ABSTRACT

A. Epithelial cell morphology and intercellular junctional tension

One morphologic hallmark of mature epithelial tissues is the establishment of the tricellular junction network and hence the hexagonal packing of cells under steady state. This hallmark is established through adhesion at intercellular junctions (17–20). D’Arcy Thompson, a pioneer in mathematic biology, was one of the earliest scientists that researched the origin of these features. In his 1917 book, On Growth and Form (21), Thompson suggested that the cobblestone shape of cells (Fig 1A) arises from the interfacial tension along the intercellular junction, akin to soap bubbles (Fig 1B). In this context, the straight ridges separating cells simply result from the minimization of “interfacial energy” (22).

Building upon this analogy between epithelial cells and soap bubbles, the epithelial topology, specifically, how the cell–cell junctions are connected and the number of neighbors each cell has, can be derived from geometry rules. In 2 dimensions, the sum of the numbers of cells (F) and vertices (V) minus the edge number (E) is always a constant, V – E + F = 2. This relationship is known as the Euler polyhedral formula, and the constant is called the Euler characteristic, which is typically 2. A direct consequence of the Euler polyhedral formula is that (1)$$\frac{1}{\langle n\rangle }+\frac{1}{\langle z\rangle }=\frac{1}{2}$$where $\langle n\rangle$ is the average number of walls per cell and $\langle z\rangle$ is the average number of edges that meet at a vertex. Detailed derivation of this equation from the Euler polyhedral formula can be found in a previous review paper (23). In a monolayer of cells, each vertex is typically joined by 3 walls, and cells are mostly hexagonal, implying that $\langle n\rangle$ and $\langle z\rangle$ are 6 and 3, respectively.

The tension force along the intercellular junctions not only determines the topology of the epithelial cell network but also impacts cell motion, as well as tissue structure and rigidity (24–27). Researchers have adapted numeric models that were originally used for simulating foams to describe the structural and mechanical properties of epithelial tissues. This simulation model, known as a vertex model, minimizes an effective energy as a function of the positions of cell vertices (28). In a general version of the vertex model, each cell stores a mechanical energy (2)$$E_{\text{cell}}=\frac{{\kappa }_A}{2}{(A-A_0)}^2+\frac{{\kappa }_P}{2}{(P-P_0)}^2$$that implies the existence of an optimal cell area and perimeter due to intercellular junction tension. Here, A, $A_0$, P, $P_0$ are the area, optimal area, perimeter, and optimal perimeter of a cell, respectively. Also, $K_A$ and $K_P$ are the parameters related to the cellular junction tension, actomyosin contraction (i.e., tensile stresses generated by myosin motor proteins pulling on cross-linked actin bundles), and cell stiffness (e.g., Young modulus and bulk modulus). Using this effective energy, scientists have been able to describe various biologic processes, such as gastrulation, appendage formation, cell migration, and vesicle growth (29). One recent important finding is the epithelial structural change associated with the jamming transition, where cell proliferation and motility dramatically slow down, as the cells crowd (30). Vertex model simulations identified a critical value of mean shape index $q=P_0/\sqrt{A_0}=3.81$. In simulation, when q is larger than 3.81, the cells are motile, capable of rearranging, and tissues are “fluidlike.” Conversely, when q becomes smaller than 3.81, the cells become immobile, and tissues are “solidlike” (31). In this perspective, the shape index acts as an order parameter that identifies the fluid–solid transition of epithelial tissues. This is akin to traditional phase transitions, including liquid freezing, the paramagnetic-to-ferromagnetic transition (i.e., change of magnetic properties of materials), and Bose–Einstein condensation (i.e., a quantum phenomenon in which a large number of bosons simultaneously occupy the system’s ground state).

