Polarized interfacial tension induces collective migration of cells, as a cluster, in a 3D tissue

Abstract

In embryogenesis and cancer invasion, cells collectively migrate as a cluster in 3D tissues. Many studies have elucidated mechanisms of either individual or collective cell migration on 2D substrates; however, it remains unclear how cells collectively migrate as a cluster through 3D tissues. To address this issue, we considered the interfacial tension at cell-cell boundaries expressing cortical actomyosin contractions and cell-cell adhesive interactions. The strength of this tension is polarized; i.e., spatially biased within each cell according to a chemoattractant gradient. Using a 3D vertex model, we performed numerical simulations of multicellular dynamics in 3D space. The simulations revealed that the polarized interfacial tension enables cells to migrate collectively as a cluster through a 3D tissue. In this mechanism, interfacial tension induces unidirectional flow of each cell surface from the front to the rear along the cluster surface. Importantly, this mechanism does not necessarily require convection of cells, i.e., cell rearrangement, within the cluster. Moreover, several migratory modes were induced, depending on the strengths of polarity, adhesion, and noise; i.e., cells migrate either as single cells, as a cluster, or aligned like beads on a string, as occurs in embryogenesis and cancer invasion. These results indicate that the simple expansion and contraction of cell-cell boundaries enables cells to move directionally forward and to produce the variety of collective migratory movements observed in living systems.

Graphical abstract

Significance

This study presents a simple mathematical model that explains how cells collectively migrate as a cluster in 3D space. Cells generate contractile and adhesive forces on their surfaces, which are simply expressed in the model by interfacial tension at cell-cell boundaries. Numerical simulations showed that interfacial tension enables cells to migrate as a cluster. The mechanism is that interfacial tension induces unidirectional flow of each cell surface from front to rear along the cluster surface. The model also explains the variety of cell migratory modes observed in embryogenesis and cancer metastasis. These results show that interfacial tension at cell-cell boundaries plays key roles in the wide range of cell migrations in living systems.

Introduction

Cells migrate directionally in three-dimensional (3D) tissues, in which cells are confined from all sides by a dense extracellular matrix (ECM) or tightly adhering cells. These migration characteristics are observed in physiological processes, such as development, immune defense, and wound healing (1, 2, 3, 4, 5, 6), as well as in cancer invasion (1, 2, 3, 4,7). Importantly, most cells move forward collectively while forming a cluster in 3D space (which is referred to hereafter as cluster migration). Over the last few decades, there has been intensive study of how cells migrate either individually or collectively on planar substrates (8, 9, 10). Recent studies have revealed a mechanism of single-cell migration in 3D space (11,12); however, little is known about how cells produce cluster migration.

Recent studies have shown how cells migrate individually as single cells. For example, individual cells are polarized such that actomyosins accumulate locally at the rear of the cell cortex, biasing cortical tension (11). The polarized tension causes the flow of cell cortex from the front to the rear, generating a propulsive force that moves the cell forward in 3D space (12). Other mechanisms may also be involved, such as those driven by pushing forces of actin polymerization and blebbing (13,14). However, although single-cell migration has gradually come to be understood, cluster migration may involve different mechanisms. Because each cell moves on a scaffold of surrounding cells, forces generated by individual cells may interfere with one another, preventing movement or breaking cell clusters apart. Cluster migration in 3D space was also investigated by numerical simulations using a particle model (15) and Voronoi model (16). However, these models involve explicit driving forces that move cells forward, such as a propulsive force acting directly on the cells (16) or a repulsive force with surrounding cells that is biased in front of and behind the migrating cells (15). Therefore, it is still unclear which specific forces, corresponding to intracellular structures and molecules, are generated by individual cells and how they are integrated through cell-cell interactions to induce cluster migration in 3D space.

3D cluster migration may also require polarized tension induced by actomyosin as a driving force, as observed in single-cell migration (11). That is, in a cell cluster, actomyosins accumulate locally at cell-cell boundaries, biasing the tension along the polarity of each cell to contract the boundaries. The polarized tension of each cell could be integrated in the entire cluster; e.g., Shellard et al. reported that neural crest cells migrate collectively on a 2D substrate by actomyosin contraction polarized throughout the entire cluster. The polarized contractility induces a flow of cells within the cluster to push the cluster forward (17,18). Moreover, the polarized force in the cluster could also be induced by the biased localization of specific cells, which can influence other cells by sensing either the mechanical or biochemical environment (7,19). In addition, adhesion through certain molecules, such as cadherins, may be involved in gathering cells as a cluster because cells with higher affinity expand their boundaries and form a cluster in a phase separation manner (20, 21, 22); e.g., simulations using a cellular Potts model suggested the importance of cell-cell adhesion in collective migration on a 2D substrate. Therefore, polarized tension and adhesion of individual cells may be integrated through cell-cell interactions and play a crucial role in 3D cluster migration (22).

The polarized tension and adhesion can be described by the interfacial tension at cell-cell boundaries; i.e., positive tension contracts the boundary, whereas negative tension expands the boundary. From a coarse-grained point of view, interfacial tension also effectively accounts for other driving forces of cell migration. For example, blebbing at the front of a cell (14) can be expressed by a relatively low tension toward the front, in which the front is pushed forward by the effect of excluded cytoplasmic volume. Thus, interfacial tension may be a unifying factor describing the forces generated in a cell population. Therefore, we attempted to reproduce cluster migration by interfacial tension.

In this study, by developing a simple mathematical model, we clarified whether polarized interfacial tension induces cluster migration. We performed numerical simulations using a 3D vertex model that describes multicellular dynamics at single-cell resolution (23,24). First, we tested whether polarized interfacial tension allows cells to migrate individually in a 3D tissue. Second, we clarified that polarized interfacial tension enables cells to migrate collectively as a cluster. Importantly, the mechanism of cluster migration was explained by the directional flow of each cell surface along the cluster surface. Last, we analyzed the dependence of migratory processes on each parameter and found several migratory modes, such as cells migrating as single cells or as a cluster. Interestingly, a new migratory mode was found, in which cells migrate aligned like beads on a string. These migratory modes vary depending on the strength of polarity, adhesion, and noise.

