Simulation of Cell Patterning Triggered by Cell Death and Differential Adhesion in Drosophila Wing

Abstract

The Drosophila wing exhibits a well-ordered cell pattern, especially along the posterior margin, where hair cells are arranged in a zigzag pattern in the lateral view. Based on an experimental result observed during metamorphosis of Drosophila, we considered that a pattern of initial cells autonomously develops to the zigzag pattern through cell differentiation, intercellular communication, and cell death (apoptosis) and performed computer simulations of a cell-based model of vertex dynamics for tissues. The model describes the epithelial tissue as a monolayer cell sheet of polyhedral cells. Their vertices move according to equations of motion, minimizing the sum total of the interfacial and elastic energies of cells. The interfacial energy densities between cells are introduced consistently with an ideal zigzag cell pattern, extracted from the experimental result. The apoptosis of cells is modeled by gradually reducing their equilibrium volume to zero and by assuming that the hair cells prohibit neighboring cells from undergoing apoptosis. Based on experimental observations, we also assumed wing elongation along the proximal-distal axis. Starting with an initial cell pattern similar to the micrograph experimentally obtained just before apoptosis, we carried out the simulations according to the model mentioned above and successfully reproduced the ideal zigzag cell pattern. This elucidates a physical mechanism of patterning triggered by cell apoptosis theoretically and exemplifies, to our knowledge, a new framework to study apoptosis-induced patterning. We conclude that the zigzag cell pattern is formed by an autonomous communicative process among the participant cells.

Introduction

A multicellular tissue often shows interesting cellular patterns constructed from its unit components (cells). To describe the pattern formation mechanism by physical dynamics, we have been constructing cell-based models of vertex dynamics (vertex dynamics cell models), in which the tissues are constructed from polygons in 2D space or polyhedra in 3D space. Owing to their simplicity and flexibility, the vertex dynamics cell models have been applied to various epithelial morphogeneses and elucidated physical mechanisms and realistic forces in respective systems (1, 2, 3). Vertex dynamics was originally proposed in physics to describe aggregated soap bubbles or polycrystalline materials (4, 5, 6, 7, 8). We have added cell deformation, cell migration, and cell growth/proliferation into the vertex dynamics (9, 10, 11, 12, 13). Here we consider programmed cell death (apoptosis). Although apoptosis is a relatively recent introduction in the long history of embryology (14), its indispensable role in the developmental process has become broadly recognized (15). Apoptosis has been brought to mathematical studies of the developmental process (16, 17, 18, 19, 20). This article showed that apoptosis is an essential cell behavior to understand formation of zigzag cell pattern in Drosophila wings. The zigzag cell patterning has been known for a long time (21), and recently investigated experimentally in detail by Takemura and Adachi-Yamada (22, 23). We show a part of their results below.

The posterior part of the Drosophila wing margin is covered with long hairs sourced from hair cells arranged in a zigzag pattern in the lateral view, shown in Figure 1, Figure 2. Figure 1, Figure 3, Figure 4, shows the photomicrographs of the formation of this zigzag pattern obtained by Takemura and Adachi-Yamada (22, 23). According to their observations, this zigzag pattern is created by selective apoptosis of wing margin cells that are spatially separated from hair cells, which rearranges the remaining wing margin cells in a well-ordered manner. The hair cells send signals to their directly neighboring cells, preventing their apoptosis.

Figure 1.

Zigzag pattern formation of wing margin hairs. All panels (1–4) were adapted from (22, 23). (1) Drosophila adult wing. (2) Higher magnification view of the posterior wing margin (the rectangular region in (1)). (3) and (4) Lateral view of the Drosophila pupal wing margin at 20 h after puparium formation (APF) and 30 h APF. Cell types are colored and/or symbolized for clarity, as explained in Fig. 3. The region between the two lines of hair cells (pink, ^∗) is the lateral part; the upper region (white cells, out of focus) is the dorsal part; and the lower region (white cells, out of focus) is the ventral part. To see this figure in color, go online.

Figure 2.

Two elementary processes of topological changes. (a) Recombination process (T1): when vertices i and j approach each other within a small distance Δ, their connections are changed from the left figure to the right figure, and vice versa. (b) Disappearance process (T2): when triangular cell A becomes small (its minimum edge 〈i j〉 becomes less than Δ), cell A shrinks into a new vertex k at the middle point of the edge 〈i j〉.

Figure 3.

