## Monday, 12 September 2011

### Mathematical biology. Morphogens.

What creates a pattern? As yet we do not know. However, this has not stopped mathematicians from suggesting mechanisms by which they form.

Although we now have many mechanisms which can produce patterns through mathematics the best understood and most used is still that of Alan Turing's diffusion-driven instability. It is a testament to his true genius that an idea that he postulated in 1952 is still applicable to today's research.

To understand his work we first need to understand the concept of a morphogen and how they are used to create patterns. You may want to have a look back at the post on diffusion as we assume that this is how many morphogens move.

What is a morphogen?
A morphogen is any substance which is able to produce a pattern.

Normally, we think of morphogens as chemicals that are able to diffuse and interact with each other creating  complex forms. However, the concept of morphogen is much broader. For instance it could be a source of nutrients for a fungus that causes the fungues to grow into patterns like that seen in the left.

For now we will simply consider the case of chemical morphogens, which are able to diffuse through animal, or plant, cells. Finally, we assume that these cells can sense the morphogens and alter their behaviour because of them.

The French flag  pattern
Suppose a system of cells has a constant source of morphogen on its left side. This morphogen will then produce a concentration profile, or gradient, that is higher on the left than the right (see row A below).

The cells on the left sense a higher concentration of morphogen and respond in some way e.g. they turn blue. The centre cells sense a middling concentration and the right-hand cells will sense a low concentration and, so, they produce different responses e.g. they turn white and red, respectively.

Hence, through simple diffusing morphogens we are able to produce the so called French flag pattern. If you are feeling more adventurous (or patriotic) you can couple multiple sources together to create the flag of the Netherlands, the Danish flag and even the Union Jack (although I've never seen this done :) ).

In rows A and B we can see that simply through diffusion we are able to produce quite complex patterns. If we now let the morphogens react with each other much more complicated structures are able to form such as spots, stripes and labyrinthine patterns as seen in row C.

It is these Turing patterns that we will be considering next week.

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References:
The original idea that diffusion could create patterns instead of just wiping them out first proposed by Alan Turing in his landmark paper, The Chemical Basis of Morphogenesis, As such Turing can be thought of as the founder of mathematical biology.
Morphogen diagram: Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation by Shigeru Kondo and Takashi Miura. Art work by S. Miyazawa