B. Nucleus size regulation

The nucleus is the largest and the most prominent cellular organelle. Since more than a century ago, scientists have found that although the size of the nucleus can vary greatly from cell to cell within a tissue, it is correlated with the size of the cell (32). In brief, the ratio between nuclear size and cellular size, the karyoplasmic or nuclear-to-cytoplasmic ratio, has been found to be a constant across numerous cell types. Despite the long history of this surprisingly simple and common finding, researchers are only beginning to understand its governing mechanism. It has long been postulated that the nucleus size is directly regulated by the DNA content (33). However, the current view posits that the cell size is the main regulator of nucleus size (34). Recent experiments showed that the main factors determining the nuclear-to-cytoplasmic ratio include the nuclear and cytoplasmic contents (e.g., proteins), the nuclear-to-cytoplasmic shuttling of proteins, nuclear membrane stiffness, and the cytoskeletal connection between the nuclear membrane and the cytoplasm (35). Specifically, recent theories have suggested that the osmotic pressure imparted by the nuclear and cytoplasmic contents may play a dominant role in deciding the nuclear-to-cytoplasmic ratio over other factors (36, 37). Through our learning module, the students will observe that the nuclear-to-cytoplasmic ratio is roughly a constant within a cell type, because these cells share similar gene expression profiles and osmotic pressure across the nuclear envelope.

C. Pedagogic background

We include a breakdown of the scientific concepts covered in this module and its corresponding teaching applicability for each education level, ranging from elementary to graduate-level education (Table 1). Suggested modifications or teaching tips for each module component will be discussed within the appropriate section throughout the text. In brief, our module contains 3 main teaching aims covering biophysics concepts, image acquisition and processing, and data analysis. Conceptually, we envision that most students at any education level can understand the epithelial–foam analogy by using visual aids. For example, because understanding phase transitions is listed as a topic within the Next Generation Science Standards for fifth grade (38), we anticipate that all students beyond elementary school will be able to understand the jamming transition as a state change from a fluidlike to a solidlike system. As a teaching tip to facilitate motivation for the module, students can experiment with some bubbles, or foam, to visually observe how the system packs. Alternatively, counting vertex-forming boundaries of foam can be issued as a prelab for more independent students. Due to its complexity and interdisciplinary nature, we recommend reserving nucleus-to-cytoplasm ratio discussions for college-level students.

Table 1.Tips for teaching: suggested teaching concepts for each education level.^a

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Regarding hands-on skills covered in the image acquisition and processing section, we anticipate that all levels above third grade will be able to operate a standard light microscope, in which there are only 2 main adjustable parameters to control the light intensity and focus plane. Because segmentation may require some quality control performed by students, we anticipate students beyond elementary school to be adequately equipped for this, and advanced high school or college-level students to be capable of performing iterative model training.

Finally, with respect to analysis, we anticipate that intercellular junction counting can be performed by students beyond elementary school, while generating AR histograms can be performed by advanced high school or college-level students. Because investigating nuclear-to-cytoplasmic correlation integrates fundamental principles from life science, physical science, mathematics, and statistics, we anticipate that only college-level students will be able to fully appreciate this analysis. We recognize it is critical to introduce image analysis to students at a young age to develop mathematic and scientific intuition, so we have included analysis adaptations that can be conducted by elementary-level students in section V of this article.

A. Histology image acquisition

In this work, we used commercial histology sections of the epidermis (outer layer) of frog skins (Triarch, Ripon, WI) to study the 2-dimensional (2D) epithelial cell morphology. Finding suitable 2D epithelial histology sections could be challenging because epithelial monolayers are usually only found in tubular (kidney) or glandular (prostate) structures, which are typically small (e.g., <100 $\text{μ}$m) and difficult to be flat mounted on microscope slides. We addressed this challenge by studying the outermost layer (i.e., stratum corneum) of frog skin, which is composed of a layer of terminally differentiated epithelial cells. As illustrated in a cross-sectional view of frog skin (Fig 2B), such a layer of cells is substantially different from and thinner than the multilayered stratum germinativum underneath it and, therefore, can be roughly considered as a 2D epithelial system.

