Ethics Statement

This study involves CD1 mice and was approved by the University at Albany, SUNY IACUC under protocol numbers 09-013 and 12-013.

The Glazier-Graner-Hogeweg (GGH) Model

The GGH model is built on the energy minimization-based Ising model, using imposed fluctuations via a Monte Carlo approach [24]. The simulation space is divided into a lattice, which may be two- or three-dimensional, and cells are represented by groups of adjacent lattice points; each lattice point has an associated energy value that is assigned based on its interactions with other lattice points. Energy is also assigned to cells based on cell-cell interactions, and the sum of energies across all lattice points and cells in the simulation space is the effective energy. The energy assignment of a lattice point is based on functions representing biological behaviors or constraints, and the effective energy of the simulation can be written as a Hamiltonian equation, where each term represents the sum contribution of a particular energy function. The model is based on the assumption that the most favorable state is the lowest energy state.

To develop the 2D GGH single cleft and tissue models of cleft progression, we used the following terms:

Contact energy represents differential adhesion between model cells of different types by assigning an energy penalty to adjacent lattice points belonging to different cells. Each possible pair of cell types (τ_a,τ_b) is enumerated and assigned an energy penalty J(τ_a,τ_b), including same-type pairs. Cell types that adhere to each other are assigned a lower energy penalty; the cell type τ of a particular lattice point i is given by τ_σ(i), where σ is the cell ID. The contact energy penalty assigned to a pair of lattice points (i,j) is therefore given as J(τ_σ(i),τ_σ(j)). To prevent lattice points within the same cell from being assigned a contact energy penalty, this is multiplied by (1−δ_σ(i),_σ(j)), where δ is the Kronecker delta. The term for contact energy in the Hamiltonian equation across all pairs of lattice points (i,j) is therefore given as:(1)for all neighboring lattice sites i, j.

Area (a) represents cell volume in two dimensions. It is a cell-based energy function that penalizes cells for deviating from a target size, simulating the biological tendency for cells to grow to and maintain a certain size. It has two constants, a target area A, and a strength factor λ. The term is thus:(2)for each cell σ and cell type τ

Perimeter (p) is a representation of surface area in two dimensions. Like area, it is a cell-based energy function, and it imposes an additional constraint on cell size based on the amount of plasma membrane available to a cell. It also uses two constants, a target perimeter P, and a strength factor λ. The energy term is given as:(3)for each cell σ and cell type τ

Focal point plasticity is a cell-based energy term that assigns an energy penalty for linked cells that deviate from a target length L, based on the distance between the cell centroids (l). Although it was developed to simulate actomyosin-dependent contractility, it is used in our model to simulate the effects of actomyosin contractility-dependent FN assembly. Since we are unable to represent the FN wedge as a physical structure, we reproduce its cleft-forming effects by exerting a separating effect on opposing cells of the cleft wall through FPP. Within the cleft, the target distances between opposing cells are assigned based on depth, and represent the shape constraint imposed by the FN structure. The λ value modulates the effect of focal point plasticity, and corresponds to the amount of actomyosin contractility present in the simulation. The energy term is:(4)for linked cells σ and σ', and cell type τ

The target distances that produce the characteristic shape of the cleft are assigned based on an inverse relationship with the depth; cells near the bottom of a cleft are assigned shorter target distances than the cells at the top of the cleft. This relationship was determined through examination of images of progressed clefts from time-lapse images of embryonic day 12 (E12) organ explants. Additionally, we used a simplified two-cell model to investigate the effects of FPP relative to Cell-Matrix (CM) contact energy, λ, and target distance, for constant cell-cell contact energy (CC) value = 10 to determine the values of λ to use in the model (Figure S4).

The full Hamiltonian equation for our simulation is thus given as the sum of these four equations:(5)

Energy minimization is carried out by choosing pairs of adjacent lattice points from different cells, and an attempt is made to copy the cell ID from the first point to the second. This copy attempt grows one cell, either by forcing another cell to shrink, or expanding into the medium. The effective energy is calculated before and after the change, and if the new energy is lower, the change is made permanent. However, if the resulting energy is higher, the change is only retained with some probability using a Boltzmann acceptance function, e^−ΔH/T. In the context of the GGH simulation, T is a constant that controls the intrinsic motility of the cell, corresponding to the amplitude of cytoskeletally derived membrane fluctuations. Using T, we have allowed a certain amount of cell motility. Allowing some amount of energy-raising lattice-copy events is important as it prevents the model from stalling at local energy minima. A single step in the GGH model actually consists of N lattice copy attempts, where N is the total number of lattice sites in the simulation space. These attempts are carried out through a Monte Carlo simulation using modified Metropolis dynamics, designated as Monte Carlo steps (MCS) [24].

Cell proliferation in the GGH model is accomplished by dividing an existing cell into two equally sized new cells. To simulate mitotic cells, a subset of cells is instructed to grow to twice their original size and divide every 100 Monte Carlo steps (MCS), mimicking the growth and mitosis of biological cells.

Simplification: Although the GGH model is able to mimic parameters such as growth factor absorption kinetics, we have omitted these from this initial study to reduce complexity and focus on the cellular behaviors. Similarly, we have simplified the basement membrane and mesenchymal compartment, which contains nerves and blood vessels [2], [32] in addition to mesenchymal fibroblasts; surrounding the epithelium into a single compartment we call “matrix” and that is often called “medium” in GGH models. The matrix compartment is essentially represented here as a single special GGH cell that is not subjected to area and perimeter constraints. We have not included apoptosis in our model since there is currently no biological data to suggest that apoptosis is important in cleft progression.

