Mathematical Study of the Role of Delta/Notch Lateral Inhibition during Primary Branching of Drosophila Trachea Development

Abstract

A wide range of cellular developmental processes employ intercellular signaling via the Delta/Notch lateral inhibitory pathway to achieve stable spatial patterning. Recent genetic experiments have shown the importance of Delta/Notch lateral inhibition for regulating the number of tip cells in the tracheal primary branching of Drosophila. To examine the role of Delta/Notch regulation in the tip-cell selection, we analyzed a mathematical model of a simple lateral inhibitory system having input signals. Mathematical and numerical analyses revealed that the lateral inhibition did not amplify the signal difference between neighboring cells over the parameter ranges in which the spatial pattern of tip selection was realized. We also show that the number of tip cells becomes less affected by a fluctuation of the input gradient signal as the lateral inhibition becomes stronger. In addition, we demonstrate that the lateral inhibitory regulation enhances the robustness of the tip-cell selection compared with a system regulated by self-inhibition, an alternative means of inhibitory regulation. These results suggest that the lateral inhibition promotes the robustness of tip-cell selection in the tracheal development of Drosophila.

Introduction

Lateral inhibitory regulation through the Delta/Notch signaling pathway plays important roles in cell differentiation in diverse cellular and developmental phenomena (1–4). Notch is a trans-membrane protein receptor for its ligand Delta that expresses on the cellular membrane. Delta on a cell surface binds to Notch on the neighboring cell and activates the Notch signaling pathway, whose downstream factors inhibit the production of Delta. This molecular interaction between adjacent cells is observed in a broad range of developmental processes for generating fine-grained spatial patterns (5–9) or for buffering biochemical noises (10,11).

Recent experimental studies have shown that the Delta/Notch lateral inhibition mechanism participates in the cell-fate determination of multiple neighboring cells in the trachea development of Drosophila embryos (12–15). In the early phase of this development, several epithelial cells differentiate as tip cells that can migrate and drive primary branching. This is called “tip-cell selection” (Fig.1, a–c). It has been considered that the tip-cell selection is induced mainly by a signal activated through a composite stimulus, including a diffusible chemoattractant Branchless (Bnl) produced from mesodermal cells surrounding each sac and a tyrosine kinase receptor Breathless (Btl) expressed on the epithelial cells. On the downstream of Bnl/Btl signal, the tip-cell signal is inhibited through Delta/Notch lateral inhibition (12,13,16). In other words, the tip-cell signal is stimulated through the Bnl/Btl signaling in a cell; the signal is also inhibited by other adjacent cells through Delta/Notch lateral inhibition (Fig. 1 d). Genetic analyses have shown that the lateral inhibition controls the selected number of tip cells. In these analyses, a gain-of-function form of Notch activity failed to perform primary branching; in contrast, a thermosensitive mutation involving an inactivation of Notch caused a larger number of tip cells than in the wild-type (13,15,17). From such analyses, it has been argued that Delta/Notch lateral inhibition functions to amplify differences in Bnl/Btl signaling between adjacent cells, leading to the tip-cell selection (15,17).

Figure 1.

Schematic representation for the tip-cell selection and model system. (a) Drosophila embryo at stage 11, 6 h after fertilization. The embryo has 10 pairs of air sacs on the lateral side. (b) Magnification view of a single air sac, corresponding to the subwindow in panel a. Bnl is expressed in the mesoderm along the air sac regularly. Some of the epithelial cells in the air sac migrate to the Bnl expression. (c) Tip-cell selection occurs when the epithelial cells receive the Bnl signal. Note that multiple neighboring cells closer to the Bnl expressing cells are selected as tip cells. (d) The tip-cell signal activated by the Bnl/Btl signal inhibits neighboring cells through Delta/Notch signaling.

Genetic experiments have been used to examine the roles of lateral inhibition in Delta/Notch, but genetic experiments alone are not enough to capture the properties of the Delta/Notch regulation. A complex biological network underlying the process of tip-cell selection may blind us to its true nature. As extensively studied by experimental and mathematical approaches, signaling properties determined by a network’s structure might be spontaneously altered in a molecular regulatory network. This alteration of the signaling modularity is called “retroactive effects” (18–23). It cannot be ruled out that Delta/Notch regulation has some retroactive effects on other signaling modules. Experimental operations derived from in vivo analyses might alter not only the Delta/Notch signal but also other signals concomitantly induced by the same signal, and there is a possibility that the tip-cell selection occurs because of those reactions. The construction and analysis of mathematical models can be useful for focusing on the roles of Delta/Notch lateral inhibitory regulation and it is helpful to combine the conclusions obtained by the mathematical model with knowledge achieved by experiments.

Mathematical models previously proposed to explain the lateral inhibition mechanism have mainly focused on the conditions for generating spatial patterning, such as checkerboard-like pattern formation during the development of Drosophila sensory bristles, sharp boundary formation of Drosophila wing veins, and angiogenic sprouting formation (11,24–31). All of these mathematical works have been based on heterogeneous spatial patterns caused by Delta/Notch lateral inhibition. Another regime of the lateral inhibitory regulation, in which interaction between an external soluble factor and a lateral inhibition factor is involved in cell-fate determination during vulval development in Caenorhabditis elegans, has been studied by Giurumescu et al. (32). They developed a mathematical model, and revealed gradient amplification mechanisms with coupling systems that guided them to cell-fate segregation.

