Fig 1.
Schematics and definitions of early embryo dynamics.
A, Schematic representation of the most common 4-cell embryo arrangements in the mouse (tetrahedron), C. elegans (diamond) and sea urchin (square). Blastomeres (small spheres) are confined by the surrounding confining envelope (pink; vitelline envelope, hard chitinous egg shell, or hyaline layer, respectively). B, Abstraction of two cells in contact, depicting the cell radius R, equilibrium distance r* and contact angle θ. C, Cell-cell interaction potential Uc(r). D, Examples of cell configurations for varying contact angles. E, Effective volume correction: the overlapping volume (gray, left) is added to each cell by increasing its radius to match the actual cell volume. F, Examples of interaction potentials, Us(r), of a cell with repulsive (left) and sticky (right) confining shells. Repulsive and attractive regions are shown in orange and blue tones, respectively. G, 3D Voronoi tessellation of neighboring cells (Methods). H, Definition of time between divisions τD and division cycle n. I, Ordered divisions indicate that cells follow specific division rules. In contrast, the cell division axis is randomly oriented for random divisions.
Table 1.
Definition of the relevant dimensionless parameters in the problem.
Fig 2.
Packing configurations and dynamics of 4-cell stage unconfined embryos.
A, Example of time evolution of cellular packings, showing how cell volume decreases by half upon division. B, Definition of possible topological arrangements at the 4-cell stage. C, Frequency of packing arrangements for slow and fast divisions (τD/τM = 0.1, 10, respectively), different contact angles θ and both ordered and random division rules (n = 103 simulation runs for each parameter set). D, Histogram of the time to reach the equilibrium tetrahedral configuration (n = 105 simulation runs for each condition). E, Frequency of packing configurations as the system relaxes to equilibrium (tE being the time to reach equilibrium) for both ordered (left) and random (right) divisions (n = 102 simulation runs for each condition). F, Top view (with cells) and cross section (without cells) showing the equipotential surface (orange tones) caused by three fixed cells on a fourth cell. G, Angular mean squared displacement (MSD) of a cell moving in the potential generated by three cells fixed in a triangle (n = 8 simulation runs), showing its diffusive nature (fit, gray line).
Fig 3.
Spherically confined cellular packings.
A, Frequency of different cell arrangements for embryos with spherical repulsive confinement as the ratio of the shell volume and total cells volume varies (random divisions; τD/τM = 10; n = 104 simulation runs for each parameter set). B, Histogram of the time to reach the equilibrium tetrahedral configuration (n = 103 simulation runs). C, Frequency of packing arrangements for embryos confined in a spherical repulsive shell (a/b = 1; Vs/Vc = 1.52), and both slow and fast divisions (τD/τM = 0.1, 10, respectively), different contact angles θ and both ordered and random division rules (n = 500 simulation runs for each parameter set).
Fig 4.
Cell arrangements for repulsive ellipsoidal confinement.
Frequency of different cell arrangements for embryos with ellipsoidal repulsive confinement of varying aspect ratio (a/b) as the ratio of the shell volume and total cells volume varies (random divisions; τD/τM = 10; n = 104 simulation runs for each parameter set). Snapshots of non-equilibrium packing configurations for large shell volumes (Vs/Vc = 6) are shown on the right. Equilibrium (t = 8000τM) packing configurations for each aspect ratio and Vs/Vc = 1.52 are shown on the left (n = 200 simulation runs for each parameter set).
Fig 5.
Cell packing arrangements in a sticky confining shell.
A-C, Frequency of packing arrangements (A, diamond; B, square; C, tetrahedron) for varying values of the shell volume to cells volume ratio, Vs/Vc, and cell-shell to cell-cell adhesion strength, (n = 5000 simulation runs for each parameter set).