Apical Junctional Fluctuations Lead to Cell Flow while Maintaining Epithelial Integrity

Abstract

Epithelial sheet integrity is robustly maintained during morphogenesis, which is essential to shape organs and embryos. While maintaining the planar monolayer in three-dimensional space, cells dynamically flow via rearranging their connections between each other. However, little is known about how cells maintain the plane sheet integrity in three-dimensional space and provide cell flow in the in-plane sheet. In this study, using a three-dimensional vertex model, we demonstrate that apical junctional fluctuations allow stable cell rearrangements while ensuring monolayer integrity. In addition to the fluctuations, direction-dependent contraction on the apical cell boundaries, which corresponds to forces from adherens junctions, induces cell flow in a definite direction. We compared the kinematic behaviors of this apical-force-driven cell flow with those of typical cell flow that is driven by forces generated on basal regions and revealed the characteristic differences between them. These differences can be used to distinguish the mechanism of epithelial cell flow observed in experiments, i.e., whether it is apical- or basal-force-driven. Our numerical simulations suggest that cells actively generate fluctuations and use them to regulate both epithelial integrity and plasticity during morphogenesis.

Introduction

Epithelia robustly maintain planar monolayer structures where cells are packed and adhere tightly to each other through adhesive cadherin-mediated contacts. This multicellular integrity protects organ and embryo shapes from external disturbances. This function requires maintaining cell configurations to be planar in epithelial sheets. Nevertheless, epithelia also cause cell flow, i.e., dynamic rearrangement of cellular configurations in the plane of the epithelial sheet (1). Typical examples of cell flow are collective cell intercalations and migrations to place cells in proper regions or to directionally deform tissues during morphogenesis (2, 3, 4, 5, 6, 7, 8, 9). Therefore, it is essential to determine how cells establish both epithelial sheet integrity and cell flow.

Historically, cell flow in epithelia has been studied in the context of morphogenesis by assuming integrity of the cell monolayer (2, 3, 4, 5, 6, 7, 8). This assumption corresponds to the two-dimensional (2D) approximation of cell configuration in the plane of the epithelial sheet. However, actual embryonic tissues are three-dimensional (3D), by which cells move in and out of the epithelial sheet plane. This additional dimension may lead to breaking the sheet structure by forming holes, extruding cells, or piling of cells. Examples of cell movements in the additional dimension are cell delamination and multicellularization, through which cells change their configuration along the normal direction of the epithelial plane (10). Therefore, understanding of the epithelial sheet integrity that underlies cell flow is lacking.

Basement membranes are key components preserving epithelial sheet integrity by continuously aligning cells on the basal side through integrin-based focal adhesions. However, there are also other examples in which epithelia maintain the sheet structure without basement membranes. For example, during the early stages of Drosophila embryogenesis before formation of basement membranes, epithelia cause large deformations such as gastrulation and convergent extension (11). During Drosophila gastrulation, epithelia exhibit dynamic cell movements while maintaining the sheet structure without basement membranes (12, 13, 14). Such dynamic cell movements with a lack of basement membrane are observed in many other situations (15, 16, 17, 18). These examples prompt the question: what are the minimal components required to generate cell flow while robustly maintaining the epithelial sheet integrity in three dimensions in the absence of a basement membrane? We address this question in this study.

Adherens junctions are a universal structure of epithelia, which align along polygonal boundaries between neighboring cells on the apical side. Actomyosin, a complex of actin and myosin, accumulates along these junctions and constitutively generates contractility. In general, the junctional contractility contributes to epithelial rigidity (19, 20, 21). Therefore, the contractile junctional structure might play a role in preserving the epithelial integrity instead of the basement membrane.

Junctional contractility also provides cell flow, i.e., apical actomyosin contracts cell-cell boundaries to rearrange neighboring cell configurations during early Drosophila embryogenesis (5, 18). These actomyosin dynamics involve oscillations (18, 22, 23, 24, 25). For example, myosin distributed on the apical side dynamically oscillates and causes pulsatile contractions of apical cell surfaces (18, 23). Junctional actomyosins generate active fluctuations of junctional lengths, which drive cell rearrangements (7). Moreover, the most recent theoretical study suggested that the active fluctuation driving cell rearrangements governs the stress-relaxation timescale of tissue (8). Thus, we hypothesized that the fluctuation of junctional contractility may be a critical factor that provides both cell flow and epithelial integrity.

To address how cells provide both cell flow and epithelial integrity simultaneously, we focused on a unidirectional flow of cells in the plane of an epithelial sheet, as observed in Drosophila genitalia rotation (6), in which multiple cells migrate around a genitalia tissue in a definite direction while maintaining a monolayer sheet structure. With respect to such studies, some reports have experimentally found directional cell rearrangements in vivo (6) and theoretically suggested that cell flow in the epithelial plane requires anisotropy of junctional contractility (25).

Vertex models have often been used in studies of epithelial cell movements (2, 3, 4, 5, 6, 7, 8). However, these models involving both 2D and 3D models explicitly employ a topological constraint that maintains the cell monolayer configuration. Therefore, these models cannot represent out-of-plane dynamics of cell topology, i.e., collapse of the epithelial monolayer by the formation of holes or piling of cells. To overcome this problem, in this study, we used a full 3D vertex model (26, 27, 28, 29) that describes the total degree of freedom of multicellular topological dynamics in 3D space (30), including collapse of the epithelial monolayer.

