Directional alignment of different cell types organizes planar cell polarity

Summary

Intercellular coupling between core PCP proteins is essential for unidirectional polarization of epithelial cells, known as planar cell polarity (PCP). While imbalances in core PCP protein levels between adjacent cells disrupt PCP in some tissues, other tissues maintain PCP despite containing multiple cell types with distinct core PCP protein abundance. How such tissues tolerate these imbalances remains unclear. Here, we analyzed how the spatial distribution of cell types contributes to PCP maintenance using a previously established mathematical model. Systematic simulations showed that the accuracy of PCP maintenance under core PCP protein imbalance strongly depends on cell-type distribution patterns. We then applied deep learning and statistical modeling to identify critical features, revealing that the orientation of cell-type alignment is a key determinant of PCP robustness. Such directional cell-type alignment was observed in the mouse oviduct. Our findings highlight an overlooked role of tissue cell-type organization in PCP maintenance.

Graphical abstract

Highlights

• •PCP can persist despite existence of cell types with different core PCP protein levels
• •Simulations showed cell-type distributions govern PCP robustness under imbalance
• •Deep learning and statistical modeling identified alignment orientation as key
• •Directional cell-type alignment was observed in the mouse oviduct

Cell biology; mathematical biosciences

Introduction

The proper function of various organs depends on planar cell polarity (PCP), the tissue-level coordination of cell polarity within the plane of epithelial tissue.^{1,2,3,4,5,6,7,8} For example, the transport of oocytes is driven by the ovary-to-uterus-directed beating of cilia in the oviduct epithelium,^9,10,11,12 and misoriented hair cells in the cochlea are associated with hearing deficit.¹³

Previous studies have identified an evolutionally conserved group of proteins, known as core PCP proteins (hereafter referred to as core proteins), that play key roles in PCP establishment.^{1,2,3,4,5,6,7} Core proteins form two types of complexes at cell edges (Figure 1A): one comprised of CELSR and FZD (FRIZZLED), and another comprised of CELSR and VANGL (Van Gogh-like; hereafter referred to as the FZD complex and the VANGL complex, respectively). In various developing organs, the FZD and VANGL complexes show asymmetric distributions at cell edges along body axes.^1,4,8 For example, in the mouse oviduct, the FZD complex is localized to the uterus side, while the VANGL complex is localized to the ovary side of the cell.¹⁰ This asymmetry is established before ciliogenesis, and the loss of CELSR1 results in misoriented ciliary beating,⁹ suggesting that the asymmetry is a prerequisite for PCP establishment. Although details remain unclear, experimental and theoretical studies suggest that the establishment of the asymmetry requires intracellular and intercellular interactions between the FZD and VANGL complexes.^14,15,16,17

Figure 1.

Mathematical modeling for evaluating effects of spatial organization of core protein imbalance on PCP

(A) A diagram illustrating the asymmetric distribution of the FZD and VANGL complexes on cell edges. (A′) Three key interactions between core-protein complexes implemented in our mathematical model: (1) autoactivation: the FZD and VANG complexes stabilize complexes of the same type. (2) Mutual inhibition: the FZD and VANGL complexes mutually destabilize complexes of the opposite type. (3) Intercellular bridge formation: the FZD and VANG complexes form bridges, intercellularly.

(B) Diagram illustrating an overview of the mathematical model. See the main text and STAR Methods for details.

(C and C”) Example analysis of how cell-type distribution affects PCP. White hexagons: high cells; gray hexagons: low cells. See (D) for the definitions of high and low cells. (C) The unidirectionally biased distribution of the FZD and VANGL complexes was reorganized by core protein imbalance (balance_LH = 0.5, compare the initial and final states). A boxed region in (C) is enlarged and shown in (C′). (C″) A bar centered in each cell represents polarity of cell-edge distribution of the FZD complexes. The bar’s length represents the magnitude of polarity, and its angle represents the orientation.

(D) Concentrations of FZD and VANGL complexes at the initial time point of our simulations in high and low cells. The number at the center of each cell indicates the concentration of the FZD or VANGL complex in the intracellular region, while the number near the cell edge represents a concentration of these complexes on each cell edge. Initial concentrations in low cells were obtained by multiplying the corresponding concentrations in high cells by balance_LH, a parameter ranging from 0 to 1, thus, “high” cells have higher levels of core protein complexes than “low” cells. See also Figures S2 and S3.

Epithelial tissues are typically composed of multiple cell types. In such cases, how is PCP formed or maintained? The mouse oviduct epithelium consists of two cell types: multiciliated cells (MCCs) and secretory cells (SCCs, Figure S1A). Notably, these two cell types differ in their levels of core proteins; CELSR1 is enriched at the cell edges of MCCs but is present at lower levels in SCCs (Figures S1B and S1B″).⁹ This imbalance in core protein levels between adjacent cells (hereafter referred to as core protein imbalance) has the potential to disrupt PCP. Indeed, when cells lacking Frizzled (Fz, a fly homolog of FZD), Van Gogh (Vang, a fly homolog of VANGLs), or Flamingo (Fmi, also known as Starry night, a fly homolog of CELSR) are experimentally produced in the Drosophila wing, polarities in surrounding wild-type cells are reorganized.^6,7,18 Nevertheless, in the oviduct, both the ciliary beating orientation and core protein distribution remain biased along the ovary-uterus axis (Figures S1B and S1B″). Such core protein imbalances have also been observed in various organs, including the mouse inner ear,¹⁹ airway,²⁰ and Drosophila eye disc.²¹ However, the mechanisms by which tissues overcome core protein imbalance to maintain PCP remain unclear.

Mathematical models have been widely used to dissect the mechanisms of PCP establishment and maintenance. Several types of PCP models exist. Mechanistic models explicitly simulate the dynamics of molecules relevant to individual core proteins or their complexes.^{14,15,22,23,24,25} On the other hand, phenomenological models abstract these molecular details and represent cell polarity as a vector that evolves according to effective parameters, such as neighbor interactions, global cues, or cell geometry.^26,27,28 These models have been used to investigate core protein functions by performing in silico perturbation experiments in which clusters of mutant cells lacking core proteins are embedded within a wild-type cell sheet. These studies primarily focused on elucidating the molecular functions of core proteins, rather than studying PCP establishment and maintenance in tissues of multiple cell-types with various spatial distributions. Therefore, these studies considered just idealized configurations of mutant cell clusters: a linear cell row, a rectangular cluster.^14,15,22 Consequently, it remains largely unclear how various spatial patterns of cell types with differing core protein levels—beyond a simple mutant cluster—affect PCP in normal tissues.

In this study, to consider the core protein imbalances, we adopted a mechanistic mathematical model in which concentrations of the FZD and VANG complexes are explicitly represented. We systematically varied the spatial patterns of core protein levels in a two-dimensional cell sheet and analyzed the resulting PCP outcomes. To identify key spatial features that promote PCP maintenance, we combined simulations with deep learning, and statistical modeling. Our analysis revealed that the directional alignment of cell types, specifically, the alignment of cell types along the tissue axis, contributes to preservation of PCP in the presence of core protein imbalance. Consistent with this theoretical finding, we found that SCCs are predominantly aligned along the ovary-uterus axis in the mouse oviduct, and PCP defects emerge in regions where this alignment is disrupted. Our study offers crucial insights into how the spatial patterning of multiple cell types influences PCP establishment and maintenance.

Results

Modeling PCP in a 2D cell sheet

To examine how distributions of cell types with different core protein levels affect PCP, we constructed a mathematical model (see also STAR Methods), which built upon previous frameworks.^15,23 Our model explicitly implements concentrations of the FZD and VANGL complexes at cell edges and within the cytoplasmic region (Figure 1B). This allows us to directly address the effects of intercellular imbalances in core protein levels—an aspect not captured by phenomenological vector-based models.^26,27,28 The epithelial sheet is represented as an array of regular hexagonal cells (Figures 1B and 1C″), where the FZD complex and the VANGL complex move between cell edges and the cytoplasmic region as follows (Figure 1B, blue arrows):

$$\begin{array}{c}{F}_{{\mathrm{C}}_k}\begin{array}{c}{K}_{{\mathrm{F}}_{ki}}\\ \begin{array}{c}\to \\ \leftarrow \end{array}\\ R_{{\mathrm{F}}_{ki}}\end{array}{F}_{{\mathrm{M}}_{ki}}\end{array}$$ (Equation 1)

$$\begin{array}{c}{V}_{{\mathrm{C}}_k}\begin{array}{c}{K}_{{\mathrm{V}}_{ki}}\\ \begin{array}{c}\to \\ \leftarrow \end{array}\\ R_{{\mathrm{V}}_{ki}}\end{array}{V}_{{\mathrm{M}}_{ki}}\end{array}$$ (Equation 2)

Here, $F_{{\mathrm{C}}_k}$ and $V_{{\mathrm{C}}_k}$ denote the concentration of the FZD and VANGL complexes in the cytoplasmic region of kth cell, respectively. The concentration of the FZD and VANGL complexes at ith cell edge in the kth cell are represented as $F_{{\mathrm{M}}_{ki}}$ and $V_{{\mathrm{M}}_{ki}}$, respectively; i = 1–6. In addition, concentrations of the FZD and VANGL complexes are assumed to be uniform within each cell edge and cytoplasmic region. Thus, each cell has seven compartments—one for each boundary and one for the cytoplasm. In our model, FZD and VANGL complexes move between cell edges and a cytoplasmic region in each cell and the kinetics of these movements is determined by the law of mass action (Figure 1B, blue arrows). The rate of FZD complex movement from cytoplasmic region to ith cell edge in kth cell is determined by a reaction rate constant $K_{{\mathrm{F}}_{ki}}\text{.}$ Similarly, the rate of FZD complex movement from ith cell edge to cytoplasmic region in kth cell is determined by a reaction rate constant $R_{{\mathrm{F}}_{ki}}\text{.}$ Therefore, the dynamics of $F_{{\mathrm{M}}_{ki}}$ is described as $d{F}_{{\mathrm{M}}_{ki}}/dt=-R_{{\mathrm{F}}_{ki}}{F}_{{\mathrm{M}}_{ki}}+K_{{\mathrm{F}}_{ki}}{F}_{{\mathrm{C}}_{ki}}$; the differential equations for other components are shown in STAR Methods.

