Biomedical Engineering Reference
In-Depth Information
Using the effective spring constant published in literature, one can give a
quantitative estimate on the magnitude of error from this source. According to the
measurement of Bustamante et al. [
22
], for a
l
-dsDNA molecule with a contour
length of 16.4
m, the force at which the Watson-Crick pairs start to break is 65 pN.
To prevent the overstretching transition such that DNA hybridization still takes place,
one expects not to exceed this value. For
m
-dsDNA, the effective spring constant
k
just before this overstretching transition takes place is about 53 pN/
l
m
m. At this
effective spring constant, the root-mean-square displacement of the bead can be
estimated, using equipartition principle, to be about (
x
B
2
)
1/2
8.8 nm. However,
one expects the actual thermal displacement of the DNA probes (
x
p
2
)
1/2
relative to the
pore is significantly smaller since the effective spring constant between the probe and
the pore is enhance by a factor of
L
/
x
, where
L
is the contour length and
x
the distance
between the probe and the pore. Here it is assumed that the DNA behaves like a linear
spring that its effective spring constant scales as
1
/
L
.
Secondly, in the limit that the entropic spring effects can be ignored, that the
DNA translocation can be viewed effectively as a biased randomwalk by the pore on
the DNA, thermal fluctuation effects will still appear as dictated by the fluctuation-
dissipation theorem. One expects that a spread in the first-arrival time (or first-
passage time) will occur. If this thermal smearing is too large (see below for what is
“too large”), the HANS sequencing method (as well as other nanopore-based
sequencing methods) will be fundamentally prohibited.
Following the analysis by Lubensky and Nelson [
23
], one can write down a
Fokker-Planck equation for the probability
P
(
x
,
t
) of finding the pore at distance
x
on the DNA as:
¼
@
P
ð
x
;
t
Þ
@t
¼ D
@
2
Pðx; tÞ
@x
2
v
@Pðx; tÞ
@x
;
(8.5)
where
D
is the diffusion constant of the DNA,
v
is the translocation velocity
(proportional to applied voltage across the nanopore). According to the Einstein
relation
,D¼ k
B
T/g
g
is the Stokes' drag coefficient of the DNA (which may
be altered by the interaction between the DNA and the wall of the nanopore during
translocation).
If the two adjacent probes are spaced at
x
apart, the Fokker-Planck equation can
predict [
24
] the mean-first-passage time of the second probe, in large
v
limit which
is satisfied for typical DNA translocation experiments in solid-state nanopores,
, where
x
v
D
v
2
;
<t>
(8.6)
and the second moment of first-passage time as:
x
2
v
2
4
D
2
v
4
<t
2
>
:
(8.7)
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