A Theoretical Model of the Wnt Signaling Pathway in the Epithelial Mesenchymal Transition

Abstract

Background: Following the formation of a primary carcinoma, neoplastic cells metastasize by undergoing the epithelial mesenchymal transition (EMT), which is triggered by cues from inflammatory and stromal cells in the microenvironment. EMT allows epithelial cells to lose their highly adhesive nature and instead adopt the spindle-like appearance, as well as the invasive and migratory behavior, of mesenchymal cells. We hypothesize that a bistable switch between the epithelial and mesenchymal phenotypes governs EMT, allowing the cell to maintain its mesenchymal phenotype even after it leaves the primary tumor microenvironment and EMT-inducing extracellular signal.

Results: This work presents a simple mathematical model of EMT, specifically the roles played by four key proteins in the Wnt signaling pathway: Dishevelled (Dvl), E-cadherin, β-catenin, and Slug. The model predicts that following activation of the Wnt pathway, an epithelial cell in the primary carcinoma must attain a threshold level of membrane-bound Dvl to convert to the mesenchymal-like phenotype and maintain that phenotype once it has migrated away from the primary tumor. Furthermore, sensitivity analysis of the model suggests that in both the epithelial and the mesenchymal states, the steady state behavior of E-cadherin and the transcription factor Slug are sensitive to changes in the degradation rate of Slug, while E-cadherin is also sensitive to the IC₅₀ (half-maximal) concentration of Slug necessary to inhibit E-cadherin production. The steady state behavior of Slug exhibits sensitivity to changes in the rate at which it is induced by β-catenin upon activation of the Wnt pathway. In the presence of sufficient amount of Wnt ligand, E-cadherin levels are sensitive to the ratio of the rate of Slug activation via β-catenin to the IC₅₀ concentration of Slug necessary to inhibit E-cadherin production.

Conclusions: The sensitivity of E-cadherin to the degradation rate of Slug, as well as the IC₅₀ concentration of Slug necessary to inhibit E-cadherin production, shows how the adhesive nature of the cell depends on finely-tuned regulation of Slug. By highlighting the role of β-catenin in the activation of EMT and the relationship between E-cadherin and Slug, this model identifies critical parameters of therapeutic concern, such as the threshold level of Dvl necessary to inactivate the GSK-3β complex mediating β-catenin degradation, the rate at which β-catenin translocates to the nucleus, and the IC₅₀ concentration of Slug needed to inhibit E-cadherin production.

Keywords: Bistable; Epithelial Mesenchymal Transition (EMT); Wnt.

Fig. 1

a β-catenin – E-cadherin relationship in a primary carcinoma tumor cell, pre-EMT. E-cadherin sequesters cytosolic β-catenin at the cell membrane where it forms a complex with other members of the catenin family to help E-cadherin attach to the cell’s cytoskeleton. b The upregulation of Dvl via Wnt signaling inhibits the degradation of β-catenin by deactivating the GSK-3β/Axin complex, allowing β-catenin to translocate to the nucleus and upregulate the transcription factor Slug. Slug suppresses the transcription of E-cadherin, which means there is less E-cadherin to sequester β-catenin at the membrane. β-catenin can continue to accumulate and translocate to the nucleus, thus completing the feedback the loop

Fig. 2

For τ = [0,20), the cell exists in the epithelial steady state where E-cadherin (e) level is high and both β-catenin (b) and Slug (s) are low. a If a small amount of Wnt signal is released by the microenvironment at τ = 20, a small amount of Dvl (d = 1.2) will accumulate at the membrane. E-cadherin will decrease slightly and both β-catenin and Slug will rise, but not enough to induce EMT. If, at τ = 50, the environment stops releasing Wnt signal, Dvl will detach from the membrane, meaning that the concentration of Dvl at the membrane will be 0 once again. Because EMT was not induced, the proteins all return to their initial epithelial values. b If enough Wnt signal is released by the microenvironment at τ = 20, enough Dvl will accumulate at the membrane (d = 5.4). The steady state values of β-catenin (b) and Slug (s) will rise and, thus, E-cadherin (e) will reach a very low steady state value. The lack of E-cadherin means that the cell will no longer be adhesive with the surrounding cells or its microenvironment, allowing it to move away from the primary tumor. If the concentration of membrane Dvl returns to 0 due to the cellular distance from the external Wnt signal at τ = 50, E-cadherin, β-catenin, and Slug will stabilize at values that allow the cell to maintain its mesenchymal state

