Size-dependent response of cells in epithelial tissue modulated by contractile stress fibers

Abstract

Although cells with distinct apical areas have been widely observed in epithelial tissues, how the size of cells affects their behavior during tissue deformation and morphogenesis as well as key physical factors modulating such influence remains elusive. Here, we showed that the elongation of cells within the monolayer under anisotropic biaxial stretching increases with their size because the strain released by local cell rearrangement (i.e., T1 transition) is more significant for small cells that possess higher contractility. On the other hand, by incorporating the nucleation, peeling, merging, and breakage dynamics of subcellular stress fibers into classical vertex formulation, we found that stress fibers with orientations predominantly aligned with the main stretching direction will be formed at tricellular junctions, in good agreement with recent experiments. The contractile forces generated by stress fibers help cells to resist imposed stretching, reduce the occurrence of T1 transitions, and, consequently, modulate their size-dependent elongation. Our findings demonstrate that epithelial cells could utilize their size and internal structure to regulate their physical and related biological behaviors. The theoretical framework proposed here can also be extended to investigate the roles of cell geometry and intracellular contraction in processes such as collective cell migration and embryo development.

Significance

As the outmost layer of most organs, epithelial tissues consisting of cells with different sizes need to react to ubiquitous mechanical stimuli constantly during embryo development and tissue morphogenesis. However, it remains unclear whether and how the response of cells is affected by their size as well as key physical factors modulating such influence. In this work, we developed a stress fiber-reinforced vertex model to answer these outstanding questions. We showed that the imposed anisotropic biaxial stretching makes the occurrence of cell rearrangement within the epithelial monolayer easier, which then leads to the size-dependent elongation of cells. At the same time, stress fibers aligning predominantly with the main stretching direction will be formed in cells, help them to resist the imposed stretching, and ultimately reduce the size dependency of their deformation. In addition to greatly enhancing our basic understanding of how epithelial cells perform different biological duties, the theoretical framework proposed here could also serve as a useful tool in investigating the roles of cell geometry and intracellular contraction in processes such as collective cell migration and organ formation.

Introduction

The densely packed epithelial tissue plays critical roles in wound healing (1,2), tissue morphogenesis (3,4), and embryo development (5,6) (Fig. 1 A). During these processes, epithelial cells within the tissue actively generate mechanical forces and undergo significant morphology changes, including buckling (3,7), shrinkage (4), and elongation (5). Interestingly, the deformability of cells was shown to be essential for the proper formation of tissues and organs. For instance, the shrinkage failure of seam cells leads to abnormal development of Caenorhabditis elegans embryos (8,9). Mechanistically, the presence of forces was found to activate a number of signaling (such as Hippo/YAP) pathways, trigger the remodeling of adherens junctions, and eventually regulate the dynamics of the epithelial monolayer (10). Recently, by adopting advanced techniques such as laser ablation (5,11), researchers also experimentally examined how alterations in the cytoskeleton (and therefore the apparent cell properties) affect the response of cells to mechanical stimuli (12). For example, the formation and evolution of intracellular actin stress fibers have been found to drive the body-axis elongation of C. elegans embryos (9) as well as to control the shape and collective migration of cells in the monolayer (13).

Figure 1.

(A) Cartoon showing different stages of Drosophila development. After pupa formation, the Drosophila pupal dorsal thorax epithelium (blue region), a monolayer consisting of cells with different sizes, will elongate under anisotropic biaxial stretching. (B) A stress fiber-reinforced vertex model was developed to describe the deformation dynamics of the epithelial monolayer. Gray arrows: anisotropic biaxial stretching forces $F_x$ and $F_y$. Blue: cell-cell contact edges. Green: stress fibers. (C) A cartoon of vertex model showing that the energy penalty $U_{cell}$ is attributed to the change of cell area $A_J$, actomyosin (green and red) contraction, and its competition with cell-cell adhesion (orange). ${\mathit{r}}_i$ refers to the location vector of i-th vertex in cell J. (D) Schematic diagrams illustrating the nucleation, peeling, merging, and breakage of stress fibers. $\theta$, opening angle of adjacent cell edges; $\phi$, angle between the opening angle bisector and the minor axis of stretching; $\alpha$, angle formed between a stress fiber and the cell edge; $v$, peeling velocity of the tip of stress fiber. To see this figure in color, go online.

