Anomalous Dynamics of Translocation

COLLABORATORS

Yacov Kantor, Tel Aviv

Jeffrey Chuang, UCSF

Mehran Kardar

MIT

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?

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

Measuring translocation of a polymer

Meller, Nivon, Branton PRL 86, 3435 (2001)

Single-stranded DNA molecules (negatively charged) are electrically driven through a pore

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 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

“Translocation” – the simplest problem

Find mean translocation time &

its distribution as a function of

N, forces, properties of the pore

Entropy of “translocating” polymer

Reviews: Eisenriegler, Kremer, Binder JCP 77, 6296 (1982); De Bell, Lookman RPM 65, 87 (1993)

s

N-s

Diffusion over a barrier – Kramers’ problem

H.A. Kramers, Physica 7, 284 (1940)

p

s

BVkT

mins maxs

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)

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

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(

Translocation vs. free diffusion

Translocation is faster than free diffusion???!

Monte Carlo model

min=2, max=101/2

Carmesin, KremerMacromol.21, 2819 (1988)

1D phantom polymer model

max=2, “w=1”

“w=3”

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)

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

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

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)

“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)

“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

“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)

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)

“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

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)

“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)

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”?