We study the passage times of a translocating polymer of length $N$ in three dimensions, while it is pulled through a narrow pore with a constant force $F$ applied to one end of the polymer. At small to moderate forces, satisfying the condition $FN^{\nu}/k_BT\lesssim1$, where $\nu\approx0.588$ is the Flory exponent for the polymer, we find that $\tau_N$, the mean time the polymer takes to leave the pore, scales as $N^{2+\nu}$ independent of $F$, in agreement with our earlier result for F=0. At strong forces, i.e., for $FN^{\nu}/k_BT\gg1$, the behaviour of the passage time crosses over to $\tau_N\sim N^2/F$. We show here that these behaviours stem from the polymer dynamics at the immediate vicinity of the pore -- in particular, the memory effects in the polymer chain tension imbalance across the pore.
Deep Dive into Passage Times for Polymer Translocation Pulled through a Narrow Pore.
We study the passage times of a translocating polymer of length $N$ in three dimensions, while it is pulled through a narrow pore with a constant force $F$ applied to one end of the polymer. At small to moderate forces, satisfying the condition $FN^{\nu}/k_BT\lesssim1$, where $\nu\approx0.588$ is the Flory exponent for the polymer, we find that $\tau_N$, the mean time the polymer takes to leave the pore, scales as $N^{2+\nu}$ independent of $F$, in agreement with our earlier result for F=0. At strong forces, i.e., for $FN^{\nu}/k_BT\gg1$, the behaviour of the passage time crosses over to $\tau_N\sim N^2/F$. We show here that these behaviours stem from the polymer dynamics at the immediate vicinity of the pore – in particular, the memory effects in the polymer chain tension imbalance across the pore.
arXiv:0706.3969v3 [cond-mat.soft] 16 Aug 2007
P
assage
Times
fo
r
P
olymer
T
ranslo
ation
Pulled
through
a
Na
rro
w
P
o
re
Debab
rata
P
anja
Institute
fo
r
Theo
reti al
Physi s,
Universiteit
van
Amsterdam,
V
al k
enierstraat
65,
1018
XE
Amsterdam,
The
Netherlands
Gera
rd
T.
Ba
rk
ema†‡
†
Institute
fo
r
Theo
reti al
Physi s,
Universiteit
Utre ht,
Leuvenlaan
4,
3584
CE
Utre ht,
The
Netherlands
‡
Instituut-Lo
rentz,
Universiteit
Leiden,
Niels
Bohrw
eg
2,
2333
CA
Leiden,
The
Netherlands
W
e
study
the
passage
times
of
a
translo
ating
p
olymer
of
length N
in
three
dimensions,
while
it
is
pulled
through
a
narro
w
p
ore
with
a
onstan
t
for e F
applied
to
one
end
of
the
p
olymer.
A
t
small
to
mo
derate
for es,
satisfying
the
ondition F Nν/kBT ≲1,
where ν ≈0.588
is
the
Flory
exp
onen
t
for
the
p
olymer,
w
e
nd
that τN
,
the
mean
time
the
p
olymer
tak
es
to
lea
v
e
the
p
ore,
s ales
as N2+ν
indep
enden
t
of F
,
in
agreemen
t
with
our
earlier
result
for F = 0.
A
t
strong
for es,
i.e.,
for F Nν/kBT ≫1,
the
b
eha
viour
of
the
passage
time
rosses
o
v
er
to τN ∼N2/F
.
W
e
sho
w
here
that
these
b
eha
viours
stem
from
the
p
olymer
dynami s
at
the
immediate
vi init
y
of
the
p
ore
in
parti ular,
the
memory
ee ts
in
the
p
olymer
hain
tension
im
balan e
a ross
the
p
ore.
I.
INTRODUCTION
Mole ular
transp
ort
through
ell
mem
branes
is
an
es-
sen
tial
me
hanism
in
living
organisms.
Often,
the
mole-
ules
are
to
o
long,
and
the
p
ores
in
the
mem
branes
to
o
narro
w,
to
allo
w
the
mole ules
to
pass
through
as
a
single
unit.
In
su
h
ir umstan es,
the
mole ules
ha
v
e
to
de-
form
themselv
es
in
order
to
squeeze
i.e.,
translo
ate
themselv
es
through
the
p
ores.
DNA,
RNA
and
proteins
are
su
h
naturally
o
uring
long
mole ules
(1;
2
;
3
;
4
;
5
)
in
a
v
ariet
y
of
biologi al
pro
esses.
