359The effect of diffusion on multispecific morphogen gradients might result in the
formation of spatial patterns. At a fundamental mathematical level, this occurs
when locally asymptotically stable equilibria become unstable with the addition of
diffusivity. This is contrary to intuition in the sense that diffusion usually results in
smoothing of concentrations, so on a first approach it seems unlikely that adding
diffusion to equilibria can cause instability. But it does, and Alan Turing was the
first to notice in 1952 (Turing 1952 ). This phenomenon has since been known as
Turing pattern formation. The most spectacular Turing patterns are observed in non-
linear systems, e.g., the reaction-diffusion system of FitzHugh-Nagumo type.
However, at a very fundamental level, Turing patterns are simply a property of lin-
ear systems (Malchow et al. 2008 ).
Consider the following systems of diffusion-reaction equations:
¶U
tDUaUaV
¶=Ñ 1 2 ++ 11 12 ,
(7.1)¶
¶=Ñ++V
tDV 2 2 aU 21 aV 22 ,
(7.2)where U and V are morphogens,
¶
¶U
t
and¶
¶V
t
represent the rate of change of themorphogens with respect to time, ∇^2 represents the Laplacian operator of second
derivatives with respect to space (i.e., acceleration of change of morphogen in every
direction), Di represents diffusion coefficients, and aij are constant coefficients.
When diffusion is negligible, then (0, 0) is a steady state, and the Jacobian of the
system is
Jaa
aa=æ
èçö
ø÷11 12
21 22.The conditions of stability are simply that both eigenvalues of the Jacobian have
negative real parts, which are equivalent to
aa^11 +<^220 ,
aa11 22->aa12 21^0.
Now, we consider what happens when we introduce diffusivity. Since we are con-
cerned only with a finite domain of size 0 <<xLx, and 0 <<yLy, resulting in
wavenumbers knnX=p/L and kmmy=p /,L then Eqs. (7.1) and (7.2) can be
solved explicitly using Fourier series so that
U
nm=å nm
. =
sin( ),
0¥
a kX
(7.3)7 Establishment of the Vertebrate Germ Layers