Various theoretical formulations at different scales, ranging from lattice based (14) to continuum level (15,16), have been proposed to describe the dynamics of multicellular systems. Among these, the so-called vertex model (17,18), originating from the Plateau model for dry foams (19), has become increasingly popular because of its capability in capturing a number of epithelial features, including tissue rigidity transition (20), epithelial folding (21), and coordinated cell movement (22). Essentially, cells are represented by polygons with straight cell-cell contact edges between them in the original vertex model. However, more important features such as chemo-mechanical feedbacks of contractility (23,24) and intercellular adhesion (25), cell polarization (22,26), and cell-cell detachment (27) have been incorporated into this formulation to make it suitable for examining phenomena like oscillatory morphodynamics, solid-to-liquid jamming transition, and fracture in the epithelial monolayer. One of the key findings from these aforementioned studies is that the collective behavior of epithelial cells is largely controlled by a single geometrical parameter—the target shape index (20,26), characterizing the capability of cells to rearrange themselves by altering their shapes. Despite these efforts, several important issues remain unsettled. First of all, most existing works were conducted under the premise that cells in the monolayer assume the same preferred area. In reality, cell size can vary by a factor of 7–8 in the same epithelial tissue (5). However, the fundamental question of how such a wide size distribution of cells influence their individual and collective behaviors remains unclear. In addition, as mentioned earlier, mounting evidence has convincingly demonstrated that the formation and evolution of subcellular stress fibers significantly affect the deformation and internal dynamics of tissues such as Drosophila pupal dorsal thorax’s epithelium (Fig. 1, A and B). Unfortunately, a theoretical framework allowing us to incorporate these factors into the simulation of cell monolayers is still lacking.

Aiming to address these outstanding issues, we developed a stress fiber-reinforced (SFR) vertex model where the nucleation, peeling, merging, and disassembling of stress fibers in individual cells were all considered. In addition, a distribution of the preferred area of cells (mimicking that observed in actual experiments) was also introduced to make the model more realistic. We then used this framework to examine the size-dependent response of cells in the monolayer under anisotropic biaxial stretching, to reveal the physical mechanisms behind, and to probe the role of stress fibers during this process.

Materials and methods

SFR vertex model

Following the classical vertex model, we represent the confluent tissue by tightly assembled polygons that are defined by a set of vertices (with position vectors ${\mathit{r}}_{\mathrm{i}}$, Fig. 1 C). In addition, stress fibers are allowed to be formed in cells in our model. Basically, the length and orientation of each stress fiber are determined by its two endpoints, located at different cell edges (Fig. 1 D). For an epithelial monolayer containing $N$ cells and $M$ stress fibers, the system energy can be written as

$$U=\sum _{J=1}^N\left[\frac{1}{2}K{\left(A_J-A_{0,J}\right)}^2+\frac{1}{2}\mathrm{\Gamma }{P_J}^2+\mathrm{\Lambda }{P}_J\right]+\sum _{k=1}^M\frac{1}{2}{\gamma }_k{\left(l_k-l_{0,k}\right)}^2\text{.}$$ (Equation 1)

Here, the first summation corresponds to the conventional energy of cells, i.e., $U_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}$, where the energy penalty due to area change implies that the cell wants to maintain its volume and height (26); the second term is attributed to actomyosin contractility; and the last one reflects the competition between cell-cell adhesion and cell edge contraction (28) (see Fig. 1 C). $K$, $\mathrm{\Gamma }$, and $\mathrm{\Lambda }$ represent the so-called areal modulus, contractile stiffness, and line tension of the cell, respectively. $A_J$ and $P_J$ are actual area and perimeter of the $J$-th cell, respectively, while $A_{0,J}$ refers to its preferred area. The second summation in Eq. 1 represents the energy stored in stress fibers, i.e., $U_{\mathrm{f}\mathrm{i}\mathrm{b}\mathrm{e}\mathrm{r}}$, where $l_k$ and $l_{0,k}$ are the current and rest length of the $k$-th fiber treated as a linear spring with a spring constant of ${\gamma }_k$. Once $U$ is known, the motion of vertex $i$ can then be determined by $\eta \frac{d{\mathit{r}}_i}{dt}=-{\nabla }_{{\mathit{r}}_i}U$, with $\eta$ being the viscous drag coefficient associated with the movement of vertices.

Although the precise relationship between the size of a cell and its properties remains largely unclear, recent reports showed that intracellular contraction apparently varies with the cell area (29,30), i.e., more assembly of F-actin at intercellular faces results in stronger contraction, eventually leading to a smaller cell area. Based on this evidence, we proceed by assuming that the contractile stiffness $\mathrm{\Gamma }$ of a cell is inversely proportional to its preferred area $A_0$, whereas the target shape index $p_0$ (defined as $p_0=-P_{0,J}/\sqrt{A_{0,J}}$, with $P_{0,J}=-\mathrm{\Lambda }/{\mathrm{\Gamma }}_J$ being the preferred perimeter of the J-th cell) is fixed among all cells. Therefore, the contractile stiffness of the $J$-th cell takes the form ${\mathrm{\Gamma }}_J=-\mathrm{\Lambda }/p_0\sqrt{A_{0,J}}$, where $\mathrm{\Lambda }$, $p_0$, and $A_{0,J}$ constitute the three independent parameters in our model that fully characterize the response of cells. Normalization of the problem with an energy scale $K{A}_{0,\mathit{min}}^2$, a length scale $\sqrt{A_{0,\mathit{min}}}$, and a timescale $\eta /K{A}_{0,\mathit{min}}$, with $A_{0,\mathit{min}}$ being the minimum target area of cells in the monolayer, leads to (recalling that ${\gamma }_k={\mathrm{\Gamma }}_J{P}_{0,J}/l_{0,k}$ if the $k$-th stress fiber is formed in cell $J$, where the contractile stiffnesses is taken to be inversely proportional to the rest length of the fiber, similar to springs connected in series)

