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.

Fungus network: http://www.cabdyn.ox.ac.uk/complexity_ukfungalnet.asp

Morphogen diagram: Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation by Shigeru Kondo and Takashi Miura. Art work by S. Miyazawa

Mathematical biology. Diffusion.

As noted in the previous post (and the zombie post), diffusion is a very important process, biologically as well as generally. Today, we take a closer look at this undervalued mechanism of motion.

To give an intuitive idea of the motion, consider a small blob of ink added to water that is not stirred or heated. Although the water is not being disturbed the ink will begin to spread out and, after enough time, it will colour all of the water. The ink particles are constantly and randomly bombarded from all sides by molecules of water, exactly like billiard balls. These random collisions cause the ink to diffuse through the water in a process called Brownian motion. 

Time series evolution of ink in water illustrating diffusion.

This post is going to contain some serious maths, which is unusual for a "fun" blog. However, I urge you to stick with it. If you have never studied maths before the symbols will appear esoteric. However, don't worry. I am going to guide you through, step by step, building up the diffusion equation until there should be no mystery any more.
  Suppose we define the concentration of a chemical at a point x and at a time t be U(x,t). This means that where we have a lot of substance U will be a high number and, conversely, where U is a small number, we will have less chemical.

Through this simple definition we find that physics tells us that the random motion has to satisfy the diffusion equation,

The diffusion equation describes the motion of any substance which can be thought to be moving randomly e.g. proteins in your body, water through soil, heat through metal and smells in the air. But where does this equation come from and what does it mean? Let us decipher these curious hieroglyphics.

The left hand side is the "derivative of U(x,t)", with respect to time, t. Explicitly,

This means that if the derivative is positive then U(x,t) is increasing at that point in time and space. If  the derivative is negative then the concentration is decreasing. This term allows us to consider how U(x,t) evolves over time. The factor D is a positive constant that controls the speed of movement.

The term on the right-hand side is a little more complex than the left, but essentially it encapsulates the idea that the chemicals are moving from high to low densities. This is illustrated with the help of the figure below.

An example of how diffusion smooths out peaks and troughs.

Initially, the concentration is higher on the left than the right. Just before the peak in density the arrow, which is the tangent to the curve, or

at this point, is pointing upwards. This means that as x increases, so does the chemical density. Hence, at this point,

Just after the peak the arrow is pointing down and, so, at this point,

Hence, at the peak, the derivative of U(x,t) with respect to x is decreasing as x increases. 

By extending the definition of derivatives that we are using, we know that

 is the rate of change of

 

as x increases and, moreover, we have just deduced that this rate is decreasing at the peak. Thus, from the diffusion equation we see that, at the peak,

this means that the peak of chemical concentration is decreasing over time.

By a similar argument we can show that U(x,t) is increasing at the trough. Overall, we see that diffusion causes the chemical to move from regions of high density to low density.

It is impossible to overstate the importance of the diffusion equation. As indicated above, it can be used in many biological and physics contexts. This week we have given you the insight that diffusion is a smoothing process. As time increases diffusion will tend to produce a uniform districbution of chemical throughout the domain as in the ink and water experiment above. Next week we consider the counterintuitive result that Turing postulated:

DIFFUSION CAN CAUSE PATTERNS TO OCCUR.

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References:

Diffusing ink: http://upload.wikimedia.org/wikipedia/commons/1/1f/Dispersion.gif

All of this work is motivated by Alan Turing's landmark paper, The Chemical Basis of Morphogenesis. As this is quite a difficult read, I've taken it upon myself to try and explain it to everyone through this blog.

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.

Turing patterns: Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation by Shigeru Kondo and Takashi Miura.

Cheetah: http://www.flickr.com/photos/wwarby/2310137233/sizes/o/in/photostream/

Poison arrow frog:
http://commons.wikimedia.org/wiki/File:2009-04-05Dendrobates_auratus151.jpg

Giraffe: http://www.photoshoptextures.com/animal-textures/animal-textures.htm

Mathematical biology. The creator.

