Hisao Honda - Mathematical Models of Cell-Based Morphogenesis: Passive and Active Remodeling

Genes are deeply related to the shape of living organisms, and elucidation of a pathway of shape formation from genes is one of the fundamental problems in biology. Information on genes inherited from ancestors defines proteins as having several functions, i.e., enzymes synthesizing and degrading materials, signaling molecules providing signal transduction and regulation, and cytoskeleton-related molecules producing force and supporting configuration for morphogenesis. Recently, much knowledge related to the shape formation of biological bodies has accumulated. For example, membrane ruffles (lamellipodia), cell protrusions (filopodia), and focal adhesion with stress fibers are organized with the actin skeleton and are regulated by members of the Rho family of small GTPases (Rac, Rho, and Cdc42). Rac regulates the accumulation of actin filaments at the plasma membrane to produce lamellipodia and membrane ruffles, Rho controls the assembly of actin stress fibers and focal adhesion complexes, and Cdc42 stimulates the formation of filopodia (Olson et al. ). However, this accumulated knowledge does not automatically lead us to an understanding of shape formation during developmental processes. We do not have methods in hand to integrate the knowledge related to the shape formation into an understanding of the formation of shapes, especially large-scale shapes. We need mathematical models for this integration.

When we view the field of physics, there are various shapes, e.g., a spherical drop of water, a polyhedral crystal of quartz, the hexagonal pattern of a snowflake, and a wind-wrought pattern on sand, which consist of constituent elements: atoms, molecules, or particles. These shapes are self-constructed in assemblages of their constituent elements, that is, spontaneously constructed under certain conditions without outside aids. These self-constructions have been described using mathematical models involving equations of motion. On the other hand, multicellular organisms consist of cells. Cells are constituent elements. We would like to consider that the shape formation of multicellular organisms is also a self-construction of cells. We attempt to describe this shape formation using mathematical models.

In this book, we establish mathematical cell models to elucidate the mechanisms underlying the formation of biological shapes. We would like to show the possibility that the formation of some biological shapes is the result of self-construction. We hope that many would agree with us that mathematical cell models in addition to conventional tools in biological sciences are indispensable to understanding shape formation. We would like to use self-construction in this book instead of self-organization that includes pattern formations via chemical reactions. We are interested in cells that construct a physico-mechanic structure via physical force.

Here, we will briefly introduce the chapters in this book. We describe two mathematical cell models to which the authors have contributed. The first is a cell center model, where there is a one-to-one correspondence between cells and points (cell centers). Cell proliferation and cell disappearance (cell loss and apoptosis) were incorporated into the cell center model (Chap. ).

The second mathematical cell model is a vertex model. The history of its completion in biological cells is described (Chap. ).

The vertex model has been ascertained to recapitulate active remodeling of tissues, during which the cells themselves produce force for active remodeling. Therefore, convergent extension (CE) of collective cells takes place, resulting in elongation of bodies along the body axis (Chap. ).