Active chemo-mechanical feedbacks dictate the collective migration of cells on patterned surfaces

Abstract

Recent evidence has demonstrated that, when cultured on micro-patterned surfaces, living cells can move in a coordinated manner and form distinct migration patterns, including flowing chain, suspended propagating bridge, rotating vortex, etc. However, the fundamental question of exactly how and why cells migrate in these fashions remains elusive. Here, we present a theoretical investigation to show that the tight interplay between internal cellular activities, such as chemo-mechanical feedbacks and polarization, and external geometrical constraints are behind these intriguing experimental observations. In particular, on narrow strip patterns, strongly force-dependent cellular contractility and intercellular adhesion were found to be critical for reinforcing the leading edge of the migrating cell monolayer and eventually result in the formation of suspended cell bridges flying over nonadhesive regions. On the other hand, a weak force-contractility feedback led to the movement of cells like a flowing chain along the adhesive strip. Finally, we also showed that the random polarity forces generated in migrating cells are responsible for driving them into rotating vortices on strips with width above a threshold value (~10 times the size of the cell).

Significance

We developed a vertex-based chemo-mechanical model to address the unsettling question of how cells can adopt different collective migration modes on patterned surfaces. Specifically, we showed for the first time that variations in the strength of force-intercellular adhesion and force-contractility feedbacks lead to the emergence of distinct migration modes, including flowing chain, suspended propagating bridge, rotating vortex, etc., of cells on strip patterns. In addition, our work also explained the observed assembly of actin cables, along with a stretched cadherin morphology, at the leading edge of the migrating cell monolayer. Findings here could provide new insights for our understanding of important processes such as tissue development and disease pathogenesis.

Introduction

During embryogenesis (1), wound healing (2), and cancer invasion (3), cells have the amazing ability to move in a coordinated manner over a time span of hours and a distance much larger than their individual size. Because of its critical role in these biological processes, intense efforts have been invested in the past few decades to understand how collective cell migration is triggered, as well as the biochemical/biophysical mechanisms behind it. For example, it is believed that specialized leader cells, exhibiting clear front-rear polarity and lamellipodial structures, need to be formed at the leading edge of the migrating group to provide guidance for other follower cells (4,5). In addition, evidence has suggested that stable cell-cell junctions not only maintain the structural integrity of the cell monolayer but also allow cells to achieve long-range physical/chemical coordination, such as plithotaxis (6,7) and wavelike propagation of cellular motion (8, 9, 10, 11, 12). Interestingly, studies also showed that the collective migration of cells can be directed by physical cues from their microenvironment geometry, i.e., grooved topography (13), long strips (14,15), circular holes (16,17), annular ring (18), tube (19), acinus-like lumen (20), and a flower-shaped pattern with concave edges (21). In particular, it was reported that, on the surface of functionalized adhesive strips or holes pattern, epithelial cells can migrate to form suspended bridges over nonadhesive areas under the driving of multicellular actin-based contractile cables or crawling forces at the leading edge (14,16,17).

Theoretically, several approaches have been proposed to describe the collective behavior of cells under geometrical confinements. For instance, Albert and Schwarz examined the migration dynamics of cells on different adhesive patterns with the Cellular Potts model where each cell was treated as an ensemble of lattices whose shape evolution was then captured by random motion of boundary lattices following the principle of energy minimization (22). On the other hand, the so-called vertex model has also been widely adopted to study collective cell migration where the cell monolayer is described by a set of vertices that defines each cell as a polygon with straight interfaces between neighboring cells (23). In this way, the motion of vertices/cells can be determined by, for example, minimizing the total energy of the system with respect to their positions. Following this picture and considering the cellular polarity, confined cells were shown to be capable of initiating rotational (24) and wavelike (11) movement patterns, something that has been found to depend on substrate curvature and topography (25) as well. Most recently, the essential role of chemo-mechanical coupling in processes like disease development (26), cell migration (12), oscillatory morphodynamics of embryo (27, 28, 29), and wound healing (30) has also been demonstrated both experimentally and theoretically.

Despite these aforementioned efforts, the fundamental question of how active cellular responses influence the coordinated movement of cells still remains poorly understood. For example, a stretched morphology of E-cadherins (14), along with prominent contractile actin cables (14,16,17), at the leading edge of the moving epithelial cell monolayer has been observed. This indicates cell-cell adhesion and cellular contractility could both be force-dependent. Several recent studies have indeed shown that the coupling between cellular stress/strain and the polarization or contractility of cells could lead to the emergence of velocity waves (9,10,12) in the cell monolayer as well as regulate its pattern of motion (31). Nevertheless, a comprehensive understanding of the underlying physical mechanisms is still lacking. Here, we develop a model to address this outstanding issue. Specifically, by focusing on the scenario of a migrating cell monolayer on strip-patterned surface (Fig. 1 a), we show how chemo-mechanical feedbacks of contractility and cell-cell adhesion interact with geometrical cues on the substrate to eventually direct the migration mode of cells as well as compare them with various experimental observations.