2012 is the year of Alan Turing's centenary. Not only was he the father of computational logic as we know it today but he also had a lesser known interest in biology. Specifically, he was interested in such questions as:

• Why is there something rather than nothing?
• How are complex structures formed from simple components?
• Why doesn't everything tend to a state of uniformity?

In 1952 he published a paper, The Chemical Basis of Morphogenesis, trying to answer these questions. His ideas were revolutionary and not fully appreciated at the time. However, his paper was the start of a whole new field of mathematics, which 60 years later still uses his ideas as their foundation.

Over the next few weeks I'll be giving you an insight into how mathematics can unravel the mysteries of biological development. Not only does this extend our understanding of the natural world but it also offers us a new way to appreciate the beauty, simplicity and diversity of the world around us.

We start next week, where I will illustrate what a mathematician interested in pattern formation can do.

Just a small selection of the patterns which Turing's theory can produce. Taken from http://www.rudyrucker.com/blog/2010/09/10/what-was-alan-turing-like/ which gives you the highlights of Turing's weirdness.

 

Maths laugh 6. Up to date satire.

Ali G. was actually an excellent mathematical philosopher!

As usual, tweet me any maths jokes to illustrate @ThomasEWoolley.

What "The Code" could have been...

On Wednesday 27th July 2011 a new three part documentary hit the British television. It was called The Code, hosted by Professor Marcus du Sautoy and had the intention of demonstrating how mathematics can be used to unravel the mysterious code that surrounds the natural and man-made world. The reaction to the shows has been mainly positive and some of the footage has truly been breathtaking (see the footage below of the bubbles creating the platonic solids, incredible!).

Whatever your thoughts or feelings are about The Code, I wanted to share with you some information about what The Code could have been and let you make your own mind up as to whether they chose the right program.

At the beginning of February 2010 I had the amazing fortune to be asked if I would like to work as a researcher on a new mathematics program called The Code. Of course I jumped at the chance. So the things I explain here are my from my own experience. They are not hearsay or rumours, just plans that never came to fruition.

The competition aspect of The Code was there from the start and it was always the intention to have the programs weave with online content. However, the style of the show had originally been very different. The plan was to team Marcus up with a different celebrity each week and challenge them to do something amazing. The celebrity would admit that they could not complete the task and the program would track the celebrities progress as Marcus tries to teach them the various, simple mathematical principles that would allow the celebrity to achieve their goal.

My job was to develop the celebrity challenges. It certainly was not easy. I had to produce an exciting result using simple mathematics. My favourite example that I came up with was:
get the celebrity to try and measure the height of a cliff standing over water. Tape measures wouldn't be possible, so, instead, they would have to learn trigonometry.
Now you may think that is extremely mundane, however, the next part of the challenge was to:
get the celebrity to jump of the cliff using a bungee cord based on their calculation.
Not only do the have to work with heights but they have to take into account the extension of the chord as well. They would literally be placing their lives into the hands of mathematics.

I created five or six such activities like that mentioned above, wrote them up in a treatment, was paid for my work and left White City, having had an amazing experience. However, on my last day I was told what would then happen to my work. The pitch would go forward and either get funded or not. If the idea was not commissioned it would be recycled until they found a format that worked. If it got funded, the project would get sent to a production team who could COMPLETELY IGNORE all the development work!

So the next time you watch an episode of The Code (the last episode is to be aired Wednesday 10th August), think about what it could have turned into and ask yourself:

Did the BBC make the right choice?

Maths laugh 5. More food based maths.

Anyone else hungry?

For some more maths based food I recommend the following links from the Evil Mad Scientist Laboratories:

Koch snowflake cupcakes

Sierpinski cookies

Here is a demonstration of how to cut a bagel into two connected pieces and, finally, turn your fruit into platonic a solid.