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polymers/proteins at membranes far from equilibrium
C. Sendner, X. Schlagberger, A. Alexander-Katz, Y.-W. Kim, TUM
1) Sedimenting rods (birefringence)
2) Driven polymers at surfaces (glycocalix deformation under shear)
3) Globules in shear flow (unfolding of proteins in blood flow)
4) Pumping with stiff polymers
Hydrodynamics
Elasticity
Thermal Fluctuations
Anomalous Electric Birefringence of Charged Polymers
Xaver Schlagberger
light
E
anisotropic
refractive index n
 birefringence
Polymers
i
bead i
Konski , Zimm (1950)
0.073 % TMV
normal birefr.
Tobacco Mosaic Virus
charge -1000 e
length 300 nm
1.4 % TMV
2.75 % TMV
anorm. birefr.
Weber 1992
fd virus :
charge -500 e
length 880 nm
normal
(parallel)
anormal
(perpendicular)
anomalous behavior for
-large polymer conc.
-low salt conc.
conclusion:
Overlapping screening clouds
do not contribute to electric
polarization??
Hydrodynamics at low Reynolds numbers
Stationary Navier-Stokes equation
,
If the Reynolds number
,
one obtains the creeping flow equation.
human
H2O:  = 0.001 Pa s;
 = 1000 kg/m3
v = 1 m/s
l =1m
bacterium
Re =
106
v = 10-5 m/s
l =1
 Re = 10-5
sinking cylinder
v ~ 10-7 m/s
l =1
 Re = 10-7
look at a rod (virus, tubulin, actin, short DNA)
ln(L /b)
 
2L
(Stokes-flow,
neglecting inertia)
force F
velocity U= F
mobility 
   /2
parallel cylinder moves faster; is there some hydrodynamic
orientation effect?
Simple Example: elastic rod under uniform force in the presence of
hydrodynamic interactions
1) orthotropic bodies (with three perpendicular planes of symmetry)
are not oriented by hydrodynamic interactions (Brenner 1964)
force
torque
2)
F  K  U  C   
T  C U   
coupling tensor C non-zero if one plane of symmetry is broken
transverse (bending) mode
longitudinal mode
flow-field due to point-force at origin:
u (r)  H  (r) f 

H (r) 


8r
1

 
ˆ
 r rˆ 
(Oseen-Tensor)
for many particles the superposition principle is valid:
u (r)   H


(r  ri ) f i

i
invert to get forces for prescribed
solvent velocity distribution !!
straightforward way to satisfy no-slip in multi-particle system
u j (rj )   H (rj  ri ) f i  0 f j  v (rj )  v(rj )




i
velocity of j-th
particle
solvent-velocity
due to other particles
solvent-velocity
at particle position
cohesive/elastic forces in objects
automatically lead to solvent flow
stagnation
Next: add thermal noise
Theoretical Framework: Position Langevin Equation
Velocity of
i-th particle:
mÝ
rÝj (t)ij  rÝi (t)  ij f j (t)  i (t)
deterministic force
 i(t) j ( t)  6 ij kB T  (t  t )
Random force
Mobility matrix:
self mobility:
f j (t)  U (t) /rj (t)  E
ij  Dij / kB T  0  ij  H(ri ,rj )
0  6R 
1
hydrodyn. interact.
equivalent to Smoluchowski equation for particle distribut. W(rj,t) :

W
  W
U / k BT

Dij
 ij f jW
with
solution:
W

e


t
rj
i, j ri 

hydrodynamic simulation of elastic rod
motion
camera moves with
central monomer
lp / L = 100,
isotrop. elast. rod
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equilibration of hydrodynamic and elastic forces
direct forces acting on monomers
direct force+hydrodynamic force
bending force in stationary case cancels
other forces
0 0
0 
C  0 0 C23 


0 C32 0 
hydrodynamic
torque for bent rod
leads to perpendicular
alignment !
Scaling laws (Xaver Schlagberger, RRN)

Hydrodynamic deformation, bending angle 
bending torque Tb = l / L
g L3 / a l
Hydrodynamic torque Th = g L2 / a

Hydrodynamic orientation, angle 
thermal noise vs orienting torque kBT = Th
low T
high T
L5 / l) (g / kBT)2
sin 
L5 / l) (g / kBT)
Electrostatic polarization orientation, angle 
high T
sin  L3 (g / kBT)2
field pulse
dynamic behavior:
n
overshoot after
switching off the field
due to bending relaxation
W. Oppermann, Makromol.
Chem. 189, 2125 (1988)
time
hydrodyn. orientation: r = L-2
polarization orient: r = L-0
hydrodyn. relaxation: r = L3
Shear-induced denaturation of von-Willebrand factor
action
Imune Response (Wound)
von-Willebrand Faktor
(fibers !!!)
Blood
Transport
Docking
Fusion
von-Willebrand Faktor (globular !!!)
Vesicels (packaged proteins)
Intracellular
monomer
(2500 aminoacids)
dimer
multimer (few hundred units)
The vWf is the biggest soluble protein in the body - why?
vWF-Release at the surface of Endothelial Cells
Schneider, LMU Munich
50 m
What‘s it‘s purpose and how is it streched ?
micro-flow-chamber
Wixforth, Schneider, Augsburg
Oberflächenwelle
(Nanopumpe)
IDT‘s mit HF Anschluß
(Quelle für OFW)
Hydrophobe Oberfläche
V = 8µl
LiNbO3
(Piezoelektrika)
200µm
40mm
1mm
Kanal mit Zellen oder Beads
vWF under Shear Flow
SAW
vWF - Globular
25μm
25μm
Functionality Test
„Super Glue“ for Blood Platelets
2.0 µm
0
50 m
12 µm
Below Critical Shear c
Above Critical Shear c
unfolding occurs
also in bulk
(without collagen
substrate)
10 µm
collapsed
10 µm
stretched
10 µm
0 ms
160 ms
320 ms
relaxation into globular state
once shear is turned off
end to end distance [µm]
a.
16
12
8
shear rate  [s-1]
4
0
100
1000
normalized rate of adhesion
10
10000
1,0
0,8
b.
0,6
0,4
0,2
0,0
10
100
shear rate  [s-1]
1000
10000
Fig. 4
transition quite abrupt as a function of shear rate
equilibrium coil-to-globule transition
Alfredo Alexander-Katz