According to previous studies, to establish the planar polarized distributions of the two complexes on cell edges, the formation of intercellular bridges composed of the two complexes (the FZD-VANGL bridges) is required, where the FZD-VANGL bridges have activities of both autoactivation and mutual inhibition (Figure 1A’).^14,15,16,17 We basically adopted the theoretical formulation in these studies to model the previous reaction rate constants as follows:

$$\begin{array}{c}{K}_{{\mathrm{F}}_{ki}}={\rho }_{\mathrm{F}}+{\omega }_{\mathrm{F}}\left(F_{{\mathrm{M}}_{ki}}{V}_{{\mathrm{M}}_{ki}}^{\prime }+d_{{\mathrm{A}}_{\mathrm{F}}}\sum _{j=i-1,i+1}{F}_{{\mathrm{M}}_{kj}}{V}_{{\mathrm{M}}_{kj}}^{\prime }\right)\end{array}$$ (Equation 3)

$$\begin{array}{c}{R}_{{\mathrm{F}}_{ki}}={\mu }_{\mathrm{F}}+{\phi }_{\mathrm{F}}\left(V_{{\mathrm{M}}_{ki}}{F}_{{\mathrm{M}}_{ki}}^{\prime }+d_{{\mathrm{I}}_{\mathrm{V}}}\sum _{j=i-1,i+1}{V}_{{\mathrm{M}}_{kj}}{F}_{{\mathrm{M}}_{kj}}^{\prime }\right)\end{array}$$ (Equation 4)

$$\begin{array}{c}{K}_{{\mathrm{V}}_{ki}}={\rho }_{\mathrm{V}}+{\omega }_{\mathrm{V}}\left(V_{{\mathrm{M}}_{ki}}{F}_{{\mathrm{M}}_{ki}}^{\prime }+d_{{\mathrm{A}}_{\mathrm{V}}}\sum _{j=i-1,i+1}{V}_{{\mathrm{M}}_{kj}}{F}_{{\mathrm{M}}_{kj}}^{\prime }\right)\end{array}$$ (Equation 5)

$$\begin{array}{c}{R}_{V_{ki}}={\mu }_{\mathrm{V}}+{\phi }_{\mathrm{V}}\left(F_{{\mathrm{M}}_{ki}}{V}_{{\mathrm{M}}_{ki}}^{\prime }+d_{{\mathrm{I}}_{\mathrm{F}}}\sum _{j=i-1,i+1}{F}_{{\mathrm{M}}_{kj}}{V}_{{\mathrm{M}}_{kj}}^{\prime }\right)\end{array}$$ (Equation 6)

where concentrations of the FZD and VANGL complexes on a neighboring cell’s edge juxtaposing edge i are denoted as $F_{{\mathrm{M}}_{ki}}^{\prime }$ and $V_{{\mathrm{M}}_{ki}}^{\prime }$,respectively (Figure 1B). The intercellular bridges are formed between the FZD and VANGL complexes: $F_{{\mathrm{M}}_{ki}}-V_{{\mathrm{M}}_{ki}}^{\prime }$ and $V_{{\mathrm{M}}_{ki}}-F_{{\mathrm{M}}_{ki}}^{\prime }$. For simplicity, the concentrations of the FZD-VANGL bridges are assumed to be proportional to the terms of $F_{{\mathrm{M}}_{ki}}{V}_{{\mathrm{M}}_{ki}}^{\prime }$ and $F_{{\mathrm{M}}_{ki}}{V}_{{\mathrm{M}}_{ki}}^{\prime }$ in the previous equations.¹⁵ The first term in the parenthesis of Equations 3 and 5 correspond to the autoactivation activity: $K_{{\mathrm{F}}_{ki}}$ and $K_{{\mathrm{V}}_{ki}}$ increase when $F_{{\mathrm{M}}_{ki}}{V}_{{\mathrm{M}}_{ki}}^{\prime }$ and $V_{{\mathrm{M}}_{ki}}{F}_{{\mathrm{M}}_{ki}}^{\prime }$ are elevated, respectively. The first term in the parenthesis of Equations 4 and 6 corresponds to the mutual inhibition activity: $R_{{\mathrm{F}}_{ki}}$ and $R_{{\mathrm{V}}_{ki}}$ increase when $V_{{\mathrm{M}}_{ki}}{F}_{{\mathrm{M}}_{ki}}^{\prime }$ and $F_{{\mathrm{M}}_{ki}}{V}_{{\mathrm{M}}_{ki}}^{\prime }$ are elevated, respectively. In addition, this model assumes a non-local effect, where the reaction rates on cell edge i is affected by core proteins on neighboring edges i-1 and i+1 (Figures S2A and S2A’; the second term in the parenthesis of Equations 3, 4, 5, and 6; see STAR Methods for details). Such a non-local effect has theoretically been shown to be essential to establish planar polarized distribution of core proteins at cell edges,^15,29 although such an effect has not been validated in vivo. In previous studies, non-local effects are implemented either by diffusion of molecules on cell membrane and/or within the intracellular region^15,29 or by simply assuming that core protein complexes on cell membrane exert autoactivation and/or mutual inhibition effects on neighboring membrane compartments²³ as in this study. A constant d determines the strength of the non-local effects (Figures S2A and S2A′). When d_A > 0, the complexes on edge i-1 and i+1elevate $K_{{\mathrm{F}}_{ki}}$ and $K_{{\mathrm{V}}_{ki}}$. Similarly, when d_I > 0, the complexes on edge i-1 and i+1 elevate $R_{{\mathrm{F}}_{ki}}$ and $R_{{\mathrm{V}}_{ki}}\text{.}$

In addition to the reaction rates that depend on concentrations of FZD-VANGL bridges, constant reaction rates, ρ and μ, that are not affected by concentrations of FZD-VANGL bridges are implemented in the model. The parameter ρ represents the rate of the FZD and VANGL complexes movements from the cell edge to the cytoplasm, while μ denotes the basal rate of the FZD and VANGL complexes movements from the cytoplasm to the cell edge. The values of parameters used in our model are listed in Table S1. We selected these values to reproduce the non-autonomous effects of mutant cells lacking core proteins, as observed in vivo (Figure S3; see details in the STAR Methods).

Setting of simulation and definition of polarity index

Before showing systematic analyses of simulations, we describe the setting of simulations. First, a cell sheet is numerically initialized with the FZD and VANGL complexes enriched at right and left ends of each cell, respectively (Figures 1C and 1D, left panel). Second, core protein levels are reduced in selected cells (low cells; gray cells in Figures 1C and 1D) while the remaining cells are designated as high cells. This setup allows us to examine how introducing patches of low cells perturbs an otherwise uniformly polarized tissue. The magnitude of the core protein imbalance is defined by a constant parameter balance_LH, which ranges from 0 to 1 (Figure 1D). When balance_LH = 1, low and high cells contain equal amounts of core proteins, whereas, when balance_LH = 0, low cells have no core proteins. Figure 1C is an example in the case that low cells (balance_LH = 0.5) are randomly distributed. In our data visualizations, we represent each cell’s polarity vector with an orange-headed bar (Figure 1C”). The orientation and the magnitude of polarity vectors were calculated as follows:

$$F_{{\mathrm{x}}_k}=\sum _{i=1}^6{F}_{{\mathrm{M}}_{ki}}\cos {\theta }_{ki}$$

$$F_{{\mathrm{y}}_k}=\sum _{i=1}^6{F}_{{\mathrm{M}}_{ki}}\sin {\theta }_{ki}$$

$${angle}_k=\arctan \left(F_{{\mathrm{y}}_k}/F_{{\mathrm{x}}_k}\right)$$

$${magnitude}_k=\frac{\sqrt{{F_{{\mathrm{x}}_k}}^2+{F_{{\mathrm{y}}_k}}^2}}{\sum _{i=1}^6{F}_{{\mathrm{M}}_{ki}}}$$

Here, angle_k and magnitude_krepresent the orientation of the FZD complex-enriched cell edges and the magnitude of the polarity of kth cell, respectively. $\overline{{\Delta angle}_{\text{high}}}$ is the mean absolute change in polarity vector angles of high cells in a cell sheet. Since the value of $\overline{{\Delta angle}_{\text{high}}}$ is 0 at the initial time point of the simulation, the value of $\overline{{\Delta angle}_{\text{high}}}$ at the final time point represents the extent of polarity reorganization. In Figure 1C, $\overline{{\Delta angle}_{\text{high}}}$ is 36.0° at the final time point, indicating that the core protein imbalance disrupts PCP.

Systematic simulations under various cell type distributions

To gain insights into which aspects of cell-type distributions influence PCP, we generated cell-type distributions by randomly placing low cells, expecting these distributions to encompass diverse features of cell-type distributions. Through our simulations, we evaluated their effects on PCP by calculating $\overline{{\Delta angle}_{\text{high}}}$ and searched for features of cell-type distributions that significantly impact PCP.

We began by generating random distributions with low-cell frequencies (freq_low) ranging from 0.05 to 0.95 in increments of 0.05. At freq_low = 0, all cells are high cells, whereas at freq_low = 1, all cells are low cells. For each value of freq_low, we generated 10 random distributions and performed simulations (Figures 2A and 2B). Figures 2A and 2A″ shows examples of our simulations at balance_LH = 0.5. freq_low significantly influenced the values of $\overline{{\Delta angle}_{\text{high}}}$ (Figure 2B; see the boxed column for these values at balance_LH = 0.5), indicating that freq_low is an essential parameter influencing PCP. Additionally, changes in balance_LH significantly altered the value of $\overline{{\Delta angle}_{\text{high}}}$ (Figures 2B, S4A and S4B’; e.g., compare Figures S4A with S4A′). We further noted that the influence of low cells on $\overline{{\Delta angle}_{\text{high}}}$ was particularly evident when balance_LH was below 0.6 (Figure 2B). This indicates that PCP is affected by low cells even when core proteins are only partially lost in these cells.

Figure 2.

