polymer translocation through nanoporesmetz/tbp4.pdfa meller, l nivon & d brenton, prl 86 (2001)...

Polymer translocation through nanopores

– Typeset by FoilTEX – 1

Theoretical and Computational Biophysics Group, University of Illinois, Urbana-Champaign 4

A Meller, L Nivon & D Brenton, PRL 86 (2001) 5

Cees Dekker group, TU Delft 6

Polymer model for translocation

1. Chain is polymer with N monomers

2. Chain is already threaded into pore

3. Reflecting B.C. @ s = 0

4. Absorbing B.C. @ s = N

5. Neglect chain-pore interactions

6. ∃ reaction coordinate:

s = #

{translocated

monomers

}

J Chuang, Y Kantor & M Kardar, PRE (2001) 7

Entropic barrier

# d.o.f. of a chain with N monomers:

ω ' µNNγG−1

s-dependence of free energy

F (s) ' (γG − 1)kBT log(

[N − s]s)

driving force needed for efficient translocation


Driving force

Driving force creates chemical potential difference ∆µ per monomer drift

1. Trans-membrane potential

2. Binding proteins

3. Cis confinement (e.g., virus)

4. Active pulling

Sung & Park, PRL 77 (1996) 9

Chaperone-driven translocation

T Ambjornsson & RM, Physical Biology (2004); T Ambjornsson, MA Lomholt & RM, J Phys Cond Mat (2005) 10


Driving: neglect polymeric d.o.f.



Forward translocation rate:

t+(m,n) = k




t+(m,n) = k

Backward translocation rate:

t−

(m,n) = k × Pr

trans binding site

site closest to

pore is vacant




t+(m,n) = k


t−

(m,n) = k × Pr

trans binding site

site closest to

pore is vacant

Chaperone unbinding rate:

r−

(m,n) = nq




t+(m,n) = k


t−

(m,n) = k × Pr

trans binding site

site closest to

pore is vacant

Chaperone unbinding rate:

r−

(m,n) = nq

Chaperone binding rate:

r+(m,n) = c0K

eq ×N

ways to add addtl

chaperone if

n bound already



1. Slow binding dynamics: purely diffusive motion τT ' N2

2. Slow unbinding dynamics: ratcheted motion τT ' N3. Fast binding/unbinding: adiabatic elimination of chaperone dynamics


Dependence of mean translocation velocity on binding strength:

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

κB=c

BK

B

eq

<v>

/v0

λB=1

λB=2

λB=12



Mean translocation time as function of chain length:

10

100

1 10

T(M

) [

k-1

]

M

λ=1, γ=1.3, κ=0.8λ=1, γ=0.4, κ=1.3λ=1, γ=0.1, κ=0.1λ=5, γ=1.3, κ=0.8λ=5, γ=0.4, κ=1.3λ=5, γ=0.1, κ=0.1

Ratchet motionSlow binding

Adiabatic, λ=1Adiabatic, λ=5

(M+1) Λmax/(Λmax-1); λ=1(M+1) Λmax/(Λmax-1); λ=5


Polymer translocation: many groups, many opinions . . .

15

Brownian approach to translocation

Translocation dynamics sufficiently slow diffusion in potential with PDF P (s, t)



Continuity equation:

∂

∂tP (s, t) = −

∂

∂sj(s, t)




∂

∂tP (s, t) = −

∂

∂sj(s, t)

Probability current:

j(s, t) = −D(∂

∂sP (s, t) +

P (s, t)

kBT

∂

∂sF (s)

)∴ D =

µ

kBT




∂

∂tP (s, t) = −

∂

∂sj(s, t)


j(s, t) = −D(∂

∂sP (s, t) +

P (s, t)

kBT

∂

∂sF (s)

)∴ D =

µ

kBT

Resulting Fokker-Planck equation (rescaled a la s→ sN & t→ tD/N2):

∂

∂tP (s, t) =

∂2

∂s2P (s, t) + (γG − 1)

∂

∂s

1− 2s

(1− s)sP (s, t)




∂

∂tP (s, t) = −

∂

∂sj(s, t)


j(s, t) = −D(∂

∂sP (s, t) +

P (s, t)

kBT

∂

∂sF (s)

)∴ D =

µ

kBT


∂

∂tP (s, t) =

∂2

∂s2P (s, t) + (γG − 1)

∂

∂s

1− 2s

(1− s)sP (s, t)

Resulting scaling of translocation time τT ' N2/D




∂

∂tP (s, t) = −

∂

∂sj(s, t)


j(s, t) = −D(∂

∂sP (s, t) +

P (s, t)

kBT

∂

∂sF (s)

)∴ D =

µ

kBT


∂

∂tP (s, t) =

∂2

∂s2P (s, t) + (γG − 1)

∂

∂s

1− 2s

(1− s)sP (s, t)

Resulting scaling of translocation time τT ' N2/D

NB: For s-independent drift: τT ' N/V where V the stationary velocity

Sung & Park, PRL 77 (1996), Muthukumar J Chem Phys 111 (1999) 16

Brownian approach to translocation dynamics?

