>c_T,

(11)

(12)

Since the molecular kinetics of β-catenin and its interaction varies between different cells, each cell within our model is considered as an individual entity which intracellular dynamics are governed by the explained equations. Motivated by the observations that cells in isolation tend to aggregate 32, we assume that an invasive cell changes into a noninvasive state again if it comes into contact with other cells to which it can attach. In this case, Eqs. (8) are recovered.

The intracellular parameters values were taken, when possible, from Lee et al. 31. Any others were chosen in a similar range to the ones they used. All the parameters values are given in Table 1.

We model each cell as an isotropic elastic object capable of migration and division and parameterize it by cell-kinetic, biophysical and cell-biological parameters that can be experimentally measured. We now describe below the key features of this modeling approach.

We assume that an individual cell in isolation is spherical. We characterize the cell shape of a spherical cell by its radius R.

In our model the cell cycle is subdivided into two phases—the interphase and the mitotic phase. We assume that during interphase the cell doubles its mass. In the mitotic phase a cell divides into two daughter cells. We model the process of cell division by replacing two cells of size R by two daughter cells of radius R/2^1/3 which then gradually grow during interphase to their original radius R. This radius value to determine the size of the new daughter cells was taken in according to the simulations performed by Galle et al. 8, where they reproduced realistic tumor growth curves using an individual force-based model of similar characteristics to ours.

With decreasing distance between the centers of two cells (e.g., upon compression), both their contact area and the number of adhesive contacts increase, resulting in an attractive interaction. On the other hand, if cells are spheroidal in isolation, a large contact area between them significantly stresses their cytoskeleton and membranes. Furthermore, experiments suggest that cells only have a small compressibility (the Poisson numbers are close to 0.5 33,34). In this instance, both the limited deformability and the limited compressibility give rise to a repulsive interaction. We model the combination of the repulsive and attractive energy contributions by a modified Hertz-model 8,10 where the potential V_ij between two cells of radius R_i and R_j is given by

(13)

(14)

(15)

We model cell movement by a stochastic equation of motion for each cell. Alternatively, the Metropolis-algorithm has been used in Drasdo and Hoehme 11. The direct use of equations of motion for the cells permits one to include more easily the limiting case of very small (or no) noise and is more intuitive. In this approach, cells move under the influence of forces and a random contribution to the locomotion which results from the local exploration of space. Moreover, in some of the scenarios explained below, a chemoattractant chemical generates a forcelike term in the Langevin equation as suggested in Stevens 36:

(16)

Inertia terms have been neglected due to the high friction of cells with their environment, and we also do not consider the existence of any memory term, as in Galle et al. 8. The value is the velocity of the cell i at time is the force of cell j on cell i (previously calculated from Eqs. (13)) and the sums are over the nearest neighbors in contact with cell i. The s-friction term determines the friction with the substrate and the cell-cell friction determines the friction with the nearest neighbors. The tensors and denote cell-cell friction and cell-substrate friction, respectively. In our model, cells are considered to be spherical and to be surrounded by a homogeneous and isotropic environment, either a gel or a homogeneous intracellular matrix, depending on the experimental situation under consideration. Under these assumptions, the cell-substrate friction tensor takes the form

The chemotaxis term is the chemotactic/haptotactic response toward a gradient of morphogen Q(t) and χ is the cell sensitivity to the chemical. This last term is only included in some of our simulations scenarios as is specified below.

In our model, the adhesion forces between cells are controlled by the density of E-cadherin in the cell membrane within the cell-cell contact zone. We could not find in the literature specific values for E-cadherin concentration per cell-cell bond. In their simulations Galle et al. 8, used an adhesion energy per unit of area of W_sϱ_m=200μNm⁻¹, so that the surface receptor density is We use this value as a maximum density of the cadherin-β-catenin complex in the membrane and define the actual density by

Fig. 2 shows the resulting force function depending on the different values. By modifying the intracellular concentration of β-catenin the cells can control the concentration of [E/β]-complexes and thereby the strength of the intercellular adhesion force.

Display large version of this figure

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.

The active decision of a cell to migrate can be triggered in different ways, all of them involving an upregulation of the soluble β-catenin which overcomes the critical threshold c_T. One case happens if the cytoplasmic concentration of β-catenin is upregulated due to a failure in the proteasome system. A further case happens if the detachment of local neighbors upregulates the soluble β-catenin concentration. In both cases, β-catenin enters the nucleus and triggers cell migration. One way that this could cause rupture of the cell-cell contacts is by physical forces that a cell that starts to migrate exerts on the cadherin bonds in the cell-cell contact area to its neighbors. In our simulations, we have chosen the last term in Eq. (16) that represents chemotaxis so large that the cells at the tumor surface were not able to detach by breaking the cell-cell contacts but they need to downregulate their adhesion molecules. However, detachment could also be triggered by an increase of the intrinsic random movement component of a cell represented by the noise term which we do not consider here.

In this section, we present the results of computational simulations carried out on our model given by Eqs. (7). In the first set of simulations in Figure 3 and Figure 4, we present results from a numerical simulation of the system of ordinary differential Eqs. (7) governing the concentrations of various forms of E-cadherin, β-catenin, and the proteasome in a single cell.