Methods

To address the mechanisms of cluster migration in 3D space, we focus on a simple situation where cells are embedded in a 3D tissue. For simplification, a cubic system box was considered within $-L/2\le \alpha \le L/2$ ($\alpha =x,y,z$), in which are packed two types of cells (i.e., cells with and without polarity, referred to hereafter as polar and nonpolar cells, respectively). A chemoattractant gradient along the $x$ axis was implicitly introduced; i.e., the interfacial tension of polar cells varies depending on the orientation of their surfaces relative to the $x$ axis (Fig. 1 a). Here, only polar cells sense the gradient of the chemoattractant and polarize in the x direction, while nonpolar cells do not. In addition, only polar cells generate noise of the interfacial tension on their surfaces, while nonpolar cells do not. The nonpolar entities may represent surrounding cells of a different type than the polar cells, or, in addition, these nonpolar entities can be interpreted as passive viscoelastic materials surrounding polar cells, such as the ECM, because they move only when subjected to external forces from polar cells. Due to the symmetry of the system, if a polar cell can move in one direction, that direction is along the x axis. Thus, a periodic boundary condition was imposed on $x=\pm L/2$ to clarify whether polar cells continuously move along the x axis. On the other hand, a fixed boundary condition was imposed at $y,z=\pm L/2$. This situation corresponds to the case where polar cells are surrounded by nonpolar entities attached to other tissues at $y,z=\pm L/2$. The box is filled by $N_{\text{t}}$ ($=432$) cells, and each cell volume is constrained to $V_0$, from which the box size is written as $L={\left(N_{\text{t}}{V}_0\right)}^{1/3}$. The system comprises $N_{\text{p}}$ polar cells and $\left(N_{\text{t}}-N_{\text{p}}\right)$ nonpolar cells.

Figure 1.

Setup of the model. (a) System box filled by cells in 3D space. Individual cell shapes are represented by polyhedrons. Polar cells have polarity along the x axis, which is expressed by an implicitly assumed chemoattractant gradient. The system comprises $N_{\text{p}}$ polar cells and $\left(N_{\text{t}}-N_{\text{p}}\right)$ nonpolar cells. (b) Cell-cell interfacial tension, ${\gamma }_{ij}$, is introduced into the $\left(ij\right)$^th polygonal boundary face shared by the $i$^th and $j$^th adjacent cells. The angle for the $i$^th cell to the $\left(ij\right)$^th face, ${\theta }_{ij}$, is defined as that from the x axis to the normal vector of the $\left(ij\right)$^th face, whose direction is toward the outside of the $i$^th cell. (c) Topological operations between patterns [H] and [I] in 3D vertex model (20,21). Cell-cell boundaries are colored in transparent gray. To see this figure in color, go online.

3D vertex model

In the 3D vertex model, the shape of each cell embedded in the 3D tissue is represented by a polyhedron whose vertices are shared by neighboring cells (Fig. 1 b). The shape and configuration of cells are described by the locations of vertices comprising cellular polyhedrons and the topological network among vertices.

In long-term cell behaviors, inertial forces can be ignored, whereas viscous forces are dominant. In 3D tissues, viscous forces create balance within the surrounding cells. Viscous forces within 3D tissues can be expressed by relative frictional forces among vertices (25). The time evolution of the i^th vertex location, represented by ${\mathit{r}}_i$, is given by

$$\eta \left(\frac{\partial {\mathit{r}}_i}{\partial t}-{\overline{\mathit{v}}}_i\right)=-\nabla U.$$ (1)

The left side of Eq. 1 is a relative friction force on the i^th vertex. Here, $\eta$ is the effective friction coefficient of vertices. Vector ${\overline{\mathit{v}}}_i$ is a velocity field, defined as the mean velocity of the surrounding vertices. The detailed description of the relative friction force is the same as that in our previous study (25). The right side of Eq. 1 is a mechanical force on the i^th vertex derived from the effective energy potential, represented by $U$. During the cell movements described by Eq. 1, individual edges and polygons in the network occasionally shrink to meet or retract from neighboring cells. According to vertex movements, the topological network is dynamically reconnected using [I]-to-[H] and [H]-to-[I] operations (23,24) (Fig. 1 c); i.e., the operations between patterns [H] and [I] are performed when the related edge lengths become infinitely small.

Effective energy function

We assume that cells have a simple free energy, given by

$$U\equiv \sum _i^{\text{cell}}\frac{1}{2}k{\left(\frac{V_i}{V_0}-1\right)}^2+\sum _{\left(ij\right)}^{\text{boundary}}{\gamma }_{\left(ij\right)}{A}_{\left(ij\right)},$$ (2)

where the first and second terms indicate volume constraint energy and interfacial energy, respectively. In the first term, the incompressibility of each cell volume is assumed as $V_i\sim V_0$, by which constant $k$ is set much higher than the interfacial energy. In the second term, ${\gamma }_{\left(ij\right)}$ and $A_{\left(ij\right)}$ describe the tension and area of the $\left(ij\right)$^th interface, respectively. The $\left(ij\right)$^th interface is the boundary face between the i^th and j^th cells (Fig. 1 b). Interfacial tension ${\gamma }_{\left(ij\right)}$ in Eq. 2 is the result of contraction and adhesion forces acting on the $\left(ij\right)$^th interface (e.g., from actomyosin contractility or adhesion through cadherin). This tension can be divided into three components: 1) polarized tension, which varies with the relative position within each cell; 2) constant tension, which varies with the combination of cell types that comprise the interface; and 3) a noise component, which fluctuates with time but has a time average of zero. Therefore, we formulate ${\gamma }_{\left(ij\right)}$ as

$${\gamma }_{\left(ij\right)}\equiv \frac{{\gamma }_i^{\left(\text{p}\right)}}{2}\left(1+\cos \left({\theta }_{ij}-\pi \right)\right)+\frac{{\gamma }_j^{\left(\text{p}\right)}}{2}\left(1+\cos \left({\theta }_{ji}-\pi \right)\right)+{\gamma }_{\left(ij\right)}^{\left(\text{c}\right)}+{\gamma }_{\left(ij\right)}^{\left(\text{n}\right)}{w}_{\left(ij\right)}.$$ (3)

