8.9 Mechanical Regulation of Morphogenesis
493
Rather than being specified, therefore, the target stress is assumed to evolve
according to the equation
˙
σ 0 = −b(σ − σ 0 ),
(8.77)
where b is a positive constant. According to this relation, an increase in stress elicits
a decrease in σ 0 ( ˙
σ 0 < 0), and vice versa. Hence, whereas growth is generally
assumed to return the stress to a constant homeostatic value, the morphogenetic
(HR) response to increased tension tends to reduce the tension below its value prior
to the perturbation.
Equations (8.76) and (8.77) are morphomechanical laws in one dimension. In
3D, these relations can be written in the tensor forms as
˙
G = A : (σ − σ 0 ) · G
˙
σ 0 = B : (σ − σ 0 ),
(8.78)
where A and B are fourth-order coefficient tensors. Alternatively, cell intercalation
could depend on changes in cell shape (strain), as in the continuum theory of
Beloussov et al. (2015).
It is important to note that the HR hypothesis is based on events that take place
relatively quickly, within a few minutes or hours, whereas cells typically take several
hours just to divide once. Thus, rather than growth, the tensor G often represents
elongation by cell intercalation, i.e., convergent-extension ( ˙
G i > 0), or shortening
by contraction ( ˙
G i < 0). 21 For simplicity, however, we herein use the term “growth”
to describe both responses.
To illustrate the fundamental behavior of the above morphomechanical laws, the
remainder of this section examines models for specific problems. It is important to
emphasize, however, that these equations do not produce the observed morphogenesis in all cases, especially those that involve a series of events following a single
mechanical perturbation. These inconsistencies and possible generalizations of the
theory are considered elsewhere (Taber 2008, 2009).
8.9.3 HR: Uniaxial Extension of a Bar
Consider two bars with the same undeformed length L 0 and cross-sectional area
A 0 that are connected end to end. The assembly is stretched and held at the overall
stretch ratio
21 Tissue contraction is not always of the actomyosin variety. It also could occur, for example, by
cell death (apoptosis), outward cell migration, or resorption of cell membranes.
493
Rather than being specified, therefore, the target stress is assumed to evolve
according to the equation
˙
σ 0 = −b(σ − σ 0 ),
(8.77)
where b is a positive constant. According to this relation, an increase in stress elicits
a decrease in σ 0 ( ˙
σ 0 < 0), and vice versa. Hence, whereas growth is generally
assumed to return the stress to a constant homeostatic value, the morphogenetic
(HR) response to increased tension tends to reduce the tension below its value prior
to the perturbation.
Equations (8.76) and (8.77) are morphomechanical laws in one dimension. In
3D, these relations can be written in the tensor forms as
˙
G = A : (σ − σ 0 ) · G
˙
σ 0 = B : (σ − σ 0 ),
(8.78)
where A and B are fourth-order coefficient tensors. Alternatively, cell intercalation
could depend on changes in cell shape (strain), as in the continuum theory of
Beloussov et al. (2015).
It is important to note that the HR hypothesis is based on events that take place
relatively quickly, within a few minutes or hours, whereas cells typically take several
hours just to divide once. Thus, rather than growth, the tensor G often represents
elongation by cell intercalation, i.e., convergent-extension ( ˙
G i > 0), or shortening
by contraction ( ˙
G i < 0). 21 For simplicity, however, we herein use the term “growth”
to describe both responses.
To illustrate the fundamental behavior of the above morphomechanical laws, the
remainder of this section examines models for specific problems. It is important to
emphasize, however, that these equations do not produce the observed morphogenesis in all cases, especially those that involve a series of events following a single
mechanical perturbation. These inconsistencies and possible generalizations of the
theory are considered elsewhere (Taber 2008, 2009).
8.9.3 HR: Uniaxial Extension of a Bar
Consider two bars with the same undeformed length L 0 and cross-sectional area
A 0 that are connected end to end. The assembly is stretched and held at the overall
stretch ratio
21 Tissue contraction is not always of the actomyosin variety. It also could occur, for example, by
cell death (apoptosis), outward cell migration, or resorption of cell membranes.