In this article, we study a mathematical model, which is of a more general framework than the model developed by Giurumescu et al. (32), to examine the roles of the Delta/Notch system in the tip-cell selection during the primary branching phase of Drosophila trachea development. We analyzed a mathematical model of a simple lateral inhibitory system along with the spatial gradient of its input stimulus (Bnl/Btl signal). As the pattern of differentiated tip cells is adjoined, we conducted the analysis for parameter ranges in which the lateral inhibition did not lead to the zig-zag patterning. We show that the lateral inhibitory regulation contributes to the robustness of the tip-cell selection when the input signal includes random noises. Furthermore, we demonstrate that the lateral inhibitory regulation enhances the robustness of the tip-cell selection compared with a system regulated by self-inhibition, an alternative means of inhibitory regulation.

Models and Results

To focus on the roles of lateral inhibitory regulation in tip-cell selection, we construct a simple mathematical model that includes a minimal number of components of chemical reactions, which can be handled analytically.

Mathematical modeling

We consider a system in which N+1 cells are arrayed in one-dimensional space with j being an index for each cell (j = −N/2, …, N/2). The Bnl signal is given from mesodermal cells surrounding the epithelia and its binding form with Btl transduces an intracellular signal (Bnl/Btl signal). The signal has its peak at a cell in the central position and is smaller for cells more distant from the center (see Fig. 1, a–c, for illustration). Specifically, we consider that the level of the Bnl/Btl signal received by cell j is given as

$$L_j=d+\exp \left(g\{1+\cos \left(\frac{2\pi j}{N}\right)\}\right)-1,$$ (1)

where d and g are the basal level of the Bnl/Btl signal at epithelia and the magnitude of Bnl/Btl signal gradient, respectively. The Bnl/Btl signal activates the expression of Delta interacting with its neighbors’ Notch that induces the suppression of its tip-cell signal, i.e., each cell is laterally inhibited via Delta/Notch signal. We regard the tip-cell signal at equilibrium as a marker signal for the tip-cell specification. In the earlier experimental studies, the expressions of different chemical species such as MAPK, Escargot, or Pointed have been adopted as tip-cell markers (16,33–35). Because there is no consensus, at the time of this writing, over which molecular species are suitable as tip-cell markers, we lump them together in the downstream of Bnl/Btl signaling with the tip-cell signal without specifying the identify of the marker. We model the dynamics of the tip-cell signal x and those of Delta/Notch signal y (Fig. 1 d) as

$$\frac{d{x}_j}{dt}=F\left(x_j,{\langle y\rangle }_j;L_j\right)=L_j\frac{1}{1+{\left(c{\langle y\rangle }_j\right)}^h}-x_j,$$ (2a)

$$\frac{d{y}_j}{dt}=G\left(x_j,y_j\right)=a{x}_j-b{y}_j,$$ (2b)

where a, b, c, and h are dimensionless parameters. (For additional reference, see The Original Model and Nondimensionalization Procedure.) The value 〈y〉_j in Eq. 2a indicates the level of Delta/Notch signal that cell j receives from its two neighbors, given by 〈y〉_j = (y_j−1 + y_j+1)/2 for j = −N/2 + 1, …, N/2 − 1. Because Bnl-expressing cells are clustered at regular intervals along the epithelial sac (Fig. 1 b), we consider the system to have periodic boundaries. We modeled the inhibitory regulation of tip-cell signal by the Delta/Notch signal as a Hill function.

We here define a system in which the differentiation state of each epithelium, i.e., whether the cell should become a tip cell or not, is determined according to the level of the tip-cell signal in the steady state ${\hat{x}}_j$ with a threshold. That is, the cell becomes a tip-cell when ${\hat{x}}_j$ is larger than the threshold, but the cell j remains a nontip-cell (or “stalk cell”) when ${\hat{x}}_j$ is smaller than the threshold, as suggested previously (34). In the following sections, we focus on the level of the tip-cell signal in the steady-state ${\hat{x}}_j$.

The tip-cell signal caused by a weak gradient of the input signal

To evaluate the response of tip-cell signal x_j caused by a gradient of the Bnl/Btl signal L_j in the current system, we first derive the solution at the steady-state ${\hat{x}}_j$ followed by a weak gradient of L_j. When there is a uniform input, L_j = d, the equilibrium solution of the tip-cell signal is also uniform. Then, assuming an input with a weak gradient, we can calculate a small deviation from the uniform solution in the steady-state ${\delta }_j^{LI}$ as

$${\delta }_j^{LI}=\frac{g\gamma }{1+2\beta }\left(1+\frac{1+2\beta }{1+2\beta \cos \left(2\pi /N\right)}\cos \left(\frac{2\pi j}{N}\right)\right),$$ (3)

where $\beta =acdh{\left(ca\hat{x}/b\right)}^{h-1}/2b{\left\{1+{\left(ca\hat{x}/b\right)}^h\right\}}^2$ and $\gamma =1/\left(1+{\left(ac\hat{x}/b\right)}^h\right)$. (For further reference, see the subsection Derivation of Eq. 3, given later in the article.)