Kramers escape theory across free energy barrier F (s) delivers diffusive scaling:

τT 'N2

D



τT 'N2

D

First remark: τT for crossing of F (s) only 20% longer than for pure diffusion



τT 'N2

D


Second remark: Equil. time of free polymer estimated by diffusion over Rg:

τR 'R2g

Dc.o.m.

'R2gN

Dmon

'N1+2ν

Dmon



τT 'N2

D



τR 'R2g

Dc.o.m.

'R2gN

Dmon

'N1+2ν

Dmon

Phantom chain (ν = 1/2): τR ' N2/Dmon marginal



τT 'N2

D



τR 'R2g

Dc.o.m.

'R2gN

Dmon

'N1+2ν

Dmon

Phantom chain (ν = 1/2): τR ' N2/Dmon marginal

SA chain (ν(2D) = 3/4 and ν(3D) = 0.59): τR longer than translocation time τT fl


Non-Brownian translocation dynamics

Extended simulations in 1D: τT ' N2

In 2D: τT ' N2.5

Kantor & Kardar suggest τT ' N1+2ν like chain relax. time τR, but with large prefactor



In 2D: τT ' N2.5


Scaling argument for anomalous dynamics:

1. Assume scaling Rg ' Nν and chain relaxation time τR ' Rzg /w dynamic expt. z



In 2D: τT ' N2.5




2. Translocation progress has variance 〈∆s2(t)〉 ' t2ζ



In 2D: τT ' N2.5





3. @ τT ' Nνz we should reach the value N2 ' 〈∆s2(τT )〉 ' τ2ζT ' N

2ζνz



In 2D: τT ' N2.5






2ζνz

4. ζ = 1/(νz)



In 2D: τT ' N2.5






2ζνz

4. ζ = 1/(νz)

5. For diffusive translocation z = (1 + 2ν)/ν and thus ζ = 1/(1 + 2ν)



In 2D: τT ' N2.5






2ζνz

4. ζ = 1/(νz)

5. For diffusive translocation z = (1 + 2ν)/ν and thus ζ = 1/(1 + 2ν)

6. ζ(2D) = 0.4 and ζ(3D) ≈ 0.46, i.e., subdiffusion


Fractional approach to translocation dynamics

Pausing events of duration t distributed as

ψ(t) 'τα

t1+α∴ 〈t〉 =

∫ ∞0

tψ(t)dt→∞



ψ(t) 'τα

t1+α∴ 〈t〉 =

∫ ∞0

tψ(t)dt→∞

Associated dynamic equation: Fractional Fokker-Planck equation

∂

∂tP (s, t) = 0D

1−αt

(−vα

∂

∂s+Kα

∂2

∂s2

)P (s, t); 0D

1−αt f(t) =

∂

∂t

1

Γ(α)

∫ t

0

f(t′)dt′

(t− t′)1−α



ψ(t) 'τα

t1+α∴ 〈t〉 =

∫ ∞0

tψ(t)dt→∞


∂

∂tP (s, t) = 0D

1−αt

(−vα

∂

∂s+Kα

∂2

∂s2

)P (s, t); 0D

1−αt f(t) =

∂

∂t

1

Γ(α)

∫ t

0

f(t′)dt′

(t− t′)1−α

Moments:

〈s(t)〉 ' vαtα!'√〈∆s2(t)〉 vα = 0 : 〈s2

(t)〉 ' Kαtα



ψ(t) 'τα

t1+α∴ 〈t〉 =

∫ ∞0

tψ(t)dt→∞


∂

∂tP (s, t) = 0D

1−αt

(−vα

∂

∂s+Kα

∂2

∂s2

)P (s, t); 0D

1−αt f(t) =

∂

∂t

1

Γ(α)

∫ t

0

f(t′)dt′

(t− t′)1−α

Moments:

〈s(t)〉 ' vαtα!'√〈∆s2(t)〉 vα = 0 : 〈s2

(t)〉 ' Kαtα

Mode relaxation:

Tn(t) = tnEα(− λn,αtα

)∼ tn

{1− tα/Γ(1 + α)

t−α/Γ(1− α)

RM & J Klafter, Biophys J (2003); Review: RM & J Klafter, Phys Rep (2000) & J Phys A (2004) 19

Agreement with simulations data: drift-free case

10−2

100

t/N2ν+2−γ

1 [MCS]

10−4

Q(t)

N=16

N=32

N=64

N=128

N=256

t−1.8

0 25000 50000

t [MCS]

0

2e−05

4e−05

6e−05

Q(t)

(a)

101

102

103

t/N2ν+2−γ

1

10−3

10−2

Q(t)

N=64

N=128

τ−1.8

101

102

N

104

106

108

<τ(N)>

N2.52 +/− 0.04

(b)