Systematic evaluation of effects of random low-cell distributions and magnitude of core protein imbalance on PCP

(A and A”) Examples showing the effects of random low-cell distributions on PCP. balance_LH = 0.5. freq_low = 0.95 (A), 0.8 (A′), and 0.95 (A″). The resulting values of $\overline{{\Delta angle}_{\text{high}}}$ are shown at the bottom of each panel. Boxes in A and A′ highlight alignment of high cells along the y axis of the cell sheet. A box in A″ highlight the misoriented polarity vectors of high cells in the vicinity of low cell clusters.

(B) For each freq_low value, 10 random distributions of low cells were generated and their effects on PCP were evaluated by simulation. Heatmap colors represent the average $\overline{{\Delta angle}_{\text{high}}}$ values over 10 random distributions for each combination of balance_LH and freq_low (unit: degrees). The boxed column represents data at balance_LH = 0.5. See also Figure S4.

Although freq_low and balance_LH appeared to be essential parameters influencing $\overline{{\Delta angle}_{\text{high}}}\text{,}$ they might not be the only determinants of $\overline{{\Delta angle}_{\text{high}}}$. For example, among the 10 simulations under freq_low = 0.4 and balance_LH = 0.5 with different cell-type distributions, the maximum and the minimum values of $\overline{{\Delta angle}_{\text{high}}}$ were 36.4 and 16.3°, respectively (variance: 40.5°; Figure S4B). Variances of $\overline{{\Delta angle}_{\text{high}}}$ at different values of freq_low and balance_LH are summarized in Figure S4C. We expected that such a variance can be explained by differences in features of cell-type distribution between cell sheets. One candidate is the orientation of the alignment of low and high cells. As highlighted by boxes in Figures 2A and 2A′, polarity vectors of high cells were strongly reoriented where these cells were aligned along the y axis of the cell sheets. In addition, we noticed that when high cells were dominant in a cell sheet, polarity vectors of high cells were reoriented only in the vicinity of low cells (see a boxed region in Figure 2A”), suggesting that differences in the frequency of edges between high and low cells underlie the variations in $\overline{{\Delta angle}_{\text{high}}}$ between cell sheets. This is consistent with a previous theoretical study, which reports that that core proteins were eliminated from cell edges between Fmi (Flamingo, a fly homolog of CELSRs)-deficient cells and wild type cells.²² We analyzed effects of these parameters on the values of $\overline{{\Delta angle}_{\text{high}}}$ in later sections.

Cell type distribution is effective on PCP maintenance

To examine whether features of cell-type distribution can explain differences in values of $\overline{{\Delta angle}_{\text{high}}}$ between cell sheets, we set out to train a neural network to predict the extent of PCP disorganization in a cell sheet using only information on cell-type distributions (Figure 3A). If the network was successfully trained, it indicates that the cell-type distribution is a crucial factor in explaining the value of $\overline{{\Delta angle}_{\text{high}}}$ in a cell sheet. Otherwise, it implies that the cell-type distribution has little impact.

Figure 3.

Exploring features of cell-type distribution affecting PCP by deep learning techniques

(A) Diagram illustrating the strategy for deep learning analysis. Cell-type distributions were provided as images for the input into the neural network. No information about polarity vectors was included in the input images. The neural network (ResNet18) was trained to predict whether a given cell-type distribution corresponds to a “top” ($\overline{{\Delta angle}_{\text{high}}}$ = high) or “bottom” ($\overline{{\Delta angle}_{\text{high}}}$ = low) classification. Finally, Grad-CAM was applied to visualize the regions in the input images that contributed to the prediction.

(B) A total of 6,000 random distributions were generated at freq_low = 0.4, and their effects on PCP were simulated at balance_LH = 0.5. The variation in the resulting $\overline{{\Delta angle}_{\text{high}}}$ values is shown as a histogram. Distributions within the top 12.5% and bottom 12.5% $\overline{{\Delta angle}_{\text{high}}}$ values were labeled “top” and “bottom”, respectively.

(C) Prediction accuracy of the neural network during training. The gray and gold lines show changes in prediction accuracy for the training and validation data, respectively.

(D and D′) Grad-CAM attention maps highlighting regions of input distributions where the network emphasized during predictions. Black and white hexagons represent low and high cells, respectively. Magenta brightness indicates attention magnitude. (D) A distribution predicted as “top”. Attention is concentrated at vertical edges between low cells and high cells. (D′) A distribution predicted as “bottom”. Attention is concentrated around low cells adjacent to high cells.

(E) Diagram illustrating the definition of L_h and L_V. Gray and white cells represent low and high cells, respectively. Blue arrow indicates the x axis of the cell sheet.

(F) Boxplots showing L_h and L_V values for top and bottom validation datasets (∗∗: p < 0.01; Student’s t test; N = 100 each). (F′) Boxplots showing frequencies of edges between low cells and high cells for the top and bottom validation datasets (∗: p < 0.05; Student’s t test; N = 100 each). Open circles represent average values. See also Figures S4 and S5.

For this end, we first generated a large number of simulation data with freq_low = 0.4 and balance_LH = 0.5. The value for balance_LH has determined based on measurement of balance_LH (≒0.5) in the oviduct (Figure S5). In addition, we considered two criteria for determining the values of the two parameters: (1) diversity in patterns of cell-type distributions and (2) variance in the values of $\overline{{\Delta angle}_{\text{high}}}$. To train the neural network to classify the cell type distributions according to the value of $\overline{{\Delta angle}_{\text{high}}}$, we considered that a dataset exhibiting a wide range of $\overline{{\Delta angle}_{\text{high}}}$ values would be preferable. We used the dataset generated for the analysis shown in Figure 2 to identify a parameter set that produced high variance in $\overline{{\Delta angle}_{\text{high}}}$ values. Notably, at balance_LH = 0.5, variance was high at around freq_low = 0.95 and 0.4 as shown in Figure S4C.

To train the neural network to classify the cell type distributions according to the value of $\overline{{\Delta angle}_{\text{high}}}$, we reasoned that a dataset exhibiting a wide range of $\overline{{\Delta angle}_{\text{high}}}$ values would be preferable. We therefore used the dataset generated for the analysis shown in Figure 2 to identify a parameter set that produced high variance in $\overline{{\Delta angle}_{\text{high}}}$ values. Notably, at balance_LH = 0.5, variance was high at around freq_low = 0.95 and 0.4 (Figure S4C). In addition, the number of possible configurations of low and high cells reaches its maximum at freq_low = 0.5. Because a dataset encompassing diverse cell-type configurations would be advantageous for identifying parameters of cell-type distributions that contribute to PCP maintenance, we selected a value around 0.5 for freq_low.

A total of 6,000 random cell-type distributions were generated, and those in the top 12.5% and bottom 12.5% of $\overline{{\Delta angle}_{\text{high}}}$ were labeled as “top” and “bottom” distributions, respectively (Figures 3A and 3B). In addition to $\overline{{\Delta angle}_{\text{high}}}$, we calculated the averaged angle of polarity vectors over low cells ($\overline{{\Delta angle}_{\text{low}}}$) and all cells ($\overline{{\Delta angle}_{\text{all}}}$). However, using $\overline{{\Delta angle}_{\text{low}}}$ or $\overline{{\Delta angle}_{\text{all}}}$ for labeling did not significantly alter the classification of each distribution as “top” or “bottom”, since these values were closely correlated with $\overline{{\Delta angle}_{\text{high}}}$ (Figures S4D and S4E).

Then, a ResNet18-based neural network³⁰ was trained to categorize distributions as either “top” or “bottom” (Figure 3A). Among the 750 “top” distributions and 750 “bottom” distributions, 100 from each group were used for validation, while the remaining distributions were used for training. After training, the model achieved a classification accuracy of 0.925 (Figure 3C), indicating that $\overline{{\Delta angle}_{\text{high}}}$ is determined by cell-type distributions.

Deep learning-based exploration of features of cell type distributions that affect PCP

To find what kind of spatial cell-type distributions are critical for PCP maintenance, we used an explainable artificial intelligence technique, Grad-CAM,³¹ by which we can visualize regions of a cell sheet where the network focused its attention to make predictions (Figure 3A). We expected that this technique would allow us to identify key features of cell-type distributions that influence $\overline{{\Delta angle}_{\text{high}}}$. Notably, in distributions predicted as “top,” Grad-CAM highlighted regions where low cells were aligned along the y axis of the cell sheet (Figure 3D). In contrast, in distributions predicted as “bottom,” Grad-CAM highlighted regions where low cells were aligned along the x axis of the cell sheet (Figure 3D’). To quantify these patterns, we measured the total lengths of horizontal and vertical alignment lengths of low cells adjacent to high cells (Figure 3E). For each pair of adjacent low cells whose shared vertex was also shared by a high cell, we measured the alignment angle θ between the two low cells (Figure 3E). Here, θ is defined as the angle between the line connecting the centroids of the two low cells and the X axis of the cell sheet (Figure 3E). We then calculated the total low-cell alignment as follows:

$$L_{\mathrm{h}}=\sum _{\text{cell}\text{sheet}}\left|\cos \theta \right|$$

$$L_{\mathrm{v}}=\sum _{\text{cell}\text{sheet}}\left|\sin \theta \right|$$

where L_h and L_v are total horizontal and vertical alignment lengths of low cells adjacent to high cells, respectively. Hereafter, we refer to alignment of low cells adjacent to high cells simply as “low-cell alignment.”

The values of vertical alignment (L_V) of “top” distributions were significantly higher than those of “bottom” distributions (p < 0.01; Student’s t test; Figure 3F). In contrast, the values of horizontal alignment (L_H) of “top” distributions were significantly lower than those of “bottom” distributions (p < 0.01; Student’s t test; Figure 3F). As discussed earlier, we expected that the frequency of edges between high cells and low cells (freq_edge) would influence $\overline{{\Delta angle}_{\text{high}}}$. Although the difference was subtle, we found that the value of freq_edge was higher in “top” distributions than in “bottom” distributions (p < 0.05; Student’s t test; Figure 3F’). As discussed later, the contribution of this parameter to $\overline{{\Delta angle}_{\text{high}}}$ is not negligible.