Ex-vivo Submandibular Salivary Gland Organ Culture

Embryos from timed-pregnant female mice (strain CD-1, Charles River Laboratories) at embryonic day 12 (E12) (with day of plug discovery designated as E0), were used to obtain submandibular salivary gland rudiments (SMGs) following protocols approved by the University at Albany, SUNY IACUC committee (protocols 09–013 and 12-013), as reported previously [6], [7], [33], [34]. E12 SMGs that contain 1 primary bud were micro-dissected from mandible slices and cultured, as described previously. For culturing ex-vivo organs, 13 mm, 0.1 µm pore size Nucleopore Track-Etch membrane filters (Whatman) were used. The SMGs were floated on top of the filters that sit on 200 µL of 1∶1 DMEM/Ham's F12 Medium (F12) lacking phenol red (Invitrogen) in glass-bottomed 50 mm microwell dishes (MatTek Corporation). The medium was supplemented with 50 µg/mL transferrin, 150 µg/mL L-ascorbic acid, 100 U/mL penicillin, and 100 µg/mL streptomycin, to make complete DMEM/F12 medium. Brightfield images were acquired on a Nikon Eclipse TS100 microscope equipped with a Canon EOS 450D digital camera at 4X (Plan 4X/0.10 NA) magnification.

Whole-Mount Immunocytochemistry and Confocal Imaging

Whole-mount immunocytochemistry was performed as previously described [6], [7], [33], [34]. E12 SMGs were fixed in 4% paraformaldehye (PFA) in 1X phosphate buffered saline (1XPBS) containing 5% (w/v) sucrose for 20 min at room temperature. SYBR Green I (1∶10000, Invitrogen) was used to detect nuclei and proliferating cells were detected using phospho-Histone H3 (pHH3) antibody (1∶100, Cell Signaling Technology). Epithelium was detected using an antibody recognizing E-cadherin (1∶250, BD Biosciences), F-actin was detected using Alexa Fluor 546 Phalloidin (Invitrogen, 1∶350), and mesenchyme was detected using an antibody recognizing PDGF receptor (R)-β (1∶100, Epitomics). Appropriate cyanine dye-conjugated AffiniPureF (ab′)2 fragments were used as secondary antibodies (Jackson ImmunoResearch Laboratories, 1∶100). SMGs were imaged on a Zeiss LSM510 confocal microscope at 20X (Plan Apo/0.75 NA), or 63X (Plan Apo/1.4 NA) magnification.

Confocal Time-Lapse Series Acquisition

E12 SMG organ explants were treated with 200 µl of Hank's balanced salt solution (HBSS lacking Ca²⁺ or Mg²⁺, Life Technologies) containing 0.4% (v/v) dispase (Life Technologies) for 25 min at 37°C, and the mesenchyme was physically removed by microdissection, as described in [10]. The epithelial rudiment was cultured in a final concentration of 6 mg/mL Matrigel (BD Biosciences) diluted in DMEM/F12 containing 20 ng/mL EGF and 200 ng/mL FGF7 (R&D Systems). The gland was imaged using time-lapse microscopy with a 20X objective lens using a Zeiss 510 Meta Confocal microscope. 120 images were captured as 5 µm sections at 10 minute intervals for a 20 hour time period using the MultiTime macro. The 543 nm laser was used to capture a near-DIC image. Images were captured at a 512×512 pixel resolution using a scan speed of 9 in line averaging mode. A total of 30 glands were imaged for 20 hours in three separate sets and 40 clefts were measured using image analysis software ImageJ [35]. The first frame and the last frame (after 20 hours) were used to measure the depth in pixels for each cleft and according to the scale, the distances were converted to micrometers (µm).

Image Processing

To enhance the contrast of the grey-scale pHH3 images and the SYBR green images, we applied the contrast-limited adaptive histogram equalization algorithm (CLAHE) [36] to the image. The CLAHE algorithm considers the image as a collection of smaller regions and applies histogram equalization on these regions. The objective of histogram equalization is to transform the image so that the intensity histogram of the output image approximately matches a specified histogram; in our case we use a curved histogram. The CLAHE algorithm evens out the distribution of used grey values and thus makes hidden features of the image more visible. Noisy regions of the images are removed by considering regions of intensity greater than a pre-determined threshold. For the E-cadherin marker images, we applied a Gaussian smoothing followed by the CLAHE algorithm, and then removed noisy regions based on a predetermined threshold. Binary masks were created for the SYBR green and pHH3 histogram equalized images by applying an OR operation on the histogram equalized image and the E-cadherin marker. The total area of the connected components in both images was calculated, and the ratio yielded the percentage of SYBR green-positive cells (total cells) that are in mitosis, or M phase, of the cell cycle.

Classification Analysis

Four values for mitosis rate (MR), six values for contractility (FPP λ), five values for cleft region adhesion (CC), and five values for cleft-matrix adhesion (CM) were chosen from the hypothesis driven individual analyses, and 40 simulations were run for each of the 600 possible combinations. Cleft simulations were classified as failed (less than 17.8 µm), non-progressive (17.8 to 30.5 µm), progressive (30.5 to 40.7 µm), and super-progressive (greater than 40.7 µm) based on minimum, first quartile, and third quartile depths of ex-vivo cleft measurements. Parameter combinations were assigned an overall class based on the cleft depths attained in a majority class within the 40 runs; in the case of a tie, the median depth was used to classify the parameter combination. This resulted in 275 failed, 188 non-progressives, 85 progressive, and 52 super-progressive results. To determine the importance of each GGH parameter in cleft progression, we formulated the problem as a supervised learning feature selection task, with each combination as a data point and the parameter values as features. Samples were created by drawing 50 random points from each class. For each of the 15 possible combinations of the four features, a 10-fold cross-validation using a radial basis kernel support vector machine (SVM) was performed on the sample, reporting the training and testing accuracies [37]. A greater decrease in classification accuracy corresponds to a more important feature. Additionally, analysis of the parameters resulting in progressive clefts was performed to confirm the importance of each parameter; parameters that were essential to progressive clefts were expected to be distributed around a particular value with low variance.

