Mathematical biology. Create your own Turing patterns.

Last week I showed some of the possible patterns that were available as solutions of mathematical equations. This week we end morphogenesis month by showing you how to create such Turing patterns for yourself.

Freely available at:

http://www.sciencemag.org/content/329/5999/1616/suppl/DC1

is a simulator (written in java) of the types of reaction-diffusion patterns that we have been considering. The supplementary information section gives a really nice overview of how Turing patterns work and how to use the software. However, for those amongst you who are too impatient to read such things click on the picture below to show a brief manual of how to use the program.

The main joy of this program is that you are able to alter the parameter values of the reaction diffusion equations, seen in the big box at the bottom, very quickly. Thus, you are able to experience different patterns being created right in front of your very eyes! Below are a few that I created in 5 minutes that easily show you the great complexity available through this simple mechanism.

Each simulation starts with a smiling face initial condition like the one seen below.

Initial condition

The first pattern occurs when the program is run with the default initial conditions. Clearly, we see that we get multiple line patterns framing the happy face. Altering the parameters slightly produces spots and stripes as seen in the second picture, giving the face a much more surprised look. Pushing the parameter values even further we generate oscillating patterns as shown in the two snapshots on the right, thus, the face changes colour rapidly. Finally, for certain patterns it is difficult to make out the initial condition at all. In  the bottom image the solution oscillates but in waves across the domain.

Click on the picture to make the images clearer.

Hopefully, you'll have some fun exploring the many different patterns you can get. Whilst you are generating complex structures think about how difficult it must have been for Alan Turing to have postulated these patterns without our moden day computers. He never saw the patterns that now bare his name, yet, he managed to construct the framework by which we understand them.


Finally, if these have entertained you and you would like to see even more peculiar patterns then try Tim Hutton's youtube channel. Through combing different types of models he was able to produce oscillating Turing patterns (seen below). Now they are really trippy!

Hopefully, you have enjoyed morphogenesis month as much as I have had making these posts. I deal with Turing's theory of morphogenesis everyday and it is such a beautiful subject that I decided everyone must know about it.

Next week, back to the normal mathematical silliness.

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References:

Java software: http://www.sciencemag.org/content/329/5999/1616/suppl/DC1
Oscillating Turing pattern: http://www.youtube.com/watch?v=QCbdYGD1m3Y&list=PL35FD96A5F8236109

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

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.