Biomedical Engineering Reference
In-Depth Information
Since nanopore experiments can make it possible to determine the concentration
of protein-ligand complexes, [PL], in solution, (
9.6
) can be used to obtain
K
a
and
K
d
for the interaction between P and L [
32
,
47
].
y ¼
½
PL
[P] + [PL]
¼
½
PL
½
L
½
L
P
o
¼
¼
þK
d
:
(9.6)
1
K
a
½
½
L
½
L
þ
is the fraction of ligand binding sites that are occupied by ligand, and [P
o
]is
the total concentration of the protein. Due to conservation of mass, [P
o
]
y
Here
¼
[P] + [PL]
and [L]
¼
[L
o
]
[PL] with [L
o
] equal to the initial concentration of the ligand.
Equation
9.6
illustrates that the dissociation constant is equal to the concentra-
tion of ligand at which half of the ligand-binding sites are occupied (i.e.
0.5).
Thus, a small value of the dissociation constant indicates that a particular ligand has
a high affinity for a particular protein. Hence
K
d
is often used to describe or
compare the affinity of a ligand for a protein or in the pharmaceutical industry for
characterizing drug-protein interactions and antibody-antigen interactions.
As stated above, in order to apply (
9.6
) for determining
K
d
, the nanopore sensing
method must be able to determine either the concentration of protein-ligand
complexes, the unbound ligand, or the unbound protein at equilibrium. The tech-
nique described in Sect.
9.3.1
[
50
] was able to calculate the number of antibodies
bound to each virus particle based on
y ¼
DI
. In a similar manner, Uram et al. applied
this analysis to data obtained by Saleh and Sohn [
38
,
47
]. In these experiments,
translocation of colloids functionalized with streptavidin through the pore resulted
in transient current fluctuations. As a function of the concentration of monoclonal
anti-streptavidin antibodies, the number of antibodies bound to the colloids could
be obtained from the peak amplitude data. Knowing the total concentration of
antibody, Uram et al
.
calculated the equilibrium concentration of free antibody
and plotted a binding isotherm (Fig.
9.4
)[
47
]. A fit to the data in Fig.
9.4
with a
variation of (
9.6
) determined
K
d
(Table
9.1
).
Whitesides and coworkers demonstrated a second method for determining the
equilibrium dissociation constant using a pore-forming peptide, alamethicin [
32
].
Alamethicin self-assembles in planar lipid bilayers to form ion channels that have
multiple, discrete conductance values depending on the number of peptides forming
the ion channel at a given time. Here the alamethicin peptides were modified with
the ligand sulfonamide, and the binding of carbonic anhydrase II to alamethicin-
sulfonamide disrupted the formation of ion channels and reduced the amount of
transported charge during the time interval of the recording. Mayer et al
.
then esta-
blished a functional relationship between the concentration of unbound alamethicin-
sulfonamide peptides and the quantity of transported charge. Thus, the authors could
calculate the concentration of unbound alamethicin-sulfonamide peptides in the
presence of various concentration of the binding protein, carbonic anhydrase II.
Consequently, the authors obtained a binding isotherm and determined the
K
a
and
K
d
values for the interaction (Table
9.1
).
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