GLM-based evaluation of factors affecting PCP

Finally, to dissect effects of orientations of low-cell alignment and other aspects of cell-type distributions on PCP, we constructed a generalized linear model (GLM). GLM is advantageous for modeling relationships between an objective variable and explanatory variables in that (1) it provides directly interpretable partial effect coefficients and (2) well-established diagnostic indices (AIC, VIF, and R²) are available for model evaluation. Here, the object variable was $\overline{{\Delta angle}_{\text{high}}}$ and candidates for explanatory variables included L_h, L_v, balance_LH, freq_low, and freq_edge. In short, by using GLM, we can expect to evaluate to what extent each explanatory variable contributes to the $\overline{{\Delta angle}_{\text{high}}}$. Based on the value of AIC (Akaike Information Criterion), we selected the model that best predicted $\overline{{\Delta angle}_{\text{high}}}$ in our simulation data (Figure 4). Data with balance_LH > 0.6 were excluded from the GLM analysis, as $\overline{{\Delta angle}_{\text{high}}}$ was weakly affected by cell distributions when the balance_LH was high. In addition, since L_h and L_v were strongly correlated (i.e., positive correlation) in the dataset, their direct inclusion in the model was problematic. To address this, we converted them into a single variable, L_h/v = L_h/L_v, which represents the anisotropy of low-cell alignment. The optimal model selected based on the value of AIC was as follows:

$$\log \overline{{\Delta angle}_{\text{high}}}=-0.834+0.615·{fre{q}^{\prime }}_{\mathrm{l}\text{ow}}-0.0390·L_{\mathrm{h}/\mathrm{v}}^{\prime }+0.327·{fre{q}^{\prime }}_{\text{edge}}-0.355·{balanc{e}^{\prime }}_{\text{LH}}$$

where values of explanatory variables were normalized so that their mean and SD were 0 and 1, respectively. Normalized variables are marked with an apostrophe (e.g., ${fre{q}^{\prime }}_{\text{low}}$). The values of $\overline{{\Delta angle}_{\text{high}}}$ predicted by this model were closely correlated with actual values of $\overline{{\Delta angle}_{\text{high}}}$ (Figure 4A; Pearson correlation coefficient: 0.859). However, the model’s accuracy decreased when the frequency of low cells was high, particularly at freq_low = 0.95 (yellow dots in Figure 4A’). Here, the absolute value of each coefficients indicates the strength of the influence of explanatory variables on $\overline{{\Delta angle}_{\text{high}}}$. The coefficients indicate that low cell frequencies (freq_low) and magnitude of core protein imbalances (balance_LH) have a significant impact on $\overline{{\Delta angle}_{\text{high}}}$, as expected. In addition, the model suggests that the frequency of edges between low and high cells (freq_edge) is a key parameter. Although the absolute value of the coefficient for the anisotropy of low-cell alignment ($L_{\mathrm{h}/\mathrm{v}}^{\prime }$) was smaller than those of the other explanatory variables, a model excluding L_h/v was less optimal according to AIC. Since the coefficient of L_h/v is negative, the model suggests that an increase in the ratio of total lengths of horizontal to vertical alignment of low cells decreases $\overline{{\Delta angle}_{\text{high}}}$. These results indicate that the frequency of low cells (freq_low), the frequency of edges between high and low cells (freq_edge), and the anisotropy of low-cell alignment (L_h/v) are key aspects of cell-type distributions that influence PCP.

Figure 4.

Regression analysis of features of cell-type distributions by generalized linear model

(A and A′) Scatterplots showing the correlation between actual $\overline{{\Delta angle}_{\text{high}}}$ values and those predicted by the generalized linear model (GLM). Each dot represents a data point shown in Figure 2B. freq_low values are color-coded in (A′). The detail of the model is shown in the right panel. Pearson correlation coefficient: 0.859. Nagelkerke’s R²: 0.882.

The direction of low-cell alignment is essential for mitigating the effect of low cells on PCP

To further address whether the direction of low-cell alignment affects PCP, we placed rectangular low-cell clusters within a cell sheet and analyzed the degree of PCP reorganization after simulations. First, low-cell clusters were aligned in a lattice at intervals of 3 cells and balance_LH was set to 0 (i.e., no core proteins in the low cells). We then examined the effects of low-cell clusters with varying values of l_h and l_v (Figures 5A and 5A′). Here, l_hand l_v represent the lengths of the cluster in the horizontal and vertical directions, respectively, relative to the x axis, which corresponds to the initial direction of PCP. Our analysis revealed that when low-cell clusters were elongated perpendicular to the x axis, the value of $\overline{{\Delta angle}_{\text{high}}}$ increased (Figure 5A, compare l_h = 3, l_v = 5 with l_h = 3, l_v = 3). In contrast, when low-cell clusters were elongated parallel to the x axis, the values of $\overline{{\Delta angle}_{\text{high}}}$ did not increase (Figure 5A, compare l_h = 5, l_v = 3 with l_h = 3, l_v = 3).

Figure 5.

Evaluating the effects of directions of low-cell alignment on PCP

(A and B′) Examples of simulations that evaluate the effects of low-cell cluster lengths (A′) on PCP. The resulting $\overline{{\Delta angle}_{\text{high}}}$ values are shown at the bottom of each panel. Clusters were arranged in a lattice (A) and were randomly shifted up to σ_max cell(s) along the x and y axis of each cell sheet (B and B′).

(C) Heatmaps showing mean values of $\overline{{\Delta angle}_{\text{high}}}$, averaged over 10 distributions for each combination of balance_LH, σ_max, l_h, and l_v. $\text{Mean}\text{values}\text{of}\overline{{\Delta angle}_{\text{high}}}$ are color coded as shown in the right lookup table (unit: degrees).

(D) Scatterplot showing a correlation between actual values of $\overline{{\Delta angle}_{\text{high}}}$ and those predicted by the GLM (right panel). Pearson correlation coefficient: 0.822. Nagelkerke’s R²: 0.759. Each dot represents a single simulation. See also Figure S6 and Table S2.

Next, we introduced stochastic shifts from the lattice points (Figures 5B and 5B′). The parameter σ_max determines the magnitude of this shift. When σ_max = 1, low cell clusters could be displaced by up to one cell along both the x and y axis. Figure 3B shows that elongation of low-cell clusters along the y axis increased the mean $\overline{{\Delta angle}_{\text{high}}}$, while elongation of low-cell clusters along the x axis did not (l_h = 3, l_v = 5 vs. l_h = 5, l_v = 3). We obtained similar results under the conditions of various parameter values, σ_max (1–3), l_h, l_v, and balance_LH (Figure 5C). When low cell clusters were regularly aligned, misorientation of polarity vectors was observed only in the immediate vicinity of low cell clusters (Figure 5A). In contrast, when clusters were shifted, misorientation of polarity vectors was not limited to the vicinity of low cell clusters (Figure 5B, middle panel), suggesting that shifting positions of clusters disrupts PCP.

These findings support our hypothesis that the direction of low-cell alignment plays a key role in mitigating the disruptive effects of low cells on PCP. However, in our analysis, modifying the lengths of low-cell clusters inevitably altered other aspects of cell-type distributions. For instance, changes in l_h and l_v also affected the frequency of low cells. To dissect effects of low-cell alignment and other aspects of cell-type distributions on PCP, we constructed a GLM. Here, the object variable was $\overline{{\Delta angle}_{\text{high}}}$ and candidates for explanatory variables included balance_LH, the frequency of low cells (freq_low), σ_max, l_h, and l_v. Based on the value of AIC, we selected the model that best predicted $\overline{{\Delta angle}_{\text{high}}}$ in our simulation data as follows.

$$\log \overline{{\Delta angle}_{\text{high}}}=-0.951+0.529·{fre{q}^{\prime }}_{\text{low}}-0.237·l_{\mathrm{h}}^{\prime }+0.289·l_{\mathrm{v}}^{\prime }-0.118·{balanc{e}^{\prime }}_{\text{LH}}+0.0418·{\sigma }_{\max }^{\prime }$$

where all explanatory variables are normalized. The Pearson correlation coefficient between the actual and predicted values of $\overline{{\Delta angle}_{\text{high}}}$ was 0.822 (Figure 5D), indicating that the model accurately described the relationship between the object variable and explanatory variables. Among all coefficients, the absolute value of the coefficient of ${fre{q}^{\prime }}_{\text{low}}$ is the largest, indicating that frequency of low cells has the most significant effect on $\overline{{\Delta angle}_{\text{high}}}$. The model shows that l_h and l_v are also essential parameters that affect PCP, as their coefficients have the second and third highest absolute values, respectively. As expected, the model shows that an increase in horizontal length (l_h) reduces $\overline{{\Delta angle}_{\text{high}}}$, an increase in vertical length (l_v) elevates $\overline{{\Delta angle}_{\text{high}}}$. In comparison, the absolute value of the coefficient for ${\sigma }_{\max }^{\prime }$ is much smaller than those of $l_{\mathrm{h}}^{\prime }$ and $l_{\mathrm{v}}^{\prime }$. Therefore, the effect of cluster distribution (σ_max) on $\overline{{\Delta angle}_{\text{high}}}$ is less significant than that of cluster lengths (l_h and l_v). These results were conserved in a different type of cell array (Figures S2B’ and S6). Taken together, we concluded that the directional low-cell alignment is a key aspect of cell-type distributions that mitigates the effects of low cells on PCP.

Orientation of PCP is correlated with the direction of low-cell alignment in vivo

To evaluate the relevance of our theoretical findings to living systems, we analyzed the mouse oviduct epithelium, which consists of only two cell types: secretory cells (SCCs) and multiciliated cells (MCCs). Because the frequency and distributions of SCCs are varied across different regions in the oviduct, we simultaneously analyzed both the direction of PCP and the distribution patterns of SCCs. We focused on two regions: infundibulum and ampulla.