attractive Lennard-Jones
potential

in shear, =2.5, =1.2
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stretching dynamics
 ~ *
Rg2
time (a. u.)
stretching response
unfolding becomes
abrupt for globular proteins
(in agreement with experiments)
Collapsed polymer under shear flow: unfolding transition
induced by single-chain excitations
 1
f  
cohesive force on protrusion
(sharp interface, diffuse interface)
from equipartition theorem lf=kBT
--> „typical“ protrusion length

 
1/ 
--> typical cohesive force on protrusion fcoh
relative velocity
sphere/solvent
shear-force on protrusion
-free draining (with slip)

# monomers

f shear  ( R)( /a) 1

-hydrodynamic case (no slip)
f shear   3 R1a11
friction coefficient
of one monomer
critical protrusion length fcoh = fshear
  a 2 / k B T
free draining
hydrodynamic


Scaling of critical shear rate:

*
 N
1/ 3
/a
3
=2kBT, N=100, =1000s-1, ----> a = 10nm !!

*
2 /

*
4 /
   
   
a /R
R /a
Glycocalix in Cells
-Glycocalix stabilizes outer leaflet of cell surface
as well as microdomains by side-by-side interactions
(electrostatic, hydrogen bonds, van der Waals)
- Controlls near surface viscosity (effects diffusion of proteins)
grafted polymers in shear flow
v
Grafted Neutral Polymers under Shear Flow (Kim/Netz)
(homogeneous sparse coverage)
lateral force at some height
generates shear flow;
flow profile calculated self-consistently,
coupled to polymer deformation.
hydrodynamic periodic boundary conditions
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shear:
persistence length:
Ý 10 7 s1

/L  5
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Ý 10 8 s1

measure time-dependent response !
Measuring the fluid-velocity profile -> stagnation length
increasing
stiffness
D
decreasing
shear velocity
DNA-grafted sphere held in laminar flow by laser trap
Gutsche, Kremer (Leipzig)
hydrodynamic
radius as function of
flow velocity
-> DNA bends
down as flow
increases
Next: diffusion of particles through layer in shear flows
produce shear with beating polymers
Ciliae
propulsion
pumping
apply asymmetric torque at the polymer base
-> measure net pumping velocity, efficiency, etc.
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pumping efficiency =
1. condition:
threshold force for
deformation:
 f /kB T 
P
net solvent velocity
power input
2. Condition: asymmetry
/L
ratio
forward torque/
backward torque
rescaled
stiffness
Symmetric motion: no net pumping
No elasticity: no net pumping (since reciprocal motion…)
Pushing grafted DNA around with vertical electric fields
Rant, Arinaga, Tornow, Abstreiter
for single-stranded (flexible) DNA
the field-driven down-motion
is faster than up-motion !?
ss-DNA different from ds-DNA
--> biosensors …
for single-stranded (flexible) DNA
the field-driven down-motion
is faster than up-motion !?
ss-DNA different from ds-DNA
--> biosensors …
hydrodynamic simulations
of DNA in electric fields
Woon Kim, RRN
range of electric field (screening)
double-stranded DNA (stiff)
up/down motion symmetric
single-stranded DNA (flexible)
up motion diffusive (slow)
down motion driven (fast)
driven polymers at solid surface
how to desorb stiff polymers from surfaces?
- by driving them laterally ! (E-field or shear)
christian sendner
no-slip surface breaks spatial symmetry and
exerts hydrodynamic torque on moving polymer
Initial angle 130o
Initial angle 110o
tendency to move away from surface !
parameters:
N = 10, 0 = 5.0E-6, persistence length l/L =100, external field E/kT = ( 100, 0, 0 )
1) free-space flow field
2) no-slip wall
3) surface „polarization forces“
4) symmetry-broken flow-field
(Blakes Greens function)
5) Elongated object receives
hydrodynamic torque
--> hydrodynamic lift
angular velocities for angles
with
x-axis () and z-axis ()
center of mass velocity Vz
perpendicular to the wall
Top view

Ex
side view
•
Vz(,=/2) = Vz(,=/2) = 0

Ex
distance from the wall: z = 2
force in x-direction:
E=1
i) orientation parallel
to external force
ii) orientation perpendicular
to external force
E /kBT = (100,0,0), z0 = 2a, lp/L=10.000
0 = 1.0E-6 2.0E5 MD-steps
to obtain average lift force, average over long simulations
and push rod down to surface (umbrella sampling)
movie for E/kBT = (1.0, 0.0, -0.025)
Probability distribution of polymer in presence of vertical pressure
Ez/kT=0.005
0.01
equilibrium
No driving field
(entropic rod-wall
repulsion)
lateral driving field
Ex/kT = 1
from difference in probability distribution obtain
hydrodynamic repulsive potential of mean force……
0.025
Hydrodynamics
Elasticity
Thermal Fluctuations