Figure 1.

Schematics of the proposed vertex model for collective cell migration on strip-patterned surface. (a) A cell monolayer migrating on striped adhesive pattern. The leader cells generate a crawling force $\mathbf{F}$ on each strip to drive the movement of follower cells. Definitions of different geometrical variables/parameters are shown in the inset. (b) The polarity force ${\mathbf{f}}_i$ acting on vertex $i$ is the average of polarity forces generated in different cells, i.e., ${\mathbf{f}}_p^J$, ${\mathbf{f}}_p^K$ and ${\mathbf{f}}_p^L$, sharing vertex $i$. (c) Variations in the position of vertex $i$ ($\delta {\mathbf{r}}_i$) lead to changes in the cell-substrate adhesion area ($\delta A_J^{sub}$, blue dots). Numerically, the adhesion area is estimated by counting the number of sampling points (that are evenly distributed with spacing $\Delta$) within the adhesive region. Red lines indicate boundaries of the adhesive pattern. To see this figure in color, go online.

Materials and methods

Chemo-mechanical model

In this study, the coordinated movement of cells on a patterned surface is analyzed within the classical vertex model framework (32,33), an approach known to be particularly suitable for examining the behavior of epithelia. Basically, each cell within the monolayer is treated as a polygon here with straight interfaces between neighboring cells (Fig. 1 a). Under such circumstance, the total energy of the system can be expressed as

$$U=\sum _J\left[\frac{1}{2}K{\left(A_J-A_0\right)}^2+\frac{1}{2}{m}_J\text{Γ}{P}_J^2+\sum _{i,j\in J}{s}_{ij}\text{Λ}{l}_{ij}+\gamma A_J^{sub}\right]$$ (1)

where $K$, $\text{Γ}$, $\text{Λ}$, and $\gamma$ represent the areal modulus of the cell, contractility stiffness, and cell-cell and cell-substrate adhesion energy density, respectively. $A_J$, $A_0$, and $A_J^{sub}$ correspond to the actual, preferred, and cell-substrate contact area of cell J, respectively. $l_{ij}$ refers to the length of cell edge between vertices $i$ and $j$ (Fig. 1 a), and the cell perimeter $P_J$ is written as $P_J=\sum _{i,j\in J}{l}_{ij}$. $m_J$ and $s_{ij}$ characterize the reinforcement of contractility and intercellular adhesion resulting from chemo-mechanical feedbacks. Once $U$ is known, the motion of vertex $i$ can then be determined by the overdamped equation

$$\eta \frac{d{\mathbf{r}}_i}{dt}=-{\nabla }_{{\mathbf{r}}_i}U+{\mathbf{f}}_i$$ (2)

with ${\mathbf{f}}_i$ representing the polarity/random force acting on vertex $i$ which locates at ${\mathbf{r}}_i$ and $\eta$ being the viscous friction coefficient. In the present study, a uniform polarity force ${\mathbf{f}}_p^J$ was assumed to be generated on all vertices of cell J along the polarization direction defined by orientation angle ${\Phi }_{\mathit{J}}$ (Fig. 1 b). After being assigned a random initial value, this angle was assumed to undergo random rotational diffusion $\frac{\partial {\Phi }_J}{\partial t}={\phi }_J\left(t\right)$, with ${\phi }_J\left(t\right)$ representing a white noise with zero mean and variance 2D, i.e., ${\phi }_J\left(t\right){\phi }_K\left(t+\Delta t\right)=2D\delta (\Delta t){\delta }_{JK}$ (34). In this way, ${\mathbf{f}}_i$ is essentially the average of polarity forces generated in all cells sharing vertex $i$ (Fig. 1 b), that is

$${\mathbf{f}}_i=\frac{1}{N_{S_i}}\sum _{J\in S_i}{\mathbf{f}}_p^J$$ (3)

where $S_i$ refers to the group of cells sharing vertex and $N_{S_i}$ is the number of cells in that group.

Note that, in light of Eq. 1, we can write the first term on the right-hand side of Eq. 2 as

$$-{\nabla }_{{\mathbf{r}}_i}U=-\sum _J\left[K\left(A_J-A_0\right){\nabla }_{{\mathbf{r}}_i}{A}_J+m_J\text{Γ}{P}_J{\nabla }_{{\mathbf{r}}_i}{P}_J+\text{Λ}\sum _{i,j\in J}{\nabla }_{{\mathbf{r}}_i}\left(s_{ij}{l}_{ij}\right)\right]-\gamma \sum _J{\nabla }_{{\mathbf{r}}_i}{A}_J^{sub}=\sum _J\left({\mathbf{F}}_i^J-\gamma {\nabla }_{{\mathbf{r}}_i}{A}_J^{sub}\right)$$ (4)

where ${\mathbf{F}}_i^J$ refers to the force applied on vertex $i$ by cell J. From geometry, it can be shown that

$${\nabla }_{{\mathbf{r}}_i}{A}_J=\frac{1}{2}(l_{i-1,i}{\mathbf{n}}_{i-1}+l_{i,i+1}{\mathbf{n}}_i)$$

