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Anomalous Driven Motion of Biopolymers

Takahiro SakaueDepartment of Physics, Kyushu University

Stretching, Compression & Rotational Dynamics

KIAS Workshop, 5/Sep./2015

supported by:

Chromosome dynamics

M-phase (yeast cell)

Courtesy of T. Sugawara & H. Nishimori (Hiroshima University)

Green: cell nucleus

White:Spindle pole body (SPB)Telomere

Tagged monomer diffusion: A paradigm of anomalous diffusion

• Time-dependent friction (diffusion coeff.)• Memory effect • Visco-elastic response

)()( tt anr

f

aNReq

Tension propagation

Rouse model

)(/1~)(~~)( 2/1

tDtttn

4/1~)(~)( tttDtr

eqRrNn ~)(~)(

sub-diffusion

2~ N

Terminal time

Strong force (non-equilibrium dynamics)

)(tr

eq

B

RTkf

f

aNReq

Drift (+ small fluctuation)

• Large deformation • Non-linear response• Tension propagation

)()()()( tsfstdstvt

)()()()( ttfsvstdst

No-linear memory kernel

1, Stretching dynamics

2, Compression dynamics

3, Rotational dynamics

collaboration with T. Saito(Kyoto, Japan)

collaboration with W. Reisner(McGill, Canada)

collaboration with E. Carlon(Leuven, Belgium)

Driven dynamics of biopolymerseq

B

RTkf

V

Stretching Dynamics

Equilibrium (at rest) Steady state

Req = a N R > Req

# Start pulling at t=0

?

Single molecule experiment, manipulation, chromosome segregation…

Transient

t

(as a paradigm of driven dynamics)

How to describe ?1, Rouse model

Analytical solutionLinear (no EV, no HI)

3, Continuum description (segment line density)Local force balance (elastic and viscous forces)Mass conservation

2, Two-phase formalismResponding and non-responding domainsSteady-state approximation for the responding domain

T. Sakaue, T. Saito and H. Wada, Phys. Rev. E 86, 011804 (2012).“Dragging a polymer in a viscous fluid: Steady state and transient”T. Saito and T. Sakaue, Phys. Rev. E 92, 012601 (2015).“Driven anomalous diffusion: an example from polymer stretching”

4, Non-linear Rouse modelLocal force balance (elastic and viscous forces) )(

2

0 tnx

nx

nD

tx n

pnn

)()(2

2

tftnxk

tx

nnn

viscous elastic

)()()(0 x

xx

xD

tx p

Two-phase picture

# Moving, and quiescent domains

# Steady-state approximationfor the moving domain

# Gross variables: size of the moving domain front of the tension propagation

Highly non-linear, non-equilibrium process in multi-dimensional space Coarse graining

fLfVt

),()(1

extension)(~)()( tLtRtL

m(m(t)

r(tL(t)

fdt

tdLt ~)()(

Steady-state dynamical Eq. of state

Force-extension (static stretching)

/1~~

Nf

gNL

P. Pincus (1976)

: critical exponent(Flory)

Alignment of blobs

Eq. of State

agfTkB /L

Steady StateDynamical Eq. of State

: force-extension

: friction-law

F. Brochard-Wyart (1993)

)(0

xTk

dxdV B

Local force balance Boundary condition (f)

)(LTkf B

)(xag

L

),(~ fNLL

),(~ fNVV

),(~ fLVV

Two-phase picture

# Moving, and quiescent domains

# Steady-state approximationfor the moving domain

# Gross variables: size of the moving domain front of the tension propagation

Highly non-linear, non-equilibrium process in multi-dimensional space Coarse graining

2/1)/1(2/~)( tftm z

2)/2(~ Nf zf

ffLVt

),()(1

extension

2/11)2/(~)( tftL z

)(~)()( tLtRtL

m(m(t)

r(tL(t)

Anomalous drift !

fdt

tdLt ~)()(

1, Rouse modelAnalytical solutionLinear (no EV, no HI)

3, Continuum description (segment line density)Local force balance (elastic and viscous forces)Mass conservation

2, Two-phase formalismResponding and non-responding domainsSteady-state approximation for the responding domain

T. Sakaue, T. Saito and H. Wada, Phys. Rev. E 86, 011804 (2012).“Dragging a polymer in a viscous fluid: Steady state and transient”T. Saito and T. Sakaue, Phys. Rev. E 92, 012601 (2015).“Driven anomalous diffusion: an example from polymer stretching”

4, Non-linear Rouse modelLocal force balance (elastic and viscous forces) )(

2

0 tnx

nx

nD

tx n

pnn

)()(2

2

tftnxk

tx

nnn

viscous elastic

How to describe ?

)()()(0 x

xx

xD

tx p

Continuum description : porous medium equation

# segment line density

# Local force balance

# segment flux

# continuity equation

1

)2(zp

2/11)2/(~)( tftL z

self-similar ansatz

),(),(),( 3

txTk

dxdtxV

atx B

z

/)1(~),( gtx

),(),(),( txVtxtxJ

),(),(),(0 tx

xtx

xD

ttx p

V(x,t)

L(t)

1, Rouse modelAnalytical solutionLinear (no EV, no HI)

3, Continuum description (segment line density)Local force balance (elastic and viscous forces)Mass conservation

2, Two-phase formalismResponding and non-responding domainsSteady-state approximation for the responding domain

T. Sakaue, T. Saito and H. Wada, Phys. Rev. E 86, 011804 (2012).“Dragging a polymer in a viscous fluid: Steady state and transient”T. Saito and T. Sakaue, Phys. Rev. E 92, 012601 (2015).“Driven anomalous diffusion: an example from polymer stretching”

4, Non-linear Rouse modelLocal force balance (elastic and viscous forces) )(

2

0 tnx

nx

nD

tx n

pnn

)()(2

2

tftnxk

tx

nnn

viscous elastic

How to describe ?