For some parameter choices, ${\hat{x}}_j$ does not have a monostable solution and the distribution can show a zig-zag patterning in the steady state: high and low values of the tip-cell signal alternate between neighboring cells. Mathematical conditions for such a result have been extensively studied (24,25). However, cell patterning with such an alternating expression of the tip-cell marker has not been observed in tracheal primary branching; the marker expression has only a single peak. Therefore, we here eliminate cases in which solutions have multiple peaks such as a zig-zag pattern and focus on a solution with a single peak that is also symmetric. According to our mathematical analysis, the pattern of our interest occurs when 2β < 1 (see Linear Stability Analysis). We consider a range j = 0, …, N/2 in the following analysis because we are interested in the symmetric spatial distribution around j = 0.

Lateral inhibition does not amplify differences in the tip-cell signals between neighboring cells

According to previous studies, the logic of tip selection is based on the effect of amplifying a small difference in the tip-cell signal between neighboring cells through lateral inhibition (12,13,15,17). To determine whether (or not) the lateral inhibition does indeed amplify the difference in the tip-cell signal, we analyzed the dependence of the difference on the intensity of lateral inhibition c.

We define the difference of the tip-cell signal between neighboring cells as $\Delta {\hat{x}}_j^{LI}\approx {\hat{x}}_j-{\hat{x}}_{j+1}$ (Fig. 2 a). Note that $\Delta {\hat{x}}_j^{LI}\approx {\delta }_j^{LI}-{\delta }_{j+1}^{LI}$ when the gradient of the input signal is small. From Eq. 3, we can obtain

$$\Delta {\hat{x}}_j^{LI}\approx g\gamma \left[\frac{cos\left(2\pi j/N\right)-cos\left(2\pi \left(j+1\right)/N\right)}{1+2\beta cos\left(2\pi /N\right)}\right],$$ (4)

and its derivative with respect to c is negative:

$$\frac{\partial \Delta {\hat{x}}_j^{LI}}{\partial c}<0.$$ (5)

The last inequality implies that the lateral inhibition always reduces the difference of ${\hat{x}}_j$ between neighboring cells. (For reference, see Derivation Procedure of Eq. 5, found later in this article.)

Figure 2.

Lateral inhibition does not amplify the difference of the tip-cell signal between neighboring cells. (a) A typical dependency of the distribution of the tip-cell signal to the intensity of lateral inhibition. The case of c = 0 is shown in blue, c = 0.1 in green, and c = 1 in red. (b) Numerical results for the dependence of ${\hat{x}}_j$ to parameter c and (c) for the dependence of $\Delta {\hat{x}}_j$ to parameter c. The lateral inhibition reduces both the level of the tip-cell signal and its difference between neighboring cells (p-value ≪ 0.01, Jonckheere-Terpstra trend test). Note that the value is normalized by 1 when c = 0 for each case. The symbols 〈⋅〉 denotes the sample average. The number of samples is ∼10,000 for each value of parameter c.

Contrary to the widespread understanding of the effect of lateral inhibition, our analysis concludes that the lateral inhibition does not amplify the difference of the tip-cell signal between neighboring cells; rather, it reduces the difference of the tip-cell signal (Fig. 2 a). This counterintuitive result was confirmed by extensive numerical analyses as well (Fig. 2, b and c; p-value ≪ 0.01, Jonckheere-Terpstra trend test).

These results indicate that the lateral inhibition would increase the susceptibility of tip-cell selection in certain cases. Let us consider the threshold-based tip-cell determination rule as described in Mathematical Modeling, and also assume that there is a fluctuation of the threshold. The fluctuation of the threshold may alter the number of tip cells when the between-cell difference of the tip-cell signal is small, although the number of tip cells can be less affected when large. The results imply that the lateral inhibition might reduce the robustness of tip-cell selection.

Lateral inhibition reduces the susceptibility of tip-cell selection to random variation in the input signal

In the last section we showed that the between-cell difference in the tip-cell signal was reduced by the lateral inhibition, and that the lateral inhibition may have contributed to the weakening of the robustness of the tip-cell selection. We here focus on another aspect of this issue—namely, that the lateral inhibition can stabilize the spatial pattern of tip cells even when there is a fluctuation of input Bnl/Btl signal L_j.

Let ${\hat{x}}_j^{\prime d}$ and ${\hat{x}}_j^{\prime g}$ be the amount of change in ${\hat{x}}_j$ caused by a unit amount of change in parameters d and g, respectively. After some calculation (for reference, see Derivation Procedure of the Expressions in Eq. 6), we can derive the following inequalities:

$$\frac{\partial {\hat{x}}_j^{\prime d}}{\partial c}<0,$$ (6a)

$$\frac{\partial {\hat{x}}_j^{\prime g}}{\partial c}<0.$$ (6b)

The expressions in Eq. 6 imply that the changes in the tip-cell signal caused by the variations in d and g are reduced by the lateral inhibitory regulation, suggesting the possible importance of the lateral inhibition in realizing the robust development.

