502
8 Morphogenesis
Constitutive Relations The theory includes both passive stress σ p and active stress
σ a , with the total Cauchy stress tensor given by
σ = σ p + σ a .
(8.92)
For the passive stress, the mixture is treated as a viscoelastic composite with the
linear constitutive relation
σ p =
E
1 + ν
+
ν
1 − 2ν
(tr )I
+ [μ 1 ˙
+ μ 2 (tr ˙
)I] ,
(8.93)
where E is Young’s modulus, ν is Poisson’s ratio, μ 1 is the shear viscosity, and μ 2
is the bulk viscosity. The first term in this expression represents the elastic response
given by Hooke’s law, while the second term provides the viscous response, which
depends on the strain rate. The linear strain tensor is given by
=
1
2 [∇u + (∇u)
T
].
(8.94)
To a first approximation, contracting cells are assumed to exert isotropic stresses
on the matrix. Experiments indicate that the traction increases with cell density
until the cells become too crowded, and then the stress decreases. This behavior
is captured by the relation
σ a =
τ n
1 + βn 2 I,
(8.95)
where the positive constants τ and β characterize the contractile strength and the
effects of cell crowding, respectively. Note that σ a → 0 as n → ∞.
Equation of Motion With inertia neglected, the equation of motion is
∇ · σ + b = 0,
(8.96)
where b is a body force per unit volume.
After substituting the above expressions for stress, strain, and the various fluxes,
Eqs. (8.86), (8.88), and (8.96) provide three equations to solve for n, ρ, and u.
Pattern Forming Mechanism To understand how Murray-Oster theory creates
pattern, consider a bar in which both cells and matrix move only in the axial
direction (Oster et al. 1983). Initially, the cells are passive (τ = 0) and distributed
uniformly in the bar (Fig. 8.77a). The ends of the bar are fixed.
Suppose now that all the cells contract equally and generate increasing amounts
of tension (τ increases). Each cell pulls the matrix and other cells toward itself, so all
cells effectively compete in a tug-of-war. For low values of τ , the cell concentration
remains uniform (Fig. 8.77a). However, when τ reaches a critical value, τ c , the
contractile stresses exerted by the cells overcome the elastic restoring force provided
8 Morphogenesis
Constitutive Relations The theory includes both passive stress σ p and active stress
σ a , with the total Cauchy stress tensor given by
σ = σ p + σ a .
(8.92)
For the passive stress, the mixture is treated as a viscoelastic composite with the
linear constitutive relation
σ p =
E
1 + ν
+
ν
1 − 2ν
(tr )I
+ [μ 1 ˙
+ μ 2 (tr ˙
)I] ,
(8.93)
where E is Young’s modulus, ν is Poisson’s ratio, μ 1 is the shear viscosity, and μ 2
is the bulk viscosity. The first term in this expression represents the elastic response
given by Hooke’s law, while the second term provides the viscous response, which
depends on the strain rate. The linear strain tensor is given by
=
1
2 [∇u + (∇u)
T
].
(8.94)
To a first approximation, contracting cells are assumed to exert isotropic stresses
on the matrix. Experiments indicate that the traction increases with cell density
until the cells become too crowded, and then the stress decreases. This behavior
is captured by the relation
σ a =
τ n
1 + βn 2 I,
(8.95)
where the positive constants τ and β characterize the contractile strength and the
effects of cell crowding, respectively. Note that σ a → 0 as n → ∞.
Equation of Motion With inertia neglected, the equation of motion is
∇ · σ + b = 0,
(8.96)
where b is a body force per unit volume.
After substituting the above expressions for stress, strain, and the various fluxes,
Eqs. (8.86), (8.88), and (8.96) provide three equations to solve for n, ρ, and u.
Pattern Forming Mechanism To understand how Murray-Oster theory creates
pattern, consider a bar in which both cells and matrix move only in the axial
direction (Oster et al. 1983). Initially, the cells are passive (τ = 0) and distributed
uniformly in the bar (Fig. 8.77a). The ends of the bar are fixed.
Suppose now that all the cells contract equally and generate increasing amounts
of tension (τ increases). Each cell pulls the matrix and other cells toward itself, so all
cells effectively compete in a tug-of-war. For low values of τ , the cell concentration
remains uniform (Fig. 8.77a). However, when τ reaches a critical value, τ c , the
contractile stresses exerted by the cells overcome the elastic restoring force provided