T
ranslo
ation
is
also
used
in
gene
therap
y
(6;
7
),
in
deliv
ery
of
drug
mole ules
to
their
a tiv
ation
sites
(9),
and
as
a
p
oten
tially
heap
er
alternativ
e
for
single-mole ule
DNA
or
RNA
sequen ing
(8;
16
).
Consequen
tly
,
the
study
of
translo
ation
is
an
a tiv
e
eld
of
resear
h:
as
a
ornerstone
of
man
y
biolog-
i al
pro
esses,
and
also
due
to
its
relev
an e
for
pra ti al
appli ations.
T
ranslo
ation
in
living
organisms
is
a
omplex
pro
ess.
T
ak
e
for
instan e
the
ase
of
gene
expression:
most
pro-
teins
are
syn
thesized
within
the
ytoplasm.
Their
subse-
quen
t
a urate
and
swift
deliv
ery
to
target
sites,
requir-
ing
energy
,
is
a
ru ial
step
in
gene
expression.
In
dier-
en
t
situations
the
energy
is
pro
vided
b
y
hap
eron
mole-
ules
(11
),
pH
gradien
t
(12
)
or
mole ular
motors
a ross
mem
branes
(13
).
These
deliv
ery
me
hanisms
an
b
e
fur-
ther
ompli ated
b
y
mem
brane
u tuations
and
some-
times
b
y
gates
that
on
trol
the
a essibilit
y
of
the
p
ores
(14
).
In
view
of
su
h
omplexit
y
,
translo
ation
as
a
bio-
logi al
or
bioph
ysi al
pro
ess
in
living
organisms
has
b
een
s rutinized
in
a
v
ariet
y
of
in
vivo
exp
erimen
tal
situations
[see
e.g.
Ref.
(15
)
and
the
referen es
therein℄.
The
exp
erimen
tal
dev
elopmen
ts
ha
v
e
b
een
follo
w
ed
b
y
a
n
um
b
er
of
mean- eld
t
yp
e
theoreti al
studies
on
p
oly-
mer
translo
ation
(20).
More
re en
tly
,
translo
ation
has
found
itself
at
the
forefron
t
of
single-mole ule-dete tion
exp
erimen
ts
(8
;
17
),
as
new
dev
elopmen
ts
in
design
and
fabri ation
of
nanometer-sized
p
ores
and
et
hing
metho
ds
ma
y
lead
to
heap
er
and
faster
te
hnology
for
the
analysis
and
dete -
tion
of
single
ma romole ules.
The
underlying
prin iple
for
these
exp
erimen
ts
is
that
of
a
Coulter
oun
ter:
mole-
ules
susp
ended
in
an
ele trolyte
solution
pass
through
a
narro
w
p
ore
in
a
mem
brane.
The
ele tri al
imp
edan e
of
the
p
ore
in reases
with
the
en
tran e
of
a
mole ule
as
it
displa es
its
o
wn
v
olume
of
the
ele trolyte
solu-
tion.
By
applying
a
v
oltage
o
v
er
the
p
ore,
the
passing
mole ules
are
dete ted
as
urren
t
dips.
F
or
nanometer-
sized
p
ores
(sligh
tly
larger
than
the
mole ule’s
ross-
se tion)
the
magnitude
and
the
duration
of
these
dips
ha
v
e
pro
v
ed
to
b
e
ee tiv
e
in
determining
the
size
and
length
of
the
mole ules.
In
the
ase
of
DNA
sequen ing
at
n
u leotide
lev
el,
usage
of
protein
p
ores
(mo
di ed α-
haemolysin,
mito
hondrial
ion
hannel,
n
u lei
a id
bind-
ing/
hannel
protein
et .),
and
et
hing
sp
e i
DNA
se-
quen es
inside
the
p
ores
(6;
18
)
ha
v
e
op
ened
up
promis-
ing
new
a
v
en
ues
of
fast,
simple
and
heap
te
hnology
for
single
ma romole ule
dete tion,
analysis
and
hara teri-
zation
[see
Ref.
(19)
for
a
re en
t
dev
elopmen
t℄.
The
sub
je t
of
this
pap
er
is
a
translo
ating
p
olymer
threaded
through
a
narro
w
p
ore
in
an
immobile
mem-
brane,
where
a
b
ead
is
atta
hed
to
one
end
of
the
p
oly-
mer,
and
the
b
ead
is
pulled
b
y
an
opti al
t
w
eezer
with
a
onstan
t
for e.
Su
h
a
setup
an
b
e
used
to
spread
apart
a
partially
unzipp
ed
dsDNA
mole ule
of
whi
h
one
strand
is
threaded
through
the
p
ore
a
pro
ess
that
an
quan
tify
…(Full text truncated)…
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