A multiscale computational model of YAP signaling in epithelial fingering behavior

Abstract

In epithelial-mesenchymal transition (EMT), cells organized into sheets break away and become motile mesenchymal cells. EMT plays a crucial role in wound healing, embryonic development, and cancer metastasis. Intracellular signaling in response to mechanical, topographic, or chemical stimuli can promote EMT. We present a multiscale model for EMT downstream of the protein YAP, which suppresses the cell-cell adhesion protein E-cadherin and activates the GTPase Rac1 that enhances cell migration. We first propose an ordinary differential equation (ODE) model for intracellular YAP/Rac1/E-cadherin interactions. The ODE model dynamics are bistable, accounting for both motile loose cells and adherent slower cells. We incorporate this model into a cellular Potts model simulation of two-dimensional wound healing using the open-source platform Morpheus. We show that, under suitable stimuli representing topographic cues, the sheet exhibits finger-like projections and EMT. Morphological differences and quantitative differences in YAP levels as well as variations in cell speed across the sheet are consistent with previous experimental observations of epithelial sheets grown on topographic features in vitro. The simulation is also consistent with experiments that knock down or overexpress YAP, inhibit Rac1, or block E-cadherin.

Significance

In normal wound healing, cells in an epithelium divide, grow, and migrate so as to seal a gap by a process called epithelial-mesenchymal transition (EMT). In some pathological states, EMT can lead to abnormal morphology, including fingering and breakage of single cells or multicellular clusters from the sheet edge. The mechanochemical control of this behavior by cell signaling circuits (YAP, Rac1, and E-cadherin) reveals how the competition between cell adhesion and cell migration contributes to the process. We use the open-source computational platform Morpheus to investigate a multiscale model for the interactions of the proteins inside cells and the resulting morphology of the cell sheet. Model predictions are consistent with experimental results in the literature.

Introduction

Epithelial-mesenchymal transition (EMT) plays an important role in normal embryonic development and wound healing, and in pathology such as cancer metastasis (1,2). In EMT, relatively static well-organized cells in epithelia lose adhesion, separate from the sheet, and migrate away individually or in clusters. The migration of single cells has been studied experimentally and computationally for decades, and, as a result, there is much information about the subcellular mechanisms that regulate and tune cell motility (3, 4, 5, 6, 7).

In collective cell migration, cells move in structures such as swarms, clusters, or sheets. Cell group properties emerge, such as polarization, morphology, or self-organization (8,9). Various intracellular signaling networks regulate the cell-cell adhesion and cell motility that are responsible for EMT. Here, we focus mainly on responses of cells to topographic features on the size scale of the extracellular matrix (ECM). The ECM, a scaffold on which cells migrate, provides topographic cues that are detected by cell-surface (integrin) receptors, stimulating a cascade of signaling and guiding directional cell migration. It was shown experimentally by Park et al. (10) that nanoridge arrays (NRAs) on the size scale of $\approx 0.8\mu \text{m}$ can mimic ECM in providing directional cues to epithelial sheet expansion. Under appropriate conditions, such cues can lead to YAP (Yes-associated protein) signaling that promotes fingering and EMT in a two-dimensional (2D) wound-healing experiment. Park et al. hypothesized that their observations could be explained by the relatively minimal signaling circuit shown in Fig. 1 B, which is reduced from the more complete circuit shown in Fig. 1 A. The minimal circuit consists of YAP, the small guanosine triphosphatase (GTPase) Rac1, and the E-cadherin adhesion protein. Our motivation here is to investigate these hypotheses by constructing a multiscale model of their experimental setup and investigating sheet morphology and other quantitative features predicted by the model.

Figure 1.