Establishing a Single-Cleft GGH Model for SMG Branching Morphogenesis

We chose to start our model at E12, when the mouse SMG undergoes the first round of branching morphogenesis. At E12, the gland is a single epithelial mass, or bud, atop a stalk, surrounded by a condensed mesenchyme (Figure 1A, 1B). Clefts initiate as indentations in the epithelium, which progressively furrow interiorly. Since cleft initiation and cleft progression are biochemically independent steps [6] and little biological information is available regarding mechanisms of cleft initiation, we chose to pre-specify an individual initiated cleft in the model and simulate only the stage of cleft stabilization and cleft progression (Figure 1E). At E12, the epithelium expresses E-cadherin (Figure 1C,1D) but later stage differentiation marker proteins are not yet expressed [38], [39]. We therefore assumed that the cell-cell adhesions present are E-cadherin-containing adherens junctions with an absence of tight junctions, as previously reported [38], [39]. The epithelium is surrounded by mesenchyme that expresses PDGFR-β, which can be used to distinguish the latter from the former (Figure 1C, 1D). Closely associated with the epithelial cells is the basement membrane, a specialized extracellular matrix (ECM) that forms a boundary between the epithelial and mesenchymal tissue compartments [5], [40]. Since we are focusing on epithelial cell parameters that control cleft progression, we modeled the basement membrane and the entire mesenchyme compartment as a simplified single cell, designated as “matrix,” which lacks area and perimeter constraints.

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Figure 1. Mouse submandibular salivary gland organ structure and cleft formation during branching morphogenesis.

Brightfield images of (a) an embryonic day 12 (E12) submandibular salivary gland (SMG) organ explant and (b) an SMG explant harvested at E12 and grown for 24 hours ex vivo with epithelium (E) and mesenchyme (M) labeled. Scale, 200 µm. Single confocal images of E12 SMGs following ICC to detect epithelium (E-cadherin, red) and mesenchyme (PDFGR, cyan) captured at (c) cleft initiation and (d) a late stage of cleft progression. Progressing clefts are indicated (white arrow head). Scale = 50 µm. (e) Diagram depicting cleft transitions.

https://doi.org/10.1371/journal.pcbi.1003319.g001

Designating Epithelial Cell Subpopulations in the CC3D Modeling Environment

At E12, there are two structurally distinct epithelial precursor cell populations [34], [38]. The outer columnar cells (OCCs) that contact the basement membrane surround a cluster of less organized inner polymorphic cells (IPCs) (Figure 2A), and this cell arrangement is maintained during 24 hours of ex-vivo culture (Figure 2B). The 6×6 pixel square cells were arranged in a homogenous grid, a simplification that approximates the initial cell distribution with OCCs labeled in dark green and IPCs in light green (Figures 3A, 3B). To calibrate the model with image data, we performed time-lapse imaging of multiple E12 mesenchyme-free SMG organ explants for 20 hours and measured the length of the resulting clefts (Figures 3C, 3D, Video S1). Clefts achieved an average depth of 36.2 µm and a median depth of 35 µm. Based on the cleft depths obtained from the time-lapse analysis, we defined normal cleft depth in the CC3D model as 36 pixels, using 6 cells per cleft, shown in light and deep blue (Figures 3A, 3B). To distinguish OCCs from IPCs, we use a baseline perimeter equivalent to the perimeter of a square for the initial cell area. Relative to this baseline, we allow a marginal increase in the target perimeter for IPCs, which encourages them to take on more irregular shapes, whereas OCCs were confined to a smaller perimeter, encouraging them to maintain a more ordered columnar shape as they do in-vivo.

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Figure 2. Cellular and cytoskeletal organization in developing salivary glands.

Epithelial cells express E-cadherin (red) during organ development and organize as polarized outer columnar cells (OCCs) and non-polarized inner polymorphic cells (IPCs) at (a) E12 and (b) retain this organization after 24 hours of growth. Cortical F-actin localization occurs during cleft formation at (c) E12 and (d) E12+24 hrs. Epithelial proliferation occurs in both outer and inner cell compartments shown with phospho-histone H3-labeled nuclei (red) relative to total nuclei with SYBR green (blue) at (e) E12 and (f) E12+24 hrs. Scale = 20 µm.

https://doi.org/10.1371/journal.pcbi.1003319.g002

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Figure 3. Construction of a GGH model of cleft formation and scope of modeling.

A six cell deep single cleft was designed having 36 pixels as the total cleft depth with predefined cleft cells (dark and medium blue). The local cleft simulation shows the other epithelial areas as polarized OCCs (dark green) and non-polarized IPCs (light green) with mitotic cells (yellow). The mesenchymal compartment (cyan) has been simplified to a single large cell. FPP links in the OCCs are shown as white lines. Spatial conversion: 1 µm = 1.06 pixels. Temporal conversion: 1 MCS = 48 sec. Single cleft model at (a) 0 Monte Carlo steps (MCS) and (b) 1500 MCS (Scale = 50 µm). Time lapse images of a mesenchyme-free E12 epithelial rudiment at (c) time 0 hr and (d) time 20 hrs with cleft measurements under 200× magnification (Scale = 20 µm). Average cleft depth = 36.2 µm. (e) Since cleft depth reaches a maximum value at 1500 MCS, this value was selected to represent the end of cleft progression. (f) The cleft depth distribution over time for the base case condition showing 34.1 pixels cleft depth after 1500 MCS, corresponding to a 20 hr growth period of a pre-defined initiating cleft through the end of cleft progression.

