Biomedical Engineering Reference
In-Depth Information
where we assume a rigid substrate (see Deshpande et al.,
2008
, for the generalization
to a deformable substrate as used here).
Those integrins not in contact with the ligand patch are assumed to be isolated.
Namely, they are unable to interact with any ligand molecules. Accordingly, we
assume that these integrins are unbounded with
Δ
i
→∞
such that the interaction
force
F
i
=
γ
. Their concentrations are then given by Eqs. (
3.12
)
and (
3.13
) with such values inserted.
0 and energy
Φ
=
3.2.3 Finite Element Framework
The contractility model for the cell behavior is implemented in the ABAQUS fi-
nite element model using a user UMAT subroutine that calculates the material re-
sponse of the cell in terms of actin polymerization and force generation. The in-
terfacial behavior between the cell and a substrate is implemented via the UIN-
TER subroutine, responsible for force equilibrium of the entire cell-substrate sys-
tem and focal adhesion development at the cell-substrate interface. Parameters for
the contractility and focal adhesion models are calibrated using the cell-on-posts
simulation presented in Sect.
3.4
. Based on this calibration study the following ref-
erence material parameters are chosen, with
T
=
310 K. The mod
ul
us
E
c
=
0
.
9kPa
a
nd the Poisson ratio
ν
c
=
0
.
3. The reaction rate constants are
k
f
=
10,
k
b
=
1,
10
−
3
s
−
1
, while the maximum tension exerted by the stress
k
v
=
6 with
ε
0
=
˙
3
.
0
×
fibers is
σ
max
=
15 kPa. For the focal adhesion model, the total concentration
ξ
o
is
taken to be 5000 integrins/µm
2
, and the difference in the reference chemical poten-
tials is taken as
μ
H
−
μ
L
=
5 kT. We choose an integrin-ligand complex stiffness
0
.
015 nNm
−
1
, and the stretch at maximum force
Δ
n
=
κ
s
=
130 nm, giving a bond
strength
2 pN. Throughout the simulations presented in this article these parame-
ters are kept constant and only the substrate properties are varied.
∼
3.2.4 Correlation Between Model Parameters and Experimental
Results
Focal adhesions are imaged in experiments by staining for the protein vinculin,
which directly correlates to the concentration
ξ
H
of the high affinity integrins. The
corresponding characterizing parameter for the stress fiber distributions is not cho-
sen so straightforwardly. Most techniques only image the dominant stress fibers.
The very fine mesh-work of actin filaments is not visible when standard epifluores-
cence or confocal microscopes are used. Thus, to correlate the observations with the
predictions we define a circular variance
Γ
η/η
max
)
, used by Pathak et al.
(
2008
), that provides an estimate of how tightly the stress fibers are clustered around
a particular orientation. Here
η
max
is the maximum polymerization level, which oc-
curs at orientation
φ
s
, while
=
1
−
(
¯
1
/π
π/
2
−
η
is an average value defined as
¯
η
¯
≡
π/
2
η
d
φ
.