Cell pattern in the dorsoventral boundary of the posterior margin of Drosophila wing (the initial state of this simulation). Cell types are designated by colors and/or symbols. Pink, ^∗, is the hair cell; yellowish green, ○, is the interhair cell; white, +, is the tooth cell; and blank areas are the preapoptotic or apoptotic cells. Numerals in the hair and interhair cells are cell numbers, which are fixed throughout this run. Letters “v” and “d” denote cells on the ventral and dorsal surfaces, respectively. To see this figure in color, go online.

Figure 4.

Basic elements of the ideal zigzag cell pattern. Symbols H, I, and T indicate cell types: hair, interhair, and tooth, respectively. Thick line C₀-C-C′-C′₁-C₁-C₂ designates dorsoventral boundary. To see this figure in color, go online.

To simulate the zigzag pattern, we inserted additional information on the Drosophila wing. As strong contraction of the basal part of wing (wing hinge) elongates the remaining wing part (wing blade), the Drosophila wing continues extending along its proximal-distal axis, resulting in oriented elongation of the epithelial wing cells (24, 25, 26). The cell alignment pattern is strengthened by cell adhesion molecules (DE-cadherin and Drosophila NEPH1/nephrin orthologs), which accumulate at the edges between the dorsal and ventral wings, and are identified depending on cell types at cell boundaries during the pupal wing formation (22, 23). The purpose of this article is to extract qualitative features of the zigzag cell patterning and find out a general framework to study apoptosis-induced patterning.

Materials and Methods

Vertex dynamics cell model

We apply the vertex dynamics cell model to the zigzag cell patterning in the Drosophila wing margin, as mentioned above. This model approximates each cell by a prism and describes an epithelial tissue as a monolayer cell sheet. The prism is expressed in terms of mathematical words (elements), namely, as vertices, edges, and faces. Because the elements are only approximated to real cells, each has a finite size of order Δ. This Δ defines a minimum length in the model, below which one cannot discuss smaller elements. In this coarse-grained model, the upper surface of the monolayer sheet is filled with polygonal cells, or vertices interconnected with straight cell boundaries.

Our system is constructed from sets (r_i, h_i) with 2D positional vector r_i and height h_i of vertex i, where i = 1, 2, ⋅⋅⋅, N (total number of vertices). In this report, the same cell height h_i is assigned to all cells. Therefore, our system is expressed by an assembly of vertices interconnected by straight cell boundaries on the 2D plane. This 2D plane expresses the apical plane of epithelial cells at the dorsoventral boundary of the posterior margin of Drosophila wing (see Fig. 1).

The velocity of vertex i is given by

$$\mathit{\eta }\frac{d{\mathit{r}}_i}{dt}=-\frac{\partial U}{\partial {\mathit{r}}_i},$$ (1)

where t is time, η is the coefficient of viscous resistance, and U is the total potential energy of the system. The term on the left-hand side of Eq. 1 expresses the dissipative force, whereas the term on the right-hand side denotes the potential force. Equation 1 guarantees that the total potential energy, U(r₁(t),r₂(t), ⋅⋅⋅), monotonically decreases or remains constant with t. The total potential energy of the system in Eq. 1 is given by

$$U=U_I+U_D,$$ (2)

where U_I is the interfacial energy of cell boundaries, and U_D is the elastic energy of cells. U_I is expressed by

$$U_I=\sum _{\langle ij\rangle }{\mathit{\sigma }}_{ij}{r}_{ij},$$ (3)

where σ_ij is the interfacial energy density of cell boundary 〈i j〉, and r_ij ≡ |r_ij|≡|r_i − r_j| is the length of boundary 〈i j〉. Denoting the area of cell μ by S_μ and its equilibrium value by S⁰_μ, U_D is written as

$$U_D=\sum _{\mathit{\mu }}^{\left(\mathrm{cell}\right)}\mathit{\rho }{\left(h^0\right)}^2{\left(S_{\mathit{\mu }}-S_{\mathit{\mu }}^0\right)}^2,$$ (4)

where ρ is the volume elasticity of the cells, and h⁰ is an equilibrium cell height—both of which are assumed to be independent of cells. Equation 4 ensures the cellular volume conservation around its equilibrium value h⁰S⁰_μ. The explicit form of Eq. 1 with Eqs. 3 and 4 is given in Eq. S1.1 in the Supporting Material.

To describe the topological changes of the network of cell boundaries, we introduce two elementary processes shown in Fig. 2 a (Recombination process T1) and Fig. 2 b (Disappearance process T2).