In this study, we performed computational simulations using the 3D vertex model and revealed the minimal components that provide both cell flow and epithelial integrity, i.e., the contractility of the apical cell perimeter and fluctuation of junctional contractility. First, by considering the fluctuating actomyosin contractility and planar cell polarity, we generated state diagrams of epithelia to reveal the minimal components that provide both cell flow and epithelial integrity. Second, we addressed the roles of cell adhesions and fluctuations on the basal side. Third, we observed the velocity field in cell sheet and 3D cell shapes, such as apicobasal inclination of individual cells, and compared them with those in cell migrations driven by traction forces via protrusions and retractions on the basal side (31, 32). Lastly, we discuss the fluid characteristics of this cell movement in terms of general collective motions in biological systems.

Materials and Methods

Full 3D vertex model of multicellular dynamics

We focused on a simple system of collective cell migration, in which epithelial cells migrate in a single direction within fixed boundaries while maintaining a monolayer epithelial cell sheet in 3D space (Fig. 1 a). This system simplifies the cell flow observed in Drosophila genitalia rotation (6). The system box is set to be within 0 ≤ x ≤ L_x, −L_y/2 ≤ y ≤ L_y/2, and −∞ ≤ z ≤ ∞, where a monolayer consisting of N cells is aligned on the xy plane under the initial condition. Although the periodic boundary condition is imposed on x = 0 and L_x, the fixed boundary condition is imposed on y = −L_y/2 and L_y/2, where all vertices that comprise the cells located across the boundaries on y = −L_y/2 and L_y/2 are fixed in space. A free boundary condition is imposed on the z-direction, by which cells can move both in and out of the plane of the epithelial sheet to form holes or cell piles. The free boundary condition represents the situation in which cells have little basement membrane on the basal side, as observed in embryogenesis (6, 16, 17, 33).

Figure 1.

3D vertex model of multicellular dynamics. (a) A geometric model of a monolayer epithelial cell sheet in 3D space is shown. Black cells located on y = −L_y/2 and L_y/2 have fixed spatial coordinates, representing the fixed boundary condition. The lower image shows a cross section of the epithelial sheet at y = 0. (b) An enlarged image of a single cell embedded in the epithelial sheet is shown. Edge colors indicate apical (red), lateral (green), and basal (blue) edges. (c) Topological operations between patterns [H] and [I] to express cell rearrangements in the full 3D vertex model are shown (26, 27, 28). Polyhedrons A–E indicate individual cells and extracellular spaces on the apical or basal side. The [H]-to-[I] operation replaces the face (bold triangle in [H]) to an edge (bold line in [I]). This operation makes neighboring polyhedrons A and B apart from each other and the surrounding polyhedrons C–E penetrate between polyhedrons A and B. The [I]-to-[H] operation is vice versa. These operations express changes in 3D cell configurations such as rearranging cells in the in- and out-plane of the epithelial sheet, forming holes through the epithelial sheet, and piling of cells on the epithelial sheet (Fig. S1). (d) A schematic illustration of the constraint force on individual apical cell perimeters (purple) is shown. (e and f) A schematic illustration of directional actomyosin contraction on individual apical edges (orange in e) according to planar cell polarity is shown. The magnitude of this force depends on the angles of individual edges from the x axis, θ_β, in the xy plane, and reaches a maximum when θ_β = θ_ref (θ_ref = π/4). (g) A schematic illustration of fluctuating actomyosin contraction on individual apical edges (green) is shown. To see this figure in color, go online.

To describe the collective migration of epithelial cells in 3D space, we used a full 3D vertex model (26, 34). In this model, individual cells are represented by polyhedrons whose vertices are shared by neighboring cells (Fig. 1 b). Therefore, the shape and configuration of cells in epithelia are described by the locations of vertices comprising cellular polyhedrons and the topological network among vertices (Fig. 1, a and b). Cell movements are expressed by changes in vertex locations. The time evolution of the i-th vertex location, represented by r_i, is given by

$${\eta }_i\left(\frac{d{\mathit{r}}_i}{dt}-{\mathit{v}}_i^{\text{f}}\right)=-\mathit{\nabla }U.$$ (1)

The left side of Eq. 1 is the friction force acting on the i-th vertex (34). Here, η_i is the friction coefficient defined as the summation of the friction coefficients from the surrounding: ${\sum }_{\alpha \left(i\right)}^{\text{cells}}{\eta }_{\text{c}}$, where η_c is the friction coefficient between cells. Vector ${\mathit{v}}_i^{\text{f}}$ is the velocity field defined as the mean velocity of the surrounding: ${\sum }_{\alpha \left(i\right)}^{\text{cells}}{\mathit{v}}_{\alpha }$, where v_α is the velocity of the α-th cell that is defined as the mean velocity of the vertices composing the i-th cell. The right side of Eq. 1 is the mechanical force acting on the i-th vertex derived from the effective energy, 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 these vertex movements, the topological network is dynamically reconnected using [I]-to-[H] and [H]-to-[I] operations (26, 27, 28) (Fig. 1 c); i.e., the operations between patterns [H] and [I] are performed when the related edge lengths become infinitely small to satisfy the energy conservation law during the operations (26). The [H]-[I] operations are different from the well-known T1 transformation used in 2D vertex models. The [H]-[I] operations express changes in 3D cell configurations, e.g., rearranging cells in the in- and out-plane of the epithelial sheet, forming holes through the epithelial sheet, and piling of cells on epithelial sheet (Fig. S1).