The YAP signaling system based on (10): YAP (Y), Rac1 (R), and E-cadherin (E) interactions. (A) Detail of relevant intermediates and downstream effects on cell protrusion and adhesion. TF, transcription factor; WT1, Wilms tumor-1; TRIO, a Rac1 GEF, i.e., activates Rac1; PAK, p21 activated kinase. (B) The simplified circuit and $Y,R,E$ mathematical model with YAP/E-cadherin mutual antagonism and YAP-Rac1 mutual positive feedback. Parameters: $k_Y$ and $C_R$ are basal rates of activation of YAP and Rac1, respectively; $\alpha =1$ for control cells; $C_E$ is basal E-cadherin production rate; $k_{YR},k_R$ are feedback-enhancement activation rates of YAP and Rac1, respectively; k_E is YAP feedback to E-cadherin inhibition ($C_E>k_E$ to avoid negative E-cadherin synthesis rate); $k_{YE}$ is the rate of E-cadherin-mediated YAP inactivation; $d_Y,d_R,d_E$ are rates of YAP and Rac1 inactivation and E-cadherin turnover, respectively. See Table S1 for parameter units and values. To see this figure in color, go online.

YAP is activated by mechanical or topographic cues such as ECM fibers or NRAs on a similar size scale. Active YAP localizes to cell nuclei, where it binds to Wilms tumor-1 (WT1) and acts as a transcription factor that suppresses E-cadherin expression. High YAP activity hence leads to loss of cell-cell adhesion. At the same time, results in Park et al. (10) suggest that E-cadherin inhibits YAP activation and nuclear localization, so that YAP and E-cadherin are in a mutual negative feedback loop.

YAP enhances the activity of the GTPase Rac1 and thereby increases cell migration. Park et al. showed that this effect is due partly to transcriptional regulation by YAP of TRIO, a Rac1 GTP-exchange factor (GEF) that activates Rac1. Since the effect of YAP on Rac1 and on E-cadherin both rely on transcription and translation, we assume that they are on a roughly similar timescale. Signaling from Rac1 via PAK and Merlin then also promotes YAP activation, indicating that YAP and Rac1 participate in mutually positive feedback. These interactions suggest that high-YAP cells should have low adhesion and high motility, whereas low-YAP cells should be relatively static and strongly adherent to one another.

We focus on several questions that arise from (10), specifically: 1) How does ECM signaling affect epithelial sheet morphology to promote EMT? 2) Can reduction of cell-cell adhesion through E-cadherin inhibition and increase of cell speed through Rac1 activation lead to fingering and cell dissemination in epithelial cell sheets? 3) What specific inputs from ECM to the YAP signaling networks could account for the experimental observations? Park et al. (10) found that there was a sharp change in YAP activity between the front and rear of the sheet when grown on NRAs. This leads to the questions: 4) What internal signaling dynamics account for this observation? 5) Are simulations of the knockdown and overexpression of YAP, E-cadherin, or Rac1 consistent with experimental manipulations of this type? 6) How do biophysical aspects such as cell stiffness or cortical tension affect sheet morphology?

To approach these questions, we first revise an ordinary differential equation (ODE) model from (10) describing the dynamics of YAP, E-cadherin, and Rac1 (Fig. 1 B). We analyze the model to determine a suitable parameter regime that yields bistable YAP activity, consistent with observations in (10). In the second step, we create a multicellular multiscale simulation whereby the YAP intracellular signaling is implemented in each cell, while minimal additional assumptions are made for cell-cell adhesion, communication, and cell motility. The model Rac1 level is linked to cell migration speed, and the model E-cadherin level sets the cell-cell adhesion in the simulated cell sheet.

There are many multicellular-migration computational models in the recent literature (see (11) for a recent review). In (12,13), vertex-based models are used to simulate wound healing. In (14), a Voronoi polygon model is used with multiple cell phenotypes to obtain finger-like projections. Monolayers in which cells are spheres or solid objects are included in (15), where the main focus was on adhesion between two distinct cell types. The role of cell-cell junctions in sheet expansion was explored in (16).

Here we used the open-source software Morpheus (17), as it makes multiscale modeling both accessible, and easy to share. The same platform can be used to build up ODE models for the internal cell circuit, to represent cell shapes, migration, and cell-cell interactions, and to visualize the collective behavior of a group of cells. The figures in the results section include links to the Morpheus.xml configuration files that allow readers to replicate our simulations. These files capture all parameters, conditions, and assumptions used. Furthermore, as we demonstrate later in this paper, the same framework can be easily adapted to study wound healing governed by other signaling systems, such as transforming growth factor β (TGF-β).

