Monday, 5 September 2011

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:



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.

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