$$\overline{U}=\sum _{J=1}^N\left[\frac{1}{2}{\left({\overline{A}}_J-{\overline{A}}_{0,J}\right)}^2-\frac{1}{2}\frac{\overline{\mathrm{\Lambda }}}{p_0\sqrt{{\overline{A}}_{0,J}}}{{\overline{P}}_J}^2+\overline{\mathrm{\Lambda }}{\overline{P}}_J\right]+\sum _{k=1}^M\frac{1}{2}{\left(-\frac{\overline{\mathrm{\Lambda }}}{{\overline{l}}_{0,k}}\right)\left({\overline{l}}_k-{\overline{l}}_{0,k}\right)}^2\text{,}$$ (Equation 2)

where ${\overline{A}}_J=A_J/A_{0,\mathit{min}}$, ${\overline{A}}_{0,J}=A_{0,J}/A_{0,\mathit{min}}$, ${\overline{P}}_J=P_J/\sqrt{A_{0,\mathit{min}}}$, $\overline{\mathrm{\Lambda }}=\mathrm{\Lambda }/K{A_{0,\mathit{min}}}^{\frac{3}{2}}$, ${\overline{l}}_k=l_k/\sqrt{A_{0,\mathit{min}}}$, and ${\overline{l}}_{0,k}=l_{0,k}/\sqrt{A_{0,\mathit{min}}}$. Here, for simplicity, the overbar symbols will be neglected in the remaining text.

Next, we shift our attention to stress fibers whose formation can be triggered by force-assisted actin polymerization (31,32). Interestingly, recent experiments showed that the cortical layer formed at cell periphery can detach from the cell edge and transform into stress fibers (5). Furthermore, it has been found that such nucleation/initiation of stress fibers predominantly takes place at tricellular junctions (TCJs) when the tissue is under stretching. If we assume that the cortical layer adheres perfectly to the kinked cell edges around the TCJ (with opening angle $\theta$, refer to Fig. 1 D), then nucleation of stress fibers is expected to occur when detachment (from cell edges) reduces the energy stored in the stressed cortical layer, that is,

$$-\mathrm{\Lambda }\left[\frac{1}{2}\frac{P_J}{P_{0,J}}\left(1+\mathit{sin}\frac{\theta }{2}\right)-1\right]\left(1-\mathit{sin}\frac{\theta }{2}\right)>E_a\text{,}$$ (Equation 3)

where $E_a$ represents the cortex-membrane adhesion energy density. Clearly, anisotropic stretching of cells will cause the increase of $P_J/P_{0,J}$ and therefore could trigger the nucleation of stress fibers. Note that, after nucleation, the tension level inside the $k$-th stress fiber (formed in the $J$-th cell) reduces from ${\mathrm{\Gamma }}_J\left(P_J-P_{0,J}\right)$ to $f_k=d{U}_{fiber}/d{l}_k={\gamma }_k\left(l_k-l_{0,k}\right)$. Driven by this tension, the stress fiber can be further peeled away from the cell edge by overcoming cortex-membrane adhesion, a scenario that is very similar to the peeling of elastic strips considered in the Kendall model (33,34). Specifically, such continuous detachment could happen as long as $f_k\left(1-\mathit{cos}\alpha \right)>E_a$, with the peeling speed of stress fiber tip given by

$$v=\frac{f_k\left(1-cos\alpha \right)-E_a}{\frac{4}{3}\pi l_k{\mathit{sin}}^2\alpha }\text{,}$$ (Equation 4)

where $\alpha$ is the angle formed between the stress fiber and cell edge (Fig. 1 D). Note that, here, we assumed that the released energy of stress fiber (after detachment) is completely consumed by viscous dissipation (see section A of the supporting material). Next, if the moving tip meets another moving one in the same cell, the two stress fibers will merge with each other. However, the moving tip will be anchored/trapped if it meets an unpeeled fiber or TCJ, and disassembly of the stress fiber is assumed to take place when both tips are anchored (refer to section A of the supporting material for details). It must be pointed out that since the peeling and breakage of fibers usually take about 30 min (5), which is much longer than the timescale of actin turnover (∼$1-10$ s), we assume the cortex has enough time to reassemble/reform during this process, and therefore corresponding parameters such as $\mathrm{\Lambda }$, $p_0$, and $A_{0,J}$ all remain unaffected by the fiber dynamics.

