Dynamics of Confined Polymer in Flow
陳彥龍 Yeng-Long Chen
([email protected])
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
Diameter: 7nm
Persistence length : ~10 mm
Endothelial Cell
F-Actin
DNA
Rg
Nuclei are stained blue with DAPI
Actin filaments are labeled red with phalloidin
Microtubules are marked green by an antibody
3.4 nm
Persistence length : ~ 50 nm
xp
Organ Printing
Mironov et al. (2003)
Boland et al. (2003)
Forgacs et al. (2000)
• Cells deposited into gel matrix
fuse when they are in proximity
of each other
• Induce sufficient vascularization
Organ printing and cell assembly
• Embryonic tissues are
viscoelastic
• Smallest features ~ O(mm)
From Pancake to Tiramisu
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!”
Edible Paper
Moto restaurant
Chicago
Confining Macromolecules
Fluid plug reactor from Cheng group, RCAS
Advantages of microfluidic chips
Channel dimension ~ 10nm - 100 mm
• High throughput
• Low material cost
• High degree of parallelization
Efficient device depends on
controlled transport
Theory and simulations help us
understand dynamics of macromolecules
Multi-Scale Simulations of DNA
Multi-component systems : multiple scales for different components
Atomistic
Coarse graining
Nanochannels
1 nm
10 nm
100 nm
Persistence length ≈ 50nm
C-C bond length
Essential physics :
3.4 nm
2 nm
Microchannels
1 mm
10 mm
Radius of gyration
l
DNA flexibility
Solvent-DNA interaction
F1
Entropic confinement
F2
Our Methods
Molecular Dynamics
Monte Carlo
Cellular Automata
- Model atoms and
molecules using
Newton’s law of motion
- Statistically samples
energy and configuration
space of systems
- Complex pattern
formation from simple
computer instructions
Polymer configuration
sampling
Sierpinksi gasket
Large particle in a
granular flow
-If alive, dead in next step
-If only 1 living neighbor, alive
Coarse-grained DNA Dynamics
DNA is a worm-like chain
2a
f ev(t)
l-DNA 48.5 kbps
f W(t)
f S(t)
DNA as Worm-like Chain
L = 22 mm
Ns = 10 springs
Nk,s = 19.8 Kuhns/spring
Marko and Siggia (1994)
Model parameters are matched
to TOTO-1 stained l-DNA
Parameters matched in bulk
are valid in confinement !
Chen et al., Macromolecules (2005)
Exp
t
Brownian Dynamics
v1
v2
v3




f (t  dt )
U (t )
dU (t  dt ) 
dt
dR(t  dt ) 
dt
m
m
 




f  f ev  fWLC  f wall  f fric  f fluc
How to treat solvent molecules ??
Explicit inclusion of solvent molecules on the micron
scale is extremely computational expensive !!
solvent = lattice fluid (LBE)



f f   (U p  U f ( x))
Brownian motion through fluctuation-dissipation


 f fluc (r, t ) f fluc (r ' , t ' )  2kBT (t  t ' ) (r  r ' )
: particle friction coef.
Ladd, J. Fluid Mech (1994)
Ahlrichs & Dünweg, J. Chem. Phys. (1999)
Hydrodynamic Interactions (HI)
Particle motion perturbs and contributes to the overall velocity field
   
  
  
  
v(r , r0 , f 0 )  v s (r  r0 )  v W (r , r0 )  Ω(r , r0 )  f 0
Free space
Wall correction
Force
Stokes Flow

