510
8 Morphogenesis
Fig. 8.84 Pattern formation
in a stretched row of
epithelial cells (Problem 8.11)
x
k
u L
(a) Draw a free-body diagram for a differential element of the bar. Then,
derive the governing equation
d 2 u
dx 2 − α
2 u = 0
for the axial displacement u(x), where
α
2
=
k
E
.
Here, E is Young’s modulus and k is the spring stiffness per unit volume
of the bar.
(b) Determine u(x) and the axial strain (x). For L = 10 and u L = 0.1, plot
the strain as a function of x/L for α = 1, 10, 20. Describe and explain
the computed behavior.
(c) Suppose all the cells are in a state of contraction on the descending limb
of the active stress-strain curve (i.e., past the peak in Fig. 5.7b, page 220).
In this case, E is negative and α is imaginary. Let
α = i ¯
α,
where
¯
α =
−
k
E
is real. Using the identity sinh iz = i sin z, write u(x) and (x) in terms
of ¯
α. For L = 10 and u L = 0.1, plot as a function of x/L for ¯
α =
1, 10, 20. Describe and explain the computed behavior.
References
Abu-Issa R, Kirby ML (2008) Patterning of the heart field in the chick. Dev Biol 319:223–233
Alberts B, Johnson A, Lewis J, Morgan D, Raff M, Roberts K, Walter P (2014) Molecular biology
of the cell, 6th edn. W.W. Norton, New York