The formulation was then implemented to simulate the response of cell monolayers, each containing 256 cells that have randomly assigned target areas, under anisotropic biaxial stretching. Specifically, an equal morphogenetic force $F_x=F_y=2$ was applied to both the horizontal and vertical boundaries of the monolayer initially. After that, $F_y$ was fixed in all simulations, while $F_x$ was increased with a constant ramping rate of $2\times {10}^{-3}$ until the designed force level is reached (Fig. 1 B). An adaptive time step scheme was employed in our simulation to limit the maximum (normalized) movement of any vertex to less than $5\times {10}^{-2}$ in each step. Note that the size of the monolayer constructed here is comparable to that of the stretching region monitored in the Drosophila pupal dorsal thorax’s epithelium (5), and therefore no periodic boundary condition was enforced in simulations. Estimations of different parameters were conducted in a way similar to that in our previous study (25). Specifically, given that the areal membrane-cortex adhesion energy density was reported to be in the range of $1.3-9.8\times {10}^{-6}\mathrm{J}/{\mathrm{m}}^2$ (35) and the stress fiber should have a thickness ∼0.5 $\mu \mathrm{m}$ (5), we estimate $E_a$ to be around ${10}^{-2}$ here.

Finally, we characterize the deformed cell shape by the second moment of area with respect to its centroid,

$$I=\left[\begin{array}{cc}{I}_{xx}& I_{xy}\\ I_{yx}& I_{yy}\end{array}\right]\text{,}$$ (Equation 5)

and define cell elongation $\lambda$ as the ratio of matrix diagonal, i.e., $I_{xx}/I_{yy}$ (since the monolayer is stretched more significantly along the horizontal direction). Discussions on this choice and its comparison with other alternative definitions are provided in section B of the supporting material.

Materials and data availability

The model was implemented in CHASTE (University of Oxford, Oxford, UK). The customized codes have been deposited to Github and could be accessed through https://github.com/SoftMatterMechanics/ApicalStressFibers.git.

Results

Size-dependent cell elongation emerges in the epithelium under anisotropic stretching

We first examine the correlation between the size and deformation of cells in the absence of any stress fiber. Interestingly, as shown in Fig. 2 A, our simulations indicated that cells within the monolayer undergo inhomogeneous deformations. Specifically, on average, the degree of cell elongation was found to increase with the cell area (Fig. 2 B), and such a trend becomes more pronounced under high morphogenetic force ratios (MFRs), i.e., large values of $F_x/F_y$. To further examine this, we quantify the linear correlation between cell elongation and area $A$ as

$$\lambda \propto \beta A\text{,}$$ (Equation 6)

where the coefficient $\beta$ characterizes the level of size dependency. After that, 10 independent simulations were conducted for each combination of the target shape index $p_0$ and line tension $\mathrm{\Lambda }$ of cells, which were varied from 2.5 to 3.8 (i.e., the monolayer remains solid-like) and −0.1 to −0.6, respectively, leading to 84 different combinations. Under such circumstances, $\beta$ as a function of MFR is shown in Fig. 2 C. We can see that, in almost all cases, the value of $\beta$ is rather insensitive to MFR until it reaches a critical level, beyond which $\beta$ increases rapidly with MFR. This indicates that certain force-induced events must be triggered by the large value of MFR, which then amplifies the size-dependent response of cells.

Figure 2.

Size-dependent elongation of cells in stretched epithelial monolayers. (A) A representative color map from our simulations showing the inhomogeneous cell deformation. Larger cells usually elongate more in the same monolayer. (B) Cell elongation grows with the cellular area as morphogenetic force ratio (i.e., $F_x/F_y$) increases. The solid black line refers to the best linear fit, and its slope $\beta$ characterizes the significance of the size effect. Results shown here are based on 1997 cells from simulations where ${\mathrm{p}}_0$ = 3 and $\mathrm{\Lambda }$ = −0.5. Error bars represent the standard error of the mean (SEM). (C) The value of $\beta$ as a function of the morphogenetic force ratio (MFR) revealed from our simulations (corresponding to 84 combinations of $p_{\mathit{0}}$ and $\mathrm{\Lambda }$ that were varied from 2.5 to 3.8 and −0.6 to −0.1, respectively). Each line represents the summary of 10 independent simulations with the same ${\mathrm{p}}_0$ and $\mathrm{\Lambda }$ but different randomly assigned preferred areas of cells. To see this figure in color, go online.