As described in Eq. 1, cell movements are driven by effective energy, U, in which we introduce apical actomyosin contractility in addition to basic cell mechanics as follows:

$$U=\sum _{\alpha }^{\text{cells}}\frac{k_{\text{v}}}{2}{\left(\frac{V_{\alpha }}{V_{\text{ref}}}-1\right)}^2+\sum _{\alpha }^{\text{cells}}{\kappa }_{\text{s}}{s}_{\alpha }+\sum _{\alpha }^{\text{apical faces}}\frac{k_{\text{p}}}{2}{\left(\frac{p_{\alpha }}{p_{\text{ref}}}-1\right)}^2+\sum _{\beta }^{\text{apical edges}}{\gamma }_{\beta }{l}_{\beta },$$ (2)

where k_v, κ_s, and k_p are positive constants. The first term in Eq. 2 expresses volume elastic energy, where the volume of the α-th cell, V_α, is conserved to be a preferable volume, V_ref, under the large k_v. The second term in Eq. 2 expresses the cortical tension energy of individual cell surfaces, where s_α is the total surface area of the α-th cell. The third term in Eq. 2 expresses the elastic energy of the apical perimeter of each apical face (Fig. 1 d). Here, the apical perimeter of the α-th cell, p_α, is maintained around a preferable length, $p_{\text{ref}}=2\surd \left(\pi L_x{L}_y/N\right)$. The last term in Eq. 2 expresses the energy of apical actomyosin contractility, which is conserved in cell-cell boundary edges on the apical side, where γ_β is the contraction force acting on the β-th edge on the apical side and l_β is the length of the β-th edge.

Fluctuation and asymmetry of apical actomyosin contractility is introduced via γ_β as follows:

$${\gamma }_{\beta }={\gamma }_{\text{str}}+{\gamma }_{\text{dir}}\text{cos}2\left({\theta }_{\beta }-{\theta }_{\text{ref}}\right)+{\gamma }_{\text{flc}\left(\text{a}\right)}{\xi }_{\beta },$$ (3)

where γ_str, γ_dir, and γ_flc(a) are non-negative constants. The first term in Eq. 3, γ_str, is a steady force of actomyosin contractility on apical edges. Although this constant force can be included as p_ref in the third term of Eq. 2, we define γ_str independently from p_ref to separately address active and passive cell behaviors; i.e., the third term in Eq. 2 reflects a passive force that minimizes the apical perimeter energy in the epithelial monolayer. The fourth term in Eq. 2 reflects active forces, in which positive γ_str generates residual stress to ensure the apical surface is under a tensile condition. The second term in Eq. 3 is a direction-dependent force of actomyosin contractility on apical edges because of the planar polarity (Fig. 1 e), where θ_β is the angle between the positive x axis and the β-th edge projected onto the xy plane (Fig. 1 f). Constant θ_ref is the angle of planar polarity that maximizes γ_β when θ_β = θ_ref. We set θ_ref = π/4 to maximize the magnitude of velocity of in-plane cell flow as reported previously (25). The last term in Eq. 3 is a pulsatile force of actomyosin contractility on apical edges, where ξ_β provides the colored Gaussian noise (Fig. 1 g). Noise ξ_β satisfies $\langle {\xi }_{\beta }\left(t\right)\rangle =0$ and $\langle {\xi }_{\beta }\left(t_1\right){\xi }_{\mathit{\varrho }}\left(t_2\right)\rangle ={\mathit{\delta }}_{\mathrm{\beta \varrho }}\text{exp}\left(-\left|t_1-t_2\right|/\tau \right)$, where δ_βρ is the Kronecker delta and τ is the correlation time of the noise. The colored Gaussian noise has been used to express the fluctuation of junctional contraction to provide cell fluidity and flow in 2D models (6, 7, 25).

Hereafter, using V_ref, η_c, and κ_s, we nondimensionalize all parameters by units: length ($V_{\text{ref}}^{1/3}$), time ($40{\eta }_c{V}_{\text{ref}}^{1/3}/k_{\text{v}}$), and energy (k_v). We summarize all physical parameters for the computational simulations in Table 1. Numerical implementation and calculations are described in Appendix A of the Supporting Materials and Methods.

Table 1.