Display large version of this figure

Figure 3

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

Display large version of this figure

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≈0.4min. 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.

To illustrate the response of possible malfunctions in the intracellular control on the β-catenin concentration, we study simulations for different attachment/detachment scenarios. If the cell remains adhered to its neighbors, almost all of the β-catenin remains bound to the E-cadherin complexes at the cell membrane. If a cell detaches, then the concentration of soluble β-catenin increases.

Fig. 3 shows the concentration of the intracellular variables of a single cell when it attaches other cells; as can be seen from the figure, soluble β-catenin (dotted line) is rapidly sequestered from the cytoplasm by the cadherins to form the [E/β] complex (solid line). As long as the contacts are maintained, the soluble β-catenin concentration remains at a low level. If some of the neighboring cells detach, then the concentration of E-cadherin forming bonds will be partially reduced.

Fig. 4 shows concentrations of the intracellular variables for two different detachment scenarios. In the figure on the left, we assume that the cell loses all its bonds with the neighbors at t≈0.4, which triggers a dramatic increase of the β-catenin concentration in the cytoplasm. This soluble β-catenin enters the nucleus in excess of the threshold concentration necessary to initiate migration and promote cell movement via transcription. On the right figure, at time t ≈ 0.4, the cell has lost only one-quarter of its bonds with the neighbors and the soluble β-catenin concentration is insufficient to cause cell migration.

We implement the intracellular dynamics model explained above in every single cell of the individual-force-based model. The advantage of using this type of modeling approach is that not only does it enable us to explicitly include the influence of intracellular pathways, but also provides a realistic approach to model the biophysical properties of individual cells which cannot be neglected when studying tissue organization. We reproduce in silico different scenarios of relevance in cancer growth and invasion and study the behavior of detachment waves in epithelial layers and how it can produce the epithelial-mesenchymal transition. We also study the β-catenin distribution in small tumors and how its upregulation can induce invasion.

Figure 5 and Figure 6 show the spatio-temporal dynamics in a hexagonal lattice of the cells soluble β-catenin concentration, similar to the natural configuration of an epithelial layer. We choose here a tissue architecture where we have a layer of cells, with each cell in the layer being attached to each neighboring cell, and initial values of E-cadherin and β-catenin at the steady state for the intracellular model. The intracellular concentration of β-catenin is denoted by the color of the cell: white denotes high concentration of soluble β-catenin and black low. We note that as we have assumed that high concentrations of β-catenin induce cell-cell detachment, this occurence of colors is equivalent to saying that black denotes strong cell adhesion and white denotes weak cell adhesion. Within the cell layer, we insert a cell which has no regulation activity in the β-catenin pathway (we force its intracellular dynamics to rise [β]>c_T after a certain time and therefore detach from its nearest neighbors). Fig. 5 shows how this malfunction produces a wave caused by the intracellular upregulation of β-catenin in the nearest neighbors. In the way this wave front is moving, it can be observed how it induces a temporal cell-cell detachment within the whole epithelial layer. A particular feature of these waves that should be highlighted is the fact that they do not satisfy the principle of superposition.

Display large version of this figure

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.

Display large version of this figure

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.

Fig. 7 shows how the three-dimensional spatio-temporal dynamics influence cell migration in a scenario where cells migrate away from a two-dimensional epithelial layer to produce the epithelial-mesenchymal transition. The so-called epithelial-mesenchymal transition is a process where a well-polarized layer of cells become diffuse and lose their initial structure and compactness. This transition occurs in a similar way at the tumor surface when invasion occurs—the mass of outer cells lose contact and invade further into the tissue. To model the epithelial-mesenchymal transition, we set as initial conditions a layer of cells where the outermost cells are fixed (i.e., unable to move). We include a constant force term as if it were a constant source of chemoattractant (term 5 in Eq. (9)) that diffuses toward the tissue in the form of the equation

Display large version of this figure

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 and Figure 9 show the distribution of soluble catenin in a two-dimensional cross section of a three-dimensional tumor, and a two-dimensional aggregation process of a culture of metastatic cells on a petri dish. The process of tumor growth involves different cellular interactions which alter and remodel the E-cadherin configuration at the cell surface with a subsequent impact on β-catenin concentration. In a proliferative mass of cancer cells, cell detachment, via internal signaling or physical forces, is a necessary process to release the stress and allow the proliferation.

Display large version of this figure

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).

Display large version of this figure

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.

Fig. 8 shows a section of the tumor (radius≈100μm) where we can observe the different distribution of the intracellular catenin. The outer rim of the growing tumor has a higher number of cells with nuclear β-catenin concentration, while, in the center of the tumor, cells downregulate the intracellular β-catenin concentration when it is sequestered by the E-cadherins forming bonds at the cell membrane. These simulations are in very good agreement with the findings of Brabletz et al. 4.

