Anomalous Dynamics of Translocation
Mehran Kardar
MIT
COLLABORATORS
Yacov Kantor, Tel Aviv
Jeffrey Chuang, UCSF
Supported by
OUTLINE
 Polymer dynamics – biological examples and
technological applications
 Translocation as an “escape” problem
 Anomalous dynamics of free translocating polymers
 Translocation under influence of force
 Conclusions
Dynamics of polymers in confined geometries
 Accumulation of exogenous DNA in host cell nucleus:
• viral infection
• gene therapy
• direct DNA vaccinations
 Motion of DNA through a pore can be used to read-off
the sequence
 Motion of polymer in random environments
 DNA gel electrophoresis or reptation
A reconstituted nucleus being dragged after a 3-µm-diameter bead, linked by a molecule of DNA.
The time interval between measurements in the first and second images is 532 sec, between the
second and third, 302 sec. Note the shortening of the maximum distance between bead and nucleus.
Salman, H. et al. (2001) Proc. Natl. Acad. Sci. USA 98, 7247-7252
What is a pore in a membrane?
Alpha-hemolysin secreted by the human pathogen
Staphylococcus aureus is a 33.2kD protein (monomer);
It forms 232.4kD heptameric pore
Song, Hobaugh, Shustak, Cheley, Bayley, Gouaux
Science 274, 1859 (1996)
Measuring translocation of a polymer
Single-stranded DNA molecules (negatively charged)
are electrically driven through a pore
Meller, Nivon, Branton PRL 86, 3435 (2001)
Measuring translocation of a polymer (cont’d)
Voltage driven translocation
Method of measuring translocation times
“in the absence of driving force”
Bates, Burns, Meller
Biophys.J., 84,2366 (2003)
Translocation through a solid membrane
Can we use translocation to read-off a DNA sequence? (“B”-real trace; “C”- “cartoon”)
Computer simulations of complicated problems
Trapped polymer chain in
porous media
Baumgartner, Muthukumar,
JCP 87, 3082 (1987)
Polymer escape from a spherical cavity
N=60, t=50,350,450,1000,4850,25000
Muthukumar, PRL 86, 3188 (2001)
“Translocation” – the simplest problem
Find mean translocation time &
its distribution as a function of
N, forces, properties of the pore
Entropy of “translocating” polymer
N-s
s
Reviews: Eisenriegler, Kremer,
Binder JCP 77, 6296 (1982);
De Bell, Lookman RPM 65, 87
(1993)
Diffusion over a barrier – Kramers’ problem
p
V
k BT
s m in
H.A. Kramers, Physica 7,
284 (1940)
s max
s
Is there a “well” in the entropic problem?
Free energy for N=1000 as a function
of translocation coordinate s
Chuang, Kantor, Kardar,
PRE 65, 011892 (2001)
Sung, Park, PRL 77, 783 (1996);
Muthukumar, JCP 111, 10371 (1999)
Is there a “well” in the entropic problem? (contn’d)
Distribution of escape times with (dashed)
and without (solid) barrier
Chuang, Kantor, Kardar,
PRE 65, 011892 (2001)
Smoluchowski equation vs. simulation
the case of 3D phantom chain
Distribution of translocation coordinate
n for 3 different times (N=100);
continuous lines represent fitted solutions
of Smoluchowski equation (D=0.011)
S.-S. Chern, A.E. Cardenas, R.D. Coalson JCP 115,
7772 (2001)
Translocation vs. free diffusion
Translocation is faster than free diffusion!???