We used a transmitted light microscope (Olympus CKX41, objective 40× numerical aperture [NA] 0.55, Olympus, Breinigsville, PA) to image the histology slide. An example image is shown in Figure 2C. We used a relatively high NA objective to suppress light scattering from background cells and extracellular matrix. Although the background suppression, hence the use of high NA objectives, may not be essential, it can potentially improve image qualities and image segmentation robustness. Images acquired by using different objectives and modalities of transmitted light microscopy are shown in Figure 2D–F. We also found that it is unnecessary to capture a large field of view by using low-magnification lenses, because the skin sample is often wrinkled, and only up to approximately 50 cells can be focused simultaneously. Based on these observations, we envision that any other student compound microscope (e.g., AmScope B120, Optika B-60, Meiji MT-30, or Leica DM300) would produce a similar image quality. Also, because we capture only approximately 50 cells per field of view, a 1-MP camera can readily resolve the morphology of cells and nuclei. Finally, we have uploaded 50 example images (40× phase contrast) used for analysis in this work as supporting materials (40). These images not only enable students without access to a microscope to complete the module but also serve as a reference for segmentation testing.

B. Image segmentation

In this work, we perform image segmentation that effectively extracts cell or nuclear contours by outputting the coordinates of the pixels that include the outline of the feature of interest. We use segmentation in order to later perform morphologic quantification. To perform image segmentation (Fig 3A), the acquired histology images were loaded into an AI-based segmentation software, Cellpose, version X (41). Although Cellpose is compatible with many imaging modalities (e.g., fluorescence, bright field, and phase contrast), we used phase contrast images as our input because this modality is widely accessible and can contrast the nuclear and cellar boundaries with ease. Cellpose currently has 2 versions: Cellpose 1.0 and 2.0. Briefly, Cellpose uses computational neural networks to analyze the imported images, and a numeric function of the corresponding image pixels is created to predict the outlines of features, such as cells and organelles. In Cellpose 1.0, the users use preestablished AI models for segmenting the images, whereas Cellpose 2.0 allows the users to either use preestablished models or train and correct new AI models. This AI-based segmentation tool has several advantages over traditional parametric methods. Cellpose is capable of accurately segmenting cells in a variety of imaging modalities. In addition, its deep learning approach reduces the need for manual intervention and can be trained with relatively few manual examples if the preset algorithms are not geared for the images being segmented. This image segmentation platform has been widely used in current research (42–44) and combined with many other cell characterization tools (45).

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We used images saved as Joint Photographic Experts Group files to reduce the file size, although Cellpose is compatible with most bitmap file types. Once our image was imported, we adjusted the cell diameter parameter in the Segmentation module in the graphical user interface (GUI) to roughly match the diameter of the cell or nucleus, as indicated by the top yellow arrow in Supplemental Figure S1. All other parameters were left at the default setting, including the flow_threshold and the cellprob_threshold. Next, we executed the segmentation by choosing the appropriate compartment model from the model zoo in the GUI. For example, if cell boundaries are to be extracted, the cyto model, indicated by the bottom yellow arrow in Supplemental Figure S1, should be selected. Following this, the results were displayed as a mask overlay on the input image. To clean up data, we removed falsely detected masks and added masks that may have been missed, by control clicking and right clicking, respectively. After mask clean up (Fig 3B), we saved the outline boundary coordinates as a text file that would later be read into ImageJ, which is a common open-source software for image analysis (39). This step will be discussed further in the following. Analyses can also be performed by using other platforms, such as Python and MATLAB. We presented the ImageJ example in this work due to its user-friendly interface with no coding experience required.

We found that the nuclear segmentation results were suboptimal when using the Cellpose nuclear model Figure 3C. This may be attributed to the fact that Cellpose is often performed on fluorescence images, in which the nuclei are the only feature in each image with a high contrast. On the other hand, histology slides contain many other features with a lower contrast between them. To overcome this issue, we trained our own nuclear model. To do so, we first manually outlined $\sim$50 nuclei in an image (Supplemental Fig S2A) and saved the masks and images as a seg.npy file (Supplemental Fig S2B) and repeated this process for 4 additional images. To train our model, we selected “train new model with image + masks in folder” under the models tab of the GUI (Supplemental Fig S2B). This procedure resulted in the trained model appearing as an option under the custom models menu, which was then used for segmenting other images (Fig 3A). We uploaded our model for students to use because we recognize training can be challenging for most kindergarten to grade 12 students (40). Finally, the instructor can emphasize the importance of data verification. Students can manually segment a manageable number of cells (e.g., 10 to 50) and compare the results with those from Cellpose. This validation activity not only ensures the quality of data for further analyses but also emphasizes the importance of rigor in biophysics research.