$${\nabla }_{{\mathbf{r}}_i}{P}_J={\nabla }_{{\mathbf{r}}_i}\left(l_{i-1,i}+l_{i,i+1}\right)={\mathbf{t}}_{i-1}-{\mathbf{t}}_i$$

$$\sum _{i,j\in J}{\nabla }_{{\mathbf{r}}_i}\left(s_{ij}{l}_{ij}\right)={\nabla }_{{\mathbf{r}}_i}\left(s_{i-1,i}{l}_{i-1,i}+s_{i,i+1}{l}_{i,i+1}\right)=s_{i-1,i}{\mathbf{t}}_{i-1}-s_{i,i+1}{\mathbf{t}}_i$$

with ${\mathbf{n}}_i$ and ${\mathbf{t}}_i=({\mathbf{r}}_{i+1}-{\mathbf{r}}_i)/l_{i,i+1}$ being the unit outward normal and counterclockwise tangent vector of the edge pointing from vertex $i$ to vertex $i+1$, respectively (Fig. 1 a). The term representing cell-substrate adhesion in Eq. 4 depends on the location of the cell: 1) if the cell is fully attached to the adhesive pattern, ${\nabla }_{{\mathbf{r}}_i}{A}_J^{sub}={\nabla }_{{\mathbf{r}}_i}{A}_J$; 2) if the cell is completely hanging over the nonadhesive area, ${\nabla }_{{\mathbf{r}}_i}{A}_J^{sub}=0$; 3) if the cell is partially adhered to the pattern, the gradient of cell-substrate adhesion area can be calculated as

$${\nabla }_{{\mathbf{r}}_i}{A}_J^{sub}=\frac{\delta A_J^{sub}}{\delta {\mathbf{r}}_i}$$ (5)

where $\delta A_J^{sub}$ is the variation of the cell-substrate adhesion area of cell J if vertex $i$ is moved with a small distance $\delta {\mathbf{r}}_i$ (Fig. 1 c). Numerically, $A_J^{sub}$ was estimated by counting the number of evenly distributed sample points located within the adhesive region ($N_J^{sub}$), that is $A_J^{sub}\approx N_J^{sub}{\Delta }^2$ with $\Delta$ being the spacing between sample points (Fig. 1 c). Note that, because $\gamma$ is negative, cells will prefer to attach and move along adhesive strips to maximize $A_J^{sub}$ in the absence of other competing factors.

Next, recall that possible enhancement or weakening of myosin contractility in cell J is described by the dimensionless variable $m_J$ in Eq. 1. Indeed, numerous studies have convincingly demonstrated that myosin motors can be recruited to the cell edge upon application of mechanical stretch (35). As a result, the elevated myosin activity promotes cell contraction to shorten the cell perimeter (i.e., to negate the applied stretch) (36), refer to Fig. 2 a. Similarly, it was also reported that the presence of stretching forces can trigger the unfolding of $\alpha$-catenin, increase its binding with vinculin, and ultimately enhance cadherin-actin interactions (37). Therefore, cell-cell adhesion could actually be strengthened (represented by an increased $s_{ij}$ in Eq. 1) by the load acting at intercellular adherens junctions, see Fig. 2 a. We proceed by adopting the well-established Hill's function (27) to describe these two aforementioned chemo-mechanical feedbacks as

$$\frac{d{m}_J}{dt}=\alpha \frac{{\lambda }_J^n}{K_m+{\lambda }_J^n}-\beta m_J$$ (6a)

$$\frac{d{s}_{ij}}{dt}=\rho \frac{{\sigma }_{ij}^q}{K_s+{\sigma }_{ij}^q}-\xi s_{ij}$$ (6b)

where ${\lambda }_J=P_J/P_0$ with $P_0=-\frac{\text{Λ}}{\text{Γ}}$ being preferred perimeter of the cell and ${\sigma }_{ij}=\frac{1}{2}\left(\right|{\sigma }_J^n|+|{\sigma }_K^n\left|\right)/{\sigma }_0$ representing the average stress sustained by edge $ij$ (shared by cell J and cell K, refer to Fig. 2 a). Here, ${\sigma }_0$ is a reference stress, while ${\sigma }_J^n$ and ${\sigma }_K^n$ correspond to the effective normal stresses acting on edge $ij$ in cell J and K, respectively. Following Nestor-Bergmann et al. (38), each cell was assumed to possess a uniform stress state taking the following form (using cell J as an example):

$${\mathbf{\sigma }}_J=-\frac{1}{A_J}\sum _{i\in J}{\mathbf{r}}_i\otimes {\mathbf{F}}_i^J.$$ (7)

Figure 2.