Vertex dynamics in the cell patterning in Drosophila wing margin

Fig. 3 illustrates the initial state of our present computer simulations, which imitates in essence Fig. 1 3 just before cell apoptosis events occur. The x axis is directed along the dorsoventral boundary and the y axis is perpendicular to it. The y axis points from the ventral side (y < 0) to the dorsal side (y > L_Y), where L_Y denotes the width of our system of interest. Now we apply the vertex dynamics cell model mentioned above to the zigzag cell patterning in the Drosophila wing margin, taking account of its peculiar aspects described below.

Interfacial energy densities of cell boundaries

The interfacial energy density σ_ij is determined by the cell types on both sides of cell boundary 〈i j〉. Although this term originates from the molecular interactions between cell membranes, it is modeled as a physical quantity averaged over a small length Δ. According to (22), we assume four cell types in the epithelial tissue of the Drosophila wing margin: hair cells, interhair cells (surviving cells between hair cells), tooth cells (cells adjacent to a hair cell, which are expected to survive), and apoptotic cells (cells that are going toward death or shrinking), as shown in Fig. 3. In the following, we express the four cell types by H, I, T, and A, respectively. As the system is symmetric (σ_μν = σ_νμ), we have 10 possible types of the interfacial energy density: σ_Hν(ν = H, I, T, A) for hair cells, σ_Iν(ν = I, T, A) for interhair cells, σ_Tν(ν = T, A) for tooth cells, and σ_Aν(ν = A) for apoptotic cells. Furthermore, cells apart from hair cells, which are waiting for apoptotic process, are called “preapoptotic” cells and they have no difference from tooth cells until their apoptotic processes start.

Based on their experimental results, Takemura and Adachi-Yamada extracted a model for establishing an interlocking cell pattern of the posterior wing margin, shown in Fig. 4 (hereafter called the “ideal zigzag cell pattern”). The ideal zigzag cell pattern can be constructed by repeating the basic unit of the cell boundary A-B-C-C′-B′-A′, as shown in Fig. 4. It should be noted that three cells (vertical H-T-I) properly interlock each other in the ideal zigzag cell pattern. The thick solid line C₀-C-C′-C′₁-C₁-C₂ in Fig. 4 represents the dorsoventral boundary that separates the dorsal and ventral parts into upper and lower sides, respectively. The cell adhesion molecules above mentioned, strongly accumulated around this boundary.

Now we can estimate the interfacial energy densities in our model by finding stability conditions of the ideal zigzag cell pattern. It is stable if in the basic unit three interfacial forces acting on each of vertices A, B, and C balance with each other under zero elastic forces. Therefore, we have six simultaneous equations. As mentioned later, we set the mirror-symmetrical boundary condition in the y axis direction, so that the hair cell extends mirror-symmetrically beyond line AA₁ into the external region of the ventral side (Line AA₁ is not a cell-boundary but the system-boundary in Fig. 4). As a result, the y component of the force equation at vertex A automatically satisfies the balance condition, which reduces the number of equations from six to five. Assuming that β = γ, which approximates the experimental results (22), we solve the five simultaneous equations with respect to angle β and dimensionless quantities σ^∗_μν/σ^∗_TT (μ,ν = H, I, T, E), where σ^∗_TT is the interfacial energy densities of the T-T boundary at the dorsoventral boundaries (see Fig. 4), and we obtain (see the Supporting Material)

$$\frac{{\mathit{\sigma }}_{HI}}{{\mathit{\sigma }}_{TT}^{\ast }}=\frac{{\left(1-{\mathit{\epsilon }}^2/4\right)}^{1/2}}{\sin \mathit{\alpha }},$$ (5a)

$$\frac{{\mathit{\sigma }}_{HT}}{{\mathit{\sigma }}_{TT}^{\ast }}=\frac{\mathit{\epsilon }}{2}+\frac{{\left(1-{\mathit{\epsilon }}^2/4\right)}^{1/2}}{\tan \mathit{\alpha }},$$ (5b)

$$\frac{{\mathit{\sigma }}_{IE}}{{\mathit{\sigma }}_{TT}^{\ast }}=2\frac{{\left(1-{\mathit{\epsilon }}^2/4\right)}^{1/2}}{\tan \mathit{\alpha }},$$ (5c)

and

$$\frac{{\mathit{\sigma }}_{IT}}{{\mathit{\sigma }}_{TT}^{\ast }}=1,\mathit{\beta }=\mathit{\gamma }={\cos }^{-1}\left(-\mathit{\epsilon }/\mathrm{2}\right),$$ (6)

where 0 < α < 90° and ε ≡ σ^∗_IT/σ^∗_TT for 0 < ε < 2. Here, σ^∗_IT and σ_IE denote the interfacial energy density of I-T boundary at the dorsoventral boundary (see Fig. 4) and that of I-E boundary A₀A in Fig. 4 (E is external cell below A₀A), respectively.

