Biomedical Engineering Reference
In-Depth Information
3.2 A Biochemomechanical Model for the Cell
We envisage a well-spread, approximately two-dimensional cell, thickness
b
, lying
on a substrate in the
x
1
-
x
2
plane. The cell model comprises two major elements:
(i) a constitutive model for the formation of stress fibers, their contractility and their
spontaneous attachment to cell adhesion complexes; (ii) a cell adhesion model caus-
ing the cell to adhere to a substrate. In (i), following an activation signal, the model
predicts the development of contractile, actin-myosin stress fibers by polymeriza-
tion, subject to their spontaneous connection to transmembrane adhesions, and con-
sistent with traction or displacement conditions imposed by these adhesions at the
interface between the cell and its substrate. The outputs of this feature of the model
are the spatial (position
x
i
) and temporal (time
t
) distributions of the stress fiber con-
centration,
η(x
i
,φ,t)
and the Cauchy stresses
Σ
ij
(x
i
,t)
generated by the resulting
stress fiber contractility, where
φ
is the angle of orientation of a given family of
stress fibers. In (ii), the stresses generated by the stress fiber model apply tractions
to the focal adhesions to which the stress fibers are attached, and, thereby, control
the spatial and temporal development of such adhesions, as parameterized through
the high affinity integrin concentration on the cell membrane at the interface with
the substrate. Such high affinity integrins are the transmembrane proteins, bound to
stress fibers in the cell and substrate ligands outside it, that are the most important
molecules forming the adhesion between the cell cytoskeleton and the substrate to
which it is attached. Note that there are two main sources of mechano-sensitivity
in the model as described below; tension in the stress fibers tends to inhibit their
depolymerization, and tractions transmitted through adhesion complexes stabilizes
them, encouraging formation of transmembrane integrins bound to ligands on the
substrate.
The mechanical equilibrium equations for the cell are written as
b
∂Σ
11
b
∂Σ
12
∂Σ
12
∂x
2
∂Σ
22
∂x
2
∂x
1
+
=
∂x
1
+
=
ξ
H
F
1
,
ξ
H
F
2
,
(3.1)
where
ξ
H
(x
i
,t)
, the concentration of high affinity, bound integrins, is their number
per unit current area of the cell membrane, and
F
i
is the force per high affinity
integrin applied by the cell to the substrate.
3.2.1 Stress Fiber Formation and Contractility
Stress fiber formation is initiated by a nervous impulse or a biochemical or mechan-
ical perturbation that triggers a signaling cascade within the cell. We model this
signal,
C
(which may be thought of as the concentration of Ca
2
+
) as a sudden rise
to unity followed by an exponential decay, given by
C
=
exp
(
−
t
i
/θ),
(3.2)
