Fig 1.
Blastula and gastrula illustration.
(A) Blastula. Illustration of a blastula of a diploblastic organism. The endodermal plate (approximately one fourth of the total number of cells) is located on the future oral side (animal pole) indicated with salmon. Opposite, is the aboral (vegetal) pole. The axis between the poles forms the primary oral-aboral axis. The rest of the blastoderm is formed by the ectodermal cells (colored beige), that together with the endoderm encloses the blastocoel (internal space). The epithelialized cells (blastomeres) are laterally adhered together, forming the spherical hollow blastula. The apical area of the cells is directed outward, while the basal area is directed toward the blastocoel. (B) Gastrula. Illustration of an invaginated gastrula. After invagination the endodermal plate has moved into the blastocoel and aligned with the ectoderm, forming two germ layers. The salmon colored cells form the endodermal layer that enclose the archenteron (primitive gut). The ectodermal layer forms the outside of the embryo. Where the ectodermal layer has curled into the opening, it forms the blastoporal lips (future pharynx), that enclose the blastoporal (oral) opening. The blastocoel has completely disappeared due to the basal alignment of the two cell layers.
Fig 2.
Biological images of invagination.
(A–D) Confocal z-stacks of Nematostella vectensis invagination. The images show cross sections of phalloidin stained embryos of Nematostella vectensis at different developmental stages during gastrulation (time of development in hours in left bottom corner). The endodermal plate is directed to the right. (A) The endodermal plate starts to constrict. (B) Invagination continues, the endoderm first aligns laterally and moves aborally. (C) The endodermal and ectodermal layers are fully attached. The embryo has become spherical with a closed off blastoporal opening. (D) After alignment, the endoderm has spread out and reduced its height. Images modified from [31]. (E–I) Favites abdita (Stony coral) invagination. The images show invaginating Favites abdita embryos. (E), (F). Embryos, with oral view of the roundish blastoporal opening, indicated with an asterisk (*). (G). Cross section showing mid-gastrula stage. (H). Cross section showing a fully invaginated embryo with a bowl-like shape and reduced blastoporal opening. (I). The embryo has become more spherical again, and the endodermal layer has reduced its height. Images modified from [9]. (J and K) Initial invaginating plate shapes. Cross sections through invaginating embryos. (J). Flat invaginating endodermal plate shape seen in stony coral (Dipsastraea (Favia) speciosa). The asterisk (*) indicates endodermal plate. (K). Concave invaginating plate shape seen in Aurelia aurita embryo. bl=blastoporal lip. The asterisk (*) indicates the invaginated endodermal plate. Image (J) modified from [9], Image (K) modified from [32]. (L and M). Drosophila ventral furrow invagination. (L). Cross section through a Drosophila embryo and (M). Ventral view of Drosophila embryo. Images (L) modified from [33] Images (M) modified from[34].
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.
Fig 4.
The images show time series of detached endodermal plates and whole blastulas with different constriction modes. The adhesion regions are located apico-lateral (20-65%), the constriction factor is 0.1, and the cell stiffness is uniform in all cells (k = 0.5). (A– C) Detached endodermal plates with 83 cells. (D– G) Blastulas with 256 cells. The endoderm cells are colored salmon. (A) Simultaneous constriction of all cells. (B) Edge cells (brown gradient) constrict first, with time interval between constricting cells, 100 units. (C) Center cell (brown gradient) constricts first, with time interval between constricting cells, 100 units. (D) Simultaneous constriction of all the endoderm cells (58) in a blastula. (E) Simultaneous constriction of all the endodermal cells (36) in a blastula. (F) Ring constriction of the endodermal cells (58) in a blastula. Edge cells constrict first. (G) Ring constriction of the endodermal cells (58) in a blastula. Center cell constricts first. Simultaneous constriction in a detached endodermal plate (A) caused the edge cells to curl up first. Because all cells constricted simultaneously, the center of the plate flattened faster, than when the edge cells constricted first (B). Constricting the center cell first (C), created a dip in the center of the plate before the edge cells curled up. Constricting endodermal plates in an embryo showed similar results as with a detached endodermal plate. When cells constricted simultaneously (D), this caused the edge cells of the plate to move inwards first, which pushed the center of the endodermal plate outwards. With fewer endodermal cells (E), this effect was less pronounced and the plate became flatter sooner before invaginating. When the edge cells constricted first (F), the top of the endodermal plate formed an extreme bulge, where the cells were almost pushed out before invaginating in. Constricting the center cell first (G) and continuing ring by ring created a concave shape in the center of the plate right at the beginning of invagination. These simulation experiments show that the endodermal plate shape during the invagination process reveals something about the timing and the constriction mechanism in the endodermal plate.