Fig. 5 shows ε-dependence of the interfacial energy densities given by (5a), (5b), (5c) and 6 for α = 37° fixed. This value of α was obtained by measuring the zigzag cell pattern in the micrograph in Fig. 1 4. In this article, we used ε as a fitting parameter and found the best set (α = 37°, ε = 1.5) resulting in the ideal zigzag cell pattern (see Appendix B).

Figure 5.

Interfacial energy densities of the ideal zigzag cell pattern calculated by (5a), (5b), (5c) and 6. Dependence on ε for α = 37° is fixed. Symbols μν: IE, HI, ⋅⋅⋅, denote interfaces between μ-type cell and ν-type cell. Symbols H, I, T, and E denote cell types: hair, interhair, tooth, and external cell, respectively. Symbol IT^∗ denotes I-T interface at the dorsoventral boundary. The best fitting with the ideal zigzag cell pattern is obtained with set (α = 37°, ε = 1.5).

Apoptotic process

Based on experimental findings (22), we specify N_A(0) cells having no or small contact with hair cells in Fig. 3 as preapoptotic cells, and define time T_A as the time required for the preapoptotic cells to almost complete their apoptosis. The apoptosis starting time t_Aμ of preapoptotic cell μ is randomly selected within T_A using uniform random numbers. After t_Aμ, the equilibrium area of apoptotic cell μ is then given by

$$\begin{array}{l}{S}_{A\mathit{\mu }}^0\left(t\right)=S_T^0\text{exp}\left(-\frac{t-t_{A\mathit{\mu }}}{{\mathit{\tau }}_A}\right),\\ \text{for}t>t_{A\mathit{\mu }},\end{array}$$ (7)

where τ_A ≈ T_A/N_A(0) denotes the lifetime of the apoptotic cell. Equation 7 means that each preapoptotic cell is indistinguishable from a tooth cell before its apoptosis starting time, and enters the apoptotic process after that time. The cell decreases its equilibrium area from its initial value S⁰_T and vanishes in ∼τ_A. Equation 7 is inferred from simulation results of wound closure in epithelial tissues, in which the wound area decreases exponentially (13). We assume that a cell apoptosis decreases the total system area S_TOT(t) by S⁰_T, which ensures that the average cell area does not change after the apoptotic cell has vanished. It should be noted, in essence, that we have postulated above the one-hair-cell, one-tooth-cell contact in the initial state, which is based on both the experiment (22) and a theory of intercellular signaling (27). Here, we have defined tooth cells as cells that touch hair cells and can receive an apoptosis-inhibition signal, and preapoptotic cells as cells that are distant from hair cells and cannot receive an apoptosis-inhibition signal (see Item 3 in Analysis of the Model of Zigzag Cell Patterning Proposed in this Article).

Dynamic deformation of outer frame

According to the experimental results (22) (see Fig. 1), cells at the posterior margin of the wing are elongated along the wing margin (x direction). Consequently, the aspect ratio of a cell, λ_C ≡ ℓ_X/ℓ_Y (where ℓ_X and ℓ_Y are the maximum lengths of the cell in the x and y directions, respectively), increases during the wing formation. This cell elongation must originate from the morphological change of the whole wing. In fact, as mentioned previously, several researchers showed that as the wing hinge (the basal part of the wing) contracts in the pupal stages, the wing-blade continues to elongate in the direction of the proximal-distal axis (24, 25, 26). The elongation of the wing-blade epithelial tissue can be modeled as deformation of the outer frame (system boundaries (L_X × L_Y)) in our case, which is described by a time-dependent aspect ratio λ(t) ≡ L_X/L_Y, given by

$$\mathit{\lambda }\left(t\right)\equiv \frac{L_X}{L_Y}=\left({\mathit{\lambda }}_0-{\mathit{\lambda }}_{\infty }\right)\text{exp}\left(-\frac{t}{{\mathit{\tau }}_F}\right)+{\mathit{\lambda }}_{\infty },$$ (8)

where λ₀ and λ_∞ denote the aspect ratios at t = 0 and t = ∞, respectively, with λ₀ < λ_∞, and τ_F denotes a characteristic time of the outer frame deformation.

