Fig 1.
Epithelial mechanics and workflow outline.
(A) Apical surface of epithelial cells within the Drosophila wing imaginal disc that are marked by E-cadherin tagged with fluorescent GFP (DE-cadherin::GFP). Multiple cells within the displayed region are undergoing mitotic rounding with a noticeable decrease in fluorescent intensities of E-Cadherin. (B) Experimental image of cross-section of wing disc marking levels of actomyosin (Myosin II::GFP). (C) Cartoon abstraction of epithelial cells, which are polarized with apical and basal sides. Actomyosin and mechanical forces during mitotic rounding are primarily localized near the apical surface. (D) At the molecular scale, the boundary between cells consists of a lipid bilayer membrane for each cell, E-cadherin molecules that bind to each other through homophilic interactions, and adaptor proteins that connect the adhesion complexes to an underlying actomyosin cortex that provides tensile forces along the rim of apical areas of cells. (E) The graphical workflow of the computational modeling setup, calibration, verification and predictions. Arrows indicate mitotic cells. Scale bars are 10 micrometers.
Fig 2.
Diagram of the underlying physical basis of model simulations.
(A) Intracellular and intercellular interactions between different elements of the model. Symbols and notations are indicated in the legend. (B) Implementation of the simulation of cell cycle in the model.
Table 1.
Potential energy functions in the Epi-Scale model.
Table 2.
Energy function parameters.
Fig 3.
Initial conditions and sample simulation output.
(A) Initial condition of a simulation with seven initially non-adherent circular cells. Each cell starts with 100 membrane elements and 20 internal elements. (B) Initial formation of an epithelial sheet after cells adhere to each other. An equilibrium distribution of internal nodes is reached for each cell. (C) Epithelial sheet after 55 hours of proliferation. (D) Enlarged view of the selected region showing different cell shapes and sizes due to interactions between cells. The large cell is undergoing mitotic rounding (MR).
Fig 4.
Calibration of model parameters through simulations.
(A-A″) Calibration test to determine parameters for cell elasticity, analogous to experimental single cell stretching tests [66], (A) Initial condition t = 0, (A′) 6 minutes after simulation with no force applied, (A″) after 72 minutes cell is completely on tension (B-B″) Cell adhesivity test, analogous to experimental tests [69] for calibrating the level of cell-cell adhesion between adjacent cells. (B) Initial condition t = 0, (B′) 6 minutes after simulation begins with no force applied, (B″) after 72 minutes, 15 nN force is applied. (C) Stress versus strain for single cell calibration (red line) and stress versus strain for calibrating the level of adhesivity between the two cells (blue line) [69,70]. Initial negative strain in adhesivity test is due to strong adhesion between two cells. (D) Force and strain as a function of time for adhesivity test. (E) Tissue growth rate calibration by comparing with the experimental data by Wartlick et al. [56]. The 95% confidence interval for the growth rate results is shown in grey color.
Fig 5.
(A-A‴) Time-lapse confocal images of cell undergoing mitosis in the wing disc with E-cadherin:GFP-labeled cell boundaries. Scale bar is 5 μm. Arrows indicate daughter cells. (B-D‴) Time series from Epi-Scale simulation of a cell undergoing mitosis and division with illustration of: (B-B‴) adhesive spring stiffness, (C-C‴) cortical spring stiffness, and (D-D‴) internal pressure, respected to their interphase values. (E-F) Comparison of size and roundness of mitotic cells with experimental data for the Drosophila wing disc. Arrow represents mitotic cell in B-D. A t-test comparing the means of computational simulations and experiments result in p = 0.72 for cell area ratio and p = 0.76 for normalized roundness of mitotic cells.
Table 3.
Implementation parameters.
Fig 6.
Emergence of tissue-level statistics from model simulations.
(A) Sample simulation output showing cells with different numbers of neighbors as different colors (B) Simulations initiated from seven cells reaches steady-state polygon-class distribution after approximately 35 hours of cell proliferation. (C) Comparison of polygon class distributions obtained by Epi-Scale model with various biological systems (data extracted from [79]) and a vertex based model by Farhadifar et al. [38]. (D) Polygon class distribution of cells at different stages of growth, and comparison of mitotic cells distribution with Drosophila wing disc experimental data [63]. (E) Polygon class distribution of cells at different level of cell’s elasticity. The results do not show sensitivity in the range of reported elasticity of epithelial cells [67]. (F) Average relative area (), and average polygon class of neighboring cells verifying that simulation results satisfy Lewis law and Aboav-Weaire law. A is the apical area of cell and
is the average apical area of the population of cells.
Fig 7.
Response surface method analysis of mechanical properties on regulating mitotic expansion and mitotic rounding.
(A) Schematic of initial full factorial design (FFD) for exploring parameter space, and subsequent central composite design (CCD) for developing the response surface models shown in (C, D). (B) Pareto front indicating computational model parameter values with lowest difference with experimental data for area ratio and normalized roundness. The parameter range defined by the CCD (Run 2) spans parameter variation where the error between experiments and simulations is within the propagated uncertainty of measurements and simulations. Error bars are the standard error of means of the normalized deviation between experiments and simulations. (C-D) Contour plots for FFD experiment where (C) shows the area ratio (Aratio = Amit/Ainter) and (D) shows the normalized roundness (Rnorm). (E-F) Contour plots for CCD experiment where (E) shows the area ratio (Aratio) and (F) shows the normalized roundness (Rnorm).
Fig 8.
Quantitation of relative sensitivity of mitotic area expansion and roundness to adhesion, stiffness and pressure changes within the physiological property space.
Sensitivity estimation of (A) (Amit/Ainter) and (B) Rnorm to small perturbation in the three mitotic parameter set points, ,
, and ΔP. Sensitivity was estimated from the reduced RSM model described in Fig 7C–7F after stepwise model regression (p-value cutoff of 0.01). (C) Proposed mechanical regulatory network defined for “physiological ranges” within the parameter ranges defined by the CCD (Run 2, Fig 7A) that summarizes the local sensitivity analysis. Cell adhesivity, an increase in
, slightly inhibits area expansion and strongly inhibits roundness. Membrane stiffness,
inhibits area expansion and promotes roundness. Mitotic area expansion is most sensitive to variation in the mitotic pressure change (ΔP), but pressure has little effect on roundness over the calibrated physiological ranges.