Fig. 3

These bifurcation diagrams show a concentration-response curve of E-cadherin (Fig. 3 a), β-catenin (Fig. 3 b), and Slug (Fig. 3 c) with respect to membrane Dvl (d). If the cell starts in the epithelial steady state (d = 0), and the level of Dvl is steadily increased, the cell remains in the epithelial steady state until d = 1.33 (vertical blue dashed line). At this point, the cell undergoes EMT and transitions abruptly into a mesenchymal-like state. Once in the mesenchymal-like state, the cell (and its protein levels) will stay there, even after the level of membrane Dvl is decreased back to its initial value (d = 0). The bifurcation diagrams illustrate the bistable switch underlying the transition

Fig. 4

Sensitivity analysis was carried out using Latin Hypercube and Pearson’s Ranked Correlation Coefficient to understand the relationship between the steady state behavior of E-cadherin, β-catenin, and Slug and the system parameters at different levels of Dvl. The dimensional model was explored in (a and b) while the nondimensional model was explored in (c and d). A system without Wnt activation is shown in (a  and c). (Fig. 4 a) Only β ₂ is significantly correlated (correlation coefficient (ρ) < −0.45, p-value <0.05) with the steady state behavior of β-catenin (B), while β ₃ is significantly correlated (correlation coefficient (ρ) < −0.45 or correlation coefficient (ρ) > 0.45, p-value <0.05) with the steady state behavior of E-cadherin (E) and Slug (S). Additionally, IC _S is significantly correlated with the steady state behavior of E-cadherin (E). (Fig. 4b): With the activation of the Wnt pathway, the rate at which β-catenin (B) translocates to the nucleus and activates Slug (S), k ₂, becomes significantly correlated with the steady state value Slug (s). (Fig. 4c) Only C ₂ is significantly correlated (correlation coefficient (ρ) < −0.45, p-value <0.05) with β-catenin (b)‘s steady state behavior, while C ₃ is significantly correlated (correlation coefficient (ρ) < −0.45 or correlation coefficient (ρ) > 0.45, p-value <0.05) with the steady state behavior of E-cadherin (e) and Slug (s). (Fig. 4d) With the activation of the Wnt pathway, the nondimensional rate at which β-catenin (b) translocates to the nucleus and activates Slug (s), F ₂, becomes significantly correlated with the steady state values of E-cadherin (e) Slug (s). This sensitivity of E-cadherin (e) to F ₂ but not k ₂ in (Fig. 4b) indicates that the steady state behavior of E-cadherin may be sensitive to the ratio of the rate at which β-catenin activates Slug to the IC₅₀ value of Slug needed to inhibit E-cadherin production

Fig. 5

In each subfigure, Dvl (d) is varied along the x-axis. Along the y-axis, the nondimensional parameters A ₁ (a), A ₂ (b), A ₃ (c), C ₁ (d), C ₂ (e), C ₃ (f), F ₁ (g), F ₂ (h), n ₁ (i), n ₂ (j), n ₃ (k), n ₄ (l) are varied one at a time. For those parameter values where the cell begins in Region I and remains in Region I, or crosses L1 into Region II, as Dvl (d) is increased, the cell is committed to the epithelial steady state. For those parameter values where the cell begins in Region III and remains in Region III, or crosses L2 into Region II, as Dvl changes, the cell is committed to the mesenchymal steady state. For parameter values that allow the cell to begin in region two, the starting steady state depends on the system’s initial conditions. If the cell begins in the epithelial steady state and crosses L2 into Region III with a change in Dvl (d), the cell will switch to the mesenchymal steady state. If the cell begins in the mesenchymal steady state in Region II and crosses L1 into Region I with changes in Dvl (d), the cell will switch to the epithelial steady state. The parameter values used in this system are marked with a dashed line. In this model, the cell begins in Region II with epithelial initial conditions