Modeling the Influence of the E-Cadherin-[Beta]-Catenin Pathway in Cancer Cell Invasion: A Multiscale Approach
By Ramis-Conde, Ignacio Drasdo, Dirk; Anderson, Alexander R A; Chaplain, Mark A J
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.
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 cellmatrix 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 beta-catenin degradation system after mutations that affect the wnt-pathway, belong to the most studied examples (2-4). 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 (8,10-12) and lattice-based models where each lattice site can at most be occupied by one single cell (13-15) (for general reviews of individual-based models, see (16-19)). 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 unking 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-beta-catenin interactions and the physical forces on the cells.
THE BIOLOGICAL BACKGROUND
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, beta-catenin, and beta-catenin. The alpha- catenin and beta-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, beta-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 beta-catenin concentration is important in preserving the tissue architecture. Upregulation of free beta-catenin (also known as soluble beta-catenin) is related to cell migration and the epithelialmesenchymal transition (22,23), a process where a wellordered and polarized layer of cells changes into an unstructured configuration to facilitate collective cell migration. Sufficiently large concentrations of soluble beta- 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 beta-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 beta-catenin. Wong and Gumbiner (22) used matrigel chambers and different cell chimeras to show how upregulation of soluble beta- 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 beta-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 beta-catenin pathway have been proposed (24-26). In particular, Kemler et al. (27) proposed that beta-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 beta-catenin as a consequence of being released from the E-cadherin-beta-catenin complex, and the biological consequences of the existence of a beta-catenin threshold which may downregulate cell adhesion during the epithelial- mesenchymal transition. Instead of considering the pathways that involve beta-catenin in every detail known, we focus on a simplified beta-catenin pathway which captures the key features of the cell adhesion process. The main interactions of beta-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 beta-catenin degradation. We include soluble beta-catenin and the E-cadherin- beta-catenin complex as the main variables of our model. Upregulation of soluble beta-catenin is assumed to interact with transcription factors in the nucleus, and induce cell migration (26).
THE beta-CATENIN KINETICS
We first present our model of the intracellular beta-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.
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 beta-catenin. When two cells become in contact, the cadherin travels to the membrane determined by the function c^sub i^(i), it binds, between other molecules, to beta-catenin and forms a bond with the neighbor cell. When detachment occurs, the complex beta-cateninE-cadherin is broken at a rate governed by the function d^sub i^(t).
Since the molecular kinetics of beta-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 and 9 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. TABLE 1 Parameter values used in the intracellular simulations
THE BIOPHYSICAL MODEL OF A SINGLE CELL
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 phasesthe 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^sup 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.
The chemotaxis term is the chemotactic/haptotactic response toward a gradient of morphogen Q(t) and chi is the cell sensitivity to the chemical. This last term is only included in some of our simulations scenarios as is specified below.
COUPLING OF CELL PARAMETERS TO INTRACELLULAR MOLECULE CONCENTRATIONS
Fig. 2 shows the resulting force function depending on the different o^sup ij^^sub m^, values. By modifying the intracellular concentration of beta-catenin the cells can control the concentration of [E/beta]-complexes and thereby the strength of the intercellular adhesion force.
The active decision of a cell to migrate can be triggered in different ways, all of them involving an upregulation of the soluble beta-catenin which overcomes the critical threshold c^sub T^. One case happens if the cytoplasmic concentration of beta-catenin is unregulated due to a failure in the proteasome system. A further case happens if the detachment of local neighbors upregulates the soluble beta-catenin concentration. In both cases, beta-catenin enters the nucleus and triggers cell migration. One way that this could cause rupture of the cellcell 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.
FIGURE 2 The left plot shows the force function between two cells, variables are distance between the centers of the cells (in [mu]m) and adhesion energy per unit of area in contact (in [mu]N/ 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.
In this section, we present the results of computational simulations carried out on our model given by Eqs. 7-12 and 16. In the first set of simulations in Figs. 3 and 4, we present results from a numerical simulation of the system of ordinary differential Eqs. 7-12 governing the concentrations of various forms of E- cadherin, beta-catenin, and the proteasome in a single cell.
To illustrate the response of possible malfunctions in the intracellular control on the beta-catenin concentration, we study simulations for different attachment/detachment scenarios. If the cell remains adhered to its neighbors, almost all of the beta- catenin remains bound to the E-cadherin complexes at the cell membrane. If a cell detaches, then the concentration of soluble beta- 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 beta-catenin (dotted line) is rapidly sequestered from the cytoplasm by the cadherins to form the [E/beta] complex (solid line). As long as the contacts are maintained, the soluble beta-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 [asymptotically =] 0.4, which triggers a dramatic increase of the beta-catenin concentration in the cytoplasm. This soluble beta- 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 [asymptotically =] 0.4, the cell has lost only one-quarter of its bonds with the neighbors and the soluble beta-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 beta-catenin distribution in small tumors and how its upregulation can induce invasion.