Fig. 9 shows the same patterns found when we study the aggregation process in two-dimensional layers. These findings show how our model can reproduce not only the epithelial-mesenchymal transition but also recover the compactness of the subsequent distant metastatic clusters via aggregation and growth. If the threshold value c_T is small enough, the same traveling wave patterns as those found in the previous scenarios can be observed within the tumor. For threshold values high enough (c_T>E_T), the waves can be avoided but the β-catenin distribution observed by Brabletz et al. 4 remains.

In Fig. 10, we show results from the same growing tumor scenario stimulated by a source of chemoattractant. If cell detachment occurs, then cells will migrate toward the source of attractants. The proteasome functionality has been downregulated to the point where migration occurs. It can be observed that the outer cells migrate and detach from the main tumor mass; the new cells at the outer rim lose part of their E-cadherin bonds, upregulate soluble β-catenin, and enhance their migration and invasion. These findings suggest how invasion can be a gradual process produced by subsequent layers of cells that detach the tumor surface.

Display large version of this figure

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.

We performed a steady-state and a sensitivity analysis of the intracellular reaction equations. Table 2 reports the result of the sensitivity analysis. Shown is the percentage of the soluble β-catenin in relation to the steady state obtained for the parameters of Table 1 (e.g., “100” means the corresponding value has not changed). The intracellular dynamics is very robust with regard to changes of most parameters. The largest sensitivity has been found for variations of the β-catenin production rate (k_m) and in the proteasome degradation rate (k₂).

The steady states of the β-catenin-proteasome complex and soluble β-catenin are

(17)

and

(18)

respectively. From Eq.

We studied the dependency of the multiscale dynamics on the intracellular β-catenin degradation rate (k₂). We simulated an invasion assay of similar characteristics to the ones performed by experimentalists in matrigel chambers 22. The initial conditions were taken as in the scenario simulated in Fig. 10 where a growing tumor is stimulated by a source of chemoattractant. We performed the migration assay for three different degradation rates: fast (k₂=10min⁻¹), medium (k₂=1min⁻¹), and no degradation (k₂=0min⁻¹), and plot over time the number of cells achieving escape from the initial tumor to a distance of 150 μm. Fig. 11 shows the results of the migration assay. It can be seen that the intensity of degradation activity of the proteasome determines the capacity of invasion of the malignant cells.

Display large version of this figure

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 150μm away from the principal tumor over time. It can be observed that as the β-catenin degradation rate is decreased (k₂=10, 1, and 0min⁻¹), the malignant cells become more invasive.

In this section, we have studied the intracellular and extracellular dynamics that would cause a possible soluble β-catenin upregulation via the release of E-cadherin bonds. As a framework, we used a similar approach to the experiments performed by Brabletz et al. 4, and we have shown the same patterns of intracellular catenin distribution under a growth process and an aggregation process. Brabletz et al. 4, looking at the intracellular concentration of β-catenin, postulated how tumor progression was driven by interactions with the tumor environment. In our findings, we have shown how this interaction may be mainly driven by the tumor cells themselves. We have shown how downregulation of β-catenin can be mainly driven by cell-cell contacts and how this fact gives an invasive advantage to the cells that are positioned at the outer rim of the tumor. We have simulated the different main steps involved in an invasion process and shown how the epithelial-mesenchymal transition can be achieved (migration) and reversed (aggregation and growth) depending on the regulation of soluble β-catenin by local contacts.

More intriguing is how these simple dynamics can create waves of temporal cell-cell detachments. When considering the structure of a human tissue, we have to bear in mind how it is exposed to continuous mechanical stress and remodeling. Cell migration, apoptosis, and cell mitosis are, probably, the three most important events that can alter the physical configuration of the layer. Shimamuram and Takeichi 29 have shown how E-cadherin expression was transient in mouse embryonic brain morphogenesis. They stained different zones of the brain and neural tube and showed how E-cadherin expression followed particular patterns of expression where positive E-cadherin cells were isolated in different configurations from negative E-cadherin cells. To maintain the natural tissue configuration and the stress levels under a threshold that permits cell survival, a local mechanism of signaling and migration of neighboring cells is necessary. This mechanism must allow cells, in attaching and detaching, to find an optimal position such as is seen in an epithelial configuration, but must also be efficient enough to avoid unnecessary detachments and disorganization of the tissue. We do not claim that our model of the internal cell dynamics is the exact model of what actually occurs, but we do wish to highlight how a simple mechanism can show transient catenin/cadherin expression (recall that, in our model, cadherin is surface-expressed when forming bonds) and differentiate the tissue into two separate parts (cadherin/catenin positive and cadherin/catenin negative). In our simulations, we have shown how a mechanism biregulated by cell-cell junctions and an independent degradation system can be sufficient to do this work. When a cell needs to migrate and releases the bonds with its neighbors, a “traveling wave” of detachment happens. This wave facilitates the cell layer to reaccommodate and release the stress caused by the new movement. Furthermore, this model produces a self-regulation mechanism—when two waves collide, both of them vanish. This helps to prevent tissue disorganization caused by an excess of signaling. If many cells in the layer are sending detachment signals at the same time, the whole layer will become detached and chaotic; however, if the detachment waves vanish when they collide, each single cell, at one time, releases its bonds as if there were only a single wave.