Biomedical Engineering Reference
In-Depth Information
where
θ
is the decay constant and
t
i
is the time after the onset of the most recent
activation signal. We treat the signal as uniform throughout the cell, on the basis
that diffusion of the signaling ions and proteins in the cytosol is fast enough to be
non-rate limiting, as determined by Pathak et al. (
2011
).
The formation of stress fibers is parameterized by an activation level, desig-
nated
η
(0
1), defined as the ratio of the concentration of the polymerized
actin and phosphorylated myosin within a stress fiber bundle to the maximum con-
centrations permitted by the bio-chemistry. The evolution of the stress fibers at an
angle
φ
to the
x
1
axis is characterized by a first-order kinetic equation
≤
η
≤
1
η(φ)
k
b
=
1
η(φ)
Ck
f
σ(φ)
σ
0
(φ)
˙
−
θ
−
−
η(φ)
θ
,
(3.3)
where the overdot denotes time-differentiation at a fixed material point in the cell. In
this formula,
σ(φ)
is the tension in the fiber bundle at orientation
φ
, while
σ
0
(φ)
=
ησ
max
is the corresponding isometric stress at activation level
η
, with
σ
ma
x
being
th
e isometric stress at full activation (
η
1). The dimensionless constants
k
f
and
k
b
govern the rates of stress fiber formation and dissociation, respectively. Note that
mechano-sensitivity is present in the depolymerization term in (1), since a tensile
stress
σ
will reduce the rate of dissociation of stress fibers; furthermore, a stress
σ
equal to
σ
0
eliminates stress fiber depolymerization completely.
The stress
σ
in stress fibers is related to the fiber contraction/extension rate
=
ε
(positive during extension) by the cross-bridge cycling between the actin and
myosin filaments. The simplified (but adequate) version of the Hill-like equation
employed to model these dynamics is specified as
˙
⎧
⎨
η
k
v
,
ε
˙
˙
0
ε
0
<
−
σ
σ
0
=
k
v
η
(
˙
ε
η
˙
ε
˙
1
+
ε
0
)
−
k
v
≤
ε
0
≤
0
,
(3.4)
⎩
˙
˙
ε
˙
1
ε
0
>
0
,
where the rate sensitivity coefficient
k
v
is the fractional reduction in fiber stress
upon increasing the shortening rate by
ε
. A 2D constitutive description for the stress
fiber assembly uses the axial fiber strain rate
˙
˙
ε
at angle
φ
related to the strain rate
ε
ij
by
˙
ε
11
cos
2
φ
ε
22
sin
2
φ
ε
˙
=˙
+˙
+˙
ε
12
sin 2
φ.
(3.5)
The average stress generated by the fibers follows from a homogenization analysis
as
σ(θ)
cos
2
φ
d
φ.
S
11
S
12
S
21
S
22
π/
2
σ(θ)
2
1
π
sin 2
φ
=
(3.6)
σ(θ)
2
sin 2
φσ
sin
2
φ
−
π/
2
The constitutive description for the cell is completed by including contributions
from passive elasticity, attributed to intermediate filaments of the cytoskeleton at-
tached to the nuclear and plasma membranes. These act in parallel with the active