Materials and methods

The YAP-Rac1-E-cadherin model

We formulated a model for YAP (Y), Rac1 ($R)$, and E-cadherin (E) along the lines of (10), but as a single 3-variable system (18). YAP and Rac1 each satisfy an equation of the form $dC/dt=\left(\text{activation rate}\right)\times C_I-\left(\text{inactivation rate}\right)\times C$. The activities of YAP and Rac1 (but not their total amounts) were quantified over time by (10). Both overexpression of YAP and its knockdown do not affect total Rac1, only Rac1 activity (Fig. 5 B in (10)). For model simplicity, we assumed that total YAP and total Rac1 are constant in each cell ($Y_{\text{tot}}$, $R_{\text{tot}}$, with units of μM), at least on the timescale of the experiments, so that inactive forms can be eliminated from the model using $R_I=R_{\text{tot}}-R$ and $Y_I=Y_{\text{tot}}-Y$. The model equations 1a–1c are shown in Fig. 1 and supporting material (Eq. 1), and all parameters are defined in Table S1.

Key experiments in (10) included YAP knockdown (KD) and YAP overexpression (OE). We modeled these manipulations by lowering (KD) or raising (OE) the total YAP pool, $Y_{\text{tot}}$. To mimic the experiments in which the Rac1-GEF TRIO was inhibited, we take $0\le \alpha <1$. Model equations were solved numerically in XPPAUTO (19), and bifurcation plots were created using the built-in AUTO feature. The same equations were then coded in Morpheus as a preliminary step (see supporting material and Fig. S2).

Multiscale model implementation

Here, we went beyond the primarily experimental paper by Park et al. (10). We first assembled the ODE model for intracellular YAP signaling (Figs. S1 and S2) and then embedded the model within individual cells in Morpheus. A description of Morpheus, its settings, and other details such as boundary conditions and initialization are provided in the supporting material.

The following assumptions were made in assembling the multiscale model.

• 1.The NRA (10), not explicitly modeled, is represented as a directional cue. The directed motion of cells is assumed to be along the axis of the ridges in the x-axis direction.
• 2.Intracellular signaling components are assumed to be well mixed inside each cellular Potts model (CPM) cell. Cell speed is influenced by Rac1 activity, and cell-cell adhesion depends on E-cadherin according to

$$\text{Directed motion speed }=C_1+C_2\frac{R}{C_3+R},\text{Ecad dependent adhesion}=A_2\frac{E}{A_3+E},$$ (1)

with $C_i,A_i\ge 0$. Here $C_2,A_2$ are magnitudes, $C_3,A_3$ are the ¹/₂-max parameters, and $C_1$ is basal migratory speed (see supporting material “adhesion and directed motion” section, and Table S2 for details, values, and definitions).

• 3.All cells start in the basin of attraction of the low-YAP state. Over a limited time span ($t<t_{lim}$ MCS), a few cells at the sheet edge are stochastically “induced” by elevating their Rac1 activation rate to $C_R=0.1$, placing them into the high-YAP basin of attraction (Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6C, blue bifurcation curve). The induction probability is kept low for better visualization of the morphology (see supporting material for details and parameter values).
• 4.The elevated Rac1 activation rate of induced cells is assumed to spread locally to neighbors. With a small probability per time step, the $C_R$ value of a cell is augmented by a small fraction of the average local $C_R$ value in the neighborhood, up to some maximal value $C_R=0.1$ (for details, see supporting material).

Figure 2.

Formation of fingers in the multiscale simulation of a control cell sheet on topographic (NRA) substrate. Time sequence (top to bottom, $t=\mathrm{250,500,1000,1250}$ MCS) showing sheet morphology colored for YAP (left) and cell speed (right) (100 MCS $\approx 1$ h; see supporting material). Cells near the front of the sheet have high YAP. In the middle of the sheet, there is a mixture of cell states. The shapes of cells close to the left edge (longer and rectangular) stem from a boundary effect, and should not be overinterpreted. See also Fig. S5. Morphology is comparable with Fig. 2 C of (10). Parameter values as in Table S1. Morpheus file and video: YREsheetNRA.xml. To see this figure in color, go online.