In the infundibulum of the adult mouse oviduct, previous studies reported that SCCs comprise 24% (freq_low = 24%) of the epithelial cells (11 weeks old),^9,11 where distributions of core proteins are biased along the ovary-uterus axis, and that CELSR1 levels are lower in SCCs than in MCCs (Figures S1B and S1B”).⁹ First, we quantitatively evaluated the imbalance in CELSR1 levels between SCCs and MCCs. To this end, we measured the intensities of CELSR1 immunofluorescent signals at cell edges. We found that CELSR1 signals at edges between MCCs were approximately twice as high as those at edges between SCCs (Figures S5A–S5C′, balance_LH = 0.5). We also validated this conclusion by using transgenic mice expressing EGFP-fused VANGL2 ubiquitously (Figures S5D and S5E′). These observations suggest that SCCs have fewer levels of core proteins than MCCs and do not establish the polarized distribution of core proteins at cell edges.

Next, we analyzed the distribution of SCCs in the infundibulum. For this purpose, we used previously published images in which cell edges and MCCs were fluorescently labeled (Figures 6A and 6A’; images were adopted from Shi et al.¹¹). In the oviduct epithelium, SCCs form small clusters and these clusters scatter around the epithelium (Figures 6A and 6A′). We measured the length of each SCC cluster by placing a bounding rectangle whose horizontal edges were aligned parallel to the ovary-uterus axis (Figure 6A’). The mean l_h was 2.45 cells (Standard deviation, SD: 1.16), which was significantly larger than the mean l_v, 1.85 cells (SD: 0.862; p < 0.01, Wilcoxon rank-sum test, Figures 6B and 6B′). This result indicates that SCCs preferentially align along the direction of PCP in the infundibulum.

Figure 6.

Direction of alignment of secretory cells was correlated with the orientation of core protein distribution

(A and B′) Analysis of SCC distribution in the infundibulum region of the oviduct. (A) Oviduct stained for ZO1 (a marker for cell edges) and γ-TUBULIN (a marker for MCCs). White and yellow arrowheads point to an SCC and MCC, respectively. (A′) Segmented image: gray lines denote cell edges; white cells are MCCs; black cells are SCCs. The ovary-uterus (O-U) and circumferential (C) lengths of each SCC cluster (l_h and l_v, respectively) were measured using a bounding rectangle (gold box).

(B and B′) l_h and l_v values were analyzed using a scatterplot (B) and a boxplot (B′). The dotted line in (B) represents l_h = l_v. Open circles in B′ indicate average values. ∗∗: p < 0.01 (Wilcoxon signed-rank test). Isolated SCCs were excluded from the analysis. N: 200 clusters; 3 mice.

(C and C”) The ampulla region of the oviduct was opened longitudinally and stained for acetylated-TUBULIN (green, a marker for cilia) and F-actin (magenta, a marker for cell edges). The boxed regions in C and C′ are shown in C′ and C″ respectively. The direction of the ovary-uterus axis is indicated by an arrow in C’. (C″) SCCs are filled with magenta. VANGL2:EGFP signals are shown in green. Asterisks mark examples of circumferential SCC alignment. White arrowheads mark VANGL2:EGFP signals enriched at cell edges running along the ovary-uterus axis. These enrichments were observed in MCCs adjacent to SCCs aligned circumferentially. In the same view field, enrichments of VANGL2:EGFP signals at cell edges running circumferentially were also observed (yellow arrowheads). These enrichments were found near low cells aligned along the ovary-uterus axis (e.g., compare yellow arrowheads with white arrowheads in C″). Yellow bars in C″ represent polarity of VANGL2:EGFP distributions at cell edges. The length and angle of each bar indicate the magnitude and direction of VANGL2:EGFP polarity, respectively (see STAR Methods for details). Compare the orientations of bars near the white arrowheads with those near the yellow arrowheads.

(D and D″) Quantification of VANGL2:EGFP polarity in the infundibulum (D′) and the ampulla (D″). Bar angles relative to the O-U axis (D) were calculated and plotted as rose diagrams (circular histograms). See also Figure S7.

In the ampulla, another region of the oviduct, MCCs are sparsely distributed (Figures 6C and 6C″). Although the distribution of SCCs was varied, we sometimes found regions where SCCs were circumferentially aligned. In these regions, we analyzed the distribution of VANGL2:EGFP in MCCs adjacent to these SCCs, and found that VANGL2:EGFP was enriched at cell edges running along the ovary-uterus axis in adjacent MCCs (white arrowheads in Figure 6C”). Note that, in regions where SCCs were aligned along the ovary-uterus axis, VANGL2:EGFP in MCCs was enriched at cell edges oriented circumferentially (yellow arrowheads in Figure 6C”). Quantitative analysis confirmed that VANGL2:EGFP localization in the ampulla was misaligned relative to that in the infundibulum (Figures 6D and 6D″). Together with the results in the infundibulum, the direction of PCP in the oviduct was consistent with our theoretical outcomes.

Discussion

Although various tissues comprised of multiple cell-types with different levels of core proteins, how PCP is established and maintained in these tissues are still unclear. In this study, we performed systematic theoretical analysis and found that spatial distribution of the low cells, as well as their core protein expression levels and their population, affects proper PCP maintenance. Moreover, deep learning-based analyses using Grad-CAM revealed the impact of spatial distribution: PCP is less affected when low cells are aligned parallel to the tissue polarity. Finally, through generalized linear modeling (GLM), we successfully quantified the contribution of spatial distribution to PCP maintenance. Our results showed that the alignment of low cells has significant impact on PCP maintenance, similar to that of expression levels and population size. From these results, we propose that the spatial distribution of the low cells can be an essential factor for maintaining polarity of tissues composed of multiple cell types. Consistent with our theoretical findings, PCP orientations in distinct regions of the mouse oviduct varied in accordance with the alignment of low cells.

Mathematical models have been used for dissecting mechanisms of how core protein segregate into asymmetric complexes. However, the effect of spatial organization of cell types that differ in core protein levels on PCP were not evaluated in these studies. Building upon these previous modeling frameworks,^15,23 we addressed a different aspect of PCP: the mechanisms by which tissues tolerate core protein imbalance and maintain PCP, with a particular focus on the effects of cell-type distributions. Our theoretical analysis revealed that the impact of low cells on PCP is strongly influenced by their spatial distribution. This finding suggests that spatial configuration acts as a regulatory factor in PCP maintenance, functioning cooperatively with molecular interactions between core proteins. In other words, cell-type distribution itself can be considered an integral component of the PCP formation system. Previously we reported that loss of CELSR1 disrupts cilia beating orientation, cell shape elongation, and epithelial folding pattern in the mouse oviduct.⁹ In addition to these defects, we observed that anisotropy of SCC alignment was lost in Celsr1 mutant (Figure S7), suggesting that CELSR1 is required for proper low-cell alignment along the O-U axis. If CELSR1 directly regulates SCC alignment, there may exist feedback regulation between core protein polarity and cell-type distribution.

Our theoretical analysis offers an important theoretical perspective that provides a testable prediction for future in vivo studies—namely, that enforcing a specific alignment direction of low cells should alter PCP stability. The mouse oviduct would be a plausible candidate for testing the prediction, since, in this study, we found that orientation of PCP is correlated with the direction of low-cell (SCCs) alignment. One ideal experimental condition would be to align SCCs or, alternatively, cells lacking CELSR1 vertically relative to the O-U axis in the infundibulum of the oviduct. Although conducting such an experiment presents technical challenges, optogenetic approaches using mice expressing photoactivatable Cre recombinase^32,33 could be a useful strategy to induce mutant cells in a spatially restricted manner.

To examine how the degree of core protein imbalance influences PCP, we introduced a parameter, balance_LH, which quantifies the relative level of core proteins in low versus high cells. This allowed us to move beyond binary mutant/wild-type modeling and analyze graded imbalances in protein levels. While our study focused on a biologically relevant case in which both FZD and VANGL complex levels are equally reduced in low cells, our model framework is flexible and can accommodate other combinations of imbalance, for example, unequal reductions in the two complexes or mixed populations of different low-cell types that might be found in vivo. A full exploration of these parameter spaces would require extensive simulations and is beyond the scope of the current study. However, our approach can be readily extended to such scenarios in future work.

Because our mathematical model is deterministic, the outcome of each simulation is determined by the initial configuration of cell types. This feature allowed us to apply deep learning techniques to systematically identify spatial patterns that influence PCP maintenance. Specifically, we used a neural network classifier to screen diverse configurations and subsequently employed generalized linear modeling (GLM) to evaluate the contributions of interpretable spatial features, such as alignment of low cells, to polarity outcomes. Although the focus of this study is the relationship between cell-type distribution and PCP maintenance, this hybrid computational framework may be broadly applicable to other problems in tissue patterning. For example, it could be adapted to investigate the effects of spatial heterogeneity in signaling molecule levels beyond core proteins, offering a general strategy for bridging complex spatial input patterns with emergent tissue-level patterning.

While our study focused on cell-type distribution, we do not exclude the possibility that additional mechanisms influence the effects of core protein imbalance on PCP.

Global cues, such as Wnts, the atypical cadherins Fat and Dachsous, and mechanical forces are known to align distribution of core proteins with body axis.⁵ Therefore, these factors may act antagonistically to the effects of low cells. Although, the “power balance” between global cues and local effects of low cells in directing core protein distribution may differ among tissues, the Drosophila wing provides a useful context for discussing this balance. In the Drosophila wing, generating mutant cell clusters lacking Fz or Vang (members of core proteins) reverses wing hair orientation in neighboring wild type cells. In this situation, the global cues are expected to remain intact, yet they fail to maintain orientations of cell polarity, particularly near the mutant clusters. Furthermore, in large mutant cell clusters lacking a global cue, Fat, the non-autonomous effects of Fz overexpressing cells extend over a greater distance than in wild-type tissue.³⁴ These observations suggest that core protein imbalance and global cues engage in a “tug-of-war” to determine the spatial organization of core protein distributions in these experimental conditions. Consequently, there may be tissues in which global cues alone are sufficient to mask the effects of core protein imbalance.