Two chemo-mechanical feedbacks introduced in the model. (a) Illustration of chemo-mechanical feedbacks regulating the development of cellular contractility (left) and cell-cell adhesion (right). (b) Evolution of myosin activity ($m_J$) or intercellular adhesion ($s_{ij}$) reinforcement under different $K_m$ or $K_s$ values, refer to Eqs. 6a and 6b. Essentially, the strength of each feedback is found to be controlled by $K_m$ or $K_s$, respectively. (c) Evolution of myosin activity or intercellular adhesion reinforcement under different $\beta$ or $\xi$ values. Clearly, these two parameters determine how fast the cellular contractility or intercellular adhesion feedback takes place. Note that, (b) and (c) were plotted for a cell ($m_J$) or a contact edge ($s_{ij}$) under the condition where ${\lambda }_J$ or ${\sigma }_{ij}$ was suddenly increased to 2. The values of $\beta \text{and}\xi$ were fixed at 0.01 in (b) while $K_m$ and $K_S$ were set as 1 in (c). To see this figure in color, go online.

The value of ${\sigma }_J^n$ can then be determined easily from the Mohr's circle based on the orientation of edge $ij$.

We want to emphasize that, in Eqs. 6a and 6b, $\alpha$ and $\rho$ represent the association rates, $\beta$ and $\xi$ stand for the dissociation rates, and $n$ and $q$ are the so-called Hill coefficients. To see how these parameters influence the feedbacks, recall that $m_J=1$ and $\frac{d{m}_J}{dt}$ = 0 if the cell is unstretched (i.e., ${\lambda }_J=1$). Under such circumstance, we have $\alpha =(1+K_m)\beta$. Therefore, the steady-state value of $m_J$ is found to be independent of $\alpha$ and $\beta$ as

$$m_J^s=\frac{\alpha }{\beta }\frac{{\lambda }_J^n}{K_m+{\lambda }_J^n}=\left(1+K_m\right)\frac{{\lambda }_J^n}{K_m+{\lambda }_J^n}$$ (8)

This means the largest $m_J$ the cell can reach (i.e., when a cell is severely stretched ${\lambda }_J\gg 1$) is approximately $(1+K_m)$, refer to Fig. 2 b. Similarly, the maximum $s_{ij}$ that can be achieved is $(1+K_s)$, independent of $\rho$ and $\xi$. However, as shown in Fig. 2 c, $\beta$ and $\xi$ determine how fast the contractility and intercellular adhesion feedbacks take place (i.e., how fast $m_J$ and $s_{ij}$ can reach their maximum).

Finally, given that on the strip-patterned surface (with strip width $W$ and gap $G$), some cells spontaneously emerge as specialized leader cells with large lamellipodial structures and clear front-rear polarity to guide the migration of other follower cells, a crawling force $\mathbf{F}$ (along the strip axis) was applied to the front-most cell on each strip in our simulations, see Fig. 1 a. We believe such treatment is consistent with the observation that follower cells will recoil immediately after laser ablation of leader cells moving on the strip (14).

Nondimensionalization and estimation of parameters

To better present/interpret the results, we proceed by normalizing the problem with the characteristic length scale $\sqrt{\frac{A_0}{\pi }}$ (i.e., effective radius of a “rest” cell), force scale $K{\left(\frac{A_0}{\pi }\right)}^{\frac{3}{2}}$ and time scale $\frac{\pi \eta }{K{A}_0}$. In this way, the movement of cells is largely controlled by the following dimensionless parameters $\overline{W}=W\sqrt{\frac{\pi }{A_0}}$, $\overline{G}=G\sqrt{\frac{\pi }{A_0}}$, $\overline{\text{Γ}}=\frac{\pi \text{Γ}}{K{A}_0}$, $\overline{\text{Λ}}=\frac{\text{Λ}}{K}{\left(\frac{\pi }{A_0}\right)}^{\frac{3}{2}}$, $\overline{\text{γ}}=\frac{\pi \gamma }{K{A}_0}$, ${\overline{\mathbf{f}}}_p^J=\frac{{\mathbf{f}}_p^J}{K}{\left(\frac{\pi }{A_0}\right)}^{\frac{3}{2}}$ and $\overline{\mathbf{F}}=\frac{\mathbf{F}}{K}{\left(\frac{\pi }{A_0}\right)}^{\frac{3}{2}}$, along with the so-called target shape index (34), defined as $p_0=\frac{P_0}{\sqrt{A_0}}=-\frac{\overline{\text{Λ}}}{\overline{\text{Γ}}\sqrt{\pi }}$.