To determine whether the influence of lateral inhibition inferred by the expressions in Eq. 6 is sufficiently effective, we performed numerical analysis of the original nonlinear dynamics. We set standard parameters p^∗ =(c^∗, d^∗, g^∗) and examined quantities measuring the change of values of a parameter that evaluated the robustness of tip-cell selection (Fig. 3 a). We adopted two different methods for evaluating the robustness. The first method is to use the fraction of the cases in which the number of selected cells n does not change. Here we examined cases of two different levels for the threshold: n = 3 (3cells-threshold) and n = 5 (5cells-threshold). Those thresholds are relatively determined by the tip-cell signals given by a certain parameter set p^∗. The 3cells-threshold is defined as

$${\varphi }_{n=3}\left({\mathbf{p}}^{\ast }\right)=\frac{\left({\hat{x}}_1+{\hat{x}}_2\right)}{2}$$

and the 5cells-threshold is defined as

$${\varphi }_{n=5}\left({\mathbf{p}}^{\ast }\right)=\frac{\left({\hat{x}}_2+{\hat{x}}_3\right)}{2}.$$

Let us now consider the tip cells are selected by 3cells-threshold ϕ_n=3(p^∗). Taking an example as shown in Fig. 3 b, we examine the number of cells whose tip-cell signal over ϕ_n=3(p^∗) becomes 1 from 3 when a single parameter changes to p⁻. Note that the number stays when the parameter changes to p⁺.

Figure 3.

Numerical simulation for evaluating the robustness of tip-cell selection to the intensity of inhibition. (a) Illustration of numerical examination. One of the parameters among c, d, and g changes with some variation, denoted as [var], although the others are fixed. (b) A demonstration that the variation of the parameter may change the number of selected tip cells. (c–h) Quantities evaluating robustness for the tip-cell selection increase for the intensity of inhibition c. Robustness index (3cells-threshold/5cells-threshold) is the fraction of cases where the tip-cell number does not change when a particular parameter fluctuated. We show the results for 3cells-threshold and 5cells-threshold separately. (c–e) Cases for the fluctuation of parameter d. (f–h) Cases for the fluctuation of parameter g. For the increase of the intensity of inhibition c, the robustness for the tip-cell selection increases in the lateral inhibitory regulation (black), whereas it does not increase in the self-inhibitory regulation (white). The number of samples is ∼10,000 for each value of c. The error bars are too small to be seen.

We use the mean deviation of the tip-cell signal from 1 in the standard parameter sets, i.e.,

$$\Omega =\left(\frac{\sum _{j=-N/2}^{N/2}{\hat{x}}_j^{\prime m}}{\left(N+1\right)}\right)$$

for m = d, g. We call those deviations the “ robustness indexes”. The standard parameters were sampled randomly from a fixed plausible parameter range (for reference, see Conditions for Numerical Calculation: Ranges and Sampling of Parameters for Numerical Analysis). The variance is denoted var; we examined var = 1.2, 1.4, and 1.6.

We found that these quantities for parameter sensitivity monotonically increased as the intensity of lateral inhibition c increased (Fig. 3, c–h). Note that we plot the inverse of the sample average of Ω, and this inverse value indicates the magnitude of robustness. With respect to the fluctuation of parameter d, for any value of variance we examined, the robustness almost always increased as c increased (Fig. 3, c–e). For the fluctuation of parameter g, 1/〈Ω〉 clearly increased with c, and the ratio of 3cells-threshold and 5cells-threshold for some values of var hardly changed even when c varied greatly (Fig. 3, f–h). Note that, for all cases, the quantities for evaluating the robustness of tip-cell selection never decrease with increasing c.

In short, the quantities used to measure the robustness of tip-cell selection tend to increase along with the increase of the intensity of lateral inhibition. Indeed, numerical analyses support the notion that the lateral inhibition reduced the susceptibility of tip-cell selection to the variance of the input signal level.

Lateral inhibitory regulation achieves a more-robust tip-cell selection than self-inhibitory regulation

To determine the properties of the lateral inhibitory regulation more clearly, we here compared the lateral inhibitory regulation with an alternative regulation: a self-inhibitory regulation, composed of Eqs. 1 and 2 in which we set 〈y〉_j = y_j. For example, the main regulator for the negative feedback loop in this regime can be considered as Sprouty or Hes/her genes that relate to spatio-temporal patterning (36–39). Note that this system has the same uniform solution as produced in the lateral inhibitory system with a spatial uniform input signal. We compared two regulatory systems with respect to both the robustness of tip-cell selection for fluctuation of the threshold level and that for fluctuation of the input level.

Using the same approach as explained in Derivation of Eq. 3, the deviation of the tip-cell signal from the uniform solution in the self-inhibition system ${\delta }_j^{SI}$ can be obtained as

$${\delta }_j^{SI}=\frac{g\gamma }{1+2\beta }\left(1+\cos \left(\frac{2\pi j}{N}\right)\right).$$ (7)

For the purpose of evaluating the robustness of the self-inhibitory regulation relative to the lateral inhibitory regulation when the threshold level fluctuates, we here introduce the ratio $\Delta {\hat{x}}_j^{LI}/\Delta {\hat{x}}_j^{SI}$. When the gradient of the input signal is small, we can calculate this quantity analytically as

$$\frac{\Delta {\hat{x}}_j^{LI}}{\Delta {\hat{x}}_j^{SI}}\approx \frac{1+2\beta }{1+2\beta \cos \left(2\pi /N\right)}\ge 1$$ (8)

for N ≥ 4. This inequality clearly indicates that the difference of ${\hat{x}}_j$ between neighboring cells in lateral inhibitory regulation is larger than that in self-inhibitory regulation. We confirmed this result by extensive numerical analyses: $\Delta {\hat{x}}_j^{LI}/\Delta {\hat{x}}_j^{SI}$ was never less than 1, and $\Delta {\hat{x}}_j^{LI}/\Delta {\hat{x}}_j^{SI}$ increased with the increase of the parameters c, d, and g, except for the edge region j = 3, in which tip-cell selection does not occur (Fig. 4, a–c). The discrepancy between the analytical result and the numerical calculation results from the input gradient profiles (see the Section S1 of the Supporting Material for more details). In conclusion, the robustness of tip-cell selection for the fluctuation of the threshold level is enhanced in lateral inhibitory regulation compared with that in self-inhibitory regulation, and the degree of enhancement becomes larger with the increase in the intensity of inhibition c, the increase in the basal level of input d, or the increase in the intensity of the gradient of input g.

