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Transcript of Anomalous Dynamics of Translocation COLLABORATORS Yacov Kantor, Tel Aviv Jeffrey Chuang, UCSF Mehran...
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Anomalous Dynamics of Translocation
COLLABORATORS
Yacov Kantor, Tel Aviv
Jeffrey Chuang, UCSF
Mehran Kardar
MIT
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?
Song, Hobaugh, Shustak, Cheley, Bayley, Gouaux Science 274, 1859 (1996)
Alpha-hemolysin secreted by the human pathogen Staphylococcus aureus is a 33.2kD protein (monomer);
It forms 232.4kD heptameric pore
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Measuring translocation of a polymer
Meller, Nivon, Branton PRL 86, 3435 (2001)
Single-stranded DNA molecules (negatively charged) are electrically driven through a pore
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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)
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Translocation through a solid membrane
Can we use translocation to read-off a DNA sequence? (“B”-real trace; “C”- “cartoon”)
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Computer simulations of complicated problems
Trapped polymer chain inporous mediaBaumgartner, Muthukumar,JCP 87, 3082 (1987)
Muthukumar, PRL 86, 3188 (2001)
Polymer escape from a spherical cavityN=60, t=50,350,450,1000,4850,25000
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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)
s
N-s
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Diffusion over a barrier – Kramers’ problem
H.A. Kramers, Physica 7, 284 (1940)
p
s
BVkT
mins maxs
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Is there a “well” in the entropic problem?
Chuang, Kantor, Kardar, PRE 65, 011892 (2001)
Free energy for N=1000 as a functionof translocation coordinate s
Sung, Park, PRL 77, 783 (1996); Muthukumar, JCP 111, 10371 (1999)
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Is there a “well” in the entropic problem? (contn’d)
Chuang, Kantor, Kardar, PRE 65, 011892 (2001)
Distribution of escape times with (dashed)and without (solid) barrier
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Smoluchowski equation vs. simulationthe case of 3D phantom chain
Distribution of translocation coordinaten for 3 different times (N=100) ;
continuous lines represent fitted solutionsof Smoluchowski equation (D=0.011)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
min=2, max=101/2
Carmesin, KremerMacromol.21, 2819 (1988)
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1D phantom polymer model
max=2, “w=1”
“w=3”
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Translocation time of 1D phantom polymer
Translocation time of 1D phantompolymers averaged over 10,000 cases
Chuang, Kantor, Kardar, PRE 65, 011892 (2001)
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
Scaled translocation times of 1D phantom polymers as a function pf pore width waveraged ofver10,000 cases (N=3,4,6,8,128,181)
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Translocation time of 2D polymer
Translocation time of 2D phantom& self-avoiding polymers (averaged over
10,000 cases(
Chuang, Kantor, Kardar, PRE 65, 011892 (2001)
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 d=2, 1+22.5
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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
Scaled inverse translocation time in 2D self-avoiding polymer as a function of scaled force.
Distribution of translocation times for N=128and values of Fa//kT=0, 0.25 and infinity, for 2D self-avoiding polymer. 250 configurations.
Kantor, Kardar (2002)
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“Infinite” force applied at the end
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.
Translocation of 2d self-avoiding polymerunder influence of infinite force att=0, 60,000, 120,000 MC time units
Kantor, Kardar (2002)
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“Infinite” force applied to phantom polymer
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)]
“Snapshots” of spatial configuration oftranslocating 1D phantom polymer (N=128)under influence of infinite force at several stages of the process
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“Infinite” force applied to free phantom polymer
“Snapshots” of spatial configuration of 1D phantom polymer (N=128) movingunder influence of infinite force. The position of first monomer was displaced to x=0.
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 N=8,16,32,…,512.
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.
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
Scaled inverse translocation time in 2D self-avoiding polymer as a function of scaled U.
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.
Kantor, Kardar (2002)
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“Infinite” chemical potential difference
Translocation of 2D self-avoiding polymerunder influence of infinite chemical potential difference at t= 10,000, 25,000 MC time units.
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.
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 N~1000.
We presented “bounds” for diffusion under influence of large forces. Are they the “real answer”?