Figure 3.

Trajectories for cell sheets grown on flat (top) and NRA (bottom) substrates. Left: experimental results from Fig. 1 B of Park et al. (10) (licensed under a Creative Commons Attribution 4.0 license), scale bar = 100 μm. Right: simulations, with speed scaled to experimentally observed range. Simulation shown for t = 1800 MCS, in a domain of size 100 $\times$500 pixels. Parameter values as in Table S1. Morpheus simulation files: top, flat substrate: CellTrajCntr_Flat.xml and CellTrajCntr_Flat.avi; bottom, NRA substrate: CellTrajCntr_NRA.xml and CellTrajCntr_NRA.avi. To see this figure in color, go online.

Figure 4.

Distribution of YAP and cell speeds across the migrating sheet. Right: in the control NRA simulation, YAP activity is highest at the front of the sheet and low in the back. Left: experimental cell speed versus distance for YAP knockdown (KD), control (C), and overexpression (OE) from Fig. 2 B of Park et al. (10) (licensed under a Creative Commons Attribution 4.0 license). Center: simulation results. $Y_{\text{tot}}=0.5,1.2,2$ for KD, control (C), and OE, respectively. The curves were generated in Excel using a third-degree polynomial fit of the simulation data points, where each point represents a single cell’s speed and its distance from the sheet edge at $t=1000$ MCS. Other parameter values as in Table S1. Morpheus file: nra.xml. To see this figure in color, go online.

Figure 5.

Effect of cell-cell E-cadherin-dependent adhesion (A2) and Rac1-dependent cell speed (C2) on sheet morphology. All images at $t=1500$ MCS. Produced with C2vsA2.xml (with cell division restricted to cell volume $>40$, see supporting material). Color depicts the relative level of E-cadherin in each cell. To see this figure in color, go online.

Figure 6.

The effect of the preferred shape index, $q_0=p_0/\sqrt{a_0}$ and the Rac1-dependent cell speed, C2, on sheet morphology. The color map depicts the individual cells’ shape index q. At lower left, most cells appear to be “jammed” (purple hue). At top right, the sheet is fluid, forming finer, longer fingers. The horizontal scans were carried out by setting the aspherity to $\phi =0.8,1,1.2$ (left to right), as explained in the supporting material. The default simulation has $q_0=3.5$, C2 = 4. All results are at $t=1500$ MCS. Produced with qvsC2.xml. See also the same qualitative results in Fig. S11, where cell division is restricted to prevent high densities in the jammed case. To see this figure in color, go online.

Parameter values

We started with initial parameters and units provided in the Park et al. (10) supplemental information. Model parameters were tuned to obtain YAP bistability in the control cultures, and to represent reasonable KD and OE results, as shown in Fig. S1.

Before finalizing the full wound-healing computation, we ran tests with single cells and cell pairs to optimize the adhesion and cell speed parameters for various cell states (see Fig. S3). Multiscale parameters $C_1,C_2,A_2$ and so forth were appropriately tuned (Table S2). Cell speeds $C_1$ and $C_2$ are chosen large enough that collective migration is driven primarily by cell motion rather than pressure from cell division at the back of the sheet. Excessive cell speed leads to premature breakup of the sheet. Adhesion ($A_2$), cell-cell, and cell-medium contact energies were set so as to maintain a cohesive sheet, with few disseminations, but without entirely overpowering cell migration (Table S3). Morphology was highly sensitive to the balance between cell division, cell removal, and cell speed. A parameter study undertaken to explore the effect of such tuning on morphology is discussed in the supporting material.

