Dynamics of Confined Polymer in Flow

陳彥龍 Yeng-Long Chen

(yenglong@phys.sinica.edu.tw)

Institute of Physics and Research Center for Applied Science

Academia Sinica

To understand and manipulate the structure and

dynamics of biopolymers with statistical physics

Fish schooling Blood flow

Micro- and Nano-scale Building Blocks

Nuclei are stained blue with DAPIActin filaments are labeled red with phalloidin Microtubules are marked green by an antibody

Endothelial Cell

F-Actin

DNA

Diameter: 7nm Persistence length : ~10 m

3.4 nm

Persistence length : ~ 50 nm

Rg

p

Organ Printing

Mironov et al. (2003)

Boland et al. (2003)

Forgacs et al. (2000)

Organ printing and cell assembly

• Cells deposited into gel matrix fuse when they are in proximity

of each other

• Induce sufficient vascularization

• Embryonic tissues are viscoelastic

• Smallest features ~ O(mm)

From Pancake to Tiramisu

Edible Paper

Moto restaurant

Chicago

Inkjet printer used as food processor

Food emulsions printed onto edible paper

Edible Menus

Not too far into the future :

“We had to go out for dinner because the printer ran out of ink!”

• High throughput

• Low material cost

• High degree of parallelization

Advantages of microfluidic chips

Efficient device depends on controlled transport

Channel dimension ~ 10nm - 100 m

Fluid plug reactor from Cheng group, RCAS

Confining Macromolecules

Theory and simulations help us understand dynamics of macromolecules

Multi-Scale Simulations of DNA

10 nm

2 nm

3.4 nm

1 nm

Atomistic

C-C bond length

100 nm

Persistence length ≈ 50nm

Nanochannels

Essential physics :

DNA flexibility

Solvent-DNA interaction

Entropic confinement

1F

2F

1F

2F

1F

2F

1F

2F

1 m 10 m

Radius of gyration

Coarse grainingMicrochannels

Multi-component systems : multiple scales for different components

Molecular Dynamics

- Model atoms and molecules using

Newton’s law of motion

Monte Carlo

- Statistically samples energy and configuration

space of systems

Cellular Automata

- Complex pattern formation from simple computer instructions

Large particle in a granular flow

Polymer configuration sampling

Sierpinksi gasket

-If alive, dead in next step

-If only 1 living neighbor, alive

Our Methods

Coarse-grained DNA Dynamics

DNA as Worm-like ChainL = 22 m

Ns = 10 springsNk,s = 19.8 Kuhns/spring

Marko and Siggia (1994)

2a

f S(t)

f ev(t) f W(t) -DNA 48.5 kbps

DNA is a worm-like chain

Model parameters are matched to TOTO-1 stained -DNA

Parameters matched in bulk are valid in confinement !

Expt

Chen et al., Macromolecules (2005)

Brownian Dynamics

))(( xUUf fpf

dtm

tUdttRd

)()(

Explicit inclusion of solvent molecules on the micron scale is extremely computational expensive !!

solvent = lattice fluid (LBE)

How to treat solvent molecules ??

dtm

dttfdttUd

)()(

: particle friction coef.

v1

v2

v3

Brownian motion through fluctuation-dissipation

flucfricwallWLCev ffffff

)'()'(2)','(),( rrttTktrftrf Bflucfluc

Ladd, J. Fluid Mech (1994)

Ahlrichs & Dünweg, J. Chem. Phys. (1999)

Hydrodynamic Interactions (HI)

Free space Wall correction

Particle motion perturbs and contributes to the overall velocity field

Stokes Flow

000W0s00 ),(),(v)(v),,(v frrrrrrfrr Ω

W

W2

v0

v0

ηp

Solved w/

Finite Element Method

For Different Channels

Force

z

velocityfluid.max

velocityDNAavg.fR

)2//(max Hv

Sugarman & Prud’homme (1988)

25 m

Detection points at 25 cm and 200 cm

detector

-DNA in microcapillary flow

Parabolic Flow

DNA Separation in Microcapillary

Longer DNA higher velocity

Chen et al.(2005)

40m

T2 DNA after 100 s oscillatory Poiseuille flow

relaxWe

v

h

V(y,z)

Dilute DNA in Microfluidic Fluid Flow

Chain migration to increase as We increases

-DNA Nc=50, cp/cp*=0.02

We=( relax)

eff = vmax / (H/2)

