The emergence of geometric order in proliferating metazoan epithelia is an article published in 2006 in Nature by Gibson and colleagues showing a new mathematical model to model the development of epithelia in Metazoan, which predicts several emergent behaviors seen in live cells. Emergent phenomena are defined as properties of a system (here, a tissue) which can only be explained as a result of examining both its parts and their relationships.
The authors open their paper with the following:
The organisation of cells into epithelial sheets is an essential feature of animal design.
Order is found all around us, whether it is to minimize surface energy or maximize space-filling, order and structure is found in biological and non-biological systems: from the coins on a tabletop to the eyes of Drosophila to the hexagonal organization of honeycombs. The ways by which these structures are achieved biologically and mechanically form the foundations of both the laws or physics and underpin some of the most important dicoveries in Biology. However, the authors point out that little work has been done to try to understand the dynamics of the geometric shapes observed in proliferating epithelia in metazoa (i.e., the outer layer of any animal’s or insect’s surface).
In essence, Gibson and colleagues set out to present a mathematical model (specifically, a discrete Markov model) to understand how the distribution of the polygons formed by the proliferating epithelia is achieved and note how despite the irregularity of these shapes and rapid cell division, this model enables the maintenance of the structural integrity of the entire tissue.
Post-mitotic relationship between daughter cells. Most of then share one side, and do not resort.
The authors first note the geometric patterns that they see when cells are dividing - taking Drosophila Melanogaster as an example. They observe that the vast majority of cells still share an edge once they have divided. This is used later on to derive some of the properties of the model.
After discussing briefly a recurrence system, which I will not discuss here due to its limited relevance, they derive a discrete Markov model to demonstrate that the polygon distribution is “not a result of biological regulation or cell packing, but a direct consequence of cell proliferation”. They validate this model with experimental data collection from multiple organisms, and find a distribution of polygons that closely matches that of the predicted equilibrium that arises from the Markov model.
Polygon distribution of observed cells in metazoa superimposed by the predicted distribution by the discrete Markov model.
Another compelling feature of the model is the presence of a “shift” matrix, which accounts for the fact that cells undergoing mitosis have, on average, one more side than parent cells in proliferating tissue, which is again confirmed by experimental data. The experimental data is shown below.
Polygon distribution between mitotic cells, the predicted distribution and the distribution for non-mitotic cells.
In another experiment, they show that their model accurately and exclusively describes the dynamics of the polygon distribution in dividing cells, for that the polygon distribution in cells forcefully undergoing mitosis by insertion of a protein called string surrounded by otherwise quiescent tissue does not exhibit the same polygon distribution, as seen in the figure below.
Polygon distribution of stg positive cells (cells forcefully going through
mitosis and the rest of the epithelia.
Together, these findings are quite powerfully describing how an emerging behavior can be exhibited in proliferating tissue and that probabilistic rules directly influence the topology of cells in proliferating tissue . Still, one should keep in mind that it describes an (albeit frequently occurring) idealized version of the system and that any departure from the cleavage place’s random nature will result in a different distribution of polygons.
You can find the slides of my seminar talk as well as the rest of the literature and source code for the presentation can be found here.