Results

Bistability in the YRE equations

We first analyzed the model (Eqs. 1a–1c in Fig. 1) using standard methods of dynamical systems (see supporting material). We found that the number of steady states varied with parameter values (Table S1). For values of $Y_{\text{tot}}<0.86$, there is a single stable steady state with low YAP/Rac and high E-cadherin. For $Y_{\text{tot}}>5$, there is also a single stable steady state, but with high YAP/Rac and low E-cadherin. Between these $Y_{\text{tot}}$ values, the system is bistable.

We used the values $Y_{\text{tot}}=1.2,0.5,2.0$ μM to represent control, KD, and OE levels, respectively. The control state is bistable, the KD state is monostable, and the OE state is bistable, but dominated by the upper steady state (largest basin of attraction). In Fig. S2, we show typical time-dependent solutions $Y\left(t\right),R\left(t\right),E\left(t\right)$ of the model.

The basal rate of Rac1 activation, $C_R$, was used as a bifurcation parameter (Fig. S1 C), as it later represents a property that varies across the cell sheet. We find that control cells can have high YAP wherever $C_R$ is sufficiently high, e.g., at the sheet front, and a regime of bistability elsewhere behind the leading edge of the sheet.

Cell-sheet simulations

We asked whether the YAP model and basic further assumptions about cell-cell interactions would recapitulate experimental observations of (10). To investigate this question, we implemented a cell-sheet simulation (Figs. 2 and S5). Each cell has the same internal well-mixed YAP-Rac1-Ecad signaling model, and each cell has its individual state and trajectory.

Additional assumptions needed to scale up from one cell to many are described in materials and methods. For example, cell speed was linked to Rac1 activity (since Rac1 promotes F-actin-driven cell protrusion), and cell-cell adhesion was linked to the level of E-cadherin. Experimental evidence that only cells close to the leading sheet edge induce nuclear (active) YAP justifies our assumption that only edge cells can be “induced” to respond with an elevated Rac1 activation rate $C_R$. The “induction” is stochastic, with probability chosen so as to generate one or two fingers in the domain over the simulation time. As leading cells migrate outward, they create tension on followers. In real cells, this tension inactivates Merlin (not explicitly modeled), a protein that inhibits Rac1. In the computations, we assume that the rate of Rac1 activation ($C_R$) can locally spread (see materials and methods and supporting material).

Fig. 2 shows the YAP level (left) and cell speed (right). At the sheet edge a few small bumps grow into fingers, with occasional dissemination of clusters. YAP activity and speed are high at the front and low at the back. Rac1 activity coincides with YAP, whereas E-cadherin has reciprocal distribution (see Fig. S5). The middle of the sheet consists of both cell types, with a few high-YAP cells that originate from neighbor activation (elevated $C_R$). Only some cells are sufficiently affected to transit to the high-YAP steady state. The simulation morphology and distribution of cell speeds are qualitatively consistent with experimental results shown in Fig. 2 C of (10).

We compared cell trajectories for cells on flat (top) versus NRA substrates (bottom) in Fig. 3. For growth on a flat substrate, we simply assumed that no cells become “induced.” The sheet edge then remained smooth and no fingering occurred (top panels in Figs 3 and S4.) Cell speeds are slow and uniform on the flat substrate, but on the NRAs some leading cells move faster. Cell trajectories are consistent with experimental observations of (10).

YAP knockdown and overexpression

KD and OE experiments in (10) were represented by adjusting the total YAP level, $Y_{\text{tot}}=1.2,0.5,2.0$ (control, KD, OE). Morphological results are shown in Figs. S6 and S7. In brief, in the YAP KD simulation (Fig. S6) the sheets expand slowly, with very low active YAP throughout. In YAP OE (Fig. S7), multiple fingers form and grow early as cells rapidly jump to the high-YAP state. These observations are consistent with experiments in (10), although our fingers are wider in the YAP KD case.

We can understand the results from the underlying YAP model. For high $Y_{\text{tot}}$ (YAP OE) the high-YAP steady state dominates for most cells (Fig. S1), implying that YAP activity should be high for most of the sheet. For low $Y_{\text{tot}}$ (YAP KD) there is a single low-YAP steady state, so YAP is low everywhere. For moderate $Y_{\text{tot}}$ (control) there is a transition from low YAP (at the back) to high YAP (at the front), given that $C_R$ is graded from front to back due to neighbor activation (Fig. S1 C). Fig. 4 (right) shows active YAP versus distance from the front edge in control cells. YAP activity is high at the front and low at the back, generally consistent with experimental findings in Fig 7 A of Park et al. (10).