Regarding mechanical forces, our study implies an additional route of PCP regulation by mechanical forces: organizing core protein distribution via aligning low cells. Moreover, core proteins themselves may influence the mechanical forces, as the loss of CELSR1 alters cell shape and tissue architecture.⁹ Although the present model isolates the contribution of local molecular interactions between different cell types, it could be extended to incorporate mechano-chemical coupling by integrating it into frameworks such as the vertex model.³⁵ Such a model would be valuable for dissecting how mechanical forces could be integrated into the system that maintains PCP in tissue exhibiting core protein imbalance.

In addition, core protein-independent mechanisms may also rectify PCP beyond the effect of core protein imbalance. In the Drosophila notum, the misorientation of bristles (a hair-like sensilla) in fz mutant is rescued by a directional tissue movement and a “comb” like effect of apical extracellular matrix.³⁶ In the mouse airway, the frequency of MCCs (0.37 on average³⁷) is much lower than in the oviduct (0.76 on average⁹). If non-ciliated cells in the airway are low cells, our model suggests that PCP maintenance would not be feasible in the airway epithelium. Fluid flow may therefore act as a compensatory mechanism to overcome core protein imbalance, as it has been shown that fluid flow aligns cilia beating orientation in cultured airway epithelial cell.³⁸

Limitations of the study

Our study has several limitations that could be addressed in future work. First, we focused on intercellular coupling of core PCP proteins and did not explicitly incorporate additional determinants of PCP, such as global cues, realistic cell geometries, or mechanochemical feedback, any of which could modulate robustness under protein-level imbalance. Second, although deep learning and statistical modeling identified cell-type alignment orientation as a key feature, these data-driven inferences may depend on the simulated parameter ranges and training data. Finally, our theoretical predictions have not been validated in vivo. Experimental perturbations that selectively alter cell-type distribution patterns or core PCP protein levels will be important to test causality and assess generality across tissues.

Resource availability

Lead contact

Requests for further information and resources should be directed to and will be fulfilled by the lead contact, Toshihiko Fujimori (fujimori@nibb.ac.jp).

Materials availability

This study did not generate new unique reagents.

Data and code availability

• •All data reported in this article will be shared by the lead contact upon request.
• •Code source files are available at: https://doi.org/10.6084/m9.figshare.30451130 and are publicly available as of the publication date.
• •Any additional information required to analyze the data reported in this paper is available from the lead contact upon request.

Acknowledgments

We thank Dr. Goshi Ogita for discussion; members of the T.F. laboratory for discussion and technical assistance. This work was supported by JSPS KAKENHI grant Nos, 21H02494, 24K02039, and 22H05168 (to T.F.), 22K15130 and the Takeda Science Foundation (to M.A.).

Author contributions

M.A., H.K., and T.F. designed research; M.A. performed research; M.A. analyzed data; and M.A., H.K., and T.F. wrote the paper.

Declaration of interests

The authors declare no competing interests.

Declaration of generative AI and AI-assisted technologies in the writing process

Gemini (Google) was used for writing R, Fiji, and Python scripts. ChatGPT (Open AI) was used for improving language and readability of the manuscript. After using these tools, we reviewed and edited the content of the scripts and the manuscript.

Star★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER
Antibodies
Mouse monoclonal anti-acetylated-TUBULIN	Sigma-Aldrich	Cat#T7451; RRID: AB_609894
Goat polyclonal Goat anti-Mouse IgG (H + L) Cross-Adsorbed Secondary Antibody, Alexa Fluor™ 647	Thermo Fisher Scientific	Cat#A-21235; RRID: AB_2535804
Deposited data
Code files for the simulation	This paper; figshare	https://doi.org/10.6084/m9.figshare.30451130
Experimental models: Organisms/strains
Mouse: R26-VANGL2-EGFP	Shi et al.9,10	–
Software and algorithms
Octave	Eaton et al.39	https://www.gnu.org/software/octave/doc/v8.2.0/
Fiji	Schindelin et al.40	https://imagej.net/software/fiji/
PyTorch	Paszke et al.41	https://pytorch.org/
R	R Core Team (2024)	https://www.r-project.org/
Cellpose2	Pachitariu and Stringer42	https://github.com/MouseLand/cellpose
QuantifyPolarity	Tan et al.43	https://github.com/Sara-Tan/QuantifyPolarity

Experimental model and study participant details

Mouse

Female adult mice (R26-VANGL2-EGFP transgenic mice¹⁰) were used to study VANGL2:EGFP distributions in wild-type oviducts. Animal care and experiments were conducted following the Guidelines of Animal Experimentation of National Institutes for Natural Sciences. All animal experiments were approved by the Animal Research Committee of National Institutes for Natural Sciences (approval number: 25A008). Animals were maintained in a light- and temperature-controlled room using a 12 h:12 h light:dark cycle at 23 ± 2°C.

Method details

Simulation

We used Octave³⁹ for numerical simulations and visualized the data by using Fiji.⁴⁰ The epithelial tissue was modeled as an array of regular hexagonal cells, with a width and height of 24 cells each. The boundary conditions were periodic in all directions. See below for details on our mathematical modeling and simulation.

The mathematical model

In our mathematical model, epithelial tissue is represented by an array of regular hexagonal cells. In each hexagonal cell, the FZD and VANGL complexes are distributed on the cell edges and within the intracellular region. Temporal changes in the concentration of the FZD complex are described by ordinary differential equations.

$$\begin{array}{c}\frac{d{F}_{{\mathrm{C}}_k}}{dt}=-\sum _i{K}_{{\mathrm{F}}_{ki}}{F}_{{\mathrm{C}}_k}+\sum _i{R}_{{\mathrm{F}}_{ki}}{F}_{{\mathrm{M}}_{ki}}\end{array}$$ (Equation 7)

$$\begin{array}{c}\frac{d{F}_{{\mathrm{M}}_{ki}}}{dt}=-R_{{\mathrm{F}}_{ki}}{F}_{{\mathrm{M}}_{ki}}+K_{{\mathrm{F}}_{ki}}{F}_{{\mathrm{C}}_k}\end{array}$$ (Equation 8)

where $F_{{\mathrm{C}}_k}$ and $F_{{\mathrm{M}}_{ki}}$ represent concentrations of the FZD complex in the intracellular region and on cell edge i of the kth cell, respectively (Figure 1B). The FZD complex in the intracellular region moves to cell edge i at a rate of $K_{{\mathrm{F}}_{ki}}$, while the FZD complex on cell edge i moves into the intracellular region at a rate of $R_{{\mathrm{F}}_{ki}}$ (Figure 1B, blue arrows). Similarly, temporal changes in the concentration of the VANGL complex are described as follows.

$$\begin{array}{c}\frac{d{V}_{{\mathrm{C}}_k}}{dt}=-\sum _i{K}_{{\mathrm{V}}_{ki}}{V}_{{\mathrm{C}}_k}+\sum _i{R}_{{\mathrm{V}}_{ki}}{V}_{{\mathrm{M}}_{ki}}\end{array}$$ (Equation 9)

$$\begin{array}{c}\frac{d{V}_{{\mathrm{M}}_{ki}}}{dt}=-R_{{\mathrm{V}}_{ki}}{V}_{{\mathrm{M}}_{ki}}+K_{{\mathrm{V}}_{ki}}{V}_{{\mathrm{C}}_k}\end{array}$$ (Equation 10)

where $V_{{\mathrm{C}}_k}$ and $V_{{\mathrm{M}}_{ki}}$ represent concentrations of the VANGL complex in the intracellular region and on cell edge i of the kth cell, respectively (Figure 1B). The VANGL complex in the intracellular region moves to the cell edge i at a rate of $K_{{\mathrm{V}}_{ki}}$, and the VANGL complex on cell edge i moves into the intracellular region at a rate of $R_{{\mathrm{V}}_{ki}}$ (Figure 1B, blue arrows). To implement autoactivation, mutual inhibition, and intercellular coupling (Figure 1A’), the reaction rate constants are described as follows;

$$\begin{array}{c}{K}_{{\mathrm{F}}_{ki}}={\rho }_{\mathrm{F}}+{\omega }_{\mathrm{F}}{F}_{{\mathrm{M}}_{ki}}{V}_{{\mathrm{M}}_{ki}}^{\prime }\end{array}$$ (Equation 11)

$$\begin{array}{c}{R}_{{\mathrm{F}}_{ki}}={\mu }_{\mathrm{F}}+{\phi }_{\mathrm{F}}{V}_{{\mathrm{M}}_{ki}}{F}_{{\mathrm{M}}_{ki}}^{\prime }\end{array}$$ (Equation 12)

$$\begin{array}{c}{K}_{{\mathrm{V}}_{ki}}={\rho }_{\mathrm{V}}+{\omega }_{\mathrm{V}}{V}_{{\mathrm{M}}_{ki}}{F}_{{\mathrm{M}}_{ki}}^{\prime }\end{array}$$ (Equation 13)

$$\begin{array}{c}{R}_{V_{ki}}={\mu }_{\mathrm{V}}+{\phi }_{\mathrm{V}}{F}_{{\mathrm{M}}_{ki}}{V}_{{\mathrm{M}}_{ki}}^{\prime }\end{array}$$ (Equation 14)

where $F_{{\mathrm{M}}_{ki}}^{\prime }$ and $V_{{\mathrm{M}}_{ki}}^{\prime }$ represents concentrations of the FZD and VANGL complexes on the cell edge of an adjacent cell juxtaposed to cell edge i of the kth cell (Figure 1B). Intercellular bridge formation was implemented by assuming that the autoactivation and mutual inhibition activities of the FZD and VANGL complexes depend on the formation of intercellular FZD-VANGL complex (FZD-VANGL bridge, Figure 1A’). The concentration of the FZD-VANGL bridge on cell edge i in kth cell is assumed to be proportional to $F_{{\mathrm{M}}_{ki}}{V}_{{\mathrm{M}}_{ki}}^{\prime }$ (or $V_{{\mathrm{M}}_{ki}}{F}_{{\mathrm{M}}_{ki}}^{\prime }$). This assumption, along with the mass-action terms, is adopted from a previous study.¹⁵ The second term of Equation 11 represents the autoactivation activity of the FZD-VANGL bridge. Note that increases in $F_{{\mathrm{M}}_{ki}}$ and $V_{{\mathrm{M}}_{ki}}^{\prime }$ on cell edge i elevate the value of $K_{{\mathrm{F}}_{ki}}$, a rate of the FZD complex movement from the intracellular region to cell edge i. Similarly, the second term of Equation 12 represents the mutual inhibition activity of the FZD-VANGL bridge, where increases in $V_{{\mathrm{M}}_{ki}}$ and $F_{{\mathrm{M}}_{ki}}^{\prime }$ on cell edge i elevate the value of $R_{{\mathrm{F}}_{ki}}$. ω and φ are constants that regulate the strength of autoactivation and mutual inhibition activities, respectively. ρ is a constant that represents the basal rate of the FZD and VANGL complex movements from the intracellular region to a cell edge, while μ is a constant that represents the basal rate of the FZD and VANGL complex movements from a cell edge to the intracellular region.

