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)
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