Physical Parameters of Computational Simulations in the 3D Vertex Model

Symbol	Unit	Value	Description	Equation
Lx	$V_{\text{ref}}^{1/3}$	9.31	system size along the x axis	–
Ly	$V_{\text{ref}}^{1/3}$	10.7	system size along the y axis	–
N	None	100	number of cells in system	–
κs	$k_{\text{v}}/V_{\text{ref}}^{2/3}$	0.0–0.15	cell volume elasticity	2
kp	kv	0.0–0.5	apical cell perimeter elasticity	2
pref	$V_{\text{ref}}^{1/3}$	$2\sqrt{\pi \left(L_{\text{x}}{L}_{\text{y}}/N\right)}$	preferred apical cell perimeter	2
γstr	$k_{\text{v}}/V_{\text{ref}}^{1/3}$	0.0–0.015	steady junctional contraction	3
γdir	$k_{\text{v}}/V_{\text{ref}}^{1/3}$	0.0–0.04	direction-dependent junctional contraction	3
θref	Rad	π/4	reference angle of cell polarity	3
γflc(a)	$k_{\text{v}}/V_{\text{ref}}^{1/3}$	0.0–0.015	fluctuation magnitude of apical junctional contraction	3
γflc(b)	$k_{\text{v}}/V_{\text{ref}}^{1/3}$	0.0–0.015	fluctuation magnitude of basal edge	–
τ	$40{\eta }_c{V}_{\text{ref}}^{1/3}/k_{\text{v}}$	3.00	correlation time of apical or basal fluctuation	–
η(b)	40ηc	0–10	friction coefficient between cells and basement membrane	–
f(b)	$k_{\text{v}}/V_{\text{ref}}^{1/3}$	0.006	traction force by basal protrusion	–

Results

Apical junctional fluctuation provides cell flow while maintaining epithelial integrity

To clarify the minimal components ensuring cell flow and epithelial sheet integrity, we performed numerical integrations of Eq. 1 and obtained state diagrams of epithelia with respect to parameters k_p, γ_dir, and γ_flc(a), as shown in Fig. 2, a and b. We classified physical states of the epithelial sheet by flow, nonflow, and collapsed states. Although a monolayer sheet structure was maintained in flow and nonflow states, it was not in the collapsed state because holes were formed or cells piled up. Flow and nonflow states were distinguished by observing the average magnitude of cell velocity along the x axis, represented by v, in the epithelial sheet plane, flow state (v > v_th), and nonflow state (v < v_th), where v_th is a threshold. The threshold v_th was set as v_th = 2k_v/(40η_c), which corresponds to the velocity that all cells move for the distance of about a single cell’s size during the entire simulation time period, corresponding to a few days in real time (see Appendix A in the Supporting Materials and Methods).

Figure 2.

Epithelial cell dynamics driven by apical actomyosin contractions in 3D space. (a and b) State diagrams of an epithelial sheet with respect to perimeter elasticity, k_p, apical fluctuation, γ_flc(a), and directional contraction, γ_dir, are shown. Colors indicate states of epithelia: flow (red), nonflow (blue), and collapse (blank), in which flow and nonflow states are distinguished by the average magnitude of cell velocity, v. (c) Time-incremental images in the flow state (i) and collapsed state with piling of cells (ii) and a collapsed state with holes (iii) are shown. In (c), bottom images show cross sections of cell sheets at the dashed lines in the top images. In the bottom images, edge colors indicate apical (red), lateral (green), and basal (blue) edges. These dynamics are also shown in Videos S1, S2, and S3. (d) Average magnitude of cell velocity, v, is shown as a function of perimeter elasticity, k_p. (e) Cell motility μ (= v/γ_dir) as a function of diffusion coefficient D. v is the average velocity over all cells when γ_dir = 0.005. D is the diffusion coefficient of cells in the x axis, defined as D = x²/2t where γ_dir = 0. v and D vary depending on γ_flc(a) in Eq. 3. The motility μ is positively related to D, which is similar to the fluctuation-dissipation relationship held in equilibrium, although this system is in nonequilibrium. In the simulations, we set κ_s = 0.02 and γ_str = 0.05. To see this figure in color, go online.

Epithelia exhibited flow (red), nonflow (blue), and collapsed (blank) states as shown in Fig. 2, a and b. In both flow (red) and nonflow (blue) states, cells maintained the monolayer structure. Specifically, in the flow state (red), cells provided directional flow while maintaining integrity, as shown in Fig. 2 c(i) and Video S1. Even in the nonflow state (blue), cells occasionally underwent nondirectional movements with frequent rearrangements. In the collapsed state (blank), epithelia formed holes or piled cells, as shown in Fig. 2 c(ii) and (iii) and Videos S2 and S3. The state diagrams shown in Fig. 2, a and b indicated that the steady actomyosin contractions of apical cell perimeters, represented by k_p, provided the epithelial sheet integrity in both flow and nonflow states. Based on the robust structure, the pulsatile actomyosin contractions of apical cell-cell junctions, represented by γ_flc(a), provided nondirectional cell movement via rearranging the in-plane cell configuration. In addition to the integrity and nondirectional cell movement, the planar cell polarity, represented by γ_dir, provided the directional cell flow.

The flow state emerged under conditions of both high γ_dir and low γ_flc(a) and high γ_flc(a) and low γ_dir (Fig. 2 b), suggesting that γ_flc(a) complements the driving force to produce the flow when γ_dir is not sufficient and vice versa. Thus, contractions γ_dir and γ_flc(a) complemented each other to produce the flow. The flow state with high γ_dir occupied a narrow region in the state diagram to easily transit to a nonflow or collapsed state when γ_dir and γ_flc(a) were changed. In contrast, the flow state with high γ_flc(a) occupied a wide region to be robust against γ_dir and γ_flc(a). This indicates that the combination of frequent fluctuation and slight polarity of apical actomyosin contractility stably provide both cell flow and epithelial sheet integrity.