Figure 4.

The lateral inhibitory regulation enhances the robustness for the tip-cell selection compared with the self-inhibitory regulation. (a–c) Dependence of $\Delta {\hat{x}}_j^{LI}/\Delta {\hat{x}}_j^{SI}$ on the parameters c, d, and g. The horizontal axis is (a) parameter c, (b) parameter d, and (c) parameter g, and the vertical axis is $\langle \Delta {\hat{x}}_j^{LI}/\Delta {\hat{x}}_j^{SI}\rangle$. The symbols 〈⋅〉 denotes the sample average. Note that $\langle \Delta {\hat{x}}_j^{LI}/\Delta {\hat{x}}_j^{SI}\rangle$ is never less than 1 for any of the parameters, and becomes larger for the increase of parameter c. (d–f) Robustness for the tip-cell selection against the fluctuation of parameter c. (Black bars) Lateral inhibitory regulation; (white bars) self-inhibitory regulation. For any variance the lateral inhibition leads to a more-robust regulation compared with the self-inhibition. The sample number is ∼10,000 for each value. The error bars are too small to be seen. (g) A typical distribution of ${\hat{x}}_j$ generated by the lateral inhibitory system (red) and by the self-inhibitory system (blue). (h) Magnification view of the subwindow in panel g. The tip-cell signal in the lateral inhibitory system becomes smaller than that in the self-inhibitory system in the distant region of the center of distribution; in the center, the opposite occurs.

Similarly, we found that the lateral inhibitory regulation system can perform a more-robust regulation against fluctuation of the input signal compared with the self-inhibitory regulation system. The robustness in the case of lateral inhibitory regulation is clearly larger than that in self-inhibitory regulation for each of parameters d, g (Fig. 3, c–h), and c (Fig. 4, d–f). Furthermore, the difference between the robustness of lateral inhibitory regulation (black) and that of self-inhibitory regulation (white) gets larger for the large intensity of inhibition (Fig. 3, c–h). These results can be explained by the comparison between the distribution of the tip-cell signal produced in lateral inhibitory regulation and that produced in self-inhibitory regulation (Fig. 4 g). Note that the tip-cell signal in lateral inhibitory regulation is larger than that in self-inhibitory regulation around the center of the domain although the opposite occurs around the edge of the domain (Fig. 4, g and h). We can give the following analytical expression corresponding to this as

$$\begin{array}{l}\frac{{\delta }_j^{LI}}{{\delta }_j^{SI}}\ge 1\text{for}0\le j\le \frac{N}{4};\\ \frac{{\delta }_j^{LI}}{{\delta }_j^{SI}}<1\text{for}\frac{N}{4}<j\le \frac{N}{2}.\end{array}$$ (9)

These results demonstrate that lateral inhibitory regulation has a property to amplify the output differences between neighboring cells compared with that of self-inhibitory regulation. We confirmed that the inequalities also hold in the nonlinear system (see the Supporting Material). By this mechanism, lateral inhibitory regulation achieves a robust tip-cell selection for the fluctuation of input level compared with self-inhibitory regulation.

Methods

The original model and nondimensionalization procedure

We modeled the dynamics of Bnl/Btl signal and Delta/Notch signal as

$$\frac{d{x}_j}{d\tau }=a_x{l}_j\frac{K^h}{K^h+{\left(C{\langle y\rangle }_j\right)}^h}-d_x{x}_j,$$ (10)

$$\frac{d{y}_j}{d\tau }=a_y{x}_j-d_y{y}_j,$$ (11)

where l_j is level of Bnl/Btl input at a cell j, K is a constant of reaction, C is the magnitude of inhibition from the neighbors, h is the Hill coefficient, and a_x (a_y) and d_x (d_y) are each positive constants for productive rate and decay rate of Bnl/Btl signal (Delta/Notch signal). Because we assume that the regulation of tip-cell signal by Bnl/Btl input signal and that by the Delta/Notch inhibition are not completely independent (i.e., they are somehow linked on the signaling pathway), the regulation is given by the product. Also, we assumed that epithelial cells equally interact only with its neighbors via the Delta/Notch signal arrayed in one-dimension with the periodic boundary. We obtained the expressions in Eq. 2 through the reduction of the number of parameters: L_j = a_xl_j/d_x, t = τd_x, c = C/K, a = a_y/d_x, and b = d_y/d_x.

Derivation of Eq. 3

We derive the small deviation of the tip-cell signal from a homogeneous solution in the steady-state ${\delta }_j^{LI}$ under an assumption that the homogeneous steady state is stable.

