# Pattern Formation in Morphogenesis. Problems and Mathematical Issues

Morphogenesis is the process of pattern formation during the development of living organisms. That is a very broad subject encompassing the growth of cells in an embryo or a cancer tumor, the regeneration of the limb of a salamander, but also the pattern formation on the wings of a butterfly or the form of leaves and how they grow on a plant. And to complete the chaos, the subject has been discussed by biologists and mathematicians alike, and they all have different approaches and different models that fit in the technical framework they are familiar with, which is either biology or mathematics. The synthesis of both is still largely missing. Surprisingly many scientists have been involved with morphogenesis. Alan Turing got interested, Aristid Lindenmayer used L-systems, René Thom used catastrophe theory, John von Neumann proposed automata, and Rupert Sheldrake has his own completely different vision.

One major obstacle for a mathematician who wants to tackle this subject is the vocabulary that biologists or medical doctors use that is so different from the mathematician's technical background. However the multidisciplinary aspect may exactly be an attractive and challenging aspect of the subject. The best the mathematician can do is to dive into the deep water of the available literature and learn to swim stroke by stroke.

The current book may be a good start to explore the subject. There are some 20 relatively short papers presenting different possible angles, approaches, and models for many different aspects of morphogenetic phenomena. The seeds for these papers were planted during a workshop on the subject at the IHES in Bures-sur-Yvette (France) in 2010. The idea of this workshop was precisely to create an environment in which biologists and mathematicians could communicate on this particular subject. These are not exactly the proceedings because the contributions were written as a result of the discussions and interactions during and after the workshop. The papers are collected into 3 parts: (I) Biological background, (II) Mathematical models, (III) Ideas, hypothesis, suggestions.

This subdivision may be a bit misleading because some of the originally biological papers do also include mathematics as the result of the interaction that took place after the lecture at the workshop. The mathematical reader should therefore not immediately feel the urge to jump to part II, because in many papers there is almost as much biology as there is mathematics, although in the biological papers, biology is obviously prevalent. Hence in part I, the main point of interest is the biological vocabulary and the names of the players (stem cells, morphogens, regulatory RNA,...) the assumptions made, the known mechanisms (Wnt signaling, Planar Cell Polarity, tissue-level dynamics resulting from subcellular mechanics, gene regulatory network topologies,...) which are supported and illustrated by experiments and case studies.

In part II of course the emphasis is more on all kinds of possible models. A model for oscillator synchronization in cell communication, a hybrid model for cell population, cell complexes and L-systems in multicellular structures, the role of stability and hysteresis mechanisms in pattern formation, dynamics of cancer growth, and population dynamics of cells. The models are only sketched at a superficial level, and one should not hope to find here an in-depth worked-out model for whatever process that is described. Neither are these models well established and well understood, or completely explaining the experimental results. They still have an aura of speculation around them. This makes the transition to part III on "ideas, hypothesis, suggestions", not an abrupt one.

Part III has 5 more papers on different subjects: How interactions of different pattern forming reactions may have contributed to the evolution from a radial symmetric (hydra) to a bilateral symmetric (vertebrates) body; the use of hyperbolic geometry to describe cell division; a formal description system of cellular architectures in plants; geometry and the morphogenetic field concept; and a plea for taking into account the interaction at different scales in biological problems (multiscale modelling).

As a conclusion, one may say that this collection gives a broad, but not so deep insight into some aspects of morphogenesis. If the reader is actually interested in the hard mathematics of PDE modelling like in reaction-diffusion or nonlinear advection-diffusion, or stochastic delay-differential equations and simulation of multiscale models including all the associated numerical aspects, then this book is way off course and it is certainly not the right place to start. If you just want to start exploring the field, you might pick up some ideas from this collection to engage in reading more about the subject at other places, e.g. starting with the ample list of references provided with each paper.

**Submitted by Adhemar Bultheel |

**3 / Dec / 2012