The need for a concept of shape homeostasis

A major step forward for developmental biology will be accomplished when someone figures out how to extend the concept of homeostasis to apply to shapes, in the sense of geometric properties of cells, tissues and organs. I propose that the biggest obstacle to this forward step is that biological researchers are not yet familiar with the properties of tensor variables, as compared with scalars. This key difference is that tensor properties can and usually do have different amounts in different directions, whereas scalar properties cannot vary with direction. Examples of tensor variables include stress, strain, curvature, permeability, and stiffness. Examples of scalar properties include chemical concentrations, osmotic pressure, hydrostatic pressure, adhesiveness and electrical voltage. Even D'Arcy Thompson treated mechanical tension (which is the classic example of a tensor variable) as if it were a scalar constant. This greatly reduced the number of geometric shapes that he could explain as being directly produced by forces. For example, in order to generate cylinders, surface contractions need to be twice as strong in one direction as compared with the perpendicular direction. Unless surface contractions vary with direction, only spheres can be generated. Another example of not distinguishing tensors from scalars is the use of suction pipettes to measure stresses of cell surfaces (for example, during cytokinesis). This method of measurement inescapably lumps together directional components of two different tensors (tension and stiffness) as if they were one scalar. Yet another obstacle was that certain scientists argued persuasively, but mistakenly, that attractor basins were evidence of minimization of thermodynamic free energy. Chemical concentrations have no special ability to generate gradients. Neither do any other scalar variables. As will be discussed below, repeated local equilibration of any quantitative variable will generate at least as good a gradient as diffusion can. It is misguided to think of scalars as being in any sense more quantitative than tensors. In fact, tensor variables can convey more information than chemical gradients, often faster and with less vulnerability to disturbance.