At the outset, let us consider the deviation from the uniform external input value θ_j with the assumption of its weak gradient:

$$\begin{array}{c}{\theta }_j=\text{exp}\left(g\left\{1+\cos \left(\frac{2\pi j}{N}\right)\right\}\right)-1\\ \approx g\left\{1+\cos \left(\frac{2\pi j}{N}\right)\right\}=g\left\{1+\frac{\left(e^{i2\pi j/N}+e^{-i2\pi j/N}\right)}{2}\right\}.\end{array}$$ (12)

Next, we consider the dynamics of δ_j and ε_j, the small deviation of x_j and y_j generated by θ_j, which can be obtained by linearization of its homogeneous state, respectively, as

$$\frac{d{\delta }_j}{dt}\approx -{\delta }_j+\frac{\partial F}{\partial y}\left({\epsilon }_{j-1}+{\epsilon }_{j+1}\right)+\frac{\partial F}{\partial L_j}{\theta }_j,$$ (13a)

$$\frac{d{\epsilon }_j}{dt}\approx a{\delta }_j-b{\epsilon }_j,$$ (13b)

where $\partial F/\partial y\equiv \partial F/\partial y_{j-1}=\partial F/\partial y_{j+1}$.

The steady state of the expressions in Eq. 13 leads to the equation

$${\delta }_j+\beta \left({\delta }_{j-1}+{\delta }_{j+1}\right)=\gamma {\theta }_j,$$ (14)

where β = −(∂F/∂y)a/b, γ = ∂F/∂L_j.

Now, we convert variables using discrete Fourier transform,

$${\delta }_j=\sum _{k=0}^{N-1}{A}_k{e}^{i\frac{2\pi kj}{N}},{\epsilon }_j=\sum _{k=0}^{N-1}{B}_k{e}^{i\frac{2\pi kj}{N}},{\theta }_j=\sum _{k=0}^{N-1}{C}_k{e}^{i\frac{2\pi kj}{N}},$$ (15)

where

$${\mathit{A}}_{\mathit{m}}=\frac{1}{N}\sum _{j=0}^{N-1}{\delta }_j{e}^{-i\frac{2\pi mj}{N}},{\mathit{B}}_{\mathit{m}}=\frac{1}{N}\sum _{j=0}^{N-1}{\epsilon }_j{e}^{-i\frac{2\pi mj}{N}},{\mathit{C}}_{\mathit{m}}=\frac{1}{N}\sum _{j=0}^{N-1}{\theta }_j{e}^{-i\frac{2\pi mj}{N}}.$$

Note that j changes its range from 0 to N−1 without loss of generality.

Substituting Eq. 15 into Eq. 14 yields

$$A_m=\frac{\gamma C_m}{1+2\beta \cos \left(2\pi m/N\right)}.$$ (16)

After some calculation, we have

$$\sum _{m=0}^{N-1}{A}_m{e}^{i2\pi mj/N}=\sum _{m=0}^{N-1}\frac{\gamma C_m{e}^{i2\pi mj/N}}{1+2\beta \cos \left(2\pi m/N\right)}.$$ (17)

Substituting Eq. 15 into Eq. 17 yields

$${\delta }_j=\frac{\gamma }{N}\sum _{m=0}^{N-1}\frac{\sum _{k=0}^{N-1}{\theta }_k{e}^{i2\pi m\left(j-k\right)/N}}{1+2\beta \cos \left(2\pi m/N\right)}.$$ (18)

Using Eq. 12, we can calculate a part of Eq. 18 as

$$\sum _{k=0}^{N-1}{\theta }_k{e}^{i2\pi m\left(j-k\right)/N}=g\left[\sum _{k=0}^{N-1}{e}^{i2\pi mj/N}{e}^{-i2\pi mk/N}+\frac{1}{2}\sum _{k=0}^{N-1}{e}^{i2\pi mj/N}{e}^{i2\pi \left(1-m\right)k/N}+\frac{1}{2}\sum _{k=0}^{N-1}{e}^{i2\pi mj/N}{e}^{-i2\pi \left(1+m\right)k/N}\right].$$ (19)

The first term on the right-hand side of Eq. 19 becomes zero except when m = 0 with a geometric series. Similarly, the second and the third terms also become zero except when m = 1 and m = N−1, respectively.

Therefore, Eq. 19 becomes

$${\delta }_j=g\gamma \left(\frac{1}{1+2\beta }+\frac{\left(e^{i2\pi j/N}+e^{-i2\pi j/N}\right)/2}{1+2\beta \cos \left(2\pi /N\right)}\right),$$ (20)

which is the same as Eq. 3.

Linear stability analysis

We here consider the condition under which the homogeneous solution is stable. Substituting Eq. 15 into Eqs. 13a and 13b, without the third term of Eq. 13a, yields

$$\frac{d{A}_m}{dt}=-A_m+2\frac{\partial F}{\partial y}\cos \left(\frac{2\pi m}{N}\right)B_m,$$ (21a)

$$\frac{d{B}_m}{dt}=a{A}_m-b{B}_m.$$ (21b)

Using the expressions in Eq. 21, we get the condition for the stability in which eigenvalues of the system are negative:

$$1-2\beta \cos \left(\frac{2\pi m}{N}\right)>0.$$ (22)

Using Eq. 22, the condition in which the largest eigenvalues are negative yields 1 > 2β when j = N/2. Under this condition the homogeneous solutions are always stable and thus, the zig-zag patterning does not occur.

