361represent a given instance, thus bridging molecular and cellular scales in the study
of germ layer differentiation.
7.10 Future Prospects
The Hourglass Model is instructive because it provides a broad context in which to
understand the similarities and differences of early vertebrate development.
According to the Hourglass Model, embryos within a phylum strongly resemble
Fig. 7.12 (a) Initial conditions for a numerical implementation of Eqs. (7.1) and (7.2) in a square
domain. The initial condition was set at 0.5 in square sub-domains at different locations for the two
variables, and zero everywhere else. (b) When the conditions for stability are met, the numerical
solution to the PDE problem in Eqs. (7.1) and (7.2), with Neumann boundary conditions, after
10,000 iterations with time step size of 0.08, shows a simple gradient with D 1 =0 003. , D 2 =0 001. ,
a 11 = 003. , a 12 =- 005. , a 21 = 003. , a 22 =- 00. 4. In this case stability conditions are met with
aa 11 +< 22 0 , and aa11 22->aa12 21 0. (c) A Turing instability appears when the conditions for
stability are violated. The figure shows pattern formation with D 1 =0 003. , D 2 =0 001. ,
a 11 =- 003. , a 12 =- 005. , a 21 = 003. , a 22 = 00. 4. In this case stability conditions are violated
with aa 11 +> 22 0 , and aa11 22-<aa12 21 0
7 Establishment of the Vertebrate Germ Layers