Specifically, in the flow state, the average magnitude of cell velocity reached the peak around k_p = 0.2 k_v and decreased with increasing k_p, as shown in Fig. 2 d. However, when reducing k_p, the epithelial sheet integrity collapsed. Thus, although providing cell flow requires deformability of the cell shape, too much deformability abolished the epithelial sheet integrity. In addition, as shown in Fig. 2 d, the average magnitude of cell velocity increased with γ_flc(a), indicating that cell-intrinsic active fluctuations facilitate cell flow.

A previous report using a 2D vertex model with a perimeter energy function similar to the third term in Eq. 2 suggested that the physical property of cells transits from solid to fluid phases depending on the preferred perimeter, but not the elastic modulus of the energy function (35). Consistently, in our 3D model, the transition depended on both the junctional contraction γ_str and cortical tension κ_s (Fig. S2, a and b), which effectively reduced the preferred perimeter. However, it was apparently inconsistent that the transition depended on the elastic modulus of perimeter energy k_p (Fig. 1 a). This may be because of the three-dimensionality of cell movements in our system, through which cells can move in both in- and out-plane directions of the epithelial sheet. The out-of-plane cell movements increased p_α compared with the in-plane cell movements and increased the probability of the condition of p_α > p_ref. Under this condition, increasing k_p led to increasing tension, similarly to reducing the preferred perimeter p_ref.

Moreover, the transition from nonflow to flow states required a finite value of γ_flc(a) in this study (Fig. 1, a and b). This result also differed from a previous report (8) in which the fluidity of cells is induced even by an arbitrarily small amount of fluctuations. This may be because of the difference in the implementation of fluctuations; i.e., in the previous study, they introduced fluctuation as the frequency of T1 transformations by which an arbitrary small amount of fluctuations certainly causes cell rearrangements, leading to cell flow to some extent. However, we introduced fluctuation as the fluctuation of junctional contraction by which the fluctuation of contraction did not necessarily cause cell rearrangements. Although the difference could also be caused by the finite time period of the simulations, in this study, cell rearrangements hardly occurred in the nonflow state during the whole time period of the simulation (corresponding to a few days in real time (see Appendix A in Supporting Materials and Methods)).

These results suggest that cell motility may be relevant to junctional fluctuations. To investigate the relationship between cell motility and junctional fluctuations, we analyzed mobility μ, defined as μ = v/γ_dir, with respect to the diffusion constant D in the x direction (Fig. 2 e). Here, v is the average magnitude of cell velocity in the x axis under the condition with small constant γ_dir (= 0.005). μ can be approximately regarded as $dv/d{\gamma }_{\text{dir}}{|}_{{\gamma }_{\text{dir}}=0}$ because γ_dir is small (Fig. S2 c). D is defined as D = <x²>/2t, where brackets indicate the average of the quantity over all cells under the condition with γ_dir = 0. Numerical simulations revealed that both v and D increased with γ_flc(a), indicating that μ and D have a positive correlation (Fig. 2 e). This is because cell velocity depends on the frequency of cell rearrangements. In epithelial sheets, cell movements are limited by the surrounding cells in the packing geometry and are caused by rearranging cell configuration. The previous (8) and our studies indicate that cell rearrangements can be caused by junctional fluctuation. Therefore, the frequency of cell rearrangements reflects the diffusion coefficient of cells in equilibrium and the response of v to additional forces such as γ_dir. In this manner, our system exhibited an almost linear relationship between μ and D, similar to the fluctuation-dissipation relationship held in equilibrium, such as the μ-D relationship of a Brownian particle, although our system was in nonequilibrium (36, 37).

Roles of cell adhesions and fluctuations on the basal side in apical-force-induced flow

We demonstrated that the force regulations on the apical side ensure that cell flow and epithelial integrity coexist (Fig. 2). We next focused on the flow induced by apical force regulation and determined the roles of cell activities on the basal side. Recent studies indicate the importance of such basal cell activities for cell motility (38, 39, 40). Typical examples are the convergent extension of nematodes and frogs, to which cell protrusions on the basal side contribute (38, 39).

First, to investigate the roles of integrin-based cell adhesions in basement membranes, we added friction between the cells and substrate on the basal side as shown in Fig. 3 a. The friction between cells and basement membrane was introduced via the left term in Eq. 1, according to our previous study (34). In this term, the friction coefficient η_i is redefined as ${\sum }_{\alpha \left(i\right)}^{\text{cells}}{\eta }_c+{\mathit{\delta }}_{i\left(\text{b}\right)}{\eta }_{\left(\text{b}\right)}$, and velocity field ${\mathit{v}}_i^{\text{f}}$ is redefined as ${\sum }_{\alpha \left(i\right)}^{\text{cells}}{\eta }_c{\mathit{v}}_{\alpha }/\left({\sum }_{\alpha \left(i\right)}^{\text{cells}}{\eta }_c+{\mathit{\delta }}_{i\left(\text{b}\right)}{\eta }_{\left(\text{b}\right)}\right)$. Here, δ_i(b) is the topological indicator that is 1 when the i-th vertex is involved in the basal surface and is 0 otherwise. Constant η_(b) is the friction coefficient between the cells and basement membrane.

Figure 3.