FIGURE 3 Plot showing the concentrations over time of the intracellular variables of a cell that attaches to a group of cells at t [asymptotically =] 0.4 min. The beta-catenin is rapidly sequestered by the cadherins that travel to the cell surface to form bonds.
FIGURE 4 Plots showing die concentrations over urne of the intracellular variables under two different scenarios. On the left plot, a cell loses its contact with its neighbors at t [asymptotically =] 0.4 min. The beta-catenin concentration increases dramatically and it enhances mechanisms which promote invasion. On the right plot only a few of the neighbors are detached, soluble beta-catenin is maintained under the threshold levels (C^sub T^ – 0.5) that enhances migration.
Detachment waves in epithelial layers
Figs. 5 and 6 show the spatio-temporal dynamics in a hexagonal lattice of the cells soluble beta-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 beta-catenin at the steady state for the intracellular model. The intracellular concentration of beta- catenin is denoted by the color of the cell: white denotes high concentration of soluble beta-catenin and black low. We note that as we have assumed that high concentrations of beta-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 beta-catenin pathway (we force its intracellular dynamics to rise [beta] > C^sub T^ after a certain time and therefore detach from its nearest neighbors). Fig. 5 shows how this malftmction produces a wave caused by the intracellular upregulation of 0-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.
FIGURE 5 Plots of the spatio-temporal dynamics of [beta] 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 beta- 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.
The epithelial mesenchymal transition
FIGURE 6 Plots showing the spatio-temporal dynamics of a scenario where two cells with upregulated soluble beta-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.
Tumor growth, aggregation and invasion
Figs. 8 and 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 beta- 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. 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).
Fig. 8 shows a section of the tumor (radius [asymptotically =] 100 [mu]m) 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 0-catenin concentration, while, in the center of the tumor, cells downregulate the intracellular beta-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^sub 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^sub T^ > E^sub T^). the waves can be avoided but the beta-catenin distribution observed by Brabletz et al. (4) remains.
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 beta-catenin distribution as those found by Brabletz et al. (4). Clearer cells denote a higher beta-catenin nuclear concentration.
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 beta-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.
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 beta- 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 beta-catenin production rate (k^sub m^) and in the proteasome degradation rate (k^sub 2^).
We studied the dependency of the multiscale dynamics on the intracellular 0-catenin degradation rate (k^sub 2^). 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 (k^sub 2^ = 10 min^sup -1^), medium (k^sub 2^ = 1 min^sup -1^)( and no degradation (k^sub 2^ = 0 min^sup -1^), and plot over time the number of cells achieving escape from the initial tumor to a distance of 150 [mu]m. 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.
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 beta-catenin is uprcgulated (light gray). Unit of time is measure in minutes.
In this section, we have studied the intracellular and extracellular dynamics that would cause a possible soluble beta- 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 beta-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 beta-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 beta-catenin by local contacts.
TABLE 2 Results of the sensitivity analysis performed to study the variation of soluble beta-catenin
More intriguing is how these simple dynamics can create waves of temporal celt-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.
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 [mu]m away from the principal tumor over time. It can be observed that as the beta-catenin degradation rate is decreased (k^sub 2^ = 10, 1, and 0 min^sup -1^), the malignant cells become more invasive.
D.D. acknowledges support by BMBF Hepatosys. No 0313081F.
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Ignacio Ramis-Conde,*[dagger] Dirk Drasdo,*[double dagger] Alexander R. A. Anderson,[dagger] and Mark A. J. Chaplain[dagger]
* French National Institute for Research in Computer Science and Control (INRIA), Domaine de Voluceau-Rocquencourt, Le Chesnay, France; [dagger] Division of Mathematics, The University of Dundee, Dundee, Scotland; and [double dagger] Interdisciplinary Centre for Bioinformatics (IZBI), University of Leipzig, Leipzig, Germany
Submitted June 11, 2007. and accepted for publication January 3, 2008.
Address reprint requests to Dirk Drasdo, Tel: 1-39-63-5036; E- mail: dirk. email@example.com, and Ignacio Ramis-Conde, ignacio,firstname.lastname@example.org.
Editor: Byron Goldstein.
Copyright Biophysical Society Jul 1, 2008
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