Physical constraints on early blastomere packings

James Giammona, Otger Campàs

Preprint posted on May 29, 2020

Tetrahedral, diamond or square? - The physics underlying cell arrangement patterns in 4-cell stage embryos

Selected by Sundar Naganathan


Cells are arranged in a stereotypical fashion in early embryos. A precise arrangement is critical, as any changes often result in detrimental effects to embryonic development. This is because, in many species, already from very early stages of development, cell-cell signaling and neighbor relations are clearly defined, which ultimately determine the cell fates and different body axes of the embryo1. While attaining a specific cell arrangement is critical for a given species, it can vary substantially across species. By investigating cell arrangements in early embryos, could we understand species-specific differences in body axes establishment and cell fate determination? To provide insights into this question, there is a need to first understand what sets cell arrangements in early embryos. While physical interactions among cells and also with the confining egg shell determine cell arrangements in some species2, a comprehensive physical model that can explain species-specific differences is lacking. To tackle this problem, Giammona J et al., develop a theoretical description of cell arrangements at the 4-cell stage of embryos with and without confinement.


To simulate the 3D dynamics of blastomeres (early embryonic cells), the authors chose a particle-based representation, which effectively accounts for cell sizes in the simulation.

  • The change in position of a cell was determined by a combination of the forces cells exert on each other and the force on cells due to the presence of a confining egg shell in addition to Gaussian noise. The force that cells exert on each other was represented by an interaction potential, which takes into account the mechanical interaction between cells. Simply put, cells attract each other when far away and repulse each other when too close. When these two forces balance out, an equilibrium distance between cells is attained, characterized by the contact angle between cells. The force emerging from the confining egg shell was determined by another interaction potential that was defined by the geometry of the egg shell.
  • Cells divide at regular intervals in embryos and this was simulated by accounting for change in volume of cells upon division. Note that early embryonic divisions do not lead to cell growth and hence the term cleavage is used in general to represent these events.
  • Cells divide along specific planes in early embryos and this was simulated by considering ordered divisions, where the current division axis was imposed orthogonal to previous division axis. To counter this, random divisions were also considered, where the direction of cell divisions were randomized.
  • To robustly identify and track cell neighbors, a 3D Voronoi scheme was used, which allows for precise calculation of forces between cells as well as for cells under confinement.

Based on these rules and considerations, the authors initiated the simulation from the single-cell stage and simulated stochastic movements of the interacting blastomeres until the 4-cell stage.

Key results

The authors reported their observations by considering the following four scenarios.

Packing configuration in unconfined embryos:
It was previously known that the minimal energy configuration of four interacting particles at equilibrium is a tetrahedron. And, this was shown to be due to relaxation of particles/cells to positions that represented mechanical equilibrium. Intriguingly, the authors show here that when cell division occurs slower than the mechanical relaxation time scale, only a diamond configuration is observed. The authors characterize this discrepancy and show that the diamond configuration evolves into a tetrahedron configuration only when the mechanical relaxation time scale is at least three orders of magnitude higher than the division time scale. In embryos, however, this is unlikely and moreover, the diamond configuration is never observed at the 4-cell stage in any species, suggesting that additional factors are likely to influence the packing configurations.

Packing configuration under spherical confinement:
When cells were confined by a spherical shell, only tetrahedral packings were observed when the volume of the confining shell was comparable to the total volume of cells. This is consistent with tetrahedral packings observed in species with spherical confinement, such as mouse or some nematode species. Importantly, confinement was observed to bias cell arrangements towards tetrahedral configuration five orders of magnitude faster than in the absence of confinement, both under ordered and random division rules.

Packing configuration under ellipsoidal confinement:
As expected, packing configurations were found to be highly dependent on the aspect ratio of the ellipsoidal shell used for the simulations. At aspect ratios of one (i.e for a sphere), tetrahedral configurations were observed as noted above and at high aspect ratios, a linear arrangement of cells were observed. At intermediate aspect ratios, diamond configurations were predominantly noted, as observed in several nematode species.

Packing configuration with sticky shells:
If the confining shells were simulated to represent sticky surfaces, as in the case of sea urchin embryos, where cells adhere to the shells, both tetrahedral and square packing configurations were observed. This is in contrast to sea urchin embryos, where only square packings are observed. Thus, additional unknown factors not included in the current model, such as possible deformations of the confining shell upon cell division, could play a role.
Taken together, the authors show that the geometry of the confining shell is the overriding factor that determines cell arrangements in 4-cell stage embryos, regardless of division rules.