Roles of cell activities on the basal side in cell flow driven by apical actomyosin contractions. (a) A schematic illustration of the basal friction driven by cell adhesions on basement membranes is shown. (b) Average magnitude of cell velocity, v, is shown as a function of the friction coefficient on basal vertices, η_(b). Solid line indicates a fitting curve of Eq. 4 with c₁ = 2.96 × 10⁻³ and c₂ = 0.102. (c) A schematic illustration of the basal fluctuation driven by cell protrusions on the basal side is shown. (d and e) State diagrams of epithelia with respect to the directional contraction on apical edges, γ_dir, apical fluctuation, γ_flc(a), and basal fluctuations, γ_flc(b), are shown. Colors indicate the states of epithelia: flow (red), nonflow (blue), and collapsed (blank), in which flow and nonflow states are distinguished by the average magnitude of cell velocity, v. (f and g) The average magnitude of cell velocities, v, is shown as functions of apical and basal fluctuations, γ_flc(a) and γ_flc(b). In the simulations, we set κ_s = 0.02, k_p = 0.2, γ_str = 0.05, and γ_flc(a) = 0.008 (in the case of b). To see this figure in color, go online.

By performing numerical integrations of Eq. 1, we obtained the average magnitude of cell velocity, represented by v (Fig. 3 b). In Fig. 3 b, we found that v decreased with η_(b) as expected. The dependence of v on η_(b) was well-fitted by

$$v\left({\eta }_{\left(\text{b}\right)}\right)=\frac{c_1}{c_2+{\eta }_{\left(\text{b}\right)}},$$ (4)

where c₁ and c₂ are fitting parameters, whose values are 2.96 × 10⁻⁴ and 0.102, respectively.

In this system, γ_dir induced directed cell movements, which is symbolically referred to as ${\hat{f}}_{dir}$. This force was balanced by two types of friction forces: friction between cells and friction between the cells and substrate. The friction between cells can be simply represented by constant ${\hat{\eta }}_{\left(\text{cc}\right)}$. Moreover, because the friction between the cells and substrate was set to be proportional to η_(b), it could be represented by ${\hat{c}}_3{\eta }_{\left(\text{b}\right)}$ where ${\hat{c}}_3$ is constant. This balance gives Eq. 4, where $c_1={\hat{f}}_{\text{dir}}/{\hat{c}}_3$ and $c_2={\hat{\eta }}_{\left(\text{cc}\right)}/{\hat{c}}_3$, which fit the results well (Fig. 3 b). This characteristic response of cell movements to external friction indicated that γ_dir generated the directionality of cell movements, even though γ_dir was balanced within individual edges to satisfy the action-reaction law as described in Eq. 2.

Second, to investigate the roles of force fluctuations in the basal side such as protrusions, we simply introduced isotropic random forces along the cellular edges (Fig. 3 c). This force was added as the term ${\sum }_{\beta }^{\text{basal}\text{edges}}{\gamma }_{\text{flc}\left(\text{b}\right)}{\xi }_{\beta }{l}_{\beta }$ to U in Eq. 2, where γ_flc(b) is a non-negative constant, l_β is the length of the β-th edge on the basal side, and ξ_β is colored Gaussian noise that has the same statistical property as those of the apical actomyosin contraction.

To compare the roles of fluctuations in apical and basal sides, we constructed state diagrams of epithelia with respect to γ_flc(a), γ_flc(b), and γ_dir (Fig. 3, d and e). As shown in Fig. 3 d, the flow state (red) was much more sensitive to γ_flc(b) than to γ_flc(a). Moreover, as shown in Fig. 3, e and f, although the average magnitude of cell velocity increased with γ_flc(a), it relatively did not increase with γ_flc(b). Rather, it decreased with γ_flc(a) when γ_flc(b) was sufficiently high (Fig. 3 g). Hence, force fluctuations on the basal side, such as those in protrusion movements, also induced flow, but their efficiency was lower than that of apical fluctuations. These results indicate that the specifically localized fluctuation on the apical side is important to ensure that cell flow and epithelial integrity coexist.

Apical actomyosin fluctuation leads to directional cell flow

Although all forces described in Eq. 2 were balanced within individual cells, the forces produced a directionality of cell flow. This is because of the energy generation in the direction-dependent contractile force described by the second term, γ_dircos2(θ_β − θ_ref), in Eq. 3. The direction-dependent contraction shrank the cell-cell boundary edge along the polar direction θ_ref (Fig. 4 a). This shrinkage of the boundary edge generated pushing forces on the two neighboring cells in the normal direction to θ_ref. Although one side of the pushing force was balanced with the reaction force on the fixed boundary of the system, the other side was dissipated by moving cells. Thus, cell movements were not induced directly by the force generated by individual cells, but indirectly by the mechanical interactions among multiple cells.

Figure 4.