According to (27), the areal modulus $K$ of cells is estimated to be between ${10}^5-{10}^{7}N/m^3$ and the contractility stiffness $\text{Γ}$ is expected to be of the order of ${10}^{-4}-{10}^{-3}N/m$. We measured the cell radius be around 10 $\mu m$ in (14) and got $A_0\approx {10}^{-10}\pi m^2$. Therefore, we estimate $\overline{\text{Γ}}=\frac{\pi \text{Γ}}{K{A}_0}$ to be in the range of $0.1-100$. In addition, the cell-substrate adhesion energy density was found to be $\sim {10}^{-4}N/m$ (39), indicating that $\overline{\gamma }=\frac{\pi \gamma }{K{A}_0}=0.1-10$. Given that the friction coefficient $\eta$ was thought to be between $0.01-0.1N\cdot s/m$ (27), the time scale $\frac{\pi \eta }{K{A}_0}$ is estimated to vary from $10$ to ${10}^{4}s$. Under such circumstance, a 40 h experimental observation translates to a normalized simulation time of $14.4-14400$. Finally, previous studies have shown that the behavior of the cell monolayer is solid-like when $p_0$ is less than ~3.81, but will become more fluid-like once the target shape index is above this critical value (34). Therefore, the range of $p_0$ examined here was chosen to be from 3.5 to 4.5 so that both the solid- and fluid-like responses can be covered. For clarity, overbars of all dimensionless parameters are neglected in the remaining part of this paper.

Implementation of simulations

The model was implemented in CHASTE (University of Oxford, Oxford, UK). Specifically, we set up patterned adhesive strips and a reservoir with well-chosen geometries, i.e., the strip width $W$ and gap $G$ were set as multiples of $W_0$, the incircle radius of the cell (represented by a hexagon in the un-deformed configuration) (Fig. 1 a). In the standard scenario, $W/W_0=1$ and $G/W_0=12$ are comparable to the experimental settings in (14). As a built-in feature of CHASTE, a left-right periodic boundary condition was applied, and each period contains 6 (horizontal) $\times$ 16 (vertical) cells. Note that the cell number in each period was increased accordingly to ensure constant cell density when a larger strip width or gap was adopted. Cell neighbor exchange (i.e., T1 transition (40)) was allowed in the simulation where the normalized threshold length for such rearrangement to occur was set as ${\Delta }_{T1}=0.01$. An edge was removed once its length is below this critical value. At the same time, a new edge was created (with length 1.5 ${\Delta }_{\mathit{T}1}$) along the perpendicular bisector of the eliminated one. To incorporate Eqs. ((3), (4), (5), (6a), (6b), (7)) into classical vertex simulation, a number of self-defined subroutines were added in CHASTE. For example, an edge element was created to make the handling of edge-based cell-cell adhesion more easily. Force contribution from cell-substrate adhesion (a factor that was not considered in the original code) was also added. Finally, a subroutine for calculating and updating the feedback strength $m_J$ and $s_{ij}$ (according to Eqs. 6a and 6b) was introduced in CHASTE. In our simulations, an active crawling force $\mathbf{F}$ was assumed to be generated in the leader cell, with the magnitude depending on the cell type. Furthermore, this force was distributed evenly on each vertice of the leader cell. Since multiple leader cells emerged on a wide strip, the same crawling force was applied to each of them. Finally, because the intracellular traction force in Madin-Darby canine kidney (MDCK) cells is known to be much lower than that of keratinocyte (14,15), a smaller value of $\mathbf{F}$ was adopted when simulating their vortex formation on wide strips.

Given that the target shape index $p_0$ is defined as the ratio between cell-cell adhesion strength $\text{Λ}$ and contractility stiffness $\text{Γ}$, we varied this parameter in our simulations by changing $\text{Λ}$ while, at the same time, fixing the value of $\text{Γ}$. Normalized time step $\Delta t$ was chosen as 2.5$\times {10}^{-2}$ with a total simulation time of 400 unless specified otherwise. At the same time, the maximum vertex displacement in any single time step was constrained to be 5$\times {10}^{-2}$. The values of parameters adopted in the present work are summarized in Table S1 unless specified otherwise.

Data analysis

For each configuration or set of parameters, simulations were repeated at least three times (with different random seeds for generating polarity forces) to determine the migration mode and vortex formation in cells. The migratory phases of cells were assessed/determined by their spatial distribution on the patterned surface. Specifically, the mode was regarded as detachment if one or a small group of (usually two to three) cells separate from the monolayer and migrate by themselves (Fig. 3 a). In the mode of flowing chain, cells move out of the reservoir one-by-one and maintain their connection with each other when migrating along the strip. On the other hand, epithelial bridge was believed to occur when cells begin to migrate over nonadhesive regions between strips while maintaining the integrity of the monolayer. However, if cells between strips cannot move forward at all, we classify the monolayer as being jammed (Fig. 3 a). Cell vortex on wide strips was assumed to be formed if more than two layers of cells are rotating around a center point. In the case of epithelial bridge relaxation, 11 independent simulations were conducted and the results are presented in the form of mean $\pm$ SEM (standard error of the mean). All simulation results were imported into Paraview (Kitware Inc., New York, NY) for visualization. The velocity ($\mathit{v}$) and vorticity ($\mathit{\omega }=\nabla \times \mathit{v}$) fields of cells were calculated by a custom-designed program in MATLAB (The MathWorks Inc., Natick, MA).

Figure 3.

