5.3 Model for a Contractile Fiber
215
where σ p and σ a are defined per unit deformed area of the corresponding component, and λ ∗ = λ/K by Eq. (5.3). The quantities φ p σ p and φ a σ a represent partial
stresses per unit total area.
Because actin and myosin filaments do not change length as they slide relative
to each other, our CE model does not affect the incompressibility condition J = 1.
Consequently, if a CF is incompressible, the Lagrange multiplier required by this
constraint is associated with the passive matrix alone. In this case, we write
σ p = ¯
σ p − p/φ p ,
(5.9)
where the multiplier is defined as p/φ p . Combining Eqs. (5.8) and (5.9) yields the
total stress in the form
σ = ¯
σ − p
¯
σ = φ p ¯
σ p + φ a σ a ,
(5.10)
and Eq. (3.249) 1 gives the response functions (for J = 1)
¯
σ p = λ
∂W p (λ)
∂λ
σ a = λ
∗ ∂W a (λ ∗ )
∂λ ∗ .
(5.11)
Specific forms for the passive and active strain-energy density functions, W p and
W a , are discussed in the next section.
Example 5.1 Experiments have shown that forces transmitted through actin bundles or stress fibers can deform the nucleus (Hu et al. 2005; Dahl et al. 2008),
causing changes in gene expression (Tajik et al. 2016; Szczesny and Mauck 2017).
Although the precise link between nuclear deformation and genetic activity is not
yet understood, it is useful to study the mechanics of force transmission in cells.
Here, we use a variation of a model proposed by Wang and Suo (2005) to examine
how lateral connections to the cytoskeleton affect loads exerted on the nucleus by
contracting stress fibers.
Consider a cell with a stress fiber connecting the nucleus to the substrate through
a focal adhesion composed of integrins (transmembrane receptors) (Fig. 5.5a). The
stress fiber is represented by a bar of undeformed length L and cross-sectional area
A 0 that is composed entirely of active CEs with strain-energy density function
W a = c a (λ
∗
− 1)
2 .
(5.12)
For simplicity, passive stress is ignored. The cytoskeleton is treated as a series
of longitudinally oriented springs (stiffness k per unit undeformed fiber volume)
215
where σ p and σ a are defined per unit deformed area of the corresponding component, and λ ∗ = λ/K by Eq. (5.3). The quantities φ p σ p and φ a σ a represent partial
stresses per unit total area.
Because actin and myosin filaments do not change length as they slide relative
to each other, our CE model does not affect the incompressibility condition J = 1.
Consequently, if a CF is incompressible, the Lagrange multiplier required by this
constraint is associated with the passive matrix alone. In this case, we write
σ p = ¯
σ p − p/φ p ,
(5.9)
where the multiplier is defined as p/φ p . Combining Eqs. (5.8) and (5.9) yields the
total stress in the form
σ = ¯
σ − p
¯
σ = φ p ¯
σ p + φ a σ a ,
(5.10)
and Eq. (3.249) 1 gives the response functions (for J = 1)
¯
σ p = λ
∂W p (λ)
∂λ
σ a = λ
∗ ∂W a (λ ∗ )
∂λ ∗ .
(5.11)
Specific forms for the passive and active strain-energy density functions, W p and
W a , are discussed in the next section.
Example 5.1 Experiments have shown that forces transmitted through actin bundles or stress fibers can deform the nucleus (Hu et al. 2005; Dahl et al. 2008),
causing changes in gene expression (Tajik et al. 2016; Szczesny and Mauck 2017).
Although the precise link between nuclear deformation and genetic activity is not
yet understood, it is useful to study the mechanics of force transmission in cells.
Here, we use a variation of a model proposed by Wang and Suo (2005) to examine
how lateral connections to the cytoskeleton affect loads exerted on the nucleus by
contracting stress fibers.
Consider a cell with a stress fiber connecting the nucleus to the substrate through
a focal adhesion composed of integrins (transmembrane receptors) (Fig. 5.5a). The
stress fiber is represented by a bar of undeformed length L and cross-sectional area
A 0 that is composed entirely of active CEs with strain-energy density function
W a = c a (λ
∗
− 1)
2 .
(5.12)
For simplicity, passive stress is ignored. The cytoskeleton is treated as a series
of longitudinally oriented springs (stiffness k per unit undeformed fiber volume)