YAP affects migration speed across the sheet

We asked whether the differences in speeds observed in experimental YAP manipulations in (10) agree quantitatively with our computational results. Consequently, we plotted cell speed versus distance from the leading edge of the cell sheet. Results are shown in Fig. 4. As shown in the left panel (data from (10)), speed is always higher at the leading edge. However, the gradient of the speed is relatively shallow in both high- and low-YAP conditions, but steep in the control case. Simulations (Fig. 4, middle) are consistent with these observations. We can understand this behavior from the dynamics of the YAP-Rac1-E-cadherin model, since the control is in a bistable regime where the front edge and rear of the sheet are at distinct steady states. In the KD YAP, the single steady state changes gradually with distance from the front edge, but does not have a sharp jump. In YAP OE, most of the sheet is in the high-YAP state, so again a sharp jump is not evident. The relatively good agreement of simulation and experimental quantitative results provides a measure of confidence in the ability of the minimal signaling model to depict the behavior of the wound-healing experiments.

E-cadherin blocking and Rac1 inhibition

Blocking the adhesive function of E-cadherin with antibodies (10) led to less stable fingers and more breakage of cells and clusters than in control experiments. A simulation of this experiment (Fig. S8, reduced value of A2) led to similar predictions. The E-cadherin-blocked sheet morphology is more unstable, with thin fragile fingers and significantly increased breakage. In (10), an inhibitor (NSC 23766) that binds to the Rac GEF, TRIO, was used to suppress Rac1 activity. Simulating this inhibition (governed by α, the fraction of Rac1 activation remaining, Fig. S8) yields a sheet with fewer fingers and slower growth. At stronger inhibition, fingering is entirely abolished.

Simultaneously varying E-cadherin-dependent adhesion (A2) and Rac1-dependent cell speed (C2) has significant effect on sheet morphology (Fig. 5). Cell sheets expand even at low C2, driven by pressure from cell division at the back. Increasing cell speed (left to right) and lowering adhesion (top to bottom) correlate with longer, more fragile fingers, and massive dissemination. Stronger adhesion favors more stable, longer sheet protrusions (bottom to top). Interestingly, low cell speed (C2 = 1, left column) can be compensated by high adhesion (top left, A2 = 24) in enabling invasive growth. As this sequence indicates, a strong adhesive force between cells can help to combine weak individual cell-migratory forces into a single collective group force that pushes a large sheet protrusion outward. The two extremes (bottom right vs. top left) illustrate the dichotomy between rapid metastasis versus stable collective invasion.

Cell biophysics affects sheet morphology

We asked how biophysical properties of the cells would affect sheet morphology. The CPM Hamiltonian has terms of the form $H={\lambda }_a{\left(a-a_0\right)}^2+{\lambda }_p{\left(p-p_0\right)}^2$, where $a_0,p_0$ are target cell area and perimeter, and ${\lambda }_i$ are the strengths of each constraint (see supporting material and (20)). Effective cell pressure, $\text{Π}=-2{\lambda }_a\left(a-a_0\right)$, and cortical tension, $\gamma =2{\lambda }_p\left(p-p_0\right)$, can be varied by changing these CPM parameters (20).

Motivated by the work of (21,22) on “jamming transitions” in vertex-based cell-sheet simulations, as well as (23) on MDCK cell sheets, we adopted the “cell shape index,” $q_0=p_0/\sqrt{a_0}$, to represent the cell’s preferred shape in 2D, and we defined $q=p/\sqrt{a}$ to be the cell’s actual shape index. The value of $q_0$ is set by our choice of the CPM rest perimeter and rest area, whereas q is computed at every time step for a given cell using its actual current perimeter and area (for a circle, a hexagon, and a pentagon, $q\approx 3.54,3.72,\text{and}3.81$, respectively). For non-motile cells, a transition from solid to fluid behavior was found at $q_0\approx 3.81$ (21). For self-propelled cells, a jamming transition curve was shown for speed versus shape index (22). Accordingly, we varied both Rac1-dependent cell speed ($1<\text{C}2<8$) and preferred shape index $2.8<q_0<4.2$ (Fig. 6) to investigate whether similar transitions occur.