In this work, we assume that the time variation of the aspect ratio is caused by an external force, consistent with the above-mentioned experimental facts (24, 25, 26). The three characteristic times τ_A, T_A, and τ_F introduced in the previous and present sections must satisfy

$${\mathit{\tau }}_A<<T_A<<{\mathit{\tau }}_F.$$ (9)

Equation 9 reflects that the outer frame deformation is a larger-scale phenomenon than apoptosis.

Results

In this section, we use the dimensionless physical quantities scaled by their characteristic quantities, the initial average cell size R₀, the time τ₀ during which a vertex travels about R₀ and the interfacial energy ε_I at the T-T boundary with length R₀ at the dorsoventral boundary. Denoting the initial average cell area as $\overline{S}\left(0\right)$, these new units are defined by

$${\mathit{R}}_0\mathrm{\equiv }{\left(\overline{S}\left(0\right)\right)}^{1/2},{\mathit{\tau }}_0\equiv \left(\mathit{\eta }{R}_0\right)/{\mathit{\sigma }}_{TT}^{\ast }\mathrm{,}{\mathit{\epsilon }}_I\equiv {\mathit{\sigma }}_{TT}^{\ast }{R}_0,$$ (10)

Scaling all quantities by the new units we rewrite Eq. 1 and obtain the dimensionless equation of motion for vertices, which takes the same form as Eq. 1 except that all quantities are dimensionless and η = 1. Hereafter, we use the same symbols as the original ones for dimensionless quantities.

We performed computer simulations starting with the initial state shown in Fig. 3 (see the Supporting Material) and using anisotropic boundary conditions, i.e., the periodic boundary condition in the x direction and a mirror-symmetric boundary condition in the y direction (see Appendix A). The mirror-symmetric boundary condition was newly introduced here to describe the vertex motion constrained along the horizontal system boundaries (y = 0 and y = L_Y lines were treated as mirror-symmetric lines). Parameter values for the simulations are given in Appendix B.

Snapshots

Taking Fig. 3 as the initial condition, we solved the dimensionless equations of motion for vertices, using Eq. 1 with η = 1, using the Runge-Kutta method (28), and obtained the vertex position r_i(t), three nearest-neighbor vertices, and three neighboring cells of vertex i (i = 1, 2, ⋅⋅⋅, N = 100) at each time t. All physical quantities in this simulation are dimensionless. The time step and minimum length (threshold length) were taken as δt = 0.0001 and Δ = 0.01, respectively. Our method to choose the parameter values used in this simulation are explained in Appendix B. The resulting snapshots are shown in Fig. 6.

Figure 6.

Snapshots obtained by the simulations of the vertex dynamics cell model (1–6). N_A and λ are the total numbers of preapoptotic and apoptotic cells and the aspect ratio of the system, respectively, at time t. Cell types are differentiated by colors and/or symbols, as described in Fig. 3. To see this figure in color, go online.

Fig. 6 1 shows the initial state from Fig. 3. The preapoptotic cells (N_A(0) = 20) begin apoptotic process at their randomly selected times. At t = 5 (Fig. 6 2), one apoptotic cell has already disappeared; two apoptotic cells, which are two white cells (enclosed with a circle), are undergoing apoptosis; and the remaining 17 preapoptotic cells have not yet started the apoptotic process. The hair cells (asterisk) have already elongated in the x direction, but the interhair (open circle) and tooth (plus-sign) cells remain roundish.

Fig. 6 4 shows t = 18 is just before the last two apoptotic cells disappear, i.e., one enclosed with a circle and another enclosed with a larger circle. Just after this, nonideal interfaces, II^∗, appear there but they are transient, as the former for t = 18.0–18.6 and the latter for t = 16.7–18.7, due to high interfacial energy density of the nonideal interfaces (see Appendix B). The radial patterns seen around them (enclosed with circles) are known as the rosette pattern, which was also recognized in the experiment (22). The apoptosis had completely finished at t = 18.6 and then the cell pattern rapidly converged to its approximate final state at t = 20 (Fig. 6 5) already, resulting in the final state at t = 25 (Fig. 6 6). This means that the zigzag pattern is almost completed shortly after the apoptosis, or that the apoptosis triggers the patterning. Here, the final state has been defined as a state where all the basic units (vertical H-T-I) properly interlock each other like Fig. 4 for the first time.