Mechanism of epithelial cell rearrangements driven by apical actomyosin contraction in three dimensions. (a) Force balance in the epithelial plane is shown. Directional actomyosin contraction shrinks edges along the polar direction (green arrows). This shrinkage causes a pushing force on cells along the normal direction (black arrows). One side of the pushing force is balanced by the reaction force on the fixed boundary (blue arrow). Therefore, cells are pushed in the opposite direction. (b) 3D cell rearrangement process is shown. Apical edge 1–2 between cells B and D is replaced with a new face 1′2′3′ between cells A and E via [I]-to-[H] transition. Then, the surface 1′2′3′ expands to the basal surface. After reaching the basal surface, the boundary face between cells B and D disappears via [H]-to-[I] transition. This process is also shown in Video S4. (c and d) Frequencies of individual cell rearrangement processes for apical and basal fluctuations, respectively, are shown. Edge color illustrates apical (red), lateral (green), and basal (blue) edges. Arrows indicate individual processes of cell rearrangements. To see this figure in color, go online.

Next, we determined the role of fluctuation in the unidirectional flow of cells in three dimensions. To this end, we observed the topological process of cell rearrangement in the flow state, resulting from the computational simulations of the case with apical fluctuation. Here, we focused on the movement of an edge and the surrounding four cells, where the edge bound two neighboring cells and connected two separated cells at its ends (Fig. 4 b; Video S4). When aligning along θ_ref, this edge gradually shrank its length and was then transformed to a new face that bound the separated cells on the apical side by the [I]-to-[H] transition. This new face gradually spread from apical to basal sides and then caused the [H]-to-[I] transition on the basal side. Eventually, the two separated cells completely attached to each other on both apical and basal sides.

During cell flows in our computational simulations, we found several patterns of the topological transition process. In some cases, cell rearrangements started from the apical side but stopped before reaching the basal side and reversed to the apical side. Similarly, cell rearrangements also started from the basal side. We next calculated the frequency of these processes in cases with apical and basal fluctuations and found that cell rearrangements tended to start from the side that involved the fluctuation (Fig. 4, c and d). These results suggest that fluctuations play a role in triggering cell rearrangements.

Comparison of apical and basal-force-induced cell flows

To understand the fluid characteristics of apical-force-driven cell flow, we compared them with those driven by basal traction force in which individual cells migrated by protrusions and retractions on the basal side (31, 32). To recapitulate basal-force-driven cell flow, instead of direction-dependent actomyosin contractility (γ_dir = 0 in Eq. 3), we introduced traction force of individual cells on the basal side, represented by $-f_{\left(\text{b}\right)}{\sum }_{\alpha }^{\text{cells}}\left({\mathit{c}}_{\alpha \left(\text{b}\right)}\cdot {\mathit{e}}_x\right)$, to the potential energy U in Eq. 2 (Fig. 5 a). Here, constant f_(b) represents the strength of the traction force, e_x is a unit vector along the x axis, and c_α(b) is the position vector of the center of the basal surface of the α-th cell. Under these conditions, cells provide unidirectional flow along the x axis (see Video S5).

Figure 5.

Comparison of cell dynamics between apical and basal force-driven cell flows. (a) Schematic illustrations of apical- (left) and basal- (right) force-driven cell flows are shown. The dynamics of basal-force-driven flow is shown in Video S5. (b) Cell velocity profiles of apical- (red) and basal- (blue) force-driven flows are shown. Each velocity field is normalized to the maximal velocity of each case. Bars indicate SDs. (c) Histograms of cell inclination for apical- (red) and basal- (blue) force-driven flows. The cell inclination is defined as the x-component of the distance from the basal to apical face centers of individual cells. In the simulations, we set κ_s = 0.02, k_p = 0.2, γ_str = 0.05, γ_flc(a) = 0.008, γ_dir = 0.02 (in the case of apical-force-driven), and f_(b) = 0.006 (in the case of basal-force-driven). To see this figure in color, go online.

Although apical-force-driven cell flow was indirectly driven by multicellular interactions as described in Fig. 4 a, the basal-force-driven cell flow was driven by force generated by individual cells. This mechanistic difference may cause differences in macroscopic flow. To understand the difference in fluid characteristics, we compared the velocity profiles in the epithelial sheet plane between apical and basal-force-driven cell flows (Fig. 5 b). Although the profile observed in the basal-force-driven flow was parabolic, which is similar to that of the Hagen-Poiseuille flow, the profile driven by the apical-force-driven flow followed a piecewise linear function. To clarify the mechanisms of providing these different velocity profiles, we performed theoretical analyses based on continuum mechanics (see Appendix B in the Supporting Materials and Methods for details).

As a result, in the case of basal-force-driven cell flow, each cell bears friction forces from both substrates and adjacent cells. Because these friction forces were proportional to the relative velocity of individual cells, the friction forces exerted strong effects on cell movements, such as those in a Newtonian fluid (41). In this harsh environment, the traction force acted as a pressure gradient on cells to generate the Hagen-Poiseuille flow. Therefore, the velocity profile tended to be parabolic as follows:

$$v_x\left(y\right)\mathit{\propto }-\left(y+\frac{L_y}{2}\right)\left(y-\frac{L_y}{2}\right).$$ (5)

In the case of apical-force-driven cell flow, the direction-dependent force γ_dircos2(θ_β − θ_ref) in Eq. 3 was the driving force of cell flow, which actively generated shear stress in the epithelial sheet plane. Because the shear stress was negative in y < 0 and positive in y > 0, the total stress in the system lost balance to provide the directional flow. Moreover, because the shear stress was homogeneous in each domain of the sheet (y < 0 and y > 0), the velocity profile became piecewise linear as follows:

$$v_x\left(y\right)\mathit{\propto }\{\begin{array}{c}\begin{array}{cc}-y+\frac{L_y}{2}& \text{for}y>0\end{array}\\ \begin{array}{cc}y+\frac{L_y}{2}& \text{for}y<0\end{array}\end{array}.$$ (6)

This tendency was independent of η_(b) and γ_flc(a).