https://doi.org/10.1371/journal.pcbi.1003319.g003

Designation of Adhesion Properties

Cells exhibit differential adhesion that can drive complex tissue-level behavior [30]. The IPCs demonstrated a slightly more diffuse distribution of the adherens junction protein E-cadherin than the OCCs, suggestive of reduced adherence of the IPCs to each other [38]. To represent cell-cell adhesions in the GGH model, we start with a baseline contact energy penalty; increasing or decreasing the penalty simulates lower and higher adhesion, respectively, as explained by Eq. 1. Relative to this baseline, we designated increased cell-cell contact energy between IPCs to represent decreased adhesion properties and decreased contact energy between OCCs, simulating a possible increased adhesion that may help OCCs maintain their regular shape. During cleft progression, contact energy between the OCCs representing the cleft walls is directed to increase relative to the baseline, while contact energy between cleft cells and the matrix is decreased. This decrease in contact energy allows cell-matrix contacts to be established between the cleft cells.

The basement membrane is a dynamic structure that plays a critical role in branching morphogenesis, and cell-matrix adhesions are known to change dynamically during branching morphogenesis [5], [6], [8], [10]. In the GGH model, we represent basement membrane through the contact energy settings between the OCCs and the matrix, which is represented as a single homogenous cell not subject to area and perimeter constraints. This contact energy is designated in our model as the “cell-matrix” contact energy and behaves as defined by Eq. 1.

Designation of Cellular Contractility

The actin cytoskeleton has long been known to be required for branching morphogenesis and was specifically shown to be required to maintain initiated clefts [20], [21]. In salivary gland epithelial cells, the actin cytoskeleton is organized primarily into cortical actin filaments at the cell perimeter (Figure 2C, 2D) in an E12 organ explant grown ex vivo for 0 or 24 hours. Our subsequent work indicated that actin and non-muscle (NM) myosin II–mediated contraction are required to regulate cleft progression [6]. The current model for cleft progression assumes that actomyosin contraction is required for assembling fibronectin through integrin activation [5], [6], [7], which then stimulates local EMT through upregulation of Btbd7 and Slug and reduction of E-cadherin levels [22]. Since EMT is one of the chief factors promoting cleft progression, we utilized variable cell-cell and cell-matrix contact energies to facilitate cleft progression. Without any other energy factors affecting cleft progression, the resultant clefts were poorly formed (Video S2).

During early cleft formation, the cleft evolves as a thin opening between OCC cells, possibly primarily aided by random cell movements [5], [11] and possibly from a hypothesized force generated by FN assembly [10] pushing assembled basement membrane into the cleft opening. FN assembly, dependent on strength of actin contractility for integrin activation, might cause the two cleft-forming epithelial cell layers to separate. FN assembly also stimulates proliferation [6], presumably causing an outward force that emanates from inside the bud to counteract an inward cleft movement force produced by FN.

Since our model lacks specific structural representation of basement membrane assembly dynamics, we could not simulate the FN generated “cleft forming force” which was hypothesized to be the primary cause for progressive clefts [10]. Therefore, we attempted to simulate the effect of this FN-actomyosin dependent “cleft forming” force through an energy function called focal point plasticity (FPP). This function establishes links between selected cells and regulates the distance between them, assigning an energy penalty for deviating from a target distance. As noted in Eq. 4, the penalty varies based on the target distance, and the λ term. To replicate the wedge-shaped cells in the cleft, we paired opposite cells on each side of the cleft, and set decreasing target distances for pairs deeper within the cleft. These target distances were determined by examining cleft depths from ex-vivo time-lapse images and measuring cleft width as a function of depth (Figure S1). We found that a target distance inversely proportional to the cleft depth approximated the observed shape. Modulating the λ term adjusts the strength of this cleft-opening/maintaining effect. Due to the fundamental role of actomyosin contractility in FN assembly, it can be viewed as modulating contractility levels within the cleft cells.

In case of cleft progression, the exact roles for actin contractility in force generation during progression is unknown and although phosphorylated NM-myosin II was detected in the OCCs [6], it is not known if OCCs contract by pulling on each other through the actomyosin bundles. So, we utilized FPPλ to assign lateral FPP links in the OCC layer between adjacent cells and additional vertical links and lateral links between cleft cells (Figures 3A, 3B). These lateral links in the OCCs helped control the shape of the boundary cells along with maintaining a constant epithelial boundary. We then utilized the lateral FPP links in the cleft cells to simulate the effect of this actomyosin dependent FN “wedge.” The varying target distances in the cleft region are manipulated dynamically to simulate the effects of a “clefting force” generated by continuous actomyosin-mediated FN assembly between the cleft cells as the cleft progresses inward.

Designation of Cell Proliferation Properties

Previous work shows cell proliferation to be dispensable for cleft initiation [41], but to be required for cleft progression [6]. Although cytoskeletal contraction can induce cell proliferation [42], [43], in the CC3D environment, cell proliferation can be regulated separately from cell contractility. In the model we designated not only the percentages of mitotic cells but also their location within each epithelial cell subtype.

Temporal Calibration of the GGH Model with Image Data

We ran initial simulations for an extended number of MCS steps to determine the range of MCS steps corresponding to the time frame encompassing cleft initiation through progression (Figure 1E). A termination value of 1500 MCS steps was selected, equating to a temporal conversion of 1 MCS≈48 seconds (Figure 3E, 3F). Figures 3A and 3B show the model at time 0 hrs (0 MCS) and time 20 hrs (1500 MCS), respectively.

