Mathematical biology. Pattern formation.

Today we actually consider the types of patterns we can produce. This is a subject dear to my heart as my area of research is in pattern formation and in particular I have recently published a paper discussing the patterning found on stingrays.

The patterns found on the stingays can be produced through using a mathematical model called the BVAM equations. The BVAM system is quite a simple set of equations but they are able to lead to extremely complex behaviour and demonstrate numerous different types of patterns. For instance, not only are they are they able to give spots like those on the stingrays:

but, with minor alterations, they are also able to produce striped patterns that look like certain types of catfish.

These patterns are created through visualising the solutions that can be created through using so called "reaction-diffusion" equations. Normally, the mathematical models describe the dynamics of two chemical species that react with each other and diffuse through the domain, hence the name. By seeding a random initial state (seen at the very start of the videos) we can evolve reaction-diffusion systems through time and these simulations calculates the distribution of the chemical species throughout space, over time.

Of course the output of these systems is simply just a set of numbers which give the value of  concentration at each point in space and time. But, by associating a colour with various levels of concentrations, we are able to produce the fantastically colourful videos above. This idea of associating numbers with colours is illustrated in the videos, as on the right you see a colour bar which tells you the value of the system based on the colour. Negative numbers are nearer the blue end of the spectrum, whilst positive numbers are nearer the red. Thus, what you are seeing in both of these simulations is the response of one chemical population reacting with a complementary species and diffusing through the domain.

A mathematical solution is one thing but a physical experiment is quite another. As I have mentioned before, one of the key criticisms behind using this mathematical theory in biology is that we lack any definite evidence that morphogens can produce Turing patterns and other spatial complexity like that seen in the simulations above.

However, chemistry is there to save the day! Below is a demonstration of the Belousov-Zhabotinsky (BZ) reaction. Amazingly, it can show numerous different types of dynamics, all of which can be described through reaction-diffusion models.

The footage firstly demonstrates the colour oscillations that are hallmarks of such reactions. The footage then moves on to show the spatial patterns that the BZ reactions create. These phenomena are so peculiar that when they were first discovered by Boris Belousov in the 1950s no one believed his results and his findings were rejected by the chemistry community. This left him a broken man causing him to leave science.

Much like Turing he was a man ahead of his time. So much so that we are only really beginning to fully appreciate the importance of his work.

Next week, I bring morphogenesis month to a close as I demonstrate how to simulate mathematical patterns for yourself.

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References:
Catfish: http://de.wikipedia.org/w/index.php?title=Datei:Pseudoplatystoma_fasciatum.jpg

BZ reactions footage: http://www.youtube.com/watch?v=IBa4kgXI4Cg

To read more about the work on stingrays I direct you to the University of Oxford's science blog,
http://www.ox.ac.uk/media/science_blog/101126.html