As shown in Fig. 6, low speed and preferred shape index (lower left corner) correspond to sluggish, solid-like “jammed” behavior (purple cells have $q<3.5$) and a relatively flat leading edge. Increasing $q_0$ to 4.2 at low speed (C2 $=1$, bottom row) results in more fluid, fat fingering. Keeping $q_0$ low while increasing cell speed ($q_0=2.8$, left column) accelerates expansion, with minimal fingering. Increasing both $q_0$ and cell speed leads to thinner, faster fingers, with eventual breakage and massive dissemination (top right). Noting the buildup of high cell densities in the jammed regimes, we repeated these simulations with cell division restricted by cell volume (Fig. S11), obtaining qualitatively similar results. Figs. 6 and S11 are both in overall concordance with (22). In the supporting material, we also examined variations in the Hamiltonian area and perimeter constraint weights ${\lambda }_a$ and ${\lambda }_p$. We found that higher values of either or both (stiffer, more pressurized cells) lead to flatter sheet edges, with greater cell jamming (and lower cell shape index). Lower values of λ made for more fluid sheet morphology, with faster fingering and more breakage. See (24) for similar experimentally observed transitions and relevance to metastasis in 3D tumor spheroids. See supporting material for further details.

Discussion

While (10) was primarily an experimental paper, here we linked a minimal model for YAP, Rac1, and E-cadherin in individual cells to a mutiscale simulation of epithelial migration based on findings of (10). We investigated fingering morphologies and quantify the distribution of cell states and cell speeds across the sheet. To do so, we used the multiscale CPM capabilities of the Morpheus open-source platform and linked cell-cell adhesion and migratory capabilities to Rac1 and E-cadherin cell levels. The results agree with experimental observations of (10) for epithelia on NRA topography and various manipulations of YAP, Rac1, and E-cadherin.

Qualitative agreement was obtained once the ODE model parameters were tuned to assure bistability of the control cells, and several further assumptions were added. We had to assume, for example, that a small number of cells at the front of the sheet were stochastically stimulated by the topography, while cells at the back, where contact inhibition is greatest, were unresponsive. Elevating a single parameter ($C_R$) would then result in high YAP in the given “induced” cell, promoting it to a leader. We also had to assume that cells affect their neighbors, spreading their elevated Rac1 rate of activation ($C_R$).

As in (10), on NRAs, YAP activity switches from high to low level across the sheet (Fig. 4). A bifurcation in the YAP ODE model explains this observation (Fig. S1). The rate of Rac1 activation, $C_R$ correlates with distance from sheet edge, so the bifurcation with respect to $C_R$ implies a jump in the YAP level at some location in the sheet. In YAP KD and YAP OE (adjusting only one model parameter for the total YAP in cells, $Y_{\text{tot}}$) there is no longer bistability, so YAP distribution as well as cell speed change much more gradually from front to back of the sheet.

Based on these results, we can address several questions raised in the introduction. 1) ECM signaling topography, by elevating the activity of the mechanosensitive protein YAP, can lead to cell states with higher speed and lower adhesion, promoting EMT. 2) High-YAP states, with reduced cell-cell adhesion and increased cell speed, can lead to fingering and cell dissemination. 3) Induction of high-YAP signaling either directly or by pathways that increase Rac1 activation, can lead to fingering morphology due to positive feedback between YAP and Rac1. 4) The sharp switch in YAP activity between front and rear of a sheet can be explained by a bifurcation in the YAP signaling model. The front and rear of the sheet have cells in distinct steady states of the system. 5) Manipulations such as YAP KD, OE, and Rac1 inhibition amount to parameter changes that affect the qualitative behavior of the model, and are consistent with experiments. 6) Cell shapes, stiffness, and cortical tension affect the fluidity of the sheet and its fingering. Stiffer cells or those with smaller shape index form sluggish slow-moving sheets, whereas softer cells or those with higher shape index form more fluid, fast-fingering morphologies.