Establishing Initial Parameters for the Single Cleft Model

Within the CC3D environment, we established a set of base values for the five primary epithelial parameters included in this computational model under which cleft progression could occur (Table 1). To conduct a parametric search, we fixed the temperature (T) at 10. Due to its central role in the energy minimization step, modification of T impacts every other energy-based parameter. We vary T and select a fixed value that permits cells to fluctuate fluidly without becoming fragmented [27], consistent with previous observations that epithelial cells undergo dynamic movements during branching morphogenesis [10], [23]. This simulates a basal level of cell migration in both OCC and IPC epithelial cells. Interestingly, the random cell movement observed produces some exchange of cells between the OCC and IPC layer. With T at 10, we conducted a parametric search on these parameters: focal point plasticity (FFP λ), mitosis rate, mitosis location, cell-cell contact energy, and cell-matrix contact energies.

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Table 1. CC3D parameters that were varied in the model and their biological significance in branching morphogenesis and cleft formation.

https://doi.org/10.1371/journal.pcbi.1003319.t001

To yield a final cleft depth of 36 pixels in 1500 MCS (Video S3), we fixed these base values for the five parameters: Mitosis rate was set to 1% (per 100 MCS steps), evenly divided within OCCs and IPCs; FPP λ values in the OCCs and cleft cells was set at 10; cell-cell contact energy was set to 10 for cleft cells and 5 for all other cells; and cell-matrix adhesion in cleft cells was set to 3. Under these parameters, our model achieved an average cleft depth of 34.1 pixels, thereby yielding a spatial conversion of 1 µm = 1.06 pixels. Each simulation was run 100 times to ensure the consistency of the results given the stochastic nature of the GGH model. Figure 3F shows an example of the temporal evolution of cleft depths, achieving a 34.1 µm depth in 1500 MCS. With T value fixed at 10, we tracked 1725 individual cells in the base case simulation for 10 runs. The average net displacement was found to be 7.3 µm and the total path length was 94.6 µm (Figure S2). Thus the cell velocity was calculated to be 4.7 µm/hour and the meandering index to be 0.08 in the base model.

Quantitative Analysis of Cleft Progression

For quantitative and consistent methods to measure the quality of simulated clefts by comparison with equivalent measurements from organ explants, we developed descriptive cleft measurement indices – cleft depth, spanning angle, and tilt angle. First, the cleft center was located at the epithelial-mesenchymal boundary by examining the angle formed by each boundary point and its 8-distance neighbor on either side. As the deepest point of the cleft, the cleft center should have the lowest such angle value. The extrema are identified by using the mean-squared error (MSE) of the best-fit line for the boundary on each side of the cleft center; for each side, we progressively include points from the boundary until the MSE exceeds a predetermined threshold. The cleft center and extrema are shown in the example image in Figure 4A, 4D. Cleft depth is calculated as the distance from the cleft center to the midpoint of the line segment joining the two extrema (Figure 4B, 4E). Spanning angle is calculated as the angle formed by the line segments joining the cleft center to each extrema (Figure 4C, 4F). Clefts measuring less than 5 pixels in depth or exceeding 160° in spanning angle were discarded. The tilt angle is a measure of the perpendicularity of a cleft to the bud surface. It is calculated as the smaller of the complementary angles formed by the line segment between the extrema, and the line segment from the cleft center to the midpoint of the line segment joining the two extrema, as shown in Figure S3. Clefts with a tilt angle of less than 45° were labeled as “failed clefts”. The cleft categorization criteria were based on measured properties of clefts from multiple time-lapse images of organ explants.

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Figure 4. Quantitative analysis of cleft formation.

A MatLab function was created for tracing the border of the local cleft. (a) The two cleft extremes were labeled in green and the cleft tip in red. A MatLab tracing of a successful cleft at 1500 MCS shows a (b) high cleft depth and (c) low spanning angle (red lines). (d) MatLab tracing of a non-progressive cleft at 1500 MCS shows (e) low cleft depth and (f) high spanning angle.

https://doi.org/10.1371/journal.pcbi.1003319.g004

Building an Organ Level Model of Cleft Progression

In a developing salivary gland, multiple clefts form on the surface of the primary bud during branching morphogenesis, and they do not all form simultaneously. To determine if the progression of one cleft has an effect on adjacent clefts, we constructed a GGH-based salivary gland organ model consisting of a single bud with three equally-spaced clefts that progress simultaneously (Figure 5A) We ran 70 independent simulations with the same base case parameters that were used for the single cleft model, each for 1500 MCS (Figures 5A, 5B, Video S4). Quantitative analyses show that each individual cleft is comparable to those produced by the single local cleft model, albeit the average final cleft depth is slightly lower at (Figure 5C) 29.7 µm rather than 34.1 µm for the single cleft model. Correspondingly, marginally higher spanning angle values were observed compared to the base case. This result interestingly predicts that the behavior of clefts is somewhat dependent upon adjacent clefts. However, to focus on the cellular parameters necessary for progression of a single cleft, we used the single cleft model in all subsequent studies.

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Figure 5. Organ level model.

An organ-level simulation containing three ideally localized clefts is shown at (a) 0 MCS and (b) 1500 MCS (Scale = 50 µm). The results were quantified with (c) cleft depth and (d) spanning angle. A slight decrease in the average cleft depth and increase in the average spanning angle was observed relative to the single cleft model run under the same basal conditions.

https://doi.org/10.1371/journal.pcbi.1003319.g005

Validation of the Single Cleft Model

Having built a cellular model replicating cleft progression, we ran simulations comprising combinatorial variations of two parameters to simulate a specific biological state. As cleft progression requires ROCKI signaling, which stimulates both actomyosin contractility and proliferation [6], we simulated these cellular conditions by reducing the lateral FPP λ values in the cleft cells from 10 to 1 to simulate reduced cellular contractility and correspondingly lowered the mitosis rates in all cells from 1% to 0.5%. We performed 100 simulations and quantified cleft quality using the three cleft measurement indices. As shown in Figure 6, the cleft depths were reduced by 40.8% in simulations and 94.7% in ex-vivo studies using 10 µM Y27632 treatments for 24 hours (Figure 6, Video S5). We also simulated the effects of blebbistatin, a pharmacological inhibitor that prevents high affinity interactions between actin and myosin to inhibit cleft progression but does not affect cell proliferation [6] using an FPP λ value of 1 in the cleft cells but without changing cell proliferation rate. We observed similar trends in the reduction of cleft depths: a 48% reduction with 20 hours of in silico simulation in comparison to 88.9% reduction in organ explants treated for 24 hours with 25 µM blebbistatin (Figure 6, Video S6). Interestingly, the computational model agrees qualitatively with the experimental data that cell contractility and mitosis affect cleft progression.