C. Morphologic analysis

A. Morphologic characterizations

As discussed in section II, the cell shape and junction network are 2 imperative features to consider when characterizing epithelial morphology, because they help regulate essential tissue properties such as the barrier function and rigidity (46). By analyzing 108 vertices, we observed that all of them included 3 edges. We also found that although most of the examined vertices were well separated (Fig 4A), a few of them were paired (Fig 4B). We summarized this counting result in Figure 4C, which demonstrates that $\sim$85% of vertices are isolated vertices (i.e., 3-cell vertices), and $\sim$15% of vertices are associated vertices (i.e., 2 vertices in close proximity). Our observation confirms that most cells in the tissue obey the behavior predicted by the Euler polyhedral formula. The observed associated vertices could be related to 4-cell vertices, which have been attributed to active structural rearrangements in living epithelial tissues (47). Events such as cell movements, cell proliferation leading to tissue growth, cell death, or cell extrusion can all contribute to changes in cell packing and organization. Our finding of associated vertices is also consistent with both reported experimental observations (48, 49) and vertex model simulations (47).

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To further illustrate how the packing rule governs epithelial cell morphology, we investigated the degree of heterogeneity in our system. To do this, we calculated the probability distribution of cell AR for 500 cells (across 19 images). Here, the AR is defined as the ratio between the major and minor axes of an ellipse fitted to a cell. AR is a key morphologic indicator of epithelial phenotype, in which fully packed cells often exhibit a low AR (Fig 4D), or more rounded shape, as a result of interfacial tension energy minimization. However, because the cells are randomly packed, different cells would experience various degrees of geometric constraint created by their neighbors. As a result, it is anticipated that a small population of cells would still display a relatively high AR (Fig 4E) or elongated shape.

As shown in Figure 4F, we first confirmed that the mean AR of epithelial cells was low ( $\langle \text{AR}\rangle \sim 1.45$). We further confirmed that the local packing gives rise to a relatively large spread in AR, indicating a noticeable morphologic heterogeneity in the epithelial monolayer sample (Fig 4F). This again highlights the dynamic nature of epithelial tissues, where active cellular events can introduce variations among the population. To compare the statistics of our results with other experiments and previous simulation predictions, we normalized the AR by rescaling the x axis of Figure 4F by using $x=\frac{\text{AR}\hspace{0.17em}-\hspace{0.17em}1}{\langle \text{AR}\rangle \hspace{0.17em}-\hspace{0.17em}1}$ (Fig 4G). This rescaling effectively sets the distribution lower bound and mean to be 0 and 1, respectively. We then fitted our data to a gamma distribution, denoted by the black curve shown in Figure 4G. Here, a gamma distribution is defined as (4)$$\text{PDF}(x;\kappa )={\kappa }^{\kappa }{x}^{\kappa -1}{e}^{-\kappa x}/\Gamma (\kappa )$$where PDF is probability density function, $\Gamma (\kappa )$ is the Legendre gamma function, x is the variable, and $\kappa$ is the shape parameter. This function is often used to describe a distribution that has a sharp lower bound of zero but no upper bound, and a positive skew. Such a distribution has been used to model numerous social and biologic examples, including the size of insurance claims, the age of cancer incidence, and the interspike interval between neuron firings (50–52). From our fitting, we determined the shape parameter $\kappa =2.40$, which is similar to the measurements using dog kidney cells ( $\kappa =2.39$), Drosophila ( $\kappa =2.52$), and the vertex model prediction ( $\kappa =2.53$) (53).