)()()(0 x

xx

xD

tx p

mtmX

tmTk

mttmX

atma B

z ),(),(

),(),(2

2

Nonlinear Rouse model: P-Laplacian eq.

# X(m,t) x(n,t) : fine-graining

# force balance (coarse-grained)

1

)2(zp

2/11)2/(~)( tftL z

self-similar ansatz

(m,t)

)()(2

0 tftnx

nx

nD

tx

nnn

pnn

, z=4 Rouse model

L(t)

),( tmXChain of blobs

),( tnx

# Anomalous drift, non-linear response

# Fluctuation of the driven monomer

nn

pnn ft

nx

nx

nD

tx

)(2

0

nf

nnfnf ft

nxk

tx

)()(

2

2)()( )()(2)()( )()()( stmnTkst B

ffm

fn

)/1(2)(

/)12()(

~~

zf

f

ffkk

Gaussian approx.

)()()( txtxtx

Fluctuations

)(2)(2 txf

Tktx B

?)(/)(2 txTktxf B

T. Saito & T. Sakaue, Phys. Rev. E, 92, 012601 (2015)Driven Anomalous Diffusion: An example from polymer stretching

2/11)2/(~)( tftx z

1, Stretching dynamics

2, Compression dynamics

3, Rotational dynamics

collaboration with T. Saito(Kyoto, Japan)

collaboration with W. Reisner(McGill, Canada)

collaboration with E. Carlon(Leuven, Belgium)

Driven dynamics of biopolymerseq

B

RTkf

V

Force-compression relation of confined DNA

optically trapped bead

slit barrier

nanochanel

Optical Nonofluidic Piston

Vslit barrier

A. Khorshid, P. Zimy, D. Tetreault-La Roche, G. Massaarelli, T. Sakaue & W. ReisnerPhys. Rev. Lett., 113, 268104 (2014)

Statics Dynamics-- no-wall pushing

14.5 μmtrapped bead

Extended T4 DNA

5 μm

1 μm/s

50s

(kymograph)

Characterization of the steady state

Contour length ~ 60 m

Channel size D= 300 nm

Steady state

Control parameter: V

Dynamic Compression: Raw time traces

Continuum description : porous medium equation

# segment line density

# Local force balance

# segment flux

# continuity equation

1

)2(zp

2/11)2/(~)( tftr z

self-similar ansatz

),(),(),( 3

txTk

dxdtxV

atx B

z

/)1(~),( gtx

),(),(),( txVtxtxJ

),(),(),(0 tx

xtx

xD

ttx p

recall

V(x,t)

L(t)

Continuum description : porous medium equation

0)( DD

Vtxtxx

txDxt

tx ),(),()),((),(

V

J

Steady-state: j=0

Diffusive flux Convective flux

Cooperative diffusion

Density profileDynamical Eq. of state (f-v-r)

0.1 m/s

0.25 m/s

1 m/s

10 m/s

10 m

2 s

*V 1.4 m/s

)(~)( DxDxxs

1

Ramp profile

V > V*entire chain deformed

Vc< V < V*initial flat profilewith eq

A. Khorshid, S. Amin, Z. Zhang, T. Sakaue & W. Reisner, submitted

Compression Decompression

1, Stretching dynamics

2, Compression dynamics

3, Rotational dynamics

collaboration with T. Saito(Kyoto, Japan)

collaboration with W. Reisner(McGill, Canada)

collaboration with E. Carlon(Leuven, Belgium)

Driven dynamics of biopolymerseq

B

RTkf

V

Belotserkovskii, PRE (2014)

M. Laleman, M. Baiesi, B.P. Belotserkovskii, T. Sakaue, J-C. Walter, E. Carlon, submitted

“Torque-induced rotational dynamics in polymers: Torsional blobs and thinning”

Rotational analogue of the stretching dynamics Fundamental polymer dynamics problem Relevant to RNA transcription process

V(x,t)

L(t)

)/)(ln()(

asTksM B

# Torque- torsional blob relation

)()(

sTksf B

cf. force-tensile blob relation

)(ln

),(N

fNP

Winding angle distribution

Steady state (driven by torque)

Steady state profile Torque-angular velocity relation (dynamical eq. of state) Steady elongation

2/1)(~)( ss

Summary: Driven dynamics of biopolymers

Acknowledgements:T. Saito (Kyoto), W. Reisner (McGill), E. Carlon (Leuven)T. Sugawara, H. Nishimori (Hiroshima)

Translocation

Palyulin et. al, SoftMatter (2014)

Belotserkovskii, PRE (2014)

Transcription

(top) (side)Polymerized membrane

Mizuochi, Nakanishi, Sakaue, EPL (2014)

Confined DNA (Nanodozer)

Khorshid, Sakaue, Reisner et. al., PRL (2014)