Furthermore, Equations 11, 12, 13, and 14 were modulated as follows to incorporate non-local effects.

$$\begin{array}{c}{K}_{{\mathrm{F}}_i}={\rho }_F+{\omega }_F\left(F_{{\mathrm{M}}_i}{V}_{{\mathrm{M}}_i}^{\prime }+d_{{\mathrm{A}}_{\mathrm{F}}}\sum _{j=i-1,i+1}{F}_{{\mathrm{M}}_j}{V}_{{\mathrm{M}}_j}^{\prime }\right)\end{array}$$ (Equation 15)

$$\begin{array}{c}{R}_{{\mathrm{F}}_i}={\mu }_F+{\phi }_F\left(V_{{\mathrm{M}}_i}{F}_{{\mathrm{M}}_i}^{\prime }+d_{{\mathrm{I}}_{\mathrm{V}}}\sum _{j=i-1,i+1}{V}_{{\mathrm{M}}_j}{F}_{{\mathrm{M}}_j}^{\prime }\right)\end{array}$$ (Equation 16)

$$\begin{array}{c}{K}_{{\mathrm{V}}_i}={\rho }_V+{\omega }_V\left(V_{{\mathrm{M}}_i}{F}_{{\mathrm{M}}_i}^{\prime }+d_{{\mathrm{A}}_{\mathrm{V}}}\sum _{j=i-1,i+1}{V}_{{\mathrm{M}}_j}{F}_{{\mathrm{M}}_j}^{\prime }\right)\end{array}$$ (Equation 17)

$$\begin{array}{c}{R}_{V_i}={\mu }_V+{\phi }_V\left(F_{{\mathrm{M}}_i}{V}_{{\mathrm{M}}_i}^{\prime }+d_{{\mathrm{I}}_{\mathrm{F}}}\sum _{j=i-1,i+1}{F}_{{\mathrm{M}}_j}{V}_{{\mathrm{M}}_j}^{\prime }\right)\end{array}$$ (Equation 18)

where the second term in parentheses denotes the non-local effects. For example, in Equation 15, an increase in the concentration of the FZD-VANGL bridges on cell edge i-1 and i+1 elevate the value of $K_{{\mathrm{F}}_{ki}}$, a rate of the FZD complex movement from the intracellular region to cell edge i (Figures S2A and S2A′). d_A and d_I are constants that regulate the strength of non-local effects (Figures S2A and S2A′). In previous theoretical studies, these non-local effects are implemented either by diffusion of molecules on cell membrane and/or within the intracellular region,¹⁵ or by simply assuming that core protein complexes on cell membrane exert autoactivation and/or mutual inhibition effects on neighboring membrane compartments.²³ We adopted the latter approach, as the details of non-local effects remain unclear and our study did not require a focus on these details.

Simulations were performed by the Euler’s method with a time step Δt = 0.01. The calculations were iterated until the simulation converged.

Parameters for models

The parameter values of the model (listed in Table S1) were determined so that the model reproduces the non-autonomous effects of mutant cells lacking core proteins on surrounding wild-type cells, as observed in vivo.^18,44 Since the non-autonomous effects in the mouse oviduct have not been thoroughly assessed, we referred to findings from Drosophila wing PCP, where the non-autonomous effects have been extensively investigated. Each wing epithelial cell produces a wing hair that points to the distal end of the wing. When cells lacking Frizzled (Fz) or Van Gogh (Vang) are genetically produced in the Drosophila wing, polarities of surrounding wild-type cells are reorganized.¹⁸ In the presence of fz mutant cell clusters, wing hair orientations of wild type cells is reversed on the distal side of the cluster.¹⁸ In our model, this phenotype was reproduced by introducing a cluster of cells completely lacking the FZD complex (Figure S3A1). In contrast, near a vang mutant cell cluster, the wing hair orientations are reversed on the proximal side,¹⁸ a phenotype we replicated by implementing a cluster of cells entirely devoid of the VANGL complex (Figure S3A2). When flamingo, a fly homolog of Celsrs, was lost in a cluster of cells, no reversal of wing hair orientation was observed in surrounding wild-type cells. However, when clusters of wild-type cells were surrounded by flamingo mutant cells, the wing hairs in the wild-type island formed a swirling pattern.¹⁸ Since Flamingo is a component of both the FZD and VANGL complexes, its loss can be approximated by the loss of both complexes. In our model, when cells lacking both FZD and VANGL complexes were introduced, misorientation of polarity emerged only in the vicinity of mutant cells and no polarity reversal was observed (Figure S3A3). However, when a cluster of wild-type cells was surrounded by cells lacking both FZD and VANGL complexes, a swirling pattern of polarity coordination emerged (Figure S3A4), although the emergence of this swirling pattern appeared to depend on the initial distribution of the FZD and VANGL complexes.

We initially set all parameter values to 1 and then changed each value to identify a parameter set that qualitatively reproduced these non-autonomous effects. In addition, we tested the robustness of the model outcome by varying ω and φ by ± 50%. These parameter changes did not alter our conclusion related to Figure 5, where we tested the effects of directional alignment of low cells on PCP maintenance.

The initial state of the simulation

We considered two types of hexagonal cell geometries, which have different initial distribution of the FZD and VANGL complexes as shown in Figures S2B and S2B’ (the initial state #1 and #2). The initial concentration in the intracellular region of kth cell was determined as 2-$\sum _i{F}_{{\mathrm{M}}_{ki}}$. balance_LH is a constant that governs the magnitude of the core protein imbalance between high cells and low cells. In low cells, the initial concentrations of the FZD and VANGL complexes at cell edges and in intercellular regions were multiplied by balance_LH, thereby reducing their levels accordingly. To manage floating point errors, small random noises were added to the initial concentrations of the FZD and VANGL complexes. The value of noise ranged from 0 to 0.6∗10⁻¹⁰ in high cells and 0 to balance_LH∗0.6∗10⁻¹⁰ in low cells.

In Figure 5, clusters of low cells were shifted randomly along the X and Y axis of the epithelial sheet. Length of shift, σ, of each cluster along each axis was determined randomly as σ = floor(rand∗(σ_max∗2 + 1))-σ_max, where rand is a random number that ranges from 0<rand < 1, and floor(x) returns the largest integer not greater than x. σ_max determines the max length of the shift.

Random cell-type distributions were generated for the analyses presented in Figures 2, 3, and 5, and S6. The number of random distributions was determined based on two practical considerations unless otherwise described: (i) verifying the main conclusion by calculating statistics and (ii) available computational resources.

Polarity vectors

The polarity of the FZD-complex distribution along cell edges was represented by a bar centered within each cell. The angle and the length of the bar represent the orientation of the FZD complex-enriched cell edges and the magnitude of the polarity, respectively. The orientation and the magnitude of polarity vectors were calculated as follows:

$$F_{{\mathrm{x}}_k}=\sum _{i=1}^6{F}_{{\mathrm{M}}_{ki}}\cos {\theta }_{ki}$$

$$F_{{\mathrm{y}}_k}=\sum _{i=1}^6{F}_{{\mathrm{M}}_{ki}}\sin {\theta }_{ki}$$

$${angle}_k=\arctan \left(F_{{\mathrm{y}}_k}/F_{{\mathrm{x}}_k}\right)$$

$${magnitude}_k=\frac{\sqrt{{F_{{\mathrm{x}}_k}}^2+{F_{{\mathrm{y}}_k}}^2}}{\sum _{i=1}^6{F}_{{\mathrm{M}}_{ki}}}$$

Here, angle_k and magnitude_krepresent the orientation of the FZD complex-enriched cell edges and the magnitude of the polarity of kth cell, respectively. angle_k ranges from −180 to 180°. θ_ki is an angle of a line connecting the centroid of cell edge i and the centroid of kth cell relative to the X axis of the cell sheet. For example, θ_ki = 30° in Figure 1B.

The magnitude of PCP disorder

We quantified the distribution of the FZD complex on cell edges to evaluate the magnitude of the disorder in PCP. Since the equations describing the dynamics of the FZD complex are equivalent to those of the VANGL complex in our model, there is no essential difference whether we use the distributions of the FZD or VANGL complex to evaluate PCP disorder.

To evaluate the magnitude of PCP disorder, we calculated $\overline{{\Delta angle}_{\text{high}}}$ as follows:

$$\overline{{\Delta angle}_{\text{high}}}=\frac{1}{n_{\text{high}}}\sum _{m\in \left\{\text{high}\text{cells}\right\}}\left|{angle}_{m,\text{initial}}-{angle}_{m,\text{final}}\right|$$

n_highis a number of high cells in a cell sheet. angle_{m, initial} and angle_{m, final} are the angle of the polarity vector of mth high cell in a cell sheet at the initial and final state, respectively. Additionally, we calculated a circular variance, which represents the angular distribution of the polarity vectors, to quantify the magnitude of PCP disorder. The circular variance ranges from 0 to 1, where a value of 0 indicates that polarity vectors are unidirectionally oriented, while a value of 1 indicates that vectors are randomly oriented. We preferred $\overline{{\Delta angle}_{\text{high}}}$, as the circular variance undervalues the disorder of PCP when polarity vectors were misoriented but still unidirectionally aligned.