Derivation procedure of Eq. 5

To begin, we show the derivation procedure for Eq. 5. The equilibrium of difference in the tip-cell signal between cell j and j + 1 is expressed by using Eq. 4 as

$$\Delta {\hat{x}}_j\approx \Delta {\delta }_j=f\left(\hat{x}\left(c\right);c\right).$$ (23)

Note the differentiation of Δδ_j to c,

$$\frac{\partial \Delta {\delta }_j}{\partial c}=\frac{\partial f}{\partial \hat{x}}\frac{\partial \hat{x}}{\partial c}+\frac{\partial f}{\partial c},$$ (24)

where $\hat{x}$ is a homogeneous steady state of the tip-cell signal with uniform input.

Using the implicit function theorem,

$$\frac{\partial \hat{x}}{\partial c}=\frac{dh\hat{x}{\left(ac\hat{x}/b\right)}^h}{c\left(dh{\left(ac\hat{x}/b\right)}^h+\hat{x}{\left(1+{\left(ac\hat{x}/b\right)}^h\right)}^2\right)}<0.$$ (25)

Note that we used

$$\frac{\partial F\left(\hat{x}\left(c\right)\right)}{\partial c}=\frac{\partial F}{\partial \hat{x}}\frac{\partial \hat{x}}{\partial c}+\frac{\partial F}{\partial c}=0.$$

Because we can explicitly calculate $\partial f/\partial \hat{x}$ and ∂f/∂c, we obtain

$$\frac{\partial \Delta {\delta }_j}{\partial c}<0.$$ (26)

See the Supporting Material for the explicit calculation that we performed with the computational software program, MATHEMATICA 8.0 (Wolfram Research, Boston, MA).

Derivation procedure of the expressions in Eq. 6

For the derivation of the expressions in Eq. 6, we first define the amount of change in ${\hat{x}}_j$ caused by the unit amount of change in parameters d and g as

$${\hat{x}}_j^{\text{'}d}\equiv \frac{\partial \hat{x}}{\partial d}+\frac{\partial {\delta }_j}{\partial d}\approx \frac{\partial \hat{x}}{\partial d},$$ (27a)

$${\hat{x}}_j^{\text{'}g}\equiv \frac{\partial \hat{x}}{\partial g}+\frac{\partial {\delta }_j}{\partial g}\approx \frac{\partial {\delta }_j}{\partial g}.$$ (27b)

Because δ_j can be calculated under the condition in which g is small (i.e., ∂δ_j/∂d is much smaller than $\partial \hat{x}/\partial d$), we only consider $\partial \hat{x}/\partial d$ for Eq. 27a. As for Eq. 27b, note that the homogeneous steady state does not depend on g.

Using Eqs. 25, 27a, and 27b, we obtain

$$\frac{\partial {\hat{x}}_j^{\text{'}d}}{\partial c}\equiv \frac{\partial {\hat{x}}_j^{\text{'}d}}{\partial \hat{x}}\frac{\partial \hat{x}}{\partial c}+\frac{\partial {\hat{x}}_j^{\text{'}d}}{\partial c}<0,$$ (28a)

$$\frac{\partial {\hat{x}}_j^{\text{'}g}}{\partial c}\equiv \frac{\partial {\hat{x}}_j^{\text{'}g}}{\partial \hat{x}}\frac{\partial \hat{x}}{\partial c}+\frac{\partial {\hat{x}}_j^{\text{'}g}}{\partial c}<0,$$ (28b)

where, under j ≤ N/4, an appropriate range for tip-cell selection occurs as in Eq. 27b. See the Supporting Material for the explicit calculation performed by MATHEMATICA 8.0.

Conditions for numerical calculation: ranges and sampling of parameters for numerical analysis

Initial values of the tip-cell signal x_j were set to be almost homogeneous and that of Delta/Notch signal y_j were taken as zero. The bias of parameters in the computer simulation was sampled from a uniform distribution in logarithmic space across several orders of magnitude for each parameter : a = 0.1, b = 0.1, c = 0.01–1.0, d = 0.1–10, and g = 0.01–1 (unless otherwise stated). The Hill coefficient value is set to be h = 2. All the numerical results in the article are produced when N = 8. For other values of h and N, the tendency of results did not change. For the sampling used in the statistical analysis we chose only the single peak distribution of the tip-cell signal because the pattern reflects the phenomena of the tip-cell selection in Drosophila trachea development.

Discussion

In this article, we studied a mathematical model of Delta/Notch lateral inhibitory regulation with an input stimulus designed to identify the role of lateral inhibitory regulation on the tip-cell selection during Drosophila trachea development. Because the tip cells have a single peak, we examined parameter ranges in which zig-zag patterning was not produced. Mathematical and numerical analyses showed that the difference in tip-cell signals between neighboring cells was not amplified. Our results also showed another aspect of the lateral inhibition—namely, that it achieves robust tip-cell selection when the Bnl/Btl signal includes fluctuations. The lateral inhibition can stabilize the spatial-pattern of tip cells to make them less affected by the change of input signal. In addition, the lateral inhibitory regulation amplifies the difference in the tip-cell signals between neighboring cells compared with that produced by a corresponding self-inhibition regulation. In the presence of lateral inhibitory regulation, the number of tip cells hardly changed even when the threshold value for differentiating the tip cells varied.

