Studying gastrulation by invagination: The bending of a cell sheet by mechanical cell properties using 3D deformable cell based simulations
Fig 3
3D Deformable cell based model.
This figure introduces the 3D deformable cell based model. The model simulates a single cell, which is a distinct entity with its own local properties that can interact with other cells to form a larger structure. Each 3D cell is formed by a detailed polyhedron surrounding a conserved volume, making it freely deformable and movable. The boundary of this polyhedron consists of a triangulated mesh representing the biological cell cortex, composed of interconnected vertices and elastic elements that adapt to applied forces. The cortex can actively deform by constricting edges or passively by interaction with neighboring cells. The vertices and edges allow for different regions to be placed on the same cell to simulate cellular properties like adhesion, stiffness, and constriction of the cortex. These properties can change over time according to predefined rules encoded in a basic scripting language executed by the model code, effectively simulating genetic traits that govern cellular behavior. Multiple of such cells can be simulated in the same space to form larger entities or structures, such as blastulas. The cells can be pushed when they are adjacent but unconnected to other cells, or be pushed and pulled when they are adhered to other cells. Panels (A–F) explain the cell model concept and how multiple cells together can simulate a hollow blastula, where cells adopt a wedge shape due to the local cell-cell interactions rather than by pre-programming the shape. (A) Description of the properties of the simulated cell. (B) A single cell with: Elastic elements, Volume conservation and Constriction. Images (B1, B4) show a single cell that has a deformable cortex. Images (B2–B3) illustrate the cell cortex that consists of edge interconnected vertices. Image (B5) depicts the edges that are modeled as a restorative Elastic element with a given rest length. When the actual length deviates from the rest length, a restorative force is generated (push or pull). Image (B6) illustrates Volume conservation. Deviation from the cells rest volume results in a volume restorative force. In image (B1), the dark blue edges in the apical, top of the cell, region (0–50%) of the spherical cell (with uniform cell stiffness) are assigned to constrict. Image (B4) shows the cell after the edges have constricted. The constricted apical edges have reduced their length, resulting in a flattened apical area and expanded lateral-basal region due to the volume conservation. The cell shape still resembles the initial spherical shape. (C) Adhesion region. Image (C1) shows the adhesion region on a single cell, visualized in light blue, where the vertices within this region are adhesive. Image (C2) depicts the same cell but now the edges in the apical region have constricted (see also Image B4). The adhesion region has now moved apically, reducing the area of the adhesion region, but not the number of vertices. This also results in the expansion of the basal non-adhesive region. (D). Vertex Restorative Forces. This panel is an abstraction of the model, to highlight the different forces that can work on a single vertex: (D1) Volume Force, (D2) Elastic Element Force, (D3) Adhesion pulling Forces, and (D4) Collision Forces that resolve boundary violations. These forces together result in Ftotal, that determines the new position of a vertex (See also Methods and S1 Appendix). (E) Multi-cellular simulations. Multiple cells can be adhered together to form a larger structure, here they form a hollow blastula. Image (E1) shows 128 single cells that are assigned a cell type, endoderm (salmon) and ectoderm (beige). Image (E2) demonstrates cells that are adhered together to form a blastula. The cells here have a stiff apical (outer) area, an intermediate stiff lateral region, and a soft basal (inner) area. This pulls the apical area flat and extends the basal area, changing the cell shape from round to elongated wedge shaped. Opening up the blastula in Image (E3) shows the emerged cell shapes and the two different sizes of the adhesion regions (light blue) placed on the two different groups of cells (endoderm and ectoderm). The endoderm has a small apical band (light blue and salmon), while the ectoderm is fully adhesive (completely light blue). Image (E4) features a close up of the cells to visualize the adhesive bonds (dark blue) between the vertices in the light blue zones. The cells are pulled apart slightly to stretch the adhesive bonds, making them more visible. (F) Cell model algorithm, shown as pseudo code.