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Figure 6. Validation of the single cleft model by comparison of predictions with experimental results for manipulation of ROCK.

SMGs were cultured ex-vivo and treated with Y27632 (ROCK inhibitor) and blebbistatin (inhibitor of actomyosin contractility that lowers the affinity of myosin-actin interaction) for 24 hours. Cleft depths were measured using brightfield time-lapse confocal imaging and compared with cleft depths in GGH simulations of ROCK I knockdown (KD). ROCK I KD simulation consisted of a reduced FPP λ value (1) along with a reduced proliferation rate (0.05%). The simulation mimicking blebbistatin action was achieved using only a reduced FPP λvalue (1). Similar trends were observed in the simulations and the experimental results, indicating that the model effectively simulated the cellular effects of inhibition of signaling molecules.

https://doi.org/10.1371/journal.pcbi.1003319.g006

Use of the single cleft model to test hypotheses regarding the mechanisms of cleft progression-

Hypothesis I: Cell Proliferation Drives Cleft Progression

Cell proliferation has long been understood to occur during branching morphogenesis. An early study indicated that cell proliferation was not required for salivary gland cleft formation [41], but later work demonstrated that although cleft initiation does not require cell proliferation, the biochemically independent step of cleft progression does require cell proliferation [6]. However, it has not been possible to experimentally increase the cell proliferation rate without affecting other cellular parameters. To test the sub-hypothesis that high cell proliferation rates are required for cleft progression, we performed simulations in which we tested increasing amounts of cell proliferation by varying the value for the GGH parameter, mitosis rate (MR). For this in silico experiment, we chose to assign the dividing cells equally in the OCC and IPC epithelial cells. We ran 100 simulations for 5 different values of MR, from 0.5% to 5%. Surprisingly, we found that high MR levels were inhibitory for cleft progression and that the best conditions for promotion of cleft progression were at a MR of 1%, where the cleft depth was the highest and the spanning angle was the lowest (Figures 7A, 7B). To experimentally validate the prediction of the single cleft model that 1% cell proliferation is ideal for cleft progression, we grew organ explants for 24 hours, and fixed a subset of tissues for immunocytochemistry with pHH3 to detect cells in M phase and staining with SYBR green to detect total nuclei at 2, 8, 12, and 24 hrs. Mitotic cells were detected in both the OCC and IPC layers, and the percentages of dividing cells were calculated from single confocal images for each tissue compartment (Figure 2E, 2F). Although the mitosis rate varied over the time period of the assay, the average mitosis rate was calculated to be 0.99% in the epithelial region (Figure 7C), as predicted by the single cleft model.

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Figure 7. Effect of varying mitosis location on cleft progression.

The location of proliferating epithelial cells was varied, with 25%, 50% or 75% mitosis occurring in the OCC population, while the MR rate was also varied from 0.5–5%. Results were quantified as (a) cleft depth and (b) spanning angle. The location of proliferating cells had an effect on cleft formation with 50–75% proliferation in the OCCs generally being more effective than 25%. (c) Image segmentation and analysis of confocal images acquired from pHH3 ICC and SYBR green-stained explants at four time points shows temporal changes in mitosis rate that produce an average epithelial mitosis rate of 0.99% between 2–24 hours of ex vivo culture. Dotted line denotes average mitosis rate (0.99%), ANOVA *P<0.05, n = 6.

https://doi.org/10.1371/journal.pcbi.1003319.g007

It has not been experimentally possible to manipulate cell proliferation rates in specific regions of the gland; therefore, it is not known if there is a regional preference for cell proliferation within the epithelium during cleft progression. Using the single cleft model, it is possible to test the sub-hypothesis that the OCCs proliferate preferentially over the IPCs. We performed simulations where we both varied the epithelial location of the proliferating cells and also varied MR. MR was set at 0.5%, 1%, 2%, 3%, or 5%, with the location for mitotic cells designated as 25%, 50%, or 75% in the OCC population (Figure 7). When 25% of the proliferating cells were located in the OCCs, we found that increasing the mitosis rate from 0.5% to 1% caused a minor decrease in cleft depth (32 µm to 31.2 µm) (Figure 7A). Further increases in MR to 2%, 3% and 5% decreased cleft depths to 25.7 µm, 15.2 µm, and 8.7 µm, respectively. Interestingly, when 50% or 75% of the mitotic cells were located in the OCC region, by increasing MR from 0.5% to 1%, a slight increase in cleft depth from 32.3 µm to 34.1 µm was observed; but further increases in rates to 2%, 3%, and 5% caused progressive decrease in depths, irrespective of the location of the dividing cells. The trends in spanning angle are shown in Figure 7B. Thus, 1% MR with 50–75% of the cells located in the OCC region was found to be the optimal condition for cleft progression.