Chemo-mechanical feedbacks dictate the mode of collective cell migration. (a) Illustration of different collective cell migration modes predicted by the model, i.e., detachment ($p_0$ =3.75, $K_m$ = 0, $K_s$ = 0), chain ($p_0$ =4.5, $K_m$ = 0.5, $K_s$ = 0.5), bridge ($p_0$ =4.0, $K_m$ = 1.0, $K_s$ = 0.5) and jamming ($p_0$ =3.5, $K_m$ = 2.0, $K_s$ = 2.0). The arrows point to locations where cell detachment is about to take place. (b) Migration modes phase diagram of fluid-like cells (i.e., $p_0$ = 4.5). (c) Migration modes phase diagram of solid-like cells (i.e., $p_0$ = 3.75). (d) Migration modes phase diagram in the 3D space of shape index $p_0$ and two chemo-mechanical feedbacks strengths ($K_m$ and $K_s$). (e) Cells fail to form suspended bridges over nonadhesive regions once the strip gap $G$ is larger than a threshold value (i.e., $G/W_0>20$). Here $p_0$ = 4.0, $K_m$ = 1.0, and $K_s$ = 0.5. The red lines in (a) and (e) refer to the boundary of adhesive pattern on the surface. To see this figure in color, go online.

Results

Active chemo-mechanical feedbacks can regulate collective cell migration modes

Interestingly, by varying the target shape index $p_0$ and feedback strengths $K_m$ and $K_s$, four distinct migration modes (i.e., detachment, flowing chain, suspended bridge, and jamming, as illustrated in Fig. 3 a) of cells, on surface patterned with narrow strips (that is, $W/W_0=1$), were revealed from our simulations. Specifically, as shown in Fig. 3 b, it was found that fluid-like cells (i.e., $p_0$ = 4.5) could form bridges to fly over nonadhesive regions only when both the contractility and cell-cell adhesion feedbacks are strong. In contrast, cells will migrate like a flowing chain along the strips when either of them is weak. In the extreme case when both feedbacks are negligible (i.e., both $K_m$ and $K_s$ become very small), the cell will detach from the group and migrate along the strip one-by-one or in a small group. These results indicate that both feedbacks are essential for the formation of suspended bridges of fluid-like cells. Physically, this can be understood by realizing that a fluid-like cell can easily change its shape during migration (note that cells migrate with an elongated shape in the detachment mode without any feedback), therefore strong contractility and adhesion feedbacks are needed to stiffen the cell boundary so that cell bridges (where cells assume a relative fixed shape and cannot detach from one another) can be formed.

In comparison, if the cell monolayer is solid-like (i.e., $p_0$ = 3.75) then suspended bridges can be formed with moderate contractility feedback strengths (Fig. 3 c). However, under such circumstance, a jamming phase (i.e., cells form a bridge-like structure but can never move forward, see Fig. 3 a) emerges when the contractility feedback is very strong. On the other hand, solid-like cells will move like a chain or as detached ones when $K_m$ becomes small, presumably because a relatively weak contractility allows cells to realize shape change more easily during migration.

The combined influence of $p_0$, $K_m$, and $K_s$ on the collective movement of cells is best summarized by the 3D phase diagram shown in Fig. 3 d. It can be seen that the space is mainly occupied by the bridge and chain phase when the monolayer is fluid-like. On the other hand, the jamming phase will occur only when the monolayer is solid-like and $K_m$ is large. As for the detachment phase, it prefers to appear in solid-like cells with both feedbacks are relatively weak. Actually, these migration modes have all been experimentally observed. For instance, as mentioned earlier, migrating keratinocytes were found to form suspended bridges on patterned surfaces (14). In contrast, the MDCK cell monolayer was recently reported to behave like Maxwell viscoelastic fluid (41) and therefore can move like a chain (14) (recall that the chain migration mode is much more easily to appear for fluid-like cells, refer to Figs. 3 a and b). We must point out that significant actin cable formation and stretched E-cadherin morphology have both been observed in the propagating bridge of keratinocytes (14), indicating the strong presence of contractility and adhesion feedbacks. On the other hand, no obvious sign of enhanced intracellular contraction or cell-cell adhesion was observed in the migrating chain of MDCK cells (14,15), which is consistent with the reduced $K_m$ and $K_s$ values adopted in our simulations for this cell type. Finally, the detachment mode was also observed in 3T3 fibroblasts migrating along nanofibers (42) or narrow strips (43).

It must be pointed out that cell migration mode was also found to be regulated by the strip gap. Specifically, experiment has shown that propagating cell bridges became jammed when the strip gap $G$ is above a threshold value (14). This phenomenon was indeed captured by our model as shown in Fig. 3 e where cells could form migrating bridges when $G/W_0=12$ but fail to move when $G/W_0$ is larger than 20. This is not surprising because larger gap means fewer leader cells and therefore lower total crawling force to drive the monolayer movement. In addition, a larger gap will also lead to reduced contractile fiber formation along the leading edge of the monolayer (refer to the next section for details) and therefore make its propagation more difficult.