The first and second terms denote the polarized surface tensions of the $i$^th and $j$^th cells, respectively. Polarized tensions depend on the chemoattractant gradient, which was implicitly introduced; i.e., the amount of this tension depends on ${\theta }_{ij}$, the angle from the x axis to the normal vector of the $\left(ij\right)$^th interface, whose normal vector is toward the outside of each cell (Fig. 1 b). The constants ${\gamma }_i^{\left(\text{p}\right)}$ and ${\gamma }_j^{\left(\text{p}\right)}$ are coefficients that represent the strengths of polarity of the $i$^th and $j$^th cells, respectively. Because ${\theta }_{ij}$ becomes larger toward the rear of each cell, the polarized tension increases as well. Each of the first two terms is maximized ($={\gamma }_i^{\left(\text{p}\right)}$) when ${\theta }_{ij}=\pi$. The first and second terms in Eq. 3 reflect, for example, the polarized cortical actomyosin contraction within each polar cell. The third term ${\gamma }_{\left(ij\right)}^{\left(\text{c}\right)}$ denotes the strength of constant interfacial tension between the $i$^th and $j$^th cells. This term reflects adhesion strength between cells; i.e., lower ${\gamma }_{\left(ij\right)}^{\left(\text{c}\right)}$ indicates greater adhesion strength. The fourth term denotes noise with a strength of ${\gamma }_{\left(ij\right)}^{\left(\text{n}\right)}$ generated by polar cells, reflecting the fluctuations of actomyosin accumulations (26,27). Here, $w_{\left(ij\right)}$ satisfies and  when $\left(ij\right)=\left(lm\right)$, when $\left(ij\right)\ne \left(lm\right)$, where the angle brackets that sandwich $w_{\left(ij\right)}$ signify their statistical average, and τ is relaxation time of the noise. This noise has been used in previous 2D and 3D models of cell flow (28,29).

Parameter setting

In the interfacial tension of Eq. 3, polarity and noise terms were assigned only to polar cell surfaces; i.e., noise strength was set to ${\gamma }_{\left(ij\right)}^{\left(\text{n}\right)}={\gamma }_{\text{n}}$ when either the i^th or j^th cell is a polar cell, otherwise ${\gamma }_{\left(ij\right)}^{\left(\text{n}\right)}=0$, and ${\gamma }_i^{\left(\text{p}\right)}={\gamma }_{\text{p}}$ when the i^th cell is a polar cell, otherwise ${\gamma }_i^{\left(\text{p}\right)}=0$. Moreover, because the strength of cell-cell adhesion between polar and nonpolar cells can be equal to or lower than that between cells of the same type (i.e., polar with polar cells or nonpolar with nonpolar cells), the tension strength was set to ${\gamma }_{\left(ij\right)}^{\left(\text{c}\right)}={\gamma }_0$ between cells of the same type and ${\gamma }_{\left(ij\right)}^{\left(\text{c}\right)}={\gamma }_1$ ($\ge {\gamma }_0$) between polar and nonpolar cells.

Parameters were nondimensionalized by unit length ${\left(V_0\right)}^{1/3}$, unit time ${\eta }_{\text{c}}/10{\gamma }_0{\left(V_0\right)}^{2/3}$, and unit energy $5{\gamma }_0{\left(V_0\right)}^{2/3}$ by fixing the values of $V_0$, ${\eta }_{\text{c}}$, and ${\gamma }_0$. Volume elasticity was set to $k=50{\gamma }_0{\left(V_0\right)}^{2/3}$, much higher than the interfacial energy for the incompressibility of each cell volume. Relaxation time was set to $\tau =3{\eta }_{\text{c}}/10{\gamma }_0{\left(V_0\right)}^{2/3}$. Interfacial tension between different types of cells, polarity, and noise were varied as the control parameters in this system (${\gamma }_0\le {\gamma }_1\le 2{\gamma }_0$, $0\le {\gamma }_{\text{p}}\le 4{\gamma }_0$, and $0\le {\gamma }_{\text{n}}\le 1.25{\gamma }_0$). The number of polar cells was also varied ($1\le N_{\text{p}}\le 9$).

Evaluation of migratory cell states

Cell states were evaluated by calculating three parameters: 1) the mean velocity of polar cells, represented by $v_{\text{p}}$, which is defined as the mean of the center mass velocity of polar cells; 2) cluster size, represented by $n_{\text{max}}$, which is defined as the average maximum number of cells composing each cluster; and 3) number of contacts, represented by $n_{\text{p}}$, which is defined as the average number of contact surfaces from each polar cell to other polar cells.

Numerical procedures

Vertex velocities were implicitly calculated by solving the simultaneous equations consisting of Eq. 1 for all vertices. The time development of vertex movements was calculated by integrating Eq. 1 using the first Euler method with the time step of $\text{Δ}t$ ($=0.005\left[{\eta }_{\text{c}}/10{\gamma }_0{\left(V_0\right)}^{2/3}\right]$). The [I]-to-[H] and [H]-to-[I] operations were applied by every time step of $\text{Δ}{t}_r$ ($=1.0\left[{\eta }_{\text{c}}/10{\gamma }_0{\left(V_0\right)}^{2/3}\right]$), where the threshold length for the operations is set to $\text{Δ}l$ ($=0.05\left[{\left(V_0\right)}^{1/3}\right]$).