Non-dilute DNA in Lattice Fluid Flow

Lattice Size = 40 X 20 X 40, corresponding to 20 x 10 x 20 m3 box

As the DNA concentration increases, the chain

migration effect decreases

Nc=50, 200, 400

H = 10 m

We=100 Re=0.14

Ld

40m

oThot

oTcold

Particle Current

Soret Coefficient

y

TccD

y

cDJ Ty

)1(

y

yT

yc

ccD

DS T

T

/

/

)1(

1

Migration of a species due to temperature gradient

Mass Diffusion Thermal Diffusion

Thermal-induced DNA Migration

Thermal fractionation has been used to separate molecules

Many factors contribute to thermal diffusivity –

a “clean” measurement difficult

Wiegand, J. Phys. Condens. Matter (2004)

Hydrodynamic interactions

Experimental Observations

Colloid Particle size

DT ↑ as R ↑ (Braun et al. 2006)

DT ↓ as R ↑ (Giddings et al. 2003, Schimpf et al., 1997)

Factors that affect DT:

Solvent quality :

DT changes sign with good/poor solvent (Wiegand et al. 2003)

DT changes sign with solvent thermal expansion coef.

Polymer molecular weight

DT ~ N0 (Schimpf & Giddings, 1989, Braun et al. 2005, Köhler et al., 2002, …)

DT ↓ as N ↑ (Braun et al. 2007)

Electrostatics ?

Thermally Driven Migration in LBE

2 4 6 8 100y, m

g(y)

T=2

ThotTcold

T=0

T=10

T(y)=temperature at height y )'()'()(2)','(),( rrttyTktrftrf Bflucfluc

TH TC

Thermal migration is predicted with a simple model)(

)/ln(

0

0

TT

cc

D

DT

Thermal Diffusion Coefficient

D(m2/s) DT (x 0.1 m2/s/K)

Duhr et al. (2005)

(27bp & 48.5 kbp)1 (48.5 kbp) 4

67.9 kbp DNA 0.82 4.1±0.6

48.5 kbp DNA 1 4.0±0.6

19.4 kbp DNA 1.7 4.6 ±0.6

Simple model appears to quantitatively predict DT

DT is independent of N – agrees with several expt’s

What’s the origin of this ?

Fluid Stress Near Particles

Thot Tcold

T=4

T=0

T=2

T=7

Dissipation of Y-dependent fluctuations leads to a hydrodynamic stress in Y

))(( xUUf fpf

Momentum is exchanged between monomer and fluid

through friction

Particle Thermal Diffusion Coefficient

Diameter

(m)

D

(m2/s)

DT (m2/K/s)

dT/dy=0.2K/m

DT (m2/K/s)

dT/dy=0.4K/m

0.0385 5.6 2.3±0.4 2.1±0.3

0.0770 2.8 1.1±0.2 1.12±0.05

0.1540 1.4 0.60±0.04 0.59±0.01

DT decreases with particle size 1/R

– agrees with thermal fractionation device experiments

DT independent of temperature gradient

(Many) Other factors still to include …

Thermal and Shear-induced DNA Migration

y/H0 0.4 0.8

g(y)

1.0

1.6

0.2

T=4

y/H0 0.4 0.80.2 0.6 1.0

1.0

2.0),(

),,(

yg

TygTH TC

Thermal gradient can modify the shear-induced migration profile

Thermal diffusion occurs independent of shear-induced migration

40m

)(

)/ln(

CH

CHT

TTD

D

As N ↑, D ↓, ST↑

stronger shift in g(y) for larger polymers

T=4

Summary and Future Directions

• Shear and thermal gradient can be used to control the position of DNA in the microchannel and their average velocity

• Shear and thermal driving forces for manipulating DNA appear to have weak or no coupling => two independent control methods.

• Inclusion of counterions and electrostatics will make things more complicated and interesting.

f ev(t)

f r(t)

f vib(t)

f bend(t) ~2nm

σmHow “solid” should the polymer be when it starts acting as a particle ?

As we move to nano-scale channels, what is the valid model?

How close are we from modeling blood vessels ?

The Lattice Boltzmann Method

Replace continuum fluid with discrete fluid positions xi and discrete velocity ci

collt dt

dnnvn

ni(r,v,t) = fluid velocity distribution function

Hydrodynamic fields are moments of the velocity distribution function

Boltzmann eqn.

)],([),(),( trtrntttcrn iiii n

)()],([ eqj

jjiji nnLtr n

Lij = local collision operator

=1/ in the simplest approx.

3D, 19-vector model

Fluid particle collisions relaxes fluid to equilibrium

Ladd, J. Fluid Mech (1994)

Ahlrichs & Dünweg, J. Chem. Phys. (1999)