Discussion

In the preceding section, applying the vertex dynamics cell model we formulated the cell patterning during the pupal wing formation of Drosophila. Now we analyze our model to clarify its essential mechanical/physical picture of the cell patterning. Then we summarize several, to our knowledge, novel features added to the vertex model.

Analysis of the model of zigzag cell patterning proposed in this article

Simulation from another initial state

To examine an initial state dependence of the zigzag cell pattern, we performed another run with an initial state shown in Fig. 7 1 (called “initial state 2” hereafter), which is different from Fig. 6 1 (called “initial state 1” hereafter). In this initial state 2, hair cells on the dorsal and ventral lines are approximately confronted with each other, whereas in initial state 1, they are approximately confronted with interhair cells.

Figure 7.

Snapshots obtained by the simulations from another initial state (1) (initial state 2), where the hair cells on the ventral and dorsal lines are confronted with each other, whereas in Fig. 6 they are confronted with interhair cells. (2) An ellipse, a circle, and a dotted circle show intermittent nonideal interfaces II^∗ ⇆ TT^∗, HT^∗ ⇆ IT, and HT^∗ ⇆ IT, respectively, where the asterisk means the ventral-dorsal boundary (see the text). (3) A final state is achieved after a long time due to long-lived, intermittent nonideal edges. Note that (2) and (3) are scaled down. To see this figure in color, go online.

Using this initial state 2 with the same model parameters as Fig. 6, we carried out simulations and obtained the results shown in Fig. 7, 2 and 3. Until all apoptotic cells vanish, many anomalous cells in shape appeared and hair cells on the two horizontal lines gradually changed their relative positions, resulting in a quasi-final state shown in Fig. 7 2. The details are given in the Supporting Material.

Fig. 7 2 shows an approximate ideal zigzag pattern in the left region x < 20 (a vertical dotted line) and a nonideal pattern in the right region x > 20. The nonideal pattern is pinned by a small edge enclosed with an ellipse during t ≈ 21–30, which repeats the T1 process II^∗(nonideal) ⇆ TT^∗(ideal) where the asterisk means the v-d boundary. The II^∗ boundary survives for an extremely short time (of order 0.1) due to its high interfacial energy density. Similarly, a circle and a dotted circle indicate later such intermittent nonideal edges. These nonideal edges prevent the nonideal pattern from changing into the ideal one until their surrounding structure changes. It needed a rather long time for our system to settle to a final state shown in Fig. 7 3, t = 50, which is almost perfect.

The above results showed that the hair cells gradually avoided confrontation well and formed the ideal zigzag cell pattern. This is a cooperation between intercellular interactions and the outer frame elongation under the differential adhesion. These results suggest that our model may be potent for various cell patternings.

Roles of the apoptosis and the outer frame elongation in the zigzag cell patterning

To clarify respective roles of the apoptosis and the outer frame elongation in the zigzag cell patterning, we performed simulations in which the apoptosis only worked without the outer frame elongation for initial state 1 (Fig. 6 1) and for initial state 2 (Fig. 7 1). All conditions except no outer frame elongation were the same as those of Figs. 6 and 7.The resulting final states are shown in Fig. 8 1 for initial state 1 and Fig. 8 2 for initial state 2. There, lines connecting cell centers show the basic unit of the zigzag cell pattern, the HTI unit. Fig. 8 1 shows a pattern composed of the ideal HTI units except cell shape. On the other hand, Fig. 8 2 shows a pattern composed of the ideal HTI units except cell shape and a nonideal HTI unit (defect, an ellipse). This defect did not vanish even after t = 25. The whole pattern moved parallel to the left, keeping its shape until t = 120 at least.

Figure 8.

Snapshots obtained by the simulations under a condition that the apoptosis only works without the outer frame elongation. Lines connecting cell centers show the HTI units (1). The final state from initial state 1: the HTI units are formed except cell shape (2). The final state from initial state 2: the HTI units are almost formed except cell shape but a nonideal HTI unit is created (an ellipse) and survives for long time after this. To see this figure in color, go online.

These results clarify respective roles of the cellular events: the apoptosis creates the HTI unit, whereas the outer frame elongation arranges cell shape and helps defect annihilation as well. In fact, we carried out subsequent simulations with the outer frame elongation from t = 25 and confirmed the formation of complete zigzag cell pattern, as shown in the Supporting Material. Those movements were achieved through mechanical processes between cells under the differential cell adhesion. From the analysis mentioned above, we can describe respective roles of the two cellular events in a rather general form, as follows: the apoptosis creates a bone structure of a new pattern, and the outer frame elongation arranges cell shape under the differential cell adhesion.