Fig 5.
Cell Stiffness and constriction factor.
This figure shows cross sections through blastulas that have constricted the apices of the endodermal plate. The endodermal plate (salmon) is oriented to the left. Below each blastula is the total number of cells in the blastula (32-1024) and number of endodermal cells between parentheses (). (A) Uniform cell stiffness constriction factor 0.1. Simulating with the following parameters: Cell stiffness 0-100% k=0.5. Constriction factor 0.1 and endodermal adhesion region 20-65%. All blastulas invaginated, but the global shape was bowl-like with a large opening. In the 1024 blastula the endodermal and ectodermal layer did not establish contact. (B) Uniform stiffness with constriction factor 0.05. Simulating with the following parameters: cell stiffness 0-100% k = 0.5, constriction factor 0.05, endodermal adhesion region 20–65%, The endodermal cells elongated and expanded more basally due to the smaller edge constriction factor. This placed more strain on the smaller embryos, therefore the number of endodermal cells were reduced to allow better invagination into the blastulas. Reduction of endodermal cells strongly influenced the global shape. The final gastrula was rounder than seen in Fig 5A but now no germ layer alignment occurred. All blastulas invaginated, but the larger the embryo size, the more bowl shaped the embryo became with a larger opening. (C) Non-uniform cell stiffness. Simulating with the following parameters; constriction factor 0.05, endodermal adhesion region 20–65%. The cells are divided into three stiffness zones (apical region: 0-30% of the spherical cell, lateral region: 30–70% and basal region: 70–100%). The cell stiffness is stiffest in the apical region (k = 1) in the lateral region the stiffness is k = 0.5 and the basal region is the softest (k = 0.1). All the blastulas invaginated and the endoderm and ectoderm aligned completely in the smaller gastrulas. The larger gastrulas (512 en 1024 cells) approached alignment. The global shape became bowl-like, with a large opening. (D) Increased cell stiffness over time. The endodermal cell stiffness is increased apically and decreased basally over time. Simulating with the following parameters: Constriction factor 0.1, endodermal adhesion region 20-65%. The cell stiffness at the start of the experiment is k = 0.5. For the endoderm cells the stiffness of the apical top (0–40%) is slowly increased to k = 1.5 and the basal side (40–100%) decreased to k = 0.1. For the ectoderm cells the apical stiffness is increased to k = 1.4 and for the basal side it is decreased to k = 0.35. The cells showed a more bottle like appearance close to the ectoderm (blastoporal lip cells). The layers did not connect. The smaller gastrulas remained rounder, but the largest gastrula (256 cells) had a strong bowl-shape and large opening. (E) Number of endoderm. Image E shows the 256 celled blastula from row C, but now the number of endodermal cells is reduced from 58 to 36. This results in more blastocoel space remaining between the endodermal plate and the ectodermal layer. The smaller number of endodermal cells resulted in one more ectodermal ring that could move into the blastoporal opening.
Fig 6.