Monte Carlo model
min=2, max=101/2
Carmesin, Kremer
Macromol.21, 2819 (1988)
1D phantom polymer model
max=2, “w=1”
“w=3”
Translocation time of 1D phantom polymer
Distribution of
translocation times of
1D phantom polymers
(normalized to mean)
averaged over 10,000
cases (N=32,64,128) vs.
solution of FP equation
Translocation time of 1D phantom
polymers averaged over 10,000 cases
Chuang, Kantor, Kardar,
PRE 65, 011892 (2001)
Scaled translocation
times of 1D phantom
polymers as a function
pf pore width w
averaged ofver10,000
cases
(N=3,4,6,8,128,181)
Translocation time of 2D polymer
Ratio between
translocation times of
2D phantom and selfavoiding polymers with
and without membrane
Translocation time of 2D phantom &
self-avoiding polymers (averaged over
10,000 cases)
Effective exponents for
2D phantom and selfavoiding polymers with
and without mebrane
Note: in d=2, 1+2n=2.5
Chuang, Kantor, Kardar,
PRE 65, 011892 (2001)
Anomalous diffusion of a momomer
Kremer, Binder, JCP 81, 6381 (84); Grest, Kremer, PR A33, 3628 (86);
Carmesin, Kremer, Macromol. 21, 2819 (88)
Anomalous translocation of a polymer
Time dependence of fluctuations in
translocation coordinate in 2D self-avoiding
polymer. The slope approaches 0.80.
Y. Kantor and M. Kardar, Phys. Rev. E 69, 021806 (2002)
Translocation with a force applied at the end
Distribution of translocation times for
N=128and values of Fa//kT=0, 0.25 and
infinity, for 2D self-avoiding polymer.
250 configurations.
Scaled inverse translocation time in 2D
self-avoiding polymer as a function of
scaled force.
Kantor, Kardar (2002)
“Infinite” force applied at the end
Translocation of 2d self-avoiding polymer
under influence of infinite force at
t=0, 60,000, 120,000 MC time units
Scaled inverse translocation time in 2D
self-avoiding polymer as a function of N
under influence of an infinite force. Slope
of the line is 1.875.
Kantor, Kardar (2002)
“Infinite” force applied to phantom polymer
“Snapshots” of spatial configuration of
translocating 1D phantom polymer (N=128)
under influence of infinite force at several
stages of the process
Translocation time of 1D phantom polymer
as a function of N under influence of an
infinite force (circles) and motion without
membrane (squares). Slopes of the lines
converge to 2.00 [Kantor, Kardar (2002)]
“Infinite” force applied to free phantom polymer
“Snapshots” of spatial configuration of 1D
phantom polymer (N=128) moving
under influence of infinite force. The position of
first monomer was displaced to x=0.
Kantor, Kardar (2002)
Short time scaling
Position of the first monomer of 1D
phantom polymer as a function of scaled
time in the absence of membrane for
N=8,16,32,…,512.
Position of the first monomer of 1D
phantom polymer as a function of scaled
time during the translocation process for
N=8,16,32,…,512.
Kantor, Kardar (2002)
“Infinite” CPD – phantom polymer
Translocation time in 2D phantom polymer
as a function of N under influence of an
infinite chemical potential difference. Slope
of the line is 1.45.
Kantor, Kardar (2002)
Translocation with chemical potential difference
Distribution of translocation times for
N=64and values of U/kT=0, 0.25, 0.75 and
2, for 2D self-avoiding polymer. 250
configurations.
Scaled inverse translocation time in 2D
self-avoiding polymer as a function of
scaled U .
Kantor, Kardar (2002)
“Infinite” chemical potential difference
Translocation of 2D self-avoiding polymer
under influence of infinite chemical potential
difference at t= 10,000, 25,000 MC time units.
Kantor, Kardar (2002)
Translocation time in 2D self-avoiding
polymer as a function of N under influence
of an infinite chemical potential difference.
Slope of the line is 1.53.
Conclusions/Perspectives
 Normal diffusion “explains” only Gaussian polymers
and gives “wrong” prefactors
Anomalous dynamics provides a consistent picture of
translocation
 There is no detailed theory that will enable calculation
of coefficients
 Crossovers persist even for N~1000.
 We presented “bounds” for diffusion under influence
of large forces. Are they the “real answer”?