Hypothesis II: Cellular Contractility Drives Cleft Progression

In previous work, we demonstrated that ROCKI is required for cleft progression through modulation of actomyosin contractility [6]. ROCKI was required for phosphorylation of NM myosin II to stimulate contractility and down-regulation of cellular contractility with blebbistatin similarly reduced cleft progression. Pharmacological inhibitors cause a global reduction of actomyosin contractility, making it impossible to assess the effect of cellular contractility specifically in the cleft cells. Actin-based contractility is responsible for dynamic cell movements and FN assembly by the cleft cells, and so we modulated the strength of the lateral FPPλ links in the cleft cells to test the hypothesis that actomyosin contractility in the cleft cells is required for cleft progression. In the cleft region, we assigned vertical FPP links between cleft cells and also lateral links between the adjacent cleft cells on either side of the cleft. Also, each pair of cleft cells was assigned a different target distance, with a lower target distance being set for cells deep within the cleft. During the course of simulation, each pair of cleft cells strives to acquire the set target distance, and after 1500 MCS, under unaltered cell and matrix contact energy settings, FPP energy is solely dictated by λ values since the distance deviation for each pair of cleft cells remain almost constant for varying λ values. Thus, by altering the lateral FPPλ values in the cleft cells, we aimed to vary actomyosin-based cellular contractility specifically in these cells to assess its effect on cleft progression.

To test the hypothesis that contractility in cleft cells is required for cleft progression, we varied the lateral FPP λ values that hold the cleft cells together, between 1 and 30. A cleft depth of 34.1 µm and a spanning angle of 46.0° was found for FPP λ = 10, whereas lowering FPP λ values to 5 or 1 caused the average cleft depth to decrease to 29.0 and 19.5 µm respectively (Figure 8A) with associated increases in spanning angle (Figure 8B). This manipulation mimicked the effect of decreasing actomyosin contractility, as performed experimentally using blebbistatin. Interestingly, when we used higher values of FPP λ such as 15, 20, and 30, progressively lower cleft depths (33.5, 31.6, and 26.0 µm) and higher spanning angles (47.3, 49.3, and 71.5°) were observed, which has not been experimentally tested. When no FPP links were used, shallow clefts were formed with an average cleft depth of 12.6+/−3.94 µm and spanning angle of 99.9° (Video S2). The single cleft model thus predicts that actomyosin contractility in the cleft is essential for cleft progression and that a moderate level of this cellular contractility favors cleft progression, with low contractility being more detrimental to cleft progression than high contractility.

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Figure 8. Effect of varying cell contractility, cell-cell adhesion, and cell-matrix adhesion strength on cleft progression.

FPP λ values were varied between 1–30, and effects on cleft progression were quantified with (a) cleft depth and (b) spanning angle. FPP λ values lower or higher than 10 are detrimental to cleft progression. Cell-cell (CC) adhesion strength in the cleft region was manipulated by increasing the cell-cell contact energy to mimic E-cadherin-based adhesions and evaluated with (c) cleft depth and (d) spanning angle. Decreasing cell-cell adhesion in the cleft region (increased contact energy) resulted in deeper clefts with lower spanning angles. Modulation of cell-ECM (CM) junctional strength by increasing cell-matrix contact energy was monitored by (e) cleft depth and (f) spanning angle. Increasing cell-matrix junctional strength affects clefts quality marginally with greater cleft depth and lower spanning angle.

https://doi.org/10.1371/journal.pcbi.1003319.g008

Hypothesis III: Lowered Cleft-Cell Adhesion and Increased Cleft Cell-Matrix Adhesion Drives Cleft Progression

It was previously reported that loss of E-cadherin-containing cleft-cell adhesions and gain of fibronectin-driven cell-matrix adhesions within the cleft region are required for cleft progression, and further that an epithelial-mesenchymal transition occurs in a subset of cells at the base of the cleft to facilitate cleft progression [5], [22]. Although experimental manipulations have been performed to examine the effect of decreasing E-cadherin-based cleft-cell adhesions [10], [38] and to increase or decrease cell-matrix adhesions [5], [6] these manipulations have been performed with whole organ explants or with whole epithelial rudiments grown in an artificial matrix and the requirement for these changes specifically within the cleft region has not been possible to address. With the single cleft model, it is possible to manipulate cleft-cell and cell-matrix adhesion strengths within a subset of cells in the cleft region by running simulations at multiple values.

To recapitulate a progressive EMT occurring in the progressing cleft, we assigned a low cell-matrix adhesion value of 3 and a higher cell-cell adhesion value of 10 in the cleft cells forming the cleft wall, whereas all other cell-cell contact energies remained at 5. These values are assigned at the onset of simulation, and in each pair of cleft cells, cell-cell adhesions are replaced by epithelial-matrix adhesions during the temporal progress of cleft deepening. In order to preserve the epithelial organization of the OCC and IPC relative to the matrix, we assign a lower energy penalty between OCCs and a higher penalty for IPC-matrix. When cleft-cell adhesion was decreased (raised contact energy), cleft depths increased beyond 34.1 µm to 38.4 (CC value = 15) and 40.7 µm (CC value = 20) (Figure 8C) with corresponding spanning angle measurements that decreased from a base value of 46.0° to 39.3° and 35.6° (Figure 8D), respectively, following 100 simulations. Interestingly, increasing cleft-cell adhesions (lowering contact energy values to 5 and 1) caused shallower clefts with depths of 24.7 µm and 19.2 µm and increased spanning angles to 76.2° (C-C value = 5) and 92.4° (C-C value = 1). Thus, the single cleft model predicts that low cell-cell adhesion strengths within the cleft are most beneficial for cleft progression.

