Biomedical Engineering Reference
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closer to the linear case where the correction,
f
(
d
m
/D
p
,
l
m
/H
eff
) ~ 0. However, even
in the linear case, the variability of the amino acid side chain volume suggests that
there will be instantaneous excluded volume
L
(
t
) changes as illustrated in Fig.
6.1b
.
6.2.4 Protein Translocation Times
The time it takes for a charged protein molecule to pass a voltage biased nanopore
or the dwell time,
t
d
, involves many phenomena. To simplify the problem, here we
first assume protein molecules are rigid particles with a total charge
Q
and once the
molecules enter the pore, they move along the center line of the pore of length
H
eff
under the electrical field strength
E
=
/
H
eff
, and we further ignore complex
issues such as protein-pore interactions and electro-osmotic flow. Under these
assumptions, the total force exerted on a protein molecule is the electric driving
force opposed by a viscous drag plus a term of random force caused by collision
with molecules in solution, we can approximate the translocation time with a
1-D Langevin equation,
C
m
dv
dt
¼ F
e
ðxÞF
drag
þ kWðtÞ
(6.2)
where
v
is the velocity of the molecule,
F
e
¼Q
in
C
/
H
eff
is the driving force due to the
electric field,
F
drag
¼av
where
a
is the drag coefficient related to the diffusion
coefficient by
a¼k
b
T
/
D
,
k
is defined by the fluctuation-dissipation theorem, and
W
(
t
) is a 'noise term' or Wiener process that represents the random thermal forces
on the molecule. The variable
x
is the position of the first part of the molecule that
enters the pore.
If we assume a protein molecule translocate a nanopore with a terminal or an
average speed, d
v
/d
t¼
0, and the average dwell time (mean first passage time)
t
d
is
long, the mean value of the fluctuating force is zero, then
F
e
(
x
)
¼F
drag
. Using this
approximation, we can derive the translocation time for a uniformly charged
long chain polymer like a DNA molecule and a charged globular shaped protein
molecule.
Globular protein translocation
. If the passing protein molecule is much smaller
(
d
m
<<D
p
,l
m
<<H
eff
) than the pore, and if we assume the interaction between a
protein molecule and the pore can be neglected (i.e. free translocation), and further
we assume the protein translocation process is driven by an electrophoretic force,
F
e
¼QC
/
H
eff
, opposed by a viscous drag,
F
drag
¼av¼C
f
v
, with a terminal speed
v¼H
eff
/
t
d
, the
t
d
can be written as
t
d
¼ C
f
H
eff
QC
(6.3)
Here
is the solution viscosity,
C
f
is a constant
for a protein in a specific shape, and
Q
is the total net effective charge of a protein.
a¼C
f
is the friction coefficient,
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