The leading edge is strengthened by active chemo-mechanical feedbacks

As pointed out earlier, the formation of intracellular contractile fibers and a stretched morphology of E-cadherins (responsible for the formation of cell-cell adhesion) have been observed at the leading edge and the suspended area of migrating cell monolayers (14), indicating elevated cell contractility and intercellular adhesion there. To see whether these features were captured by our model, we plot the distribution of $m_J$ (the parameter characterizing myosin activity) and maximum $s_{ij}$ (reflecting intercellular adhesion) of each cell in Figs. 4 a and b, respectively. As expected, it was found that the strongest contractility and intercellular adhesion occur at the leading edge. Specifically, if we choose 20 cells forming the first three layers of cells at the leading edge (Fig. 4 a) and plot their $m_J$ and maximum $s_{ij}$ values, then it can be seen that both cell contractility and intercellular adhesion decrease when moving from the edge of the cell monolayer to its interior (Figs. 4 c and d). In particular, the third layer cells can hardly sense enhancement of intercellular adhesion despite that the strength of cell-cell contact reaches the maximum/saturation value $(1+K_s)$ in the first cell layer (Fig. 4 d). Finally, the acute angle that appeared in leading cells is likely because all vertices of these cells were directly pulled by the crawling force in the simulation.

Figure 4.

Contractility and cell-cell adhesion are strengthened at the leading edge of suspended bridges. Predicted distribution of (a) $m_J$ and (b) maximum $s_{ij}$ in each cell within the cell monolayer where numbers are assigned to sample cells for easy analysis. Here, parameter values were chosen as $p_0$ = 4.0, $K_m$ = 1.0, and $K_s$ = 0.5. (c) The values of $m_J$ and (d) maximum $s_{ij}$ in different sample cells marked in (a). The data are shown in mean $\pm$ SEM, n = 11. (e) After formation of the cell bridge (left), blebbistatin treatment of cells (leading to weakened myosin activity) led to its partial retraction (right), refer to the black dotted line. Strip pattern on surface is indicated by the red lines in (a), (b) and (e). To see this figure in color, go online.

At this point, it is interesting to see what would happen if active feedbacks are turned down after the formation of cell bridges. Actually, from their experiments, Vedula et al. concluded that the myosin contractility is essential for bridge formation but may be less important for its maintenance (14). Specifically, they found cells treated with blebbistatin (a myosin inhibitor) were unable to form suspended bridges. However, if cell bridge was already formed, then the addition of blebbistatin only led to slight retraction (rather than complete disappearance) of the bridge. Remarkably, these phenomena were also reproduced by our simulations where addition of the drug was assumed to trigger a 20% reduction of myosin-related parameters (i.e., $\text{Γ}$, $K_m$, $K_s$, $p_0$, $f_p$). In particular, if such reduction was introduced at the beginning of our simulation, then cells were found to migrate like a chain rather than form bridges (Video S1). Yet, only a retraction of the bridge was observed if the reduction was introduced after its formation (Fig. 4 e and Video S2). Furthermore, similar to experimental observations, it was found that the contractility and intercellular adhesion remain high in the relaxed bridge (Fig. S1). Physically, this means the final/equilibrium configuration that the cell bridge can reach is history dependent, i.e., the energy landscape of the system will be different when cellular contractility is disrupted before or after bridge formation. Interestingly, such phenomenon can also be reproduced in our simulation (Fig. S2, Videos S3 and S4) if the value of $K_s$ is decreased to be negative (i.e., $K_s=-0.5$) while other parameters remain unchanged. Note that a negative $K_s$ means that cell-cell adhesion will be weakened rather than strengthened by the applied stress. This appears to be consistent with the observation that the same history-dependent chain-bridge transition also occurs after $\alpha$-catenin (a key player in the mechano-regulation of cell-cell adhesion) was knocked down with short hairpin RNA (14).

The appearance of vortex migrating pattern of cells depends on strip width and the magnitude of active polarity forces

Another interesting finding from our simulations (Fig. S3) is that cells could move in a vortex fashion on strips with width larger than a critical value (i.e., 10 $W_0$). The regulation of the size of geometrical confinement and magnitude of polarity forces on the appearance of vortex migration pattern is summarized by the phase diagram shown in Fig. 5 a. Basically, the swirling motion of cells is easier to generate as the strip width and the magnitude of random polarity force $f_p$ increase. For instance, in Fig. 5 b, the overlapped velocity and vorticity fields clearly show the formation of an anticlockwise vortex around a local maximum of vorticity. In contrast, no vortex motion can be observed when cells are moving on narrow strips or if $f_p$ is small (Fig. 5 c). We want to emphasize that both the random polarity forces generated in cells and their interactions (24,25), i.e., lateral alignment and contact inhibition of locomotion (44,45), have been thought to contribute to the global rotation (46) or local swirling (47) of cells. Results here clearly indicate that random force alone is enough for cell vortex formation.

Figure 5.