Th

eory

pred

icti

on

MC simulations: JLA Dubbeldam, A Milchev, VG Rostiashvili & TA Vilgis, PRE R.C. (2007) 20

Agreement with simulations data: constant drift case

0 400 800 1200 1600

τ

0

0.001

0.002

0.003

0.004

Q(τ

)

f = 0.5

f = 0.8

f = 1.010

−1

100

101

102

103

104

t.f

10−1

100

101

102

103

<s(t

)> f = 0.5

f = 0.8

f = 1.0

t2/3

t0.8

f = 0.5 (MC)

f = 0.8 (MC)

f = 1.0 (MC)

(a)

10−1

100

101

102

103

104

t.f

10−1

100

101

102

103

104

105

<s2(t)>

f = 0.5

f = 0.8

f = 1.0

t4/3

t1.6

f = 0.5 (MC)

f = 0.8 (MC)

f = 1.0 (MC)

(b)

First passage time distribution:

MC simulations: JLA Dubbeldam, A Milchev, VG Rostiashvili & TA Vilgis, PRE R.C. (2007) 21

Scaling laws for driven translocation

REM: Rouse relaxation time τR ' N1+2ν



¿How does the translocation time scale with N : τT ' Nα ?




Kramers/Brownian approach: τT ' N2 unbiased, τT ' N with drift





Estimate for lower bound under driving: τT ' N1+ν[Kantor & Kardar, PRE (2004)]






Excellent agreement with simulations [Dubbeldam et al, EPL (2007); Gauthier & Slater, JCP (2007)]







From theory with memory effects: τT ' N (1+2ν)/(1+ν)[Vocks et al, J Phys Cond Mat (2008)]








Also corroborated by simulations [Vocks et al, J Phys Cond Mat (2008)]









¿ Can we reconcile these conflicting statements ?









¿ Can we reconcile these conflicting statements ?

Yes, we can . . .

K Luo, T Ala-Nissila, S-C Ying & RM, EPL (2009) 22

Slow versus fast translocation

10 100

101

102

103

In 3D interpolation between α = 1 + ν ≈ 1.59 and α′ = (1 + 2ν)/(1 + ν) ≈ 1.37


Fast translocation: nonequilibrium effects

–20 –15 –10 –5 0 5 10 15 20

–10

0

10



10 100

1

10



F ξ F/ξ α (τ ∼ Nα) δ (v ∼ Nδ) β (〈s(t)〉 ∼ tβ) αβ

Fast 10.0 0.7 14.28 1.37± 0.02 −0.79± 0.01 0.84± 0.01 1.15

5.0 0.7 7.14 1.37± 0.05 −0.79± 0.02 0.85± 0.01 1.16

Slow 5.0 3.0 1.67 1.52± 0.01 −0.94± 0.01 0.71± 0.01 1.08

2.5 3.0 0.83 1.51± 0.02 −0.95± 0.02 0.69± 0.01 1.04

0.5 0.7 0.71 1.58± 0.03 −1.01± 0.02 0.64± 0.01 1.01

10 100

10–2

10–1

100

1 10

102

103


Variations on a theme

The chain sucker

Into confinement

Out of confinement

K Luo, RM, T Ala-Nissila & S-C Ying, PRE (2009); K Luo & RM, JCP (2010), PRE (2010) 27

Translocation into a laterally unbounded confinement

R

Chain /w N monomers of diam. σ


R


σ � R� Rg: 2D SAW of nb blobs


R



g =

{#monom.

per blob

}=

(R

σ

)1/ν3D


R



g =

{#monom.

per blob

}=

(R

σ

)1/ν3D

nb ={

#blobs}

=N

g


R



g =

{#monom.

per blob

}=

(R

σ

)1/ν3D

nb ={

#blobs}

=N

g

Longitudinal size of polymer:

R‖ ' nν2Db R ' Nν2Dσ

(σ

R

)ν2D/ν3D−1

' N3/4σ

(σ

R

)0.28


R



g =

{#monom.

per blob

}=

(R

σ

)1/ν3D

nb ={

#blobs}

=N

g

Longitudinal size of polymer:

R‖ ' nν2Db R ' Nν2Dσ

(σ

R

)ν2D/ν3D−1

' N3/4σ

(σ

R

)0.28

Longit. relax. time: polymer moves by its own size (diffusivity D ' 1/N):

τ‖ 'R2‖

D' N1+2ν2DR

2(1−ν2D/ν3D) ' N2.50R−0.55

K Luo, T Ala-Nissila, S-C Ying & RM, PRE (2009) 28


τ‖ 'R2‖

D' N1+2ν2DR

2(1−ν2D/ν3D) ' N2.50R−0.55

10 100

101

102

103

=⇒

⇐= Fast/unconfined



1 10 100 1000

1

10

100


polymer translocation through nanoporesmetz/tbp4.pdfa meller, l nivon & d brenton, prl 86 (2001)...

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