The ideal zigzag cell patterning needs that one hair cell touches one tooth cell initially

To achieve the ideal zigzag cell pattern, it is necessary that one hair cell touches one tooth cell at the initial time. To show this, we performed a simulation with an initial state in which one hair cell touched two tooth cells (called a “1H2T-initial state” hereafter). This 1H2T-initial state was made from Fig. 3 by changing a preapoptotic cell (white cell), which touches hair cell 4, into a tooth cell. Here, among white cells touching hair cells we chose a cell that had the longest contact-edge.

Originally a wing margin cell should become a tooth cell by receiving apoptosis-inhibition signal from hair cells (22). A cell with longer contact-edge with a hair cell will gain more signal to be apoptosis-inhibited. In other words, a cell nearer to a hair cell is more apoptosis-inhibited.

Fig. 9 shows a final state obtained by the simulation in which all conditions are the same as those of Fig. 6 except the 1H2T-initial state. Lines connecting cell centers indicate the HTI units, as in Fig. 8. A defect appears around hair cell 4, although the HTI units are mostly formed. An incomplete HT pair (ellipse) and an isolated interhair cell (circle) are formed there. Such a defect may appear in real system.

Figure 9.

A final state obtained by the simulations from a 1H2T-initial state, where one hair cell touches two tooth cells. Lines connecting cell centers show the HTI units. An ellipse shows an incomplete HT pair and a circle shows an isolated interhair cell. To see this figure in color, go online.

Novel features added to the vertex dynamics cell model

• 1)Theoretical formulas of interfacial energy densities derived here reproduced qualitative features of the zigzag cell pattern experimentally observed, i.e., the vertical HTI units and the cell shapes horizontally elongated.
• 2)Apoptosis was described by gradually reducing the equilibrium areas of the cells, leading to gradual apoptotic processes and gradual zigzag cell patterning (apoptosis-triggered patterning). To satisfy the matter conservation law at this time, we assumed that the contents of those dead apoptotic cells were discharged through the cell membranes outside our enclosed system (15).
• 3)The outer frame of the system was elongated by an external force during the patterning, which described wing elongation during morphogenesis. The outer frame elongation promoted the formation of zigzag cell pattern. An almost same mechanism was reported in Drosophila pupal wing epithelium: the anisotropic tissue stress promotes hexagonal cell packing (29). These exemplify patterning under a moving system.
• 4)Mirror-symmetric boundary condition effectively described the vertex motion along the horizontal system boundary (see Appendix A).

Conclusions

Overall conclusion

The zigzag pattern of hair cells in Drosophila wing is formed by an autonomous communicative process/physical process among the participant cells. In the course of this study, we have established a framework to study apoptosis-triggered patterning during morphogenesis.

Possible future direction

Determining the relative values of interfacial energy densities from a micrograph

We derived five simultaneous balance equations of the forces acting on the vertices with nine unknown quantities, resulting in four independent unknown quantities (see the Supporting Material). If we can measure three angles in a micrograph experimentally obtained, we can substitute them into the balance equations. Because there remains only one unknown interfacial energy density we can obtain the relative values of all interfacial energy densities by scaling all others by the unknown one.

Relating the interfacial energy densities to the component molecules at the interfaces

During the zigzag cell patterning of Drosophila wing margin, cadherin and other cell-adhesion molecules accumulated at the cell boundaries around the hair cells (22). If the amount of these accumulated molecules can be measured at each cell boundary and related to the interfacial energy densities, we can quantify the interfacial energy densities.

Describing apoptosis by signaling from hair cells

In this article, we determined preapoptotic cells, which were distant from hair cells, as candidates for apoptotic cells initially and made them start apoptotic processes at random time points. This is a simple model that describes apoptosis-inhibition signaling by diffusion between cells, because preapoptotic cells behave as tooth cells do until they start their respective apoptotic processes at random time points. However, such apoptotic events should be described by using a theory of intercellular signaling like (27), because circumstances around hair cells vary from moment to moment until all apoptotic cells vanish. This is a challenging subject.

Appendix A: Anisotropic Boundary Conditions

Referring to Fig. 3, we explain the boundary conditions imposed on our simulation. The experimental micrographs show that, before apoptosis in the wing-margin formation stage, several hundred cells align along the x axis (Fig. 1 3 shows a part of them), whereas ∼10 cells align along the y axis (several thousand cells in total) (22). In the simulated cell system, we align ∼10 cells along the x axis and five cells along the y axis (50 cells in total), as shown in Fig. 3. To describe the real system by this small 50-cell system, we impose the periodic boundary condition in the x direction and the mirror-symmetric boundary condition in the y direction.