Numerical simulations were performed using the initial condition in which polar cells are located around the center of the cubic system box (Fig. S1 a). Under this condition, the locations of all cells are pre-randomized. The obtained results were independent of the initial condition and the system box size; i.e., the cell migratory modes observed in this condition were also observed in the simulations using an initial condition with scattered polar cells (Fig. S1 b) and the larger system box (Fig. S1 c). The simulations were performed 1–10 times for each parameter set during the time duration of $\mathrm{10,000}\left[{\eta }_{\text{c}}/10{\gamma }_0{\left(V_0\right)}^{2/3}\right]$. In the case using the condition of polar cells located around the center of the system box as a cluster, the system reached the steady state in the time duration of $5000\left[{\eta }_{\text{c}}/10{\gamma }_0{\left(V_0\right)}^{2/3}\right]$ (Fig. S1 d). Therefore, the physical quantities such as $v_{\text{p}}$, $n_{\text{max}}$, and $n_{\text{p}}$ were calculated from the last $2000\left[{\eta }_{\text{c}}/10{\gamma }_0{\left(V_0\right)}^{2/3}\right]$ time duration.

Results

Polarized interfacial tension induces single-cell migration in a 3D tissue

First, we tested whether cells move as individual cells, and found that the interfacial tension of Eq. 3 succeeded in inducing single-cell migration for appropriate parameter values (Fig. 2). To analyze polar cell movements, the mean velocity of the polar cell, $v_{\text{p}}$, was observed (Fig. 2 a), depending on the classification of cell behaviors as one of two states, i.e., migrating ($v_{\text{p}}\ge 0.001\left[10{\gamma }_0{V}_0/{\eta }_{\text{c}}\right]$) or arrested ($v_{\text{p}}<0.001\left[10{\gamma }_0{V}_0/{\eta }_{\text{c}}\right]$). Here, cells migrated in cases with or without noise, where migrations in the case with noise required lower polarity than those without noise. During single-cell migration, a cell moved unidirectionally along the positive x axis while sequentially rearranging its configuration (Fig. 2 b; Video S1).

Figure 2.

Single-cell migration in 3D space and its dependence on interfacial tension. (a) Diagrams of cell states and velocities, $v_{\text{p}}$, as functions of polarized tension (${\gamma }_{\text{p}}$), constant tension between polar and nonpolar cells (${\gamma }_1$), and noise (${\gamma }_{\text{n}}$). The states of migration and arrest were classified according to $v_{\text{p}}$. Parameters were set to $N_{\text{p}}=1$. (b) Development of single-cell migration over time (also shown in Video S1). A single polar cell is colored in red, and nonpolar cells are colored in transparent gray. (c) Geometry and tension of cell-cell interfaces during single-cell migration (also shown in Video S2). The color scale at right indicates interfacial tensions of individual surfaces of the single polar cell, some of which are tagged alphabetically (A through G). (d) Interfacial tension as a function of the angle between the x axis and the vector from the center of a polar cell to its individual surfaces. Lines in the plot indicate the average values, and band widths indicate standard deviations. Parameters in (b)–(d) were set to ${\gamma }_1={\gamma }_0$, ${\gamma }_{\text{p}}=1.5{\gamma }_0$, and ${\gamma }_{\text{n}}=0$. To see this figure in color, go online.

Video S1. Single-cell migration

3

During cell migration, the surface of each cell flowed from the front to the rear of the cell (Fig. 2 c, Video S2); e.g., the polar cell newly meets the nonpolar cell located in front of it, forming a new interface C between them (reconnected from an edge by [I]-[H] operation). Face C was initialized to have a small area with relatively low polarized tension and expanded while face C was located at the front of the polar cell (t = 60 to 160). Moreover, face C moved from the front to the rear of the polar cell as the interfacial tension increased and shrank while face C was located toward the rear (t = 240 to 360). Face C eventually reached the rear of the polar cell and disappeared as the cell of interest separated from the nonpolar cell located to its rear (reconnected to a new edge by [H]-[I] operation). The directional flow of cell-cell interfaces repeated cyclically, driving individual cells forward.

Video S2. Geometry and tension of cell-cell interfaces during single-cell migration

4

The surface tension of the polar cell was always polarized (i.e., lower in the front and higher in the rear; Fig. 2 d). The polarized tension caused the directional flow of the cell surface within the cell; i.e., the interface shrunk in the rear by the relatively high tension and expanded in the front by the relatively low tension. The mechanism of single-cell migration observed in this study is essentially the same as that proposed in previous works (10, 11, 12), in which polarized interfacial tension causes the surface to flow within each cell from front to rear, and is the driving force of single-cell movement. A difference is that our model is more abstract but expresses the effects of both cortical tension and cell-cell adhesion using the polarized interfacial tension. Moreover, when polarized tension is high, the polar cell can migrate even without noise but requires noise when polarized tension is low (Fig. 2 a). This result suggested that, when polarized tension is high, the cell continually crosses the energy barrier of cell rearrangements independent of noise but stochastically crosses it in a ratcheting manner due to noise when polarized tension is low.

Polarized interfacial tension enables cells to migrate as a cluster in a 3D tissue

Next, we increased the number of polar cells in the box to assess whether clusters of cells move collectively (Fig. 3). When ${\gamma }_{\left(ij\right)}^{\left(\text{c}\right)}$ had the same amplitude among all cells, polar cells moved forward as individual cells. On the other hand, when ${\gamma }_1>{\gamma }_0$ (i.e., cells of the same type tended to aggregate), polar cells formed a cluster that was maintained during unidirectional movement (Fig. 3 a, Video S3).

Figure 3.