Comparison of this study with previous studies would make the difference clearer concerning the roles of Delta/Notch lateral inhibition. The previous studies have focused on the pattern formation in which a key signal is expressed alternately in cells, such as a salt-and-pepper patterning, and have emphasized the amplification of signals between neighboring cells generated by the lateral inhibition (24,40). Although these studies contributed to our understanding of how a specific pattern is formed, the mechanism cannot be applied to the tip-cell selection under which a single signal-peak emerges within multiple neighboring cells. Our study addressed the conditions in which the tip-cell signals were expressed sequentially within a single peak, and demonstrated that the lateral inhibition did not amplify the between-cell differences of a key signal. We showed that increasing the intensity of lateral inhibition decreased both the level of the tip-cell signal and the differences between neighboring cells. This effect of lateral inhibition reduced the shift of the tip-cell signal by the change of input level, indicating that the level of the tip-cell signal with the lateral inhibition is less affected by fluctuation of the input gradient. Such mechanism for buffering fluctuations of input stimuli has been found in many and diverse simple systems, such as those of E. coli or Drosophila early embryo, as confirmed both experimentally and mathematically (40,41). By this mechanism, the lateral inhibition promotes the robustness of the tip-cell selection in a noisy environment such as the variation of input gradient.

In this study we dealt with a specific profile of input stimuli and showed the lateral inhibition produced a more-robust tip-cell selection than the self-inhibition did. This should be the same for the case of other profiles, such as a linear input (see the Supporting Material). Indeed, we found that the robustness of tip-cell selection in the lateral inhibitory regulation was always better than that in the self-inhibitory regulation for the analysis of linear input case as well as the case of the nonlinear input (see Fig. S2, e–g, and Fig. S3, c and f, in the Supporting Material). However, this was not the case at some c on the 3cells-threshold and 5cells-threshold (see Fig. S3, a, b, d, and e). This is because the lateral inhibitory regulation coupled with the input stimuli level depends on the input gradient profiles and thus, the robustness would change depending on a specific threshold for choosing the number of tip cells. For example, the lateral inhibition always enhances the robustness compared with the self-inhibition on 1cell-threshold. (See the Supporting Material for the detail analysis of the linear input case.) We emphasize that the Bnl/Btl signal should be a nonlinear curve as Eq. 1 rather than the input profile because of its geometry of the Drosophila trachea during its primary branching development. Also, this holds for many situations in a consideration with tube morphology. Therefore, our nonlinear input gradient profile is appropriate for the analysis of the tip-cell selection.

We here chose a modeling strategy to demonstrate the input-output relation in general context of lateral inhibition. Our model consists of a minimal number of factors for representing the lateral inhibitory system with the input stimulus, although there is a complicated chemical interaction including a positive feedback loop involving Breathless (12,13,17) and inhibitory interactions between Delta and Notch in its own cell, termed “cis-inhibition” (11,31). Moreover, our model is built on one-dimensional space for the convenience of analysis. We conjecture that the qualitative results in the case of two-dimensional analysis would be the same as those demonstrated in the analysis of this article, because the inhibitory regulations by neighboring cells are identical.

Our analysis is based on the assumption that the timescale for the lateral inhibition between neighboring cells is much shorter than that for the morphological change. Thus, we calculate the dynamics of chemical regulation with fixed Bnl/Btl values. The Bnl/Btl input distribution would have a steep gradient followed by a geometrical change of epithelial sacs, as shown in Fig. 1 b. For further understanding of branching formation theoretically, future studies will be needed to examine the influences of lateral inhibitory regulation, combining with morphological change of epithelial tube and other fundamental chemical interactions (42–44).

From a broader viewpoint, our study corroborates that intercellular communications, such as the Delta/Notch lateral inhibition, can realize a more-robust development compared with noncommunication systems. It has been argued that coupling systems between intercellular and intracellular regulation achieve robust development against extrinsic and intrinsic noises in a variety of contexts (40,45) and that those in C. elegans vulval development, for example, amplify cellular perception of the morphogen gradient (32). We consider that the result shown in Lateral Inhibition Regulation Achieves a More-Robust Tip-Cell Selection than Self-Inhibitory Regulation is one demonstration of this. That is, a coupled intercellular system such as Delta/Notch lateral inhibition enhances cell fate segregation by amplifying differences in the Bnl/Btl signaling between neighboring cells compared with an uncoupled system that is regulated by the self-inhibition of each cell. We conclude that the regulatory network in the primary tracheal branching realizes a robust developmental system utilizing the Delta/Notch lateral inhibition.

Without limiting the Drosophila trachea development, the role of Delta/Notch lateral inhibitory regulation as a noise-suppression mechanism may work in other development situations. In the early development of murine lung, for example, FGF10, a homolog of Bnl, activates Notch signal in distal epithelial cells of lung bud, and disruption of Notch signaling leads to both the expansion of distal epithelial cell fate and the induction of ectopic budding (46,47). This indicates that the patterning of the distal cell fate is determined through the lateral inhibitory regulation and it should contribute to the robust selection of distal cells as well as Drosophila trachea development. Furthermore, the significance of FGF10-Notch signaling on branching morphogenesis has been unveiled in the development of other foregut derivatives, such as the pancreas and stomach, and also in the nephron formation (48–50). We believe that the proposed mechanism of the lateral inhibitory regulation underlies in a variety of organogenesis.