It has also not been possible to determine experimentally whether it is the loss in cleft-cell adhesions or the increase in cell-matrix adhesions that occurs in progressive clefts that has the most significant effect on cleft progression. In the single cleft model, we varied the cleft-matrix contact energy from 1 to 5 and ran 100 simulations with each value. Unexpectedly, variations in the cell-matrix contact energy values had minimal effects on cleft progression. Higher cleft cell-matrix contact energy values (lowered adhesion) yielded slightly shallower clefts (30.8 µm for CM = 4 and 27.3 µm for CM = 5) than the base value of 3 (Figure 8E), with corresponding changes in spanning angles (Figure 8F). Lower cleft cell-matrix contact energies (increased adhesion) yielded marginally deeper clefts (36.1 µm for value = 2 and 35.1 µm for CM = 1), with corresponding changes in spanning angles, than obtained with the base value of 3. Thus, the single cleft model predicts that for efficient cleft progression in the cleft region, a low cell-cell adhesion value is required more so than specific levels of cell-matrix adhesion; however, higher cell-matrix adhesion levels are somewhat beneficial for cleft progression.

Relative Contribution of Cellular Parameters for Cleft Progression in Branching Morphogenesis

Although multiple studies have been performed to assess the importance of individual cellular factors in the process of branching morphogenesis, it is not possible to rank the importance of these cellular factors using experimental methods alone. Using the single cleft simulation model, we varied each of the four parameters independently for a total of 600 parameter combinations. Each parameter combination was simulated 40 times, and classified into one of four categories based on the majority result. These cleft classes were designated based on the distribution of measurements derived from time-lapse data, and labeled “failed,” “non-progressive,” “progressive,” and “super-progressive.” Failed (F) clefts did not stabilize and regressed back to the epithelial boundary and non-progressive (NP) clefts stabilized but failed to progress. Progressive (P) clefts fall within the normal size range of clefts measured from time-lapse data, whereas super-progressive (SP) clefts exceed the average size. The depths that each class corresponds to are shown in Figure 9A. Out of the 600 parameter combinations, we obtained 275 F, 188 NP, 85 P, and 52 SP combinations in each of the cleft classes.

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Figure 9. Classification analysis.

(a) Distribution of cleft measurements from SMG organ explant image data. Classes for parameter combinations were assigned as follows: failed, <17.8 µm; non-progressive, 17.8–30.5 µm; progressive, 30.5–40.7 µm; super-progressive, >40.7 µm. (b) Cross-validated classification accuracy on test set for the full parameter combination (CC, CM, FPP and MR), and exclusion of a single parameter CC, CM, FPP, or MR. A larger decrease in accuracy indicates greater importance. (c) Parameter value distributions within the progressive class. Peaks indicate higher importance for that value in progressive cleft formation.

https://doi.org/10.1371/journal.pcbi.1003319.g009

The number of stabilized versus progressive clefts, for each parameter variation have been outlined in Figure S5 and in Tables S1 and S2. The proportion of F, NP, and P clefts defined the limits of our parametric search for each hypothesis. For instance, the range of FPP λ values was chosen after assessing the number of progressive clefts (P) obtained from each variation. Figure S6B shows that at λ value 0.5, no P was obtained. Hence the lower limit for FPP λ variation was set to 1. The number of P decreased with increasing contractility and a higher value of 30 was set as the upper limit of the range. Also from Table S1, the number of F and NP increased highly at λ value 30, and beyond 30 there were almost no P, with mostly destabilized failed clefts.

To assess the relative importance of each GGH parameter, we measured how accurately a classifier could predict the cleft class of a parameter configuration, with the expectation that parameters that have a high importance in cleft progression should also serve as good predictors of cleft class (simulation outcome). Conversely, when removed as a feature, the absence of such a parameter should have a strong negative impact on the classification accuracy. Using a radial basis kernel SVM classifier, we were able to achieve 75.6% accuracy when the classifier was provided with all four GGH parameters as features. We then attempted classification using the remaining 14 possible combinations of three, two, or one parameter. The cross-validated training and testing accuracies and decrease in accuracy relative to the full parameter set are reported in Table 2. The testing accuracy for combinations where a single parameter was removed is also shown in Figure 9B. Individual removal of the three parameters MR, FPP λ, and CC resulted in similar drops in classification accuracy of 15.3%, 18.5%, and 15.6%, respectively. The removal of cellular contractility (FPP λ) had a marginally higher impact on the classification accuracy than cell-cell adhesion (CC) and mitosis rate (MR). In contrast, omitting CM resulted in a classification accuracy of 75.0%, which is only a 0.6% decrease from the 75.6% accuracy level obtained with all four features, suggesting that the contribution of cell-matrix adhesions are the least significant contributor to cleft progression in our model. Thus, our analysis that considers the drop in classification accuracy as the metric for importance suggests that FPP λ is the most significant contributor to cleft progression in our model, followed closely by CC and MR.

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Table 2. Classification accuracy table.

https://doi.org/10.1371/journal.pcbi.1003319.t002

After classifying each parameter set into the four cleft types, we questioned which specific parameter values are important for achieving clefts that fall into the data-driven normal cleft range, described as “progressive” clefts (30.5–40.7 µm ∼85 out of the 600 simulations). The distributions of FPP, CC, MR and CM values of all clefts falling into the progressive class are shown in Figure 9C. The results indicate that the conditions to form “progressive” clefts vary slightly from the conditions required to form clefts, in general: (i) the optimal FPP λ value was 5, rather than 10 (base case of 34.1 µm cleft depth), indicating that a slightly lower value of cleft cell contractility is sufficient for progressive clefts. (ii) CC values peaked at 15 and 20, showing that a lower level of epithelial adhesion favors cleft progression in the “progressive class” than in the base case category where the deepest clefts peaked at a value of 10. (iii) CM adhesion values did not peak particularly at any value, but a lower value (∼1) promoted cleft progression, whereas a value of 3 was optimal for the base case. The optimal MR was similar to that predicted by the base case cleft categorization. This analysis also indicated that CM had the lowest impact on cleft progression.