Cluster migration in 3D space. (a) Development of clustered cell migration over time (also shown in Video S3). Individual polar cells are designated with different colors, and nonpolar cells are colored in translucent gray. (b) Relay process of a boundary face between polar and nonpolar cells, tagged by S, from the front to the rear within a cluster. At t = 440, face S lies across cells (i) and (ii). (c) Interfacial tensions as a function of the angle from the x axis to the center of individual interfaces around the center of the cluster. Lines in the plot indicate the average values, and band widths indicate standard deviations. Parameters were set to $N_{\text{p}}=6$, ${\gamma }_1=1.5{\gamma }_0$, ${\gamma }_{\text{p}}=2{\gamma }_0$, and ${\gamma }_{\text{n}}=0.25{\gamma }_0$. To see this figure in color, go online.

Video S3. Clustered cell migration

5

Interestingly, the mechanism of cluster migration was analogous to that of single-cell migration. That is, single-cell migration was induced by the flow of interfaces within each cell, while cluster migration was induced by the flow of interfaces within the whole cluster. In other words, the interfaces between polar and nonpolar cells relayed along the cluster surface from cells near the front to cells near the rear of the cluster (Fig. 3 b). To describe the process more explicitly, we focus on the movement of face S, the interface between the cluster and a surrounding nonpolar cell (Fig. 3 b). Initially, only cell (i) was in contact with the nonpolar cell through face S at t = 340. While the cluster moved forward, face S spread to the adjacent cell (ii), which is located behind cell (i) in the cluster, and which newly made contact with the nonpolar cell through face S at t = 380. Face S gradually moved from cell (i) to cell (ii) around t = 440. When cell (i) separated from the nonpolar cell, face S eventually moved to cell (ii) completely at t = 500.

The relay of interfaces from cell to cell can be understood by considering the difference in interfacial tensions between neighboring cells. Let us focus on the situation in Fig. 3 b at t = 440, in which face S lies across both cells (i) and (ii). Face S on cell (i) had a large interfacial tension because it was located at the rear of cell (i) and was directed toward the negative x axis. In contrast, the face S portion of cell (ii) had a small interfacial tension because it was located at the front of cell (ii) and was directed toward the positive x axis. Because interfacial tension is equivalent to energy density, face S energetically preferred to be located on cell (ii) rather than cell (i). Although similar movements occur at the boundaries between polar cells, the movements between neighboring polar cells compete to fix the configuration of polar cells inside the cluster. The cell-to-cell relay of interfaces continued from the front end to the rear end of the cluster, and this directional flow of interface in the cluster was cyclically repeated to move the cluster forward.

To address the driving force of cluster migration, we analyzed the spatial distribution of interfacial tension in a cluster (Fig. 3 c). There are two types of interface: interfaces between polar cells inside the cluster and interfaces between polar and nonpolar cells on the cluster surface. As a result, interfacial tensions are homogeneous inside the cluster, indicating that the polarized tensions within each cell cancel one another. The cancellation arises from the form of Eq. 3; i.e., the summation of the first and second terms becomes a constant independent of the orientation of an interface inside the cluster. On the other hand, on the cluster surface, interfacial tensions are lower in the front and higher in the rear (i.e., polarized). These results indicate that the polarized tension within each cell provides the globally polarized tension on the whole cluster surface, inducing a directional flow of each cell surface along the cluster surface.

To understand the kinetic behaviors of cluster migration, velocity fields of cells and interfaces around the cluster were observed (Fig. 4), including the relative velocities of individual cells with respect to the velocity of the center of the cluster (Fig. 4 a). During cluster migration, polar cells aggregated around the center of the cluster (Fig. 4 b). Importantly, the relative velocities of cells within the cluster along the x axis were almost zero (Fig. 4 c), whereas relative velocities of interfaces were globally negative (Fig. 4 d). Namely, all polar cells moved forward with the same speed, whereas the interfaces flowed from the front to the rear of the cluster unidirectionally. Moreover, velocities of cells and interfaces along the radial axis (represented by the r axis) were positive in the front and negative in the rear of the cluster (Figs. 4 e and f), indicating that polar cells moved forward by scrambling the surrounding nonpolar cells. Additionally, we observed the trajectory of the center of each polar cell during cluster migration, which was projected on the xy plane (Fig. 4 g), showing that the configuration of polar cells was mostly maintained within the cluster during migration. These results showed that polarized tension induced directional flow of interfaces, and this cluster migration does not require cell convection (i.e., cell rearrangement) within the cluster.

Figure 4.

Flow of cells and interfaces. (a) Explanation of the coordinates. The cylindrical coordinates along the x axis were considered, and the radial axis is referred to as the r axis. The origin of the coordinates was set to the center of the cluster. The physical quantities in (b)–(f) were averaged along the rotational axis of the cylindrical coordinates. (b) Map of the mean number density of polar cell centers on the xr plane. Based on this map, the region including polar cells is surrounded by a black frame in (b)–(f). (c–d) Map of the mean relative velocities of cells and interfaces along the x axis on the xr plane. (e and f) Map of the mean relative velocities of cells and interfaces along the r axis on the xr-plane. In (c)–(e), the relative velocities were obtained with respect to the velocity of the cluster. (g) Trajectories of individual polar cells. The trajectory of each polar cell during the period of $\mathrm{5,000}\left[{\eta }_{\text{c}}/10{\gamma }_0{\left(V_0\right)}^{2/3}\right]$ was projected on the xy plane. Parameters were set to $N_{\text{p}}=6$, ${\gamma }_1=1.5{\gamma }_0$, ${\gamma }_{\text{p}}=2{\gamma }_0$, and ${\gamma }_{\text{n}}=0.25{\gamma }_0$. To see this figure in color, go online.