Why I chose this preprint?

While the role of confinement in influencing different cell arrangements is well established, a comprehensive physical model to describe packing configurations across species is still missing. Although several open questions remain (see below), this is a promising step that will eventually lead to understanding the evolution of mechanisms that sets the egg shell shape and properties in the first place.

Open questions

1. Cells are generally assumed to divide symmetrically in the simulations. In C. elegans embryos, early divisions are asymmetric, however. Does consideration of asymmetric divisions lead to a qualitative or quantitative change in model predictions?

2. In C. elegans, the two cells at the 2-cell stage do not divide simultaneously. If this is taken into account in the model, any change in model predictions is expected?

3. In C. elegans, extensive cell re-arrangements3 and rotations4,5 are also known to occur already before the 4-cell stage. Thus, active cell mechanics and movements are important that determine arrangement of cells at the 4-cell stage. However, this has not been considered in the model. It is unclear therefore, whether the model predictions hold under these circumstances.

4. A recent report6 suggests that the size of early blastocyst in mouse embryos is invariable in the presence or absence of confinement by zona pellucida. Although this was performed at later cell stages than the 4-cell stage under investigation in this preprint, it raises an important question about tissue size. How is the initial cell size of cells/particles defined in the model? Does changing this parameter lead to differential cell rearrangements?


  1. Schulze J. and Schierenberg E., Evolution of embryonic development in nematodes, EvoDevo, 2011
  2. Yamamoto K. and Kimura A., An asymmetric attraction model for the diversity and robustness of cell arrangement in nematodes. Development, 2017
  3. Pimpale L. et al., Cell lineage-dependent chiral actomyosin flows drive cellular rearrangements in early development, Biorxiv, Nov 2019
  4. Singh D. and Pohl C., Coupling of Rotational Cortical Flow, Asymmetric Midbody Positioning, and Spindle Rotation Mediates Dorsoventral Axis Formation in C. elegans, Dev. Cell, 2014
  5. Schonegg S. et al., Timing and mechanism of the initial cue establishing handed left-right asymmetry in Caenorhabditis elegans embryos, Genesis, 2014
  6. Chan JC. et al., Hydraulic control of mammalian embryo size and cell fate, Nature, 2019

Tags: cellular packings, egg shell, embryo geometry, non-equilibrium dynamics

Posted on: 9th July 2020


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  • Author's response

    Otger Campàs shared

    1. If the degree of asymmetry in the division is small, it will not affect the results qualitatively. However, for large degrees of asymmetry between daughter cells, it is well possible that new packing configurations arise. While we have not studied asymmetric division, it is likely that there is a threshold in the level of asymmetry between daughter cells over which new packing configurations arise.
    2. We are actually studying this question at the moment and its role in determining the robustness of the packings. The resulting packing configurations do not change if the divisions are not synchronous or if there is noise in the timing. In other words, the geometry of the confining shell robustly sets the packings at the 4-cell stage. This is not to say that asynchronous divisions do not have a function – they seem to have a function, just not in establishing the packing configuration of the blastomeres at this stage.
    3. Cell movements were considered throughout the simulations and active mechanics was introduced as a simple active stochastic force. It is true that the cell mechanics is more complex, especially because cell geometry can generate very complex force fields that are not accounted for in a particle-based description. In some cases, these cell shape effects may not matter, and the fact that our results agree with several observations might well indicate that they do not matter in some cases. Indeed, previous works have used particle-based simulations to reproduce the movements of cells in early C. elegans embryos. However, in some cases cell geometry is critical, as in spiral cleavage, where the intracellular structures show already a chiral cell even at the 2-cell stage. For these cases, a description that accounts for the full cell shape may be more appropriate.
    4. We defined a cell size (volume) for the fertilized egg and a different size (volume) for the egg shell. We then systematically study how varying the ration of these two sizes affects the blastomere packings. Also, since the initial volume is conserved through the first few cleavages, we impose volume conservation, meaning that cells become smaller upon division. In particular, each daughter cell has half the volume of the mother cell. Despite being a particle-based description, we introduced cell size (via an interaction potential cutoff) and impose volume conservation.


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