Biomedical Engineering Reference
In-Depth Information
−
Theworkterm
F
i
Δ
i
term in Eq. (
3.9
) has been identified previously by Shemesh
et al. (
2005
) as an important constituent of the thermodynamic state of focal adhe-
sions.
The kinetics governing the diffusion of low affinity integrins in the cell mem-
brane is typically fast compared to all other time-scales involved (Deshpande et al.,
2008
; Pathak et al.,
2011
). Consequently, we neglect the diffusion of the low affinity
integrins and the kinetics between their low and high affinity states at any location
on the plasma membrane. These simplifying assumptions imply that the concentra-
tions are given by thermodynamic equilibrium between the low and high affinity
binders, i.e. by
χ
H
=
χ
L
,
(3.11)
which specifies that the integrin concentrations are related to the force
F
i
via
ξ
0
exp
(
μ
H
−
μ
L
+
Φ
−
F
i
Δ
i
kT
ξ
H
=
1
,
(3.12)
)
+
ξ
0
ξ
L
=
1
,
(3.13)
μ
H
−
μ
L
+
Φ
−
F
i
Δ
i
exp
(
−
)
+
kT
respectively, where
ξ
0
=
ξ
L
is the fixed, total concentration of integrins. It
is evident that with decreasing
Φ
ξ
H
+
F
i
Δ
i
(which typically occurs when the tensile
force
F
i
increases), the concentration
ξ
H
of the high affinity integrins increases due
to the conversion of the low affinity integrins to their high affinity state.
To complete the thermodynamic representation it remains to specify the form of
the energy
Φ(Δ
i
)
and the associated force
F
i
in the integrin-ligand complex. Rather
than employing a complex interaction, such as the Lennard-Jones (
1931
) potential,
we utilize the simplest functional form. This is a piecewise quadratic potential ex-
pressed as (Deshpande et al.,
2008
)
−
⎧
⎨
(
1
/
2
)κ
s
Δ
e
Δ
e
≤
Δ
n
,
κ
s
Δ
n
+
(
1
/
2
)κ
s
Δ
e
=
Φ
−
2
κ
s
Δ
n
Δ
e
−
Δ
n
<Δ
e
≤
2
Δ
n
,
(3.14)
⎩
κ
s
Δ
n
Δ
e
>
2
Δ
n
,
κ
s
Δ
n
is the surface energy due to high affinity integrins,
≡
Φ(Δ
i
→∞
=
where
γ
)
the effective stretch
Δ
e
=
Δ
2
, and
κ
s
is the stiffness of the integrin-ligand
complex. The maximum force
κ
s
Δ
n
occurs at a stretch
Δ
e
=
Δ
1
+
Δ
n
. We relate the
evolution of the stretch
Δ
i
to the displacement
u
i
of the material point on the cell
membrane in contact with the ligand patch on the substrate as (Deshpande et al.,
2008
)
Δ
n
or
(
∂Φ
Δ
e
<
0
),
u
i
,Δ
e
≤
˙
Δ
i
=
∂Δ
e
(3.15)
0
,
otherwise
,