Our computational model has several limitations. First, we implemented cell division only at the back of the sheet to allow for sheet expansion. Cell division and death throughout the sheet could be easily implemented in a future version of the Morpheus simulation. Second, the CPM is a phenomenological way to track cell shapes and motion. Our implementation in Morpheus does not track physical forces due to molecular motors, nor other details of internal processes that affect cell shape. The NRAs were modeled only implicitly, providing the directional cue and enabling cell induction. We also used a shortcut to model the influence of leader cells on followers. Biologically, the protein Merlin is inactivated by the tension exerted on a cell by its moving neighbor, thereby increasing the basal activation rate of Rac1, but we did not include Merlin in the model. In future versions of the model, tension between cells and Merlin could be included explicitly.

In other studies, continuum models for epithelial fingering relate fingering wavelength to various leading-edge mode instabilities. Examples include the polar fluid and active fluid models of (14,25) that considered such effects as hydrodynamic screening length and tissue size (25) or a curvature based force at the leading edge (14), and stability analysis of the uniform sheet edge in the relevant partial differential equation representation. A discrete particle-based model of (26) with pairwise interaction potentials and velocity alignment also assumed curvature-dependent forces (Helfrich bending energy) on the epithelium edge to deduce fingering wavelength. In our computations, fingers arise with a probability and timing that we set to show the morphology of individual fingers rather than to estimate inherent fingering wavelength. Other discrete simulations include the vertex-based model of (12), and cell-center based multiscale model of (27) with interaction of E-cadherin and β-catenin. Alternative gene networks that regulate EMT such as (11,28, 29, 30) are not implemented in multiscale tissue computations, but in (31) a CPM is coupled to a finite-element model, combining intra- and intercellular signaling in wound healing and other geometries.

There is increased recent interest in “EM” cell states that are neither fully epithelial (E) nor fully mesenchymal (M) (29,30,32,33), but that are potentially more efficient at generating tumors (33). Such ideas raise the question of whether a given cell circuit that governs EMT has three states, representing E, EM, and M. Our simple circuit (Fig. 1 B) and model system is bistable but not tristable. It would be relatively straightforward to revise our model by adding feedbacks, and cascades or multimerization (Hill functions with powers $n\ge 2$) to arrive at an alternative tristable version of the model (see for example (34,35)). Here we do not go down this route. Instead, we end by demonstrating, in the supporting material, an alternative EMT circuit based on TGF-β, constructed by (29), which we implemented in the same type of cell-sheet simulation. Three cell states clearly emerge along a gradient of TGF-β from a growing sheet. Replacing the YAP-Rac1-E-cadherin circuit by an alternative signaling circuit proves to be straightforward.

Conclusion

The mechanosensitive protein YAP, by responding to topographic cues, is able to coordinate both cell-cell adhesion (via E-cadherin) and cell migration (via the GTPase Rac1). A relatively elementary model for positive and negative feedback between YAP, Rac1, and E-cadherin accounts for experimentally observed wound-healing morphology. Distinct cell states (high YAP and fast cells versus low YAP and tightly adherent cells) are predicted by the model, and their distributions in the cell sheet agree with experiments in which YAP, Rac1, or E-cadherin are manipulated. Software packages such as Morpheus, by facilitating multiscale multicellular modeling, allow us to visualize the morphology of a cell sheet, YAP-associated fingering, and cell breakage. The computation is easy to share, and is readily adaptable to a variety of signaling circuits that are known to affect EMT and growth of epithelia.

Author contributions

N.M. carried out the research and wrote the MSc thesis on which this paper is based. L.E.-K. designed the research, carried out the research, and wrote the paper. E.N.C. participated in editing the paper, providing feedback, and model analysis.

Editor: Mark Alber.