Modelling proportionate growth

An important question in biology is how the relative size of different organs is kept nearly constant during growth of an animal. This property, called proportionate growth, has received increased attention in recent years. We discuss our recent work on a simple model where this feature comes out quite naturally from local rules, without fine tuning any parameter. The patterns produced are composed of large distinguishable structures with sharp boundaries, all of which grow at the same rate, keeping their overall shapes unchanged. It is fascinating to see baby animals grow into adults. Understanding the development of different organs from a single egg cell has been the central problem in developmental biology for over a hundred years. However, there is a considerably simpler problem of understanding how a small baby animal grows to a much larger size. In the case of humans, the body weight increases by a factor of 30 or so. In the case of elephants, this factor is about 100. As the baby grows, different parts of the body grow at same rate. This is called proportionate growth. Of course, this is only a good first approximation. For example, in humans, it is well known that the head grows less than the limbs, some changes in body structure occur at puberty, etc. However, at the simplest level of description, it is useful to ignore such complications. Understanding how different organs are formed, starting from a single cell is the subject of cell differentiation and morphogenesis. The basic mechanism underlying this is believed to be the Turing instability in reaction-diffusion systems [1]. In a baby becoming an adult, all the organs are already formed, and we sidestep this more difficult question. We would like to emphasize that even the simpler question is not well-understood. The important point is that proportionate growth requires regulation, and coordination between the different growing parts. If there is no regulation, it would be very difficult to maintain the overall left-right symmetry that is seen in many animals. In a cell, all chemical reaction rates have a fair amount of fluctuation, because the number of molecules of the chemical species undergoing change is typically small. If the growth in different parts were independent, these fluctuations would lead to much larger variations in net growth than what is observed. For example, in mammals, the bilateral symmetry is maintained quite well during growth (typically to within a few percent). That proportionate growth is special is clear from the fact that examples of proportionate growth outside the biological world are difficult to find. Sure, if one takes a balloon, with some picture drawn on it, and blows it up, all parts of the picture grow proportionately. But this is not really `growth', it is just stretching. Or, consider the growth in a droplet of water suspended in air supersaturated with water vapor. As the droplet collects more water from the surrounding air, it grows in size, and keeps its roughly spherical shape. But in this case, there are no internal distinct parts, and hence this also does not qualify as proportionate growth. One can think of crystals growing from a supersaturated solution. The crystals can have nontrivial shapes, but all the growth occurs on the surface, and the structure of the internal regions, once formed remains frozen. There are many other examples of growth studied in physics literature so far, e.g. diffusion limited aggregation [2], surface growth by molecular beam epitaxy [3], Eden growth model [4], invasion percolation [5] etc. In all these cases also, the structure of inner parts gets frozen, and growth occurs only at the surface, and not everywhere.