Biomedical Engineering Reference
In-Depth Information
Consider a number
M
of available surface receptors to which
N
ligands of the
cluster are bound on average. We use the term average here to account for the fact
that fluctuations in the instantaneous number of bound ligands will randomly change
N
with time. The relationship between
M
and
N
is not trivial.
M
is defined by the
number of receptors
available
to the cluster ligands. Thus,
M
is the areal density of
receptors multiplied by the area of the surface accessible to the end of a polymer
linker within the cluster. To tune
M
at constant linker length, receptor density can be
changed; or at constant receptor density, the length of the linkers can be changed. The
bound state partition function of a single ligand-receptor pair under the influence of
a polymer in the linear force-extension approximation is given by
2π
k
B
T
κ
0
1
λ
F
2
e
(
Δ
U
0
−
/
(
2
k
poly
))
/
k
B
T
Z
1
,
on
(
F
)=
(3.87)
An unbound ligand will retract back to diffuse around the polymer's potential mini-
mum and therefore the single unbound partition function follows as
2π
k
B
T
k
poly
1
λ
=
Z
1
,
of f
(3.88)
Note that at equilibrium only the relative levels of energy minima and their cur-
vatures (harmonic approximation) govern the populations and free energy differ-
ence between bound and unbound states. The unbound state is the minimum of the
linker potential, which is arbitrarily set to zero energy. The force term appearing in
Equation 3.87 is due to the linker raising the minimum of the bound state by an
amount
F
2
(see Equation 3.20). If the cluster forms
N
bonds with the sur-
face receptors, the total partition function of the bound state is given in terms of
Equation3.87as
/
(
2
k
poly
)
M
!
N
Z
N
=
!
(
Z
1
(
F
/
N
))
(3.89)
,
on
,
on
N
!
(
M
−
N
)
2π
k
B
T
κ
0
1
λ
k
B
T
N
M
!
F
2
N
2
2
k
poly
))
/
e
(
Δ
U
0
−
/
(
=
(3.90)
N
!
(
M
−
N
)
!
where the prefactor is the binomial coefficient, which accounts for the number of
different combinations of
N
occupied receptor sites out of
M
total (see Figure 3.9).
The
N
uncoupled single-bond partition functions are multiplied, and the force expe-
rienced at the transducer is reduced by 1
N
at each bond. When all
N
ligands are
unbound, we divide by
N
! to account for each identical ligand in the construct being
indistinguishable from one another:
/
1
λ
2π
k
B
T
k
poly
N
1
N
!
Z
N
=
(3.91)
,
off
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