0  p  η v W

0    vW
2
Solved w/
Finite Element Method
For Different Channels
z
DNA Separation in Microcapillary
T2 DNA after 100 s oscillatory Poiseuille flow
detector
25 mm
l-DNA in microcapillary flow
Sugarman & Prud’homme (1988)
Chen et al.(2005)
Detection points at 25 cm and 200 cm
40mm
  vmax /(H / 2)
Parabolic Flow
Rf 
avg. DNA velocity
max. fluid velocity
Longer DNA  higher velocity
We   relax
Dilute DNA in Microfluidic Fluid Flow
V(y,z)
z
v
l-DNA Nc=50, cp/cp*=0.02
y
h
We=(  relax)
eff = vmax / (H/2)
Chain migration to increase as We increases
Non-dilute DNA in Lattice Fluid Flow
Lattice Size = 40 X 20 X 40, corresponding to 20 x 10 x 20 mm3 box
Nc=50, 200, 400
We=100 Re=0.14
As the DNA concentration
increases, the chain
migration effect decreases
Ld
40mm
H = 10 mm
Thermal-induced DNA Migration
oTcold
y
Migration of a species due to
temperature gradient
Particle Current
Soret Coefficient
oThot
Mass Diffusion
Thermal Diffusion
c
T
J y   D  DT c(1  c)
y
y
DT
1
c / y
ST 

D
c(1  c) T / y
Thermal fractionation has been used to separate molecules
Many factors contribute to thermal diffusivity –
a “clean” measurement difficult
Hydrodynamic
interactions
Wiegand, J. Phys. Condens. Matter (2004)
Experimental Observations
Factors that affect DT:
Colloid Particle size
DT ↑ as R ↑ (Braun et al. 2006)
DT ↓ as R ↑ (Giddings et al. 2003, Schimpf et al., 1997)
Polymer molecular weight
DT ~ N0 (Schimpf & Giddings, 1989, Braun et al. 2005, Köhler et al., 2002, …)
DT ↓ as N ↑ (Braun et al. 2007)
Electrostatics ?
Solvent quality :
DT changes sign with good/poor solvent (Wiegand et al. 2003)
DT changes sign with solvent thermal expansion coef.
Thermally Driven Migration in LBE
T(y)=temperature at height y


 f fluc (r, t ) f fluc (r ' , t ' )  2kBT ( y) (t  t ' ) (r  r ' )
TH
Thot
TC
Tcold
T=10
T=2
T=0
g(y)
0
2
4
y, mm
6
8
10
DT ln( c / c0 ) Thermal migration is predicted

D
(T0  T )
with a simple model
Thermal Diffusion Coefficient
D(mm2/s)
DT (x 0.1 mm2/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



f f   (U p  U f ( x))
Momentum is
exchanged between
monomer and fluid
through friction
T=7
T=4
T=2
T=0
Thot
Dissipation of Y-dependent fluctuations
leads to a hydrodynamic stress in Y
Tcold
Particle Thermal Diffusion Coefficient
Diameter
(mm)
D
(mm2/s)
DT (mm2/K/s)
dT/dy=0.2K/mm
DT (mm2/K/s)
dT/dy=0.4K/mm
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
Thermal gradient can modify the
shear-induced migration profile
TH
1.6
TC
T=4
Thermal diffusion occurs independent
of shear-induced migration
g ( y, T ,  )
g ( y,  )
2.0
T=4
g(y)
DT  ln(  H / C )

D
(TH  TC )
1.0
1.0
0.2
0
0.4 y/H
0.8
0
0.2
0.4
0.6
0.8
y/H
As N ↑, D ↓, ST↑
40mm
stronger shift in g(y)
for larger polymers
1.0
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.
How “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 ?
σm
f r(t)
f bend(t)
f vib(t)
f ev(t)
~2nm
The Lattice Boltzmann Method
Replace continuum fluid with discrete fluid
positions xi and discrete velocity ci
ni(r,v,t) = fluid velocity distribution function
Boltzmann eqn.
 dn 
 t n  v  n   
 dt coll
ni (r  ci t , t  t )  ni (r , t )  i [n(r , t )]
 i [n(r , t )]   Lij (n j n eq
j )
j
Fluid particle collisions
relaxes fluid to equilibrium
Lij = local collision operator
=1/ in the simplest approx.
3D, 19-vector model
Hydrodynamic fields are
moments of the velocity
distribution function
Ladd, J. Fluid Mech (1994)
Ahlrichs & Dünweg, J. Chem. Phys. (1999)