The periodic boundary condition represents a macroscopic system by a small number of basic units, and is generally used in statistical physics. In our case it will work well due to short-range mechanical interactions between vertices and also due to next-nearest neighbor coupling of signaling between cells (27). The mirror-symmetric boundary condition describes the experimental situation in which vertices on the horizontal system boundaries (y = 0 and y = L_Y lines) move along the boundaries due to the wing elongation (22, 24).

Appendix B: Parameter Values

The total system area at the initial time is the sum of the equilibrium cell areas of individual cell types: S_TOT(0) = N_HS_H⁰ + N_IS_I⁰ + N_TS_T⁰ + N_A(0)S_T⁰, where N_μ is the number of μ-type cells and time-independent except for μ = A. The parameter values are set as follows: L_X(0) = 20, L^Y(0) = 10; N_H = N_I = N_T = 10, N_A(0) = 20; and S_H⁰ = S_I⁰ = 4.21, S_T⁰ = 3.86. Tooth cells are defined as cells that touch hair cells and can receive apoptosis-inhibition signal from them. The remaining cells N_A(0) are preapoptotic cells, which are distant from hair cells and cannot receive an apoptotic inhibition signal. Initially, preapoptotic cells are not distinguished from tooth cells. We adopt a smaller value of S⁰_T than the other according to the experimental result, Fig. 1 4.

Although our system evolves from the initial to its final state, some of the cell boundaries deviate from those of the ideal zigzag cell pattern. Therefore, we need to assign them interfacial energy densities. First, apoptotic cells should have the same interfacial energy densities as those of tooth cells, because they derive from the same origin. Assuming that these interfacial energy densities equal σ^∗_TT, we have σ_TT = σ_TA = σ_AA = 1.00 (see Fig. 3).

The annihilation of apoptotic cells may lead to nonideal cell boundaries near the dorsoventral boundary. As those boundaries eventually disappear, they must have large interfacial energy densities. Therefore, we assume σ_HH = σ_II = σ^∗_HT = ⋅⋅⋅ = σ_MAX, where σ_MAX is larger than any interfacial energy density of the ideal zigzag cell pattern. Under this condition, a nonideal boundary disappears immediately after its emergence.

Now we calculate the interfacial energy densities of the ideal zigzag cell pattern by (5a), (5b), (5c) and 6. By measuring angle α of the zigzag cell pattern in the micrograph, Fig. 1 4, we obtained average values of the angle, α = 37°. Varying another unknown parameter ε from 0.1 to 1.9, we observed the basic unit of the zigzag pattern, the vertical HTI unit, and found the best value ε = 1.5. As a result, we have the other six quantities via (5a), (5b), (5c): σ_HI = 1.10, σ_HT = 1.63, σ_IE = 1.76, σ^∗_IT = 1.00, σ^∗_IT = 1.50, and σ_TT = 1.00. For nonideal cell boundaries, we adopt σ_MAX = 6.00, which is large enough compared with all σ-values because it gives a lifetime of order 1/2σ_MAX ≈ 0.08 for an average edge with length 1.

The threshold length of elementary processes is set to Δ = 0.1, the minimum length of our coarse-grained model, for both T1 and T2 processes in Fig. 2. In the case of T1, the new edge is placed perpendicular to the old edge at the center of the old edge. The new edge length is infinitesimally elongated as 1.1Δ to avoid occurrence of the reverse process within a few time steps.

The volume elastic constant and equilibrium cell height values in Eq. 4 are taken as ρ = 5.5 and h⁰ = 1.0, respectively, according to the values obtained in the simulation of epithelial morphogenesis (9) and wound closure (13).

The characteristic times of the apoptotic process, (τ_A, T_A), and the outer frame elongation, τ_F, are set as τ_A = 0.5, T_A = 15, and τ_F = 100, and the aspect ratios of the system (λ₀, λ_∞) as λ₀ = 2 and λ_∞ = 30, taking account of the postulate of Eq. 9.

Author Contributions

H.H. and T.N. designed the research. T.N. and H.H. performed the research. M.T. offered Fig. 1. M.T. and H.H. contributed interpretations of the experiment. T.N., H.H., and M.T. wrote the article.

Editor: Stanislav Shvartsman.