Monday, 29 August 2011

Mathematical biology. Motivation.

Firstly, I should probably answer the question of what mathematical biology actually is. As the name suggests, it is the application of mathematical techniques to biological problems. But why should we want to do such as thing, when experiments can be run? Why would be want to turn the beauty of nature into an ugly equation? Well, here are a few reasons off the top of my head:
• Experiments can identify cause and effect relationships. Mathematics can suggest mechanisms which underpin these relationships.
• By predicting the outcome of altered systems we reduce experimental waste.
• Once a system has been successfully mathematically modelled (whatever that may mean). We maybe able to understand the cause of pathological cases. This leads to theories on how to correct them.
As mentioned above, my main interest is in pattern formation. Mathematically, we have a number of models that produce qualitatively the same patterns as animal skins. You can see this below; on top you have a the skins of a cheetah, a poison arrow frog and a giraffe; all of which have distinct patterns. Below this you have the mathematical patterns which can be produced using only one mathematical theory.

An important aspect of Turing's is that it suggests many types of animals depend on the exact same mechanism to produce their individual patterns. This supports the idea that evolution has simply picked a simples mechanism, whilst mutations and various types of selection will specify how the model behaves. Interestingly, it is not just skins that are thought to use these patterns, they even appear of animal shells, as seen at the top of the page. The left picture in each couple is a real shell, whilst the right picture is a computer simulation.

The patterning systems we use tend to rely on diffusion as the key mechanism. In terms of evolution this is important as it suggests that no energy from the animal is needed to produce the pattern; only to create the reactive agents (called morphogens) which will naturally diffuse.

Next week I'll be demonstrating how we model diffusion mathematically.

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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.

The application of Turing's theory to shells can be seen in Han's Meinhardt's book, The Algorithmic Beauty of Sea Shells.