Lastly, we observed cell shapes and found a remarkable difference in cell inclinations of the epithelial sheet plane (Fig. 5 c). In the case of basal-force-driven cell flow, cells inclined as their basal side forewent. In the case of apical-force-driven cell flow, cells maintained upright alignments relative to the epithelial sheet plane. These tendencies were preserved even when cells strongly adhered to substrate (high η_(b)), as shown in Fig. S3. Because regular cell alignments preserved the sealing function of the epithelial sheet, the mechanism of apical-force-driven cell flow may also contribute to the functional robustness of the epithelial sheet.

Discussion

The flow of epithelial cells has been well studied. However, most studies have been implicitly based on the assumption that the epithelial sheet has a robust structure. For example, several studies using mathematical models suggest that the direction-dependent contraction according to planar cell polarity is important to produce cell flow (6, 33, 34), with a study suggesting that fluctuation is unnecessary, at least under the 2D approximation (25). Although the importance of direction-dependent contraction is certain, these studies employed 2D or 3D models by ignoring the possibility that the epithelial structure collapses. In these models, the contraction can be sufficiently high to produce cell flow because of the fixed structure of the epithelial sheet. In 3D space, however, the highly direction-dependent contraction probably collapses the cell monolayer sheet structure of the epithelium. In this study, we overcame this problem by developing a full 3D vertex model that allows collapsing the epithelial sheet structure by forming holes or piling cells. Remarkably, this modeling led to finding a new, to our knowledge, mechanism that establishes both plasticity and integrity of the epithelial sheet.

Our main finding is that apical junctional fluctuation leads to cell flow while robustly maintaining the epithelial monolayer structure in 3D space (Fig. 2 a). In addition, apical actomyosin fluctuation is sufficient to produce cell flow (Fig. 3 c), and this fluctuation plays a role in triggering cell rearrangements (Fig. 4 c). In addition to fluctuation, apical direction-dependent contraction according to planar cell polarity produces directionality of the cell flow (Fig. 2 b). Moreover, in the cell flow driven by apical actomyosin fluctuation, cell adhesions on basement membranes and the fluctuation force generated via propulsions play roles in both suppressing and enhancing cell flow (Fig. 3, b and d).

Both fluctuation and direction-dependent contraction often coexist in real situations, such as in convergent extension and the genitalia rotation of Drosophila embryogenesis (18, 25). Our study suggests that fluctuation and direction-dependent contraction complement each other to produce cell flow and play a key role in switching physical states of epithelia between flow and nonflow (Fig. 2 b). This complementarity is established because both fluctuation and direction-dependent contraction tend to shrink the lengths of cell-cell boundary edges to trigger cell rearrangements. Specifically, direction-dependent contraction directionally shrinks the lengths of edges to directionally cause cell rearrangements. Thus, both fluctuation and direction-dependent contraction may be employed to regulate the physical state of the epithelial sheet.

Cell flow emerges under conditions of both high bias and low fluctuation and high fluctuation and low bias (Fig. 2 b). The condition with high fluctuation produces cell flow more robustly than that with high bias. In fact, conditions with high fluctuation correspond to several real situations such as in Drosophila embryogenesis and mouse neural tube closure. Under these conditions, although actomyosin distributions along adherens junctions are certainly biased depending on junctional angles, the magnitude of the bias is slight (less than four times the intensity) (42, 43). Thus, the developmental processes may be optimized to maintain the epithelial sheet integrity, i.e., cells actively generate the fluctuation and use them stably to provide both cell flow and epithelial integrity.

Many studies have reported that the flow of epithelial cells is driven by basal traction forces (31, 32). We also investigated the difference in cell movements between apical or basal-force-driven cell flows. Although the difference in their molecular mechanisms is well known, we revealed the difference in their fluid characteristics and mechanics (Figs. 4 and 5). We found that the functional robustness of the epithelial sheet is also maintained in the apical-force-driven cell flow (Fig. 5 c). Therefore, the mechanism of apical-force-driven cell flow may have an advantage to preserve both the structural and functional robustness of the epithelial sheet.

From a general perspective, conventional studies of cell flow have focused on cell movements driven by traction forces on the basal side via integrin-based adhesions. These studies have followed collective motions in biological systems, such as the movements of birds, fish, and ants (44). From a physical viewpoint, the mechanism of the basal-force-driven cell flow is analogous to those of the conventional systems. Remarkably, in contrast, the mechanism of apical-force-driven cell flow distinctly differs from those of the conventional systems: although movements in the conventional systems are directly driven by the autonomy of individuals, the apical-force-driven cell flow in this study indirectly results from multibody effects. Therefore, this study reveals the peculiar mechanism of cell motion establishing both the plasticity and integrity of the epithelial sheet, broadening the general understanding of collective motions in biological systems.

Author Contributions

S.O. and K.S. performed computational simulations. S.O., K.S., and E.K. wrote the manuscript.

Editor: Stanislav Shvartsman.