Cell rearrangements induce the size-dependent elongation

To understand the physical origin behind the size effect of cells revealed by our simulations, we generate the heatmap (Fig. 3 A) showing the level of MFR needed for resulting in $\beta$ >0.1 (referred to as ${\mathrm{M}\mathrm{F}\mathrm{R}}_{0.1}$) under different target shape index and line tension values. Clearly, we can see that ${\mathrm{M}\mathrm{F}\mathrm{R}}_{0.1}$ reaches the maximum at the bottom left corner of the map, i.e., when $p_0$ is small and $\mathrm{\Lambda }$ is most negative. Since the maximum cell contractile stiffness in the monolayer is given by ${\mathrm{\Gamma }}_{\mathit{max}}=-\mathrm{\Lambda }/p_0$ (i.e., when ${\overline{A}}_{0,J}$ equals to 1 in Eq. 2 for the cell with the smallest target area), we check the dependency of ${\mathrm{M}\mathrm{F}\mathrm{R}}_{0.1}$ on ${\mathrm{\Gamma }}_{\mathit{max}}$. Surprisingly, as illustrated in Fig. 3 B, a strong nonlinear relationship between the two was observed where the threshold value ${\mathrm{M}\mathrm{F}\mathrm{R}}_{0.1}$ varies significantly with ${\mathrm{\Gamma }}_{\mathit{max}}$ in a wide transition regime (i.e., the shaded region in Fig. 3 B). This suggests that ${\mathrm{\Gamma }}_{\mathit{max}}$ cannot be the only factor regulating the size-dependent behavior of cells.

Figure 3.

Size-dependent cell elongation is induced by local cell rearrangement (i.e., T1 transition). (A) Phase diagram showing the threshold of MFR, as a function of ${\mathrm{p}}_0$ and $\mathrm{\Lambda }$, for triggering significant size-dependent elongation of cells (i.e., $\beta \ge 0.1$). (B) The threshold (${\mathrm{M}\mathrm{F}\mathrm{R}}_{0.1}$) exhibits a strong nonlinear dependence on the maximum cell contractile stiffness (${\mathrm{\Gamma }}_{\mathit{max}}=-\mathrm{\Lambda }/{\mathrm{p}}_0$) in the monolayer but (C) clearly falls into two distinct regimes when plotted against the average count of T1 transitions $n_T$ occurred in the monolayer: 1) ${\mathrm{M}\mathrm{F}\mathrm{R}}_{0.1}$ remains at a low and constant level, insensitive to ${\mathrm{\Gamma }}_{\max }$, when T1 transitions occur frequently (gray filled circles), and 2) ${\mathrm{M}\mathrm{F}\mathrm{R}}_{0.1}$ becomes large and increases with ${\mathrm{\Gamma }}_{\mathit{max}}$ when $n_T$ is small (red filled circles). (D) Evolution of $n_T$ as the MFR increases. In all the simulations (corresponding to different combinations of ${\mathrm{p}}_0$ and $\mathrm{\Lambda }$ as shown in A), $n_T$ remains at a very low level before undergoing a rapid increase when the MFR reaches a critical value. (E) Plot showing the relationship between the critical MFRs (${\mathrm{M}\mathrm{F}\mathrm{R}}_{\mathrm{t}}$) for triggering the sudden increase in $n_T$ (i.e., the frequency of T1 transitions shown in D) and in $\beta$ (i.e., the degree of size-dependent response of cell shown in Fig. 2C). A strong correlation ($r=$ 0.8589 and $\mathrm{p}<{10}^{-15}$) was observed. (F) The observed size-dependent elongation of cells disappears and is actually reversed (i.e., $\beta$ becomes negative) when T1 transition was turned off in our simulations. (G) Occurrence of cell rearrangement always releases the elongation of cells (left axis) participating in the T1 transition. However, the magnitude of such an elongation drop (right axis) decreases with the cell area, i.e., more elongation will be released in smaller cells. Results shown here are based on 2000 cells participating in T1 transitions in our simulations where ${\mathrm{p}}_0$ = 3.5 and $\mathrm{\Lambda }$ = −0.3. Error bars represent the SEM. To see this figure in color, go online.