Finally, we calculated the shape index $q=P_0/\sqrt{A_0}$ of each cell and found that the mean value, $\sim 4.21$, is slightly higher than the fluid-to-solid transition threshold 3.81, predicted by the vertex model. This discrepancy could be due to dynamic tissue rearrangement resulting from extrusion or apoptotic events. Specifically, as dying cells are shed off and new cells enter the stratum corneum, some cellular rearrangements could still occur and contribute to a larger measured shape index. We should, therefore, be reminded of the dynamic nature of a living tissue and the importance of considering other biologic factors when interpreting measurements of physical properties. In addition, because our analyzed frog skin epithelial cells are fully differentiated, the tissue should be completely solidlike. Such a deviation in shape index may be then due to the tortuous perimeter of our tested cell. This effectively leads to a higher shape index value, which was not considered in the original vertex model simulation. The origin of ruffled junctions (i.e., zigzag shape) and the biologic implications remain active research topics (54), which could be an interesting postlab discussion topic. Overall, we conclude that our morphologic analyses are in agreement with previous experiments and simulations.

B. Nuclear-to-cytoplasmic area ratio

In this section, we analyzed the correlation between the nucleus and cell area. This ratio has been shown to be roughly a constant within a tissue sample but can vary across cell types. In testing the cell–nucleus correlation, we also demonstrated how statistics impact the analysis results and robustness, which play an essential role in interpreting research observations. We first plotted the nucleus area as a function of cell area by using all of the data points ( $N=500$, shown in the left panel of Fig 5A). In this plot, we observed a clear positive correlation between the cell area and nucleus area. The observed correlation was confirmed by a moderate Pearson correlation coefficient $r\sim$ 0.67. Furthermore, the proportionality between the cell and nucleus areas was further confirmed by a linear fit (red line), $A_n=0.067A_c-1.883$ and its small intercept $1.883/\langle A_n\rangle \ll 1$.

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To observe how the statistics influence the correlation analysis, we systematically reduced the data points by sampling a subset of $N=100,50,$ and 10 points. As shown in Figure 5A, we found that the nuclear-to-cytoplasmic area ratio is preserved across our tested sample sizes. However, we also found that the data trend starts to vary when the sample size is less than 50, as indicated by the changing slope of fitted lines (red dashed lines). To more quantitatively characterize this finding, we calculated the Pearson correlation coefficient between cell and nucleus area for all tested sample sizes and repeated this calculation for 100 randomly sampled data subsets (Fig 5B). Consistent with our observation, although the mean value (Fig 5B, dark blue middle line) of the Pearson correlation coefficient does not depend on sample size, its standard deviation (Fig 5B, light blue band) does. In other words, the Pearson correlation coefficient remains unchanged when sampling only a few cells, although determining a trend becomes less definitive when the sample size is less than $N=20$.

Finally, we analyzed how the correlation’s P value varies in response to sample size (Fig 5C). Here, the P value is the probability of observing a correlation in the tested data when, in fact, there should be no correlation (i.e., the null hypothesis is true). This statistical analysis is important to report for any biologic experiments, which often contain notable uncertainties and variations. Typically, a P value less than 0.05 (yellow-shaded region in Fig 5C) is accepted to be significant, suggesting that the null hypothesis can be rejected. Our results showed that the observed correlation becomes statistically significant at around a sample size of N = 20. This is supported by our results of Figure 5B in which we saw a high degree of reproducibility in the Pearson correlation coefficient beginning around N = 20. Together, these results demonstrate that students should analyze at least 20 cells in future experiments to obtain statistically significant data.

In our error analysis activity, the 500 data points were collected by using images from 3 histology samples, thus, presumably 3 biologic replicates. Biologic replications are critical for quantification and allow statistical analyses of the data. A potential caveat with purchased histologic slides is that one cannot guarantee the preparations from different individual animals. Therefore, although the activity was designed to demonstrate the importance of sample size, instructors should discuss how biologic replicates are typically achieved in laboratories, the differences between technical and biologic variations, and the use of biologic replicates for assessing statistical significance.