Mechanism of single-cell and cluster migrations in 3D space

These results illustrate a common mechanism for both single-cell and cluster migrations; i.e., by regarding either a single cell or a cell cluster as an object, the object moves forward by the directional flow of the object surface from the front to the rear. In either single-cell or cluster migration, the directional flow is induced by the polarized interfacial tension on the object surface. In both cases, the surface tension of the object was distributed as a gradient against the angle from the center (Figs. 2 d and 3 c).

To analytically evaluate whether an object that corresponds to a single cell or a cluster of cells moves along the x axis due to the polarized interfacial tension of the object in the steady state, we made a simple mathematical model (see appendix for details). In this model, we represent either a single cell or a cluster of cells by a spherical object with a constant radius $R$ and give the object the surface tension whose strength depends on $\theta$, denoted by $T\left(\theta \right)$. Here $\theta$ is the angle between the x axis and the vector from the center of the object to the point at the surface under consideration. We assume that the surface of the object bears a frictional force with the medium surrounding the object; the strength of the frictional force is proportional to the speed of movement of the surface. We take the form of $T\left(\theta \right)$ as $T\left(\theta \right)=a(1-\cos \theta )$ as a representative example, where $a$ is a positive constant that represents the degree of polarized surface tension. We refer to this simple mathematical model as the continuum model hereafter. The setup of this continuum model corresponds to the situation of our vertex model because each vertex in the vertex model bears two types of forces that come from the interfacial tension and the frictional force due to movement of the vertex relative to its surroundings (see Eq. 1). In addition, the volume of each cell in the vertex model remains almost constant (see Eq. 2). If we regard the nonpolar cells surrounding the polar cells as a medium, then we can recognize that there is a correspondence between the vertex model and our continuum model. The results of the continuum model show that $T\left(\theta \right)$ induces flow of the surface of the object, and this flow generates the driving force for moving the object along the x axis through the frictional force with its surrounding medium. The velocity of the object in the steady state is given by $2a/\eta R$, where $\eta$ is the frictional coefficient between the surface of the object and the surrounding medium. That is, the speed of the object increases with increasing degree of polarity of the surface tension, $a$, and decreases with the object’s radius $R$ and the frictional coefficient $\eta$. This result coincides with that of movement of droplets in a fluid due to the Marangoni effect in the sense that speed is inversely proportional to $R$ and $\eta$ (30). Note that our model does not consider the flow of the surrounding medium and the inside of the object. This demonstration, presented in the appendix, indicates that the polarized interfacial tension enables the object to move in a definite direction in the steady state, while maintaining the force balance between the object and the surrounding medium.

Interfacial tension produces the variety of cell migratory modes

Additionally, to clarify what conditions determine single-cell and cluster migrations, we analyzed the effects of each term in Eq. 3 on cell behaviors. To evaluate whether or not polar cells migrate, we calculated the mean velocity of polar cells, $v_{\text{p}}$ (Fig. 5 a). In addition, to evaluate whether or not the polar cells form a cluster, we observed the maximum cluster size $n_{\text{max}}$ (Fig. 5 b). Moreover, to evaluate the morphology of the cluster, we calculated the number of contacts between polar cells $n_{\text{p}}$ (Fig. 5 c). Results showed that the migration velocity increased with increasing polarity ${\gamma }_{\text{p}}$ and with increasing adhesion between polar and nonpolar cells (decreasing ${\gamma }_1$) (Fig. 5 a). Cluster size, $n_{\text{max}}$, indicated that cells formed a cluster independently of ${\gamma }_1$ in the case with low noise (${\gamma }_{\text{n}}=0.25{\gamma }_0$) (Fig. 5 b). On the other hand, in the case with high noise (${\gamma }_{\text{n}}=1.25{\gamma }_0$), cells became scattered as ${\gamma }_1$ decreased. Moreover, cell-cell contacts, $n_{\text{p}}$, showed that cells formed more contacts with decreasing adhesion between polar and nonpolar cells (increasing ${\gamma }_1$) (Fig. 5 c).

Figure 5.

3D cell migration modes and their dependence on interfacial tension. (a–c) Mean velocity $(v_{\text{p}}$), cluster size ($n_{\text{max}}$), and cell-cell contacts ($n_{\text{p}}$) as functions of polarized tension (${\gamma }_{\text{p}}$), constant tension between polar and nonpolar cells (${\gamma }_1$), and noise (${\gamma }_{\text{n}}$), respectively ($N_{\text{p}}=6$). The state was classified into four states: arrested (I), single (II), cluster (III), and line (IV), depending on $v_{\text{p}}$, $n_{\text{max}}$, and $n_{\text{p}}$. (d) Snapshots of migrations as single cells (II), a cluster (III), and cells in alignment (IV). The development of these migration modes over time are shown in Videos S3, S4, and S5. Individual polar cells are designated with different colors, and nonpolar cells are colored with translucent gray. The snapshots in (d) were obtained using the following parameters: (II) ${\gamma }_1={\gamma }_0$, ${\gamma }_{\text{p}}=2{\gamma }_0$ and ${\gamma }_{\text{n}}=1.25{\gamma }_0$; (III) ${\gamma }_1=1.5{\gamma }_0$, ${\gamma }_{\text{p}}=2{\gamma }_0$ and ${\gamma }_{\text{n}}=0.25{\gamma }_0$; and (IV) ${\gamma }_1={\gamma }_0$, ${\gamma }_{\text{p}}=2{\gamma }_0$, and ${\gamma }_{\text{n}}=0.25{\gamma }_0$. (e) Mean velocity, $v_{\text{p}}$, as a function of ${\gamma }_{\text{p}}$ and ${\gamma }_{\text{n}}$. (f) Mean velocity, $v_{\text{p}}$, and cluster size as functions of ${\gamma }_1$. (g) Mean velocity, $v_{\text{p}}$, as a function of $N_{\text{p}}$. Parameters were set to $N_{\text{p}}=6$ in (a)–(f). Results in (g) were produced using five samples for each condition. To see this figure in color, go online.