Given that local rearrangement (i.e., the so-called T1 transition) of cells has been found to be critical in the motility (36) and fracture (27) of epithelial tissues, we proceed by examining whether this factor also plays a role here. Specifically, the relationship between ${\mathrm{M}\mathrm{F}\mathrm{R}}_{0.1}$ and the average count of T1 transitions (normalized by the total number of cells), i.e., ${\mathrm{n}}_{\mathrm{T}}$, that occurred in the monolayer is shown in Fig. 3 C. Surprisingly, all our data fall into two distinct regimes: 1) ${\mathrm{M}\mathrm{F}\mathrm{R}}_{0.1}$ is insensitive to ${\mathrm{\Gamma }}_{\mathit{max}}$ when T1 transition takes place frequently in the monolayer, and 2) ${\mathrm{M}\mathrm{F}\mathrm{R}}_{0.1}$ becomes strongly dependent on ${\mathrm{\Gamma }}_{\mathit{max}}$ when T1 transition is rare (see Fig. 3, B and C). Based on these findings, we hypothesize that cell rearrangement regulates the emergence of size-dependent cell elongation. Indeed, the count of T1 transitions keeps increasing during the stretching (Fig. 3 D; Video S1) and, more interestingly, exhibits a very similar trend to that of $\beta$ (refer to Fig. 2 C). To quantify such similarity, we extract the critical MFR values from Fig. 2 C and 3 D beyond which $\beta$ and $n_T$ undergo rapid increase and then plot them against each other in Fig. 3 E. As expected, a strong and significant correlation ($r$ = 0.8589, $\mathrm{p}$ < ${10}^{-15}$) was observed here. This suggests that the emergence of a size-dependent cell response is synchronized with the occurrence of T1 transitions. To further validate our hypothesis, we effectively turned off the T1 transition in our simulations by lowering the critical length for its occurrence from ${10}^{-2}$ to an extremely small value of ${10}^{-6}$. Under such circumstances, the trend is reversed, i.e., the value of $\beta$ becomes negative, as shown in Fig. 3 F, confirming the indispensable role of cell rearrangement in the observed size effect of cells. This can also be seen from the change in elongation of cells (participating in rearrangement) before and after T1 transitions (see Fig. 3 G). Clearly, the presence of T1 transition reduces the elongation of cells at every size. However, such a drop in smaller cells is bigger than that in larger ones and therefore results in a more pronounced size-dependent cell elongation.

Video S1. Cell rearrangements in the stretched epithelial

monolayer

At this point, we are in a position to explain the characteristics of $\beta$ curves shown Fig. 2 C. Basically, as the morphogenetic force anisotropy (or equivalently $F_x$) increases, more cell rearrangements will be triggered in the monolayer, eventually leading to a rapid increase of $\beta$ when T1 transitions become very frequent.

Nonlinear dependence of stretching force threshold on cell contractility originates from the energy barriers of cell rearrangement

To illustrate how monolayer stretching influences the occurrence of T1 transitions as well as to clarify the physics behind the nonlinear dependency of the MFR threshold on cell contractility, we construct a four-cell minimal model similar to those proposed in previous studies (20). Specifically, a growing morphogenetic force is applied to two cells, trying to deform them in the vertical direction (insets of Fig. 4 A). As T1 transition progresses, i.e., the center horizontal cell edge (with a negative length) gets eliminated while a vertical edge (with a positive length) gets formed, the system potential energy will change. This energy as a function of cell edge length $l$ is plotted in Fig. 4 A, which shows that such rearrangement corresponds to the transition of the system from one local minimum to another. The presence of stretching force significantly lowers the energy barrier between two minimums and, consequently, makes the occurrence of T1 transition easier and more frequent.

Figure 4.

A four-cell minimal model was developed to analyze the occurrence of T1 transitions in the cell monolayer. (A) Potential energy landscape, with/without the stretching force, associated with the T1 transition. The appearance of a stretching force reduces the energy barrier and therefore makes the occurrence of T1 transition easier. The topology change of cells during the transition is shown in the insets, where the red line represents the eliminated/newly created cell edge having length $l$ (negative and positive $l$ correspond to before and after transition, respectively). (B) Heatmap showing the dependence of the energy barrier for T1 transition on the target shape index and line tension. (C) The energy barrier shown in (B) exhibits a strong nonlinear dependence on the cell contractile stiffness ($\mathrm{\Gamma }=-\mathrm{\Lambda }/{\mathrm{p}}_0$). To see this figure in color, go online.

We then examine how the energy barrier for T1 transition varies under different combinations of $p_0$ and $\mathrm{\Lambda }$. As shown in Fig. 4 B, this energy barrier is almost zero except in the region where the target shape index is small and the line tension is most negative, i.e., the bottom left corner of the plot. Interestingly, this pattern is almost identical to the heatmap shown in Fig. 3 A, indicating that the reason for the peak of the MFR threshold that appeared in Fig. 3 A is because cell rearrangement becomes difficult in that region, and therefore a higher stretching force (in the horizontal direction) is needed to trigger their occurrence. This can also be observed from the plot illustrating the relationship between the energy barrier and the cell contractile stiffness $\mathrm{\Gamma }$ (Fig. 4 C). Basically, the energy barrier remains at a very low level before shooting off when $\mathrm{\Gamma }$ increases to a critical value, a trend that is again very similar to that shown in Fig. 3 B. As such, we conclude that MFR threshold’s nonlinear dependence on cell contractility is determined by the energy barrier of T1 transition in the monolayer.

