Anomalous Dynamics of Translocation

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Anomalous Dynamics of Translocation
Mehran Kardar MIT
COLLABORATORS Yacov Kantor, Tel Aviv Jeffrey
Chuang, UCSF
Supported by
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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

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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

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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
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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)
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Measuring translocation of a polymer
Meller, Nivon, Branton PRL 86, 3435 (2001)
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Measuring translocation of a polymer (contd)
Bates, Burns, Meller Biophys.J., 84,2366 (2003)
Method of measuring translocation times in the
absence of driving force
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Translocation through a solid membrane
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Computer simulations of complicated problems
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Translocation the simplest problem
Find mean translocation time its distribution
as a function of N, forces, properties of the pore
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Entropy of translocating polymer
Reviews Eisenriegler, Kremer, Binder JCP 77,
6296 (1982) De Bell, Lookman RPM 65, 87 (1993)
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Diffusion over a barrier Kramers problem
s
H.A. Kramers, Physica 7, 284 (1940)
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Is there a well in the entropic problem?
Free energy for N1000 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)
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Is there a well in the entropic problem?
(contnd)
Distribution of escape times with (dashed) and
without (solid) barrier
Chuang, Kantor, Kardar, PRE 65, 011892 (2001)
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Smoluchowski equation vs. simulationthe case of
3D phantom chain
Distribution of translocation coordinate n for 3
different times (N100) continuous lines
represent fitted solutions of Smoluchowski
equation (D0.011) S.-S. Chern, A.E. Cardenas,
R.D. Coalson JCP 115, 7772 (2001)
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Translocation vs. free diffusion
Translocation is faster than free diffusion!???
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Monte Carlo model
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1D phantom polymer model
max2, w1
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Translocation time of 1D phantom polymer
Translocation time of 1D phantom polymers
averaged over 10,000 cases
Chuang, Kantor, Kardar, PRE 65, 011892 (2001)
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Translocation time of 2D polymer
Ratio between translocation times of 2D phantom
and self-avoiding polymers with and without
membrane
Effective exponents for 2D phantom and
self-avoiding polymers with and without
mebrane Note in d2, 12n2.5
Translocation time of 2D phantom self-avoiding
polymers (averaged over 10,000 cases)
Chuang, Kantor, Kardar, PRE 65, 011892 (2001)
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Anomalous diffusion of a momomer
Kremer, Binder, JCP 81, 6381 (84) Grest,
Kremer, PR A33, 3628 (86) Carmesin, Kremer,
Macromol. 21, 2819 (88)
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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)
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Translocation with a force applied at the end
Distribution of translocation times for N128and
values of Fa//kT0, 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)
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Infinite force applied at the end
Translocation of 2d self-avoiding polymer under
influence of infinite force at t0, 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)
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Infinite force applied to phantom polymer
Snapshots of spatial configuration
of translocating 1D phantom polymer (N128) 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)
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Infinite force applied to free phantom polymer
Snapshots of spatial configuration of 1D
phantom polymer (N128) moving under influence of
infinite force. The position of first monomer was
displaced to x0.
Kantor, Kardar (2002)
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Short time scaling
Position of the first monomer of 1D phantom
polymer as a function of scaled time during the
translocation process for N8,16,32,,512.
Position of the first monomer of 1D phantom
polymer as a function of scaled time in the
absence of membrane for N8,16,32,,512.
Kantor, Kardar (2002)
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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)
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Translocation with chemical potential difference
Distribution of translocation times for N64and
values of U/kT0, 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)
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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)
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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 N1000.
• We presented bounds for diffusion under
  influence of large forces. Are they the real
  answer?