(A) Linear row of endodermal cells. This image shows two time points of an embryo with 512 cells that has an elongated endodermal plate shape. The simulation parameters are given in S2 Table. Constricting one row of endodermal cells mostly reduced the apical area of the cells vertically, due to the deformability of the passive surrounding cells and resistance of the constricting cells. When more rows constricted, the apical areas became rounder again, since now the force pattern changed (see S1 Graph). (B) Endodermal purse string constriction. The endodermal cells of a 512 celled blastulas with 87 endodermal cells constricted using a purse-string method. The single cell shows the center cell from this gastrula with a purse string constriction. The green zone indicates the constriction region. Due to the outward volume pressure created by the purse string constriction in the cell, the apical top stiffness also had to increase. During constriction the cell stiffness of the apical top (0–20%) was increased from k = 0.9 to k = 1.8 and the 20–60% zone from k = 0.5 to k = 1.4, to prevent excessive bulging of the apical area.
Fig 7.
The endodermal plate shape and blastoporal opening.
This figure shows different blastulas (256–512 cells) with different endodermal plate shapes. The front view shows the oral side with the initial (T = 50) and final endodermal plate shape, and the cross section at the end of the simulation. The numbers of endodermal cells depends on the plate shape; Plate shape: (A) 67 cells, (B) 64 cells, (C) 69 cells, (D) 71 cells, (E) 73 cells, (F) 58 cells, (G) 31 cells, (H) 32 cells, (I) 64 cells, (J) 64 cells, (K) 128 cells. The parameters of each blastula are given in tables S2 Table and S3 Table. In all simulations the plates of the blastulas invaginated regardless of the shape. However, not all endodermal plates connected with the ectodermal layer. All the gastrula shapes became bowl-like. The blastula in row (F), had a spherical plate with softer ectoderm and smaller adhesion region between the ectodermal cells, which helped to close the blastoporal opening more. The oral view shows an angular shape and folds in the embryo. The plate shapes in row (G), (H) and (I) allowed for a better closure of the blastoporal opening, but not a better layer alignment. Only gastrula I seemed to align the layers better, except for the blastoporal lip region, where the ectoderm interrupted the endodermal plate. Row (J) and (K) show blastulas with 512 cells in a star shaped pattern. Row (J) had a better closure of the blastoporal opening, but the layers did not connect. While the layers in row (K) (larger star shape plate) did connect, but now the opening remained large.
Fig 8.
The simulation results show the blastoporal opening of the gastrulas from Fig 5A–5C of the 256–1024 celled gastrulas. Here image A–C show the oral view of the 256, 512 and 1024 celled gastrula, with uniform cell stiffness and constriction factor 0.1. Images D–F show the oral view of the 256, 512 and 1024 celled gastrula, with uniform cell stiffness and constriction factor 0.05. Images G–I show the oral view of the 256, 512 and 1024 celled gastrula, with non uniform cell stiffness. The larger the blastula and the smaller the opening, the more the embryo gets folded around the blastoporal opening.
Fig 9.
(A) A cell with different regions. The colored bands mark different regions on a cell, which can vary in thickness. (B) Cell-cell adhesion. Two cells are adhered together by adhesion molecules (elastic element that connects two vertices) in the selected adhesion region (blue region).
Fig 10.
Collision handling examples of deformable cells: (A) Two overlapping cells, no collision handling. (B) Two equally stiff cells. Collision handled. The stiffness equality causes the cells to deform in the same magnitude. (C) Two cells which have a different stiffness. Collision Handled. The right cell is stiffer than the left cell. The left cell shapes it self around the stiffer right cell. All cell are made transparent to be able to view the effect of the collision handling.
Fig 11.
Two overlapping cells and their axis aligned bounding boxes (AABB). The overlap contains all potentially interacting elements. In this example, the overlap is exaggerated for illustrative purposes. During simulations, the collision detection will have intervened as soon as an overlap is detected and objects will not overlap this much.
Fig 12.
Model efficiency for blastula simulations with 16-2048 cells. This figure shows the wall clock time (hh:mm:ss) for 1000 iterations when the simulations are in equilibrium state. The time scales linear with the number of cells (axis are logarithmic scale). Doubling the number of cells per blastula results in twice the amount of execution time for the same amount of iterations.