Interestingly, cell behaviors could be classified into four states by $v_{\text{p}}$, $n_{\text{max}}$, and $n_{\text{p}}$ (Figs. 5 a–d): 1) arrested, 2) single, 3) cluster, and 4) line. First, polar cell states were classified into two states depending on $v_{\text{p}}$; i.e., the migrating ($v_{\text{p}}\ge 0.001\left[20{\gamma }_0{V}_0/{\eta }_{\text{c}}\right]$) and arrested ($v_{\text{p}}<0.001\left[20{\gamma }_0{V}_0/{\eta }_{\text{c}}\right]$) states (Fig. 5 a). In addition, the migrating cell state was classified into three states depending on $n_{\text{max}}$ and $n_{\text{p}}$. That is, the state was defined as single if $n_{\text{max}}<2$ and $n_{\text{p}}<1$, line if $n_{\text{max}}<2$ and $n_{\text{p}}\ge 1$, and cluster if $n_{\text{max}}\ge 2$ and $n_{\text{p}}\ge 1$ (Figs. 5 b and c). These four states have the following characteristic features (Fig. 5 d, Videos S3, S4, and S5). 1) In the case with low polarity (${\gamma }_{\text{p}}/{\gamma }_0<0.25\sim 0.5$), cells were nearly arrested in the coordinates, and did not migrate. When the polarity (${\gamma }_{\text{p}}$) was increased, the cell state changed from nonmigrating to migrating, and then the migration velocity increased. 2) In the case with high noise (${\gamma }_{\text{n}}=1.25{\gamma }_0$) and homogenous adhesion among cells (${\gamma }_1={\gamma }_0)$, cells migrated as individual cells. 3) When the adhesion between polar and nonpolar cells was lower than between the same type of cells (${\gamma }_1>{\gamma }_0)$, polar cells formed a cluster. 4) Interestingly, in the case with homogenous adhesion (${\gamma }_1={\gamma }_0$) and low noise (${\gamma }_{\text{n}}=0.25{\gamma }_0$), cells aligned like a string of beads.

Video S4. Migration of a single cell

6

Video S5. Migration of cells aligned in a line

7

More quantitatively, we observed the dependence of $v_{\text{p}}$ on polarity ${\gamma }_{\text{p}}$ and noise ${\gamma }_{\text{n}}$ under the condition with homogenous adhesion (${\gamma }_1={\gamma }_0$) (Fig. 5 e). In the case without noise (${\gamma }_{\text{n}}=0$), polar cells were arrested when ${\gamma }_{\text{p}}$ was less than a certain value and migrated when ${\gamma }_{\text{p}}$ was larger (i.e., there was a threshold for ${\gamma }_{\text{p}}$). On the other hand, in the case with noise (${\gamma }_{\text{n}}={\gamma }_0$), polar cells migrated even when ${\gamma }_{\text{p}}$ was small. This is because the noise promoted rearrangement of the cell configuration. The migration velocity was almost linearly proportional to ${\gamma }_{\text{p}}$. We also observed the dependence of $v_{\text{p}}$ and $n_{\text{max}}$ on ${\gamma }_1$ under the condition without noise (${\gamma }_{\text{n}}=0$) (Fig. 5 f). The migration velocity decreased as the adhesion between polar and nonpolar cells decreased (${\gamma }_1$ increased), whereas the cluster size increased. The decrease in speed as the cluster size increases is qualitatively consistent with the results of the analytical calculation using the continuum model. Note that $v_{\text{p}}$ decreased with ${\gamma }_1$ even while $n_{\text{max}}$ saturated (polar cells formed a cluster). This is because the shape of the cluster varied with the value of ${\gamma }_1$; i.e., the cluster became rounder as ${\gamma }_1$ increased. The rounder shape makes the cluster harder to move forward because the rounder shape has a larger cross-sectional area in the plane normal to the migration direction, so the cluster meets greater resistance from the surrounding nonpolar cells.

Furthermore, to address effects of polar cell density, we observed the dependence of $v_{\text{p}}$ on the number of polar cells, $N_{\text{p}}$, under the condition of the line state with homogenous adhesion (${\gamma }_1={\gamma }_0$) and low noise (${\gamma }_{\text{n}}=0$) (Fig. 5 g). Interestingly, a notable dependence was found; i.e., $v_{\text{p}}$ was accelerated simply by increasing $N_{\text{p}}$ from one cell to multiple cells. Under these conditions, polar cells are aligned like a string of beads during migration, so the mechanism of the dependence of $v_{\text{p}}$ on $N_{\text{p}}$ is different from that predicted by the simple model in the appendix but can be explained by the stochastic rearrangements of individual cells during cell migration. The frequency of these stochastic rearrangements depended on the noise to which polar cells were exposed. While cell rearrangements occurred frequently with greater noise, the multi-body effect was observed under the condition with low noise. Under the condition with low noise, the interaction between polar cells may provide a noise-like effect to facilitate cell rearrangements; i.e., the resultant movement of each polar cell acts as noise to facilitate the rearrangements of each other cell. Because the interaction between polar cells increases with $N_{\text{p}}$, $v_{\text{p}}$ tends to increase with $N_{\text{p}}$. This dependence is consistent with recent studies of particle-based simulations and experiments with border cells, where migratory velocity increases with cluster size (6,15). Moreover, the mechanism of inducing a line state could be explained by a similar multi-body effect between polar cells. Polar cells move forward by scrambling the surrounding nonpolar cells (Fig. 4 e), which requires a certain energy cost. For each polar cell, following the advancing polar cells costs less energy than scrambling the surrounding nonpolar cells. The line state is observed when this energy difference is dominant for cell movements (${\gamma }_{\text{n}}=0.25{\gamma }_0$) but not when the noise is larger than the energy cost for scrambling the surrounding nonpolar cells (${\gamma }_{\text{n}}=1.25{\gamma }_0$).