Aligned stress fibers modulate size-dependent cell elongation

In this section, we adopt the full SFR vertex model to investigate the role of stress fiber dynamics in the size-dependent deformation of cells. Specifically, as observed in earlier experiments, stress fibers could be nucleated at TCJs via the detachment of the cortical layer from the cell edge (Fig. 1 D). An interesting question to ask is whether such nucleation is easier at certain cell vertices than others within the monolayer. To answer this, we characterize each vertex by its opening angle $\theta$ (i.e., the angle formed between two adjacent edges at the vertex) and its bisector orientation $\phi$ with respect to the vertical axis (i.e., the minor axis of stretching) (see Fig. 1 D). Since the opening angle is more or less uniform for all bisector orientations in a randomly generated monolayer, the nucleation of stress fibers was found to be orientation independent from our simulations (see the dotted line in Fig. 5 A).

Figure 5.

Comparison between predicted stress fiber dynamics in the cell monolayer and experimental observations (5). (A) Cumulative distribution with respect to the opening angle bisector orientation $\phi$, of the nucleation ($n$ = 10,657), peeling ($n$ = 1985), and breakage ($n$ = 638) of stress fibers observed in our simulations. The stress fiber nucleation is largely uniform, but most peeling of stress fibers occurs at vertices with a bisector close to the minor axis of stretching, i.e., when $\phi <\pi /4$, and break at vertices with bisectors close to the major axis of stretching, i.e., when $\phi >\pi /4$. (B) Most long stress fibers (left subplot, $n$ = 1985) are formed along the major axis of stretching. Experimental data shown here are from (5), which agree very well with our simulation results. To see this figure in color, go online.

After nucleation, nascent stress fibers can peel from the cell edge to become longer and mature ones. Interestingly, as shown in Fig. 5 A, our simulations revealed that the peeling process is significantly orientation dependent—most long stress fibers are from vertices with a bisector close to the minor axis of stretching, i.e., when $\phi <\pi /4$, and break at vertices with bisectors close to the major axis of stretching, i.e., when $\phi >\pi /4$. As a result, most fully developed stress fibers align horizontally along the major axis of stretching (Fig. 5 B; Video S2). These are in quantitative agreement with recent observations in the dorsal thorax’s epithelium of wild-type Drosophila pupal where stress fibers emerge from TCJs, move toward the cell center, and align with the major stretching direction of the epithelium (5). Therefore, according to our model, we argue that such observed orientation dependency of stress fibers comes from their peeling rather than nucleation. It must be pointed out that the small size of nascent stress fibers likely makes them indistinguishable from the cell cortex due to limited fluorescent resolution in experiments.

Video S2. Stress fiber dynamics in the deformed

Cells

Next, we ask why the peeling process is orientation dependent. Actually, by examining the tension level of a freshly nucleated stress fiber in the monolayer, it can be shown that the driving force for peeling is

$$G=-\mathrm{\Lambda }\left(\frac{P_J}{P_{0,J}}\mathit{sin}\frac{\theta }{2}-1\right)\left(1-\mathit{sin}\frac{\theta }{2}\right)\text{,}$$ (Equation 7)

which is unlikely to overcome cortex-membrane adhesion for further detachment (see section A of the supporting material). Therefore, most nascent stress fibers could not move away from the vertices of their nucleation. However, the monolayer starts to deform when anisotropic stretching forces are applied, leading to an increase in the cell perimeter and an angle of vertices with small bisector orientations (see Fig. 6 A). Consequently, the tension stored in nascent stress fibers nucleated from those vertices increase, eventually making their peeling possible. In comparison, the angle of vertices with a bisector aligned more along the major stretching axis will decrease, therefore making the peeling of stress fibers more difficult (Fig. 6 B).

Figure 6.

Formation of contractile stress fibers modulates the size-dependent response of cells. (A) As stretching force grows, the opening angle $\theta$ increases for those with small bisector orientations but decreases for those with large $\phi$ values. Results shown here are based on 11,867 opening angles. (B) The tensions stored in nascent stress fibers ($n$ = 10,657) nucleated from vertices with a bisector aligned more along the minor stretching axis increase, making their subsequent peeling more likely. In all simulations shown here, the target shape index and line tension were fixed as ${\mathrm{p}}_0$ = 3 and $\mathrm{\Lambda }$ = −0.5. Error bars represent the SEM. (C) Under the fixed stretching force ratio (MFR = 2.66), size-dependent cell elongation is evident in the absence of stress fibers ($\mathrm{\Lambda }$ = −0.5, no stress fiber, $n$ = 1998) or when line tension is relatively high ($\mathrm{\Lambda }$ = −0.3, $n$ = 1761). In contrast, stress fibers in the cell monolayer with weak line tension ($\mathrm{\Lambda }$ = −0.5, $n$ = 1998) reduce the size dependency of cell elongation. Error bars indicate the SEM. (D) The appearance of stress fibers significantly reduces the count of T1 transitions in the cell monolayer. Results shown here are based on simulations with a fixed target shape index of ${\mathrm{p}}_0$ = 3. To see this figure in color, go online.