Deep learning analysis

Data generation and labeling

For generating random distribution of low cells in Figure 2, random numbers that range from 0 to 1 were assigned to each cell, and cells with top 100∗freq_low % values were defined as low cells. Threshold values of $\overline{{\Delta angle}_{\text{high}}}$ for labeling “top” or “bottom” were determined by the accuracy of prediction made by the trained network. More specifically, at first, 3,000 distributions were generated at freq_low = 0.4 and balance_LH = 0.5 and data that took top 25% and bottom 25% values of $\overline{{\Delta angle}_{\text{high}}}$ were labeled as “top” and “bottom”, respectively. We used these data for training the network, but the accuracy of the prediction made by the network did not exceed 0.85. We additionally generated 3,000 distributions and selected distributions that took top 12.5% and bottom 12.5% values of $\overline{{\Delta angle}_{\text{high}}}$ for deep learning.

In addition to the measurement of balance_LH (≒0.5) in the oviduct shown in Figure S5, we considered two criteria for determining the values of freq_low and balance_LH to generate data: (i) diversity in patterns of cell-type distributions and (ii) variance in the values of $\overline{{\Delta angle}_{\text{high}}}$. To identify the relevant parameters of cell-type distributions, we aimed to construct a dataset that encompassed a wide variety of distribution pattens. The number of possible configurations of low and high cells reaches its maximum at freq_low = 0.5. Therefore, we chose values around 0.5 for freq_low. To train the neural network to classify the cell type distributions according to the value of $\overline{{\Delta angle}_{\text{high}}}$, we considered that a dataset exhibiting a wide range of $\overline{{\Delta angle}_{\text{high}}}$ values would be preferable. We used the dataset generated for the analysis shown in Figure 2 to identify a parameter set that produced high variance in $\overline{{\Delta angle}_{\text{high}}}$ values. Notably, at balance_LH = 0.5, variance was high at around freq_low = 0.95 and 0.4 as shown in Figure S4C.

Although the analysis at specific values of balance_LH and freq_low might limit generality, we demonstrated that the parameters identified from this analysis contributed to predicting the extent of PCP reorganization in other parameter sets, as confirmed by GLM analysis (Figure 4).

Data augmentation is a technique to increase the number and diversity of training data. Since our simulation adopted the periodic edge condition in all directions, shifting a cell-type distribution along the X and Y axis did not affect the value of $\overline{{\Delta angle}_{\text{high}}}$. Taking advantage of this, we generated three cell-type distributions from one distribution: one shifted 12 cells along the X axis, one shifted 12 cells along the Y axis, and one shifted 12 cells along both axes.

Training a neural network

We used the PyTorch ResNet18 architecture³⁰ that was pre-trained with the ImageNet-1K dataset for training. We re-trained the network (i.e., transfer learning) by using Google colaboratory. Hyperparameters used for training were as follows: the optimizer was SGD with a learning rate of 0.01 and a momentum of 0.9. batch size, 50; epoch number, 10. Since ResNet18 accepts images as input, each cell-type distribution was converted to an RGB image with 224 pixels in height and 224 pixels in width each. Since each cell was drawn as a hexagon, the converted image had margins at edges. The red channel was assigned to high cells, the green to low cells, and the blue to the margin. Then intensities of images were normalized so that the mean and SDs of intensities of each channel matched with these of the ImageNet-1K dataset. The accuracy of the trained network was evaluated by counting the number of validation data whose labels were correctly predicted by the network.

Grad-CAM visualization

Grad-CAM is an explainable AI technique that can visualize which regions of images the network focuses on when making a classification of an input image.³¹ We used pytorch-gradcam package in Python to visualize the network attention. Layer2[-1] of PyTorch ResNet18 were selected as a target layer to compute the attention map.

Generalized linear model

The object variable $\overline{{\Delta angle}_{\text{high}}}$ takes only positive values and does not follow a normal distribution, which violates the assumptions of ordinary least-squares regression. Therefore, we employed a generalized linear model (GLM) for the regression analysis. All following procedures were performed in R. Generalized linear models (GLM) were fit to data by using glm function of stats package (version 4.4.1). Explanatory variables were standardized before fitting GLM by using scale function of base package (version 4.4.1), which enabled comparisons of coefficients of explanatory variables. Gamma distribution was selected as a probability distribution, because $\overline{{\Delta angle}_{\text{high}}}$ takes positive continuous values. The object variable was related to explanatory variables via a log link function. To avoid multicollinearity, we calculated variance inflation factors (VIF) by using vif function of car package (version 3.1–2). We selected explanatory variables so that the value of VIF does not exceed 10. Then, models were evaluated based on values of AIC (Akaike information criterion) by using dredge function of MuMIN package (version 1.48.4). The model that took the lowest value of AIC was selected and used in the manuscript. r2 function of R package performance version 0.12.4 was used for calculating Nagelkerke’s R2 values. cor.test function of R package stats version 4.4.1 was used for calculating Pearson correlation coefficient.

For the GLM analysis in Figures 5 and S6, we generated data with varying cluster intervals to minimize correlations between cluster lengths and freq_low. A complete list of parameter values used for data generation is provided in Table S2. We excluded data simulated with balance_LH > 0.5 because, in this range, ($\overline{{\Delta angle}_{\text{high}}}$) were only weakly affected by low-cell distributions. Including these data in GLM modeling reduced the predictive accuracy of the model.

Imaging fluorescent signals

Oviducts were dissected and opened longitudinally along the ovary uterus axis. For imaging VANGL2:EGFP signals, oviducts were fixed in 4% PFA in PBS (−) at 4°C overnight. After washing, samples were incubated with Blocking One (Nacalai Tesque, 03953-95) at room temperature for 30 min. The samples were then incubated with anti-acetylated-TUBULIN antibody (1/500, Sigma-Aldrich, T7451) in Blocking One at 4°C overnight. After washing, samples were incubated with phalloidin (1/500, Thermo Fisher Scientific, A12381) and anti-Mouse IgG antibody (1/500, Thermo Fisher Scientific, A21235) in Blocking One at 4°C overnight. Finally, after washing, samples were mounted in Fluoromount G (SouthernBiotech, 0100-01). All washing steps were performed as follows: 3 x rinse and 3 × 10 min washes with 0.1% Triton X-100 in PBS (−) at room temperature. Images were acquired using Nikon ECLIPSE Ti2 microscope equipped with CSUW1 spinning disk confocal unit (Yokogawa).

Low-cell distributions in vivo

In Figures 6A and B′, data adopted from Usami et al.¹¹ were reanalyzed. In Figure S7, data adopted from Shi et al.⁹ were reanalyzed. The Celsr1 mutant mice used in Figure S7 showed no immunofluorescent signal when their oviducts were stained with an anti-CELSR1 antibody, indicating that this mutant allele does not produce functional CELSR1 protein.⁹ Cluster lengths were measured as follows: First, cells were segmented using Cellpose2,⁴² and the segmented data were saved as ImageJ ROIs. ROIs marking SCCs were selected based on signals of MCC markers (γ-TUBULIN or acetylated-TUBULIN). These ROIs were then dilated and filled with white, resulting in fusions of areas of adjacent SCCs. The images were then rotated so that the ovary-uterus axis aligned with the X axis of the image. In the rotated images, white areas were converted to ROIs using the analyze particle function in Fiji. Using these ROIs, the O-U and circumferential lengths of each cluster were measured by the measurement function in Fiji. Clusters that crossed the image edges or epithelial edges were excluded from our analysis. Since oviduct epithelial cells elongate along the ovary-uterus axis,⁹ we normalized the O-U and circumferential lengths of each SCC cluster. To achieve this, we divided cluster lengths by the average length of SCCs that comprise each cluster. These values were plotted in Figures 6B and B‴ and Figures S7C and S7D.

In Figure S7, frequencies of MCCs were measured by calculating the ratio of MCCs to all cells in each image. To measure the frequency of edges between SCCs and MCCs, we counted the number of these edges and divided the value by the number of all cell edges in each image.

Core PCP protein distributions in vivo

For quantification of CELSR1 immunofluorescent signals, data from Shi et al.⁹ were reanalyzed. Before quantification, the intensity of background signals was subtracted from the images. To measure background signal intensity in each image, three regions of interest (ROIs), each measuring 30 pixels in height and 30 pixels in width, were placed in areas outside the epithelial fold (Figure S5A). The average intensity across these ROIs was defined as the intensity of background signals.

Cell edges were segmented using Cellpose2⁴² and skeletonized using the Skeletonize function in Fiji. Each cell-edge was then saved as a ROI (cell-edge ROI) using a Fiji macro. Within each ROI, the average intensity of signals was measured using the measurement function in Fiji, and the CELSR1 or VANGL2:EGFP intensity was normalized by divided it by the F-actin intensity. Cell edges were categorized as either circumferential or O-U based on the orientation of each ROI’s centroid relative to the centroid of the corresponding cell (Figure S5B). Additionally, cell edges were further classified into three groups: edges between two MCCs (MCC-MCC), edges between as MCC and as SCC (MCC-SCC), and edges between two SCCs (SCC-SCC). Cell types (MCC or SCC) were determined based on acetylated-TUBULIN immunofluorescent signals.

In Figure 6, the distribution of VANGL2:EGFP at cell edges of individual cells was quantitatively evaluated using QuantifyPolarity v2.1.⁴³ The principal component analysis (PCA) method provided by QuantifyPolarity was used to determine both the magnitude and angle of polarity in the VANGL2:EGFP distribution. The angle represents the orientation of the signal-enriched region. The magnitude reflects the degree of directional bias in the signal distribution; higher values (depicted as longer bars in Figure 6) indicate stronger enrichment in a specific direction along the cell periphery.

Quantification and statistical analysis

R packages were used for statistical analysis as follows: the Student’s t test for Figures 3F and 3F’ (stats version 4.4.1); the Wilcoxon rank-sum test for Figures S5E, S7E, and S7F (stats version 4.4.1); the Wilcoxon signed-rank test for Figure 6B’ (stats version 4.4.1) and the Steel-Dwass test for Figures S5C and S7D (NSM3 version 1.18). Sample sizes and p values are described in the figures and legends. Non-parametric tests were used for data that were not normally distributed. Multiple-comparison tests were applied where appropriate to control for multiple testing.

Published: February 11, 2026