Discussion

In summary, simulations using the 3D vertex model showed that polarized interfacial tension induces collective cell migration as a cluster in 3D space. This mechanism has analogy to the Marangoni effect, where interfacial tension induces the directional flow of each cell surface along the cluster surface. This mechanism is common for both single-cell and cluster migrations; i.e., either a single cell or a cell cluster moves forward by the directional flow of the surface from the front to the rear (Fig. 6 a). In single-cell migration, polarized tension within a cell directly induces the surface flow, while, in a cell cluster, the tension cancels out inside the cluster and is biased only on the cluster surface, along which each cell surface flows directionally (Fig. 6 b). Importantly, this mechanism does not require convection of cells (i.e., cell rearrangements) inside the cluster. A previous study suggested a conceptual model referred to as the rear-wheel-drive mechanism, which requires the flow of cells within the cluster to cause cluster migration (17,18). However, in our model, a new surface is created at the front of each polar cell, moves to the rear of the cell, and disappears repeatedly. The creation and disappearance of new faces corresponds to mass transport between the cell surface and cytoplasm within each cell, such as endocytosis and exocytosis. Cell rearrangements within the cluster are not necessary for this mechanism but can be induced incidentally by noise. Cell rearrangements within the cluster are observed in border cell migration (31). Conventional models explain cluster migration as being caused by the convection of cells inside the cluster. On the other hand, our model shows that the gradient of interfacial tension (small at the front and large at the rear) enables the cluster to continuously migrate in one direction, even without convection. Our model can be verified if experimental observations confirm a situation in which there is a gradient of interfacial tension and little or no rearrangement within the cluster.

Figure 6.

Mechanism of single-cell and cluster migrations in 3D space. (a) Schematic illustration of the common mechanism of single-cell and cluster migrations. In both cases, new interfaces are sequentially generated at the front, moved backward, and disappeared in the rear. (b) Schematic illustration of the distribution of interfacial tensions and flows in either single-cell or cluster migrations. In single-cell migration, the tension is biased within a cell, directly inducing the surface flow. In the cell cluster, the tension cancels out inside the cluster, while tension is biased on the cluster surface, along which each cell surface flows directionally. To see this figure in color, go online.

The conditions that determine single-cell and cluster migrations were also identified; i.e., the existence of cell polarity to directionally move cells forward, and heterogeneous adhesion among cells to form a cluster. Because cell-cell interfacial tension can be regulated by the accumulation of actomyosin and adhesion molecules, both single-cell and cluster migrations can be thus driven by their localized accumulations within each cell. Whether migrating cells form a cluster may be determined by the type and expression levels of adhesion molecules among the cells.

Although our results served to clarify those points, more detailed analyses are needed to address several remaining questions; e.g., how migrating cells align themselves like beads on a string despite the absence of heterogeneous adhesion among cells (Figs. 5 a–c), and how the migration velocities were accelerated by increasing the number of cells (Fig. 5 g). More detailed analyses are needed to address those issues and others; e.g., the geometry of packed cells imposes energy barriers to cell rearrangements (32) that must be overcome for cell clusters to migrate in 3D space. From our results, it is evident that cells did sequentially cross barriers in the 3D packing geometry; however, the process of crossing the energy barriers is still unclear. Spatial dimensions such as topological dimensions and Poisson effects may be important for cells to cross these energy barriers (33,34). For example, the energy barriers should differ between 2D and 3D systems. In the 2D case where cells maintain a monolayer structure, such as epithelia, the cells can move only in directions within the plane. On the other hand, in the 3D case, cells can also move in out-of-plane directions. Therefore, cells are more likely to move in three dimensions than in two. More quantitative analyses are required to understand the effects of spatial dimensions on cell rearrangements.

From a biological point of view, the migration modes and state transitions found in the simulation may be worth discussing; i.e., cells migrate as single cells, as a cluster, or aligned in a row (Fig. 5 d). Similar modes are observed in cancer metastasis, where cells migrate as single cells or form clusters, alveolar structures, or trabecular structures (the latter corresponding to the aligned migration mode) (35,36). In our results, the migration velocity of the line of cells was higher than that of the cluster (Fig. 5 a), which corresponds to the invasive potential of cancer cells (35,36). These results raise a biological question; i.e., whether collective cell migration in living systems can be explained by interfacial tension.

Author contributions

S.O. and K.S. conceived the project and wrote the manuscript.

Declaration of interests

The authors declare no competing interests.

Editor: Ben O'Shaughnessy.

Supporting material

Supplementary Figure S1. The migratory mode is independent of the initial condition and system box size

(a) Initial conditions; i.e., clustered polar cells in a cubic box with a side of length $L$, scattered polar cells in a cubic box with the same size, and clustered cells in a rectangular box with only the x axis side extended to $2L$. (b) Resulting migratory modes; i.e., single, cluster, and line. (c) Average number of cell-cell contacts, $n_{\text{p}}$, as a function of time. Parameters in (c) were set to $N_{\text{p}}=6$, ${\gamma }_1={\gamma }_0$, ${\gamma }_{\text{p}}=2{\gamma }_0$, and ${\gamma }_{\text{n}}=1.25{\gamma }_0$ for the single-cell migration mode; $N_{\text{p}}=6$, ${\gamma }_1=1.75{\gamma }_0$, ${\gamma }_{\text{p}}=2{\gamma }_0$, and ${\gamma }_{\text{n}}=1.25{\gamma }_0$ for the cluster mode; and $N_{\text{p}}=6$, ${\gamma }_1={\gamma }_0$, ${\gamma }_{\text{p}}=2{\gamma }_0$, and ${\gamma }_{\text{n}}=0.25{\gamma }_0$ for the aligned migration mode.