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Alan Turing in his original paper presents the following system of differential equations:

$\frac{\delta X_r}{\delta t} = f(X_r, Y_r) + \mu(X_{r+1} -2X_r + X_{r-1})$

$\frac{\delta Y_r}{\delta t} = g(X_r, Y_r) + v(Y_{r+1} - 2Y_r + Y_{r-1})$

for $r \in \{0, ..., N-1\}$

He talks about N cells forming a ring and each cell having two morphogens X and Y. $X_r(t)$ and $Y_r(t)$ are the concentrations of those morphogens in the r-th cell.

Then he reasons that under certain conditions the equilibrium state $f(X_r, Y_r) = g(X_r, Y_r) = 0$ is unstable and thus $X_r(t)$ as well as $Y_r(t)$ quantities will deviate in time. This is called the Turing instability.

Finally, because of the Turing instability, the animal skin can have many interesting patterns, such as stripes, spots, etc.

My questions:

1) Do I understand this correctly?

2) How to relate those differential equations to the animal skin pattern?

3) Could one assume that the concentration of the chemical $X_r$ is proportional to the "intensity" of the cell colour? Thus, those cells for which $X_r$ is high will be black, others white, and that's how the colour pattern is formed?

4) If (3) is assumed, then the pattern of the animal skin is eventually formed, thus each $X_r$ settles down to some fixed value. I.E. eventually $\frac{\delta X}{\delta t} = 0$. So does it mean that the system reaches another fixed state $\frac{\delta X_r}{\delta t} = 0$ and $\frac{\delta Y_r}{\delta t} = 0$ that is stable this time?

5) The example assumes N cells forming a ring. How to relate this to a skin, for which a plane (rather than a ring) looks like a better representation?

mercury0114
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    A quick note on your second question: the term $\mu(X_{r+1}-2X_r+X_{r-1})$ is a central difference approximation to the second (spatial) derivative. It represents diffusion of the morphogen between nodes on the ring. Then $f$ and $g$ are the reactions between those chemicals, as they diffuse around. – Sort of Damocles Jan 25 '18 at 15:41
  • Could one assume that the concentration of the chemical $X_r$ is proportional to the "intensity" of the cell colour? Thus, those cells for which $X_r$ is high will be black, others white, and that's how the colour pattern is formed? – mercury0114 Jan 25 '18 at 16:36
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    Yeah, that's a fine way to think about it, although in biology it may not be such a simple relationship. – Sort of Damocles Jan 25 '18 at 20:07

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