It is now 60 years since Francis Crick and James Watson worked out the molecular structure of DNA, with vital input from Rosalind Franklin and Maurice Wilkins. Their discovery that DNA encodes genetic information, in a form that can be copied by purely chemical processes, began the molecular revolution in biology. Instead of concentrating on the form and behavior of living creatures, biologists focused on the structure of the complex molecules that make life possible.
Initially, many biologists hoped that most of the important features of a living organism could be deduced from its DNA. But, as the revolution matured, it became clear that biology is not that simple. Instead of being some fixed blueprint, DNA was more like a list of ingredients for a recipe. An awful lot depended on precisely how those ingredients were combined and cooked.
A few mavericks tried to understand the growth and form of living creatures using mathematical techniques, but their ideas were overshadowed by the flood of results appearing in molecular biology, and looked old-fashioned in comparison. In 1952 Alan Turing published a theory of animal markings, only a year before Crick and Watson's epic work. In 1976 Jonathan Cooke and Christopher Zeeman applied new geometric techniques to the early development of the spinal column in an embryo -- the formation of a series of segments along its back, known as somites.
Turing proposed that animal markings are laid down in the embryo as a 'pre-pattern' resulting from special chemicals, which he called morphogens. The pre-pattern acts as a template for the patterns of colored pigment that appears as the creature grows, and it is created by a combination of chemical reactions and diffusion, in which molecules spread from cell to cell. Cooke and Zeeman proposed that somites arise through a wave of chemical changes, coordinated by a molecular clock.
However, biologists came to prefer a different approach to the growth and form of the embryo, known as positional information. Here an animal's body is thought of as a kind of map, and its DNA acts as an instruction book. The cells of the developing organism look at the map to find out where they are, and then at the book to find out what they should do when they are in that location. Coordinates on the map are supplied by chemical gradients: for example, a chemical might be highly concentrated near the back of the animal and gradually fade away towards the front. By 'measuring' the concentration, a cell can work out where it is.
The main difference in viewpoint was that the mathematicians saw biological development as a continuous process in which the animal grew organically by following general rules controlled by specific inputs from the genes, whereas to the biologists it was more like making a model out of chemical Lego bricks by following a plan laid out in the DNA genetic instruction book.
Important evidence supporting the theory of positional information came from transplant experiments, in which tissue in a growing embryo is moved to a new location. For example, the wing bud of a chick embryo starts to develop a kind of striped pattern that later becomes the bones of the wing, and a mouse embryo starts to develop a similar pattern that eventually becomes the digits that make up its paws. The experimental results were consistent with the theory of positional information, and were widely interpreted as confirming it.
However, in December 2012 a team of researchers led by Rushikesh Sheth at the University of Cantabria in Spain carried out more complex experiments of the same kind, but involving a larger number of digits. They showed that a particular set of genes affects the number of digits that the mouse develops. As the effect of these genes decreases, the mouse grows more digits than usual -- like a human with six or seven fingers instead of five. Their results were incompatible with the theory of positional information and chemical gradients, but made complete sense in terms of Turing's reaction-diffusion approach.
In January 2013, a team led by Volker Lauschke at the European Molecular Biology Laboratory in Heidelberg used new molecular techniques to study the formation of segments in the spinal region of a mouse. Their main result was that the number of segments, and their size, are controlled by a clock-and-wavefront process of the type proposed by Cooke and Zeeman.
Both investigations have two messages in common. The first is that rigid interpretations of genetic information cannot explain the structural changes that occur in the developing embryo. More flexible mathematical processes are also important. The second is that mathematical models become genuinely useful only when they are combined with detailed and specific information about which genes are active in a given process. Neither molecular biology nor mathematics alone can explain how developing organisms grow and change. We need both, working together.