360
V
nm=å nm
. =
sin( ),
0a
b kX
(7.4)where k=(,kknm), and X=(,xy). Substituting Eqs. (7.3) and (7.4) into (7.1) and
(7.2), we obtain
d
dtnm aDkanm nm
a
=-() 11 1 2 ab+ 12 ,d
dt
nm aDk nmb
nm=+aa 21 () 22 - 2 2 b.where K^22 =+kknm^2.
The Jacobian of the system isAaD a
aaD=æ
èçö
ø÷
11 12
12
21 22 22k
k.The conditions for stability are analogous to the previous case:ââ^11 +<^220 ,
ââ11 22->aa12 21^0.
where âa 11 =- 11 D 1 k^2 , and âa 22 =- 22 D 2 k^2.
Some general properties can be inferred immediately from this solution: (1) a
change in stability only occurs if aa12 21< 0 , i.e., if they have opposite signs. (2)
Destabilization of the asymptotically locally stable steady state is possible only if
the diffusivity coefficients D 1 and D 2 are different.
When we extend this simple 2-morphogen example to multiple morphogens, the
conditions for absolute stability are given by the Routh–Hurwitz stability criterion.
If any of these conditions are violated, then instability occurs, and a Turing pattern
is formed (Qian and Murray 2001 ).
A simple computational exercise illustrates well the differences between a stable
and an unstable system. Figure 7.12a shows an arbitrary initial condition with com-
pact support, i.e., a value greater than zero in an isolated region, and zero everywhere
else. When the conditions for stability are met, the system evolves with smooth gra-
dients (Fig. 7.12b). However, when the conditions for instability are violated, Turing
patterns appear in ways that resemble some morphological processes (Fig. 7.12c).
Turing pattern formation is a vehicle with which to study the activation of regula-
tory responses influenced by spatial diffusion of morphogens. More complex non-
linear functions can be linearized about a point of interest so that the previous
analysis can be applied. Given the simplicity in the structure of a multi-morphogen
system, Turing pattern formation can be studied both analytically and with numerical
simulations. Analysis informs fundamental patterns, and simulations are useful to
W. Tseng et al.