Vortex formation in the directionally moving cell monolayer depends on both geometrical confinement and active cellular activities. (a) Phase diagram illustrating the influence of strip width ($W$) and magnitude of random cellular polarity force ($f_p$) on the formation of cell vortex where parameter values were chosen as $p_0$ = 4.25, $K_m$ = 0.1, and $K_s$ = 0.1. Essentially, the probability of vortex formation increases as strip width or magnitude of random polarity force grows. (b) Snapshot showing the velocity field (arrow) within the moving cell layer on a wide strip ($W/W_0$ = 40). Here the magnitude of random force was set as $f_p$ = 0.4. The vortex is amplified in the inset overlapping with the local vorticity field (heat map). (c) Snapshot showing the velocity field within the moving cell layer on a narrow strip ($W/W_0$ =15). Here, $f_p$ was chosen as 0.2 in the simulation and the angle between strip axis and the velocity of a cell is defined as $\theta$. (d) Evolution of order parameter $<cos\theta >$ (characterizing the alignment of velocities of all cells) under different magnitude of active polarity force and strip width. To see this figure in color, go online.

Another way to look at this is to examine how the moving direction of cells varies on the strip. Specifically, if we define the angle between the strip axis and velocity of individual cell as $\theta$ (Fig. 5 c), then the order parameter $<cos\theta >$ (i.e., average of $<cos\theta >$ for all cells on the strip) reflects whether cells tend to move in an aligned manner or more randomly (and therefore more likely to form vortices). Note that $<cos\theta >$ = 1 means all cells move perfectly along the strip axis while $<cos\theta >$ = 0 corresponds to a totally disordered velocity field. As shown in Fig. 5 d, the order parameter was found to remain close to unit (and hence no vortex formation, refer to Fig. 5 c and Video S5) when the magnitude of random polarity force is relatively small. On the other hand, $<cos\theta >$ dropped to ~0.6 when $f_p$ increases to 0.4, which led to the swirling of cells (Fig. 5 b and Video S6). It must be pointed out that data presented here are based on $K_m=0.1\text{ and }{K}_s=0.1$. However, similar results/conclusions will also be obtained under other values of these two feedback parameters as long as they are not very large (say less than 0.5), refer to Fig. S4.

Conclusions and discussions

In this study, we systematically examined how the collective behaviors of cells moving on a patterned surface are influenced by internal cellular activities, such as chemo-mechanical feedbacks and polarization. Specifically, it was found that strong contractility and intercellular adhesion feedbacks are essential for fluid-like cells to form suspended bridges “flying” over nonadhesive areas because they help to strengthen the leading edge of the cell monolayer and maintain its integrity. In addition, phase diagrams obtained here illustrate how variations in the strength of these two feedbacks (i.e., $K_m$ and $K_s$), along with the cell shape index $p_0$, enable cells to switch among different migration modes, including detachment, flowing chain, suspended propagating bridge, and jamming, consistent with experimental observations (14,42). Finally, we also showed that the presence of random active polarity force in cells could be responsible for their swirling motion on wide strips.

We want to point out that intensive theoretical efforts have been invested in the past few decades to investigate collective cell migration (22, 23, 24,48). However, most previous studies focused on how physical cues on the extracellular matrix, such as the pattern (14, 15, 16, 17), curvature (21,49), adhesiveness (50), and topography (13), influence the movement of cells, whereas the role of active cellular responses (i.e., force-dependent chemo-mechanical feedbacks, polarization, etc.) remains poorly understood. Our study fills this gap by connecting a variety of experimental observations with possible chemo-mechanical feedbacks governing the development of contractility and intercellular adhesion within the cell monolayer. Nevertheless, various simplifications adopted in the present model warrant further investigation. For example, the pulling force was only applied to the leader cell in this study, a treatment that could suppress possible self-propulsion of cells (i.e., propelling forces might be generated spontaneously in multiple cells, not only in the leader one). The cell-substrate adhesion energy density was also taken to be a constant while in reality this quantity should depend on the spatial distribution of adhesion molecules and extracellular matrix properties (51,52). In addition, since we ignored polarity alignment of neighboring cells, the vortices observed in silico here are relatively small and transient, in contrast to the large and ordered ones obtained in (44,45). Actually, the size of vortices has been reported to be influenced by substrate compliance (47) and apparent viscosity of the monolayer (53). In this regard, the question of how these different factors work together to dictate the swirling dynamics of cells is certainly worth investigating. Details on exactly how crawling forces are generated in cells are also missing in the current formulation. For example, several lines of evidence have suggested that the magnitude of this force should be tightly regulated by factors such as actin polymerization (54, 55, 56), focal adhesion dynamics (57), and contact inhibition/following of locomotion (58). More careful modeling efforts are needed to take these important features into account. Finally, given the critical roles of active cellular forces and cell adhesion in processes like tissue morphogenesis (59), cellular volume regulation (60,61), and embryo development (62,63), it is conceivable that the formulation presented here can also be extended to investigate these phenomena.

Author contributions

Y. L. conceived and designed the study. C.F., J.Y., and Y.L. developed the model. C.F. and J.Y. performed the simulations. C.F., J.Y., Y.Z., and Y.L. drafted and edited the manuscript.

Editor: Guy M Genin.