Last, we examine the mechanical role of stress fibers in the deformed cell monolayer. Under the same applied stretching forces, it was found that either the absence of stress fibers or an increase in the value of line tension $\mathrm{\Lambda }$ (equivalent to reducing the contractile stiffness) makes the size-dependent cell elongation more pronounced in the monolayer. On the other hand, the size effect becomes very small in the control group (with $\mathrm{\Lambda }$ = −0.5) (refer to Fig. 6 C). Overall, our results suggest that a weakened cytoskeleton (i.e., reduced contractility and stress fiber formation) could make cell rearrangement easier (Fig. 6 D) and hence amplify the size-dependent response of cells in the monolayer. It is worth noting that, after tuning down the activity of actin in Drosophila pupal dorsal thorax’s epithelium, a more pronounced size dependency in the elongation of cells was indeed observed (5), consistent with our theoretical predictions.

Discussion

In this work, we examine the deformation of an epithelial tissue, consisting of cells with different sizes, under anisotropic biaxial stretching. Interestingly, it was found that cell elongation will become size dependent due to the occurrence of cell rearrangements (i.e., T1 transition) when the magnitude and anisotropy of stretching forces increase. The force threshold for triggering significant size-dependent elongation in cells exhibits a strong nonlinear correlation with their contractility. We further demonstrated that this is because the energy barrier for cell rearrangement varies nonlinearly with cell contractility. On the other hand, when the nucleation and continuous peeling of subcellular stress fibers are allowed, more long stress fibers with orientations aligned with the major stretching direction will be formed in larger cells and, consequently, modulate their elongation. These findings are in excellent agreement with recent experimental observations on Drosophila pupa.

It must be pointed out that the mechanical response of epithelium tissues under stretching has been intensely investigated both experimentally (37,38,39) and theoretically (27,40). Specifically, it has been shown that both cell proliferation (40) and local rearrangement (27) could lead to plastic deformation of the cell monolayer (under uniaxial stretching) and therefore reduce its internal stress, in good agreement with the findings here. However, the monolayer was often assumed to be composed of cells with the same size in those studies. In reality, cell size can vary by a factor of 7–8 in the same epithelial tissue (5). Furthermore, the formation and evolution of subcellular stresses have not really been carefully incorporated into existing formulations describing the behavior of multicellular systems. In this regard, the SFR vertex model developed here could serve as a useful tool in studying processes such as tissue fracture (38), adhesion- and polymerization-regulated collective migration (13,41,42,43), and curvature sensing (44) of cells, where inhomogeneity within the monolayer or active contraction by stress fibers has been shown to play key roles. For instance, observations have showed that actin cables crossing multiple cells could be formed within the monolayer and act as purse strings to facilitate healing of epithelial wounds (45). By adding a proper description of the communication between different stress fibers through intercellular adhesion, it will be very interesting to see whether the formation of such supracellular cables can be realized in our SFR model.

Finally, we want to emphasize that cells were assumed to have the same target shape index in the present study, implying that contractile stiffness is higher in smaller cells. This assumption is consistent with recent experimental observations, i.e. the assembly of more F-actin at intercellular faces generates stronger contractions and leads to smaller apical cell areas (29,30), and is actually indispensable for the emergence of size-dependent elongation (Fig. S4, A and B). Nevertheless, we want to point out that the main conclusion that T1 transitions help smaller cells to release more elongation when compared with larger ones remains valid even if the cells were assumed to possess the same contractile stiffness $\mathrm{\Gamma }$ rather than the same $p_0$ (and therefore smaller cells will have a larger target shape index and can thus undergo severe deformations more easily) (refer to section C of the supporting material for details). In addition, it is conceivable that cellular strain/elongation can also be released via the division and apoptosis of cells within the tissue (46). However, these factors were neglected in the present study and certainly warrant future investigations. Besides, some of our assumptions, such as nucleated nascent stress fibers being indistinguishable from the cell cortex and the inverse relationship between cell area and contractility, could be carefully tested by techniques like superresolution light (47) or confocal (48) microscope and laser ablation, which will definitely further enhance our understanding of these processes.

Author contributions

Y.L. conceived the study. C.F. and Y.L. built the model. C.F. and X.S. performed the simulations. C.F., X.S., Y.T., and Z.C. analyzed data. C.F. and Y.L. wrote the manuscript.

Editor: Dimitrios Vavylonis.