Rephrased: Exploring the Mathematical Aspects of the Roughness of Cell Boundaries

Researchers have discovered the mathematical and biological mechanism behind the bumpy structures at the cell borders of tissues such as the kidneys and nasal glands. The team hopes their new insights will help develop new ways to treat comorbidities and build better biological models for future study.

Our cells come in all kinds of shapes and sizes. From the nerve cells that span the central nervous system, to the spherical white blood cells that protect us from infection, cell shape and structure are critical to their function in the body. Structures can differ between cells as well, and likewise hold a crucial benefit.

One such interstitial structure is the “firm” or “wave” pattern commonly found among epithelial cells, the type of cell that covers your skin and most other organs and blood vessels. Under a microscope, these patterns can look quite wild, but for Professor Takashi Miura of Kyushu University’s School of Medical Sciences, this is an intriguing topic.

explains Miura, who led the study published in iScience. “Many cells have intercellular boundaries. For example, renal foot cells that act as filters to generate urine have very complex interaction patterns. Plant leaf epidermal cells look like a jigsaw puzzle in order to reduce the mechanical stress on the cell walls.”

A critical function of epithelial cells is to facilitate the transport of molecules and fluids between said cell boundaries, a process known as paracellular transport. Recent work suggested that overlapping boundaries enhances transfer efficiency. However, exactly how these structures form—and their physiological significance—is still not fully understood.

“We set out to study overlap in MDCK cells, which are a type of epithelial cell originating from the kidney and commonly used to study phenotype formation,” says Miura. “We found something unexpected when we cracked a mathematical calculation of the pattern of cell-cell boundaries. It turns out that these seemingly random structures are not random at all, and are in fact mathematically scaled. In other words, the pattern has self-similarity — if you scale up the boundaries, they hold the same original style properties

The team then explored well-established mathematical models to understand how and why the similarity patterns have this distinctive shape. After a number of working hypotheses, they settled on a model called the Edwards-Wilkinson model.

“It uses the Edwards-Wilkinson model to mathematically simulate random vibration boundaries with a function of minimizing the length of those boundaries. The cell boundary scale we found fits this model,” Miura continues. “Then, our next step was to find the molecular mechanism responsible for these dynamics.”

The team focused on the role of actomyosin, the actin-myosin protein complex responsible for nearly everything that requires force in cellular activities. Observing closely, they identified specific myosin proteins that would lie at the curved cell borders.

Miura explains that their new findings give them a better understanding of the fundamentals of cell dynamics, and contribute to the larger direction of developing the mathematical foundations of biology.

He concludes, “Mathematics has always been closely associated with the fields of chemistry and physics. The mathematic analysis of fundamental processes in biology is still a relatively new field that has grown significantly over the past twenty years.” “I think it shows that the field of biology is maturing. And as we develop the field, it will give us new perspectives on the fundamentals of life, and the beauty of biological patterns.”