Spontaneous ordering of semiflexible polymers on nanotubes and nanospheres
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Spontaneous ordering of semiflexible polymers on nanotubes and nanospheres
Simcha Srebnik
Chemical Engineering
Technion
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Why study semiflexible polymers?
• Biopolymers– double-stranded DNA– unstructured RNA– unstructured polypeptides (proteins).
• Semiflexible Polymers– aromatics – bulky side groups
Unlike the ideal chain, there is no consistent model that describes their behavior
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Polymer statistics
• The semiflexible chain– N=104, lp = 1 (ideal), 6.5 (e.g., polyacrylamide), 500
(α-helix)
For flexible chains,
220
20
2
22
where nlR
Rll
nnlnlnR p
pppp
-50
0
50
-20020406080-80
-60
-40
-20
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-1000
100200
300 -500
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-100
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lR 2212 10 lR 2212 106
lR 4212 10
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The wormlike chain model
• Kratky-Porod chains– the orientation correlation function for a worm-like
chain follows an exponential decay
ii
i–1
pxxii
xii lxlexp~cos
ssss
plL
pp
LL
elLl
dndnlR
12 2
00
22 ssRR
Kratky and Porod, Recl. Trav. Chim. Pays-Bas 68 (1949) 1106
si
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Scaling of semiflexible chains
• The KP model accurately predicts end-to-end distance for the entire range of chain flexibility
– Drawback• Cannot obtain end-to-end
distance distribution for comparison with experiments (S(k))
• Other exact theories exist, but solution is numerical and extension to other related problems (e.g., external forces, geometrical constraints) is difficult.
flexible
rigid
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Coarse-grained simulation
• Use simplified models of ‘pearl necklace’ polymer chains
– Ideal (ghost particles)
– excluded volume (hard sphere)– Lennard-Jones (soft sphere)
2
1
21cosN
ii
BTkU
0
1
2
3
4
-3 -2 -1 0 1 2 3
U/k
bT
Polymer lp/l0 |Poly(ethylene oxide) 2.5 5Poly(propylene) 3 6Poly(ethylene) 3.5 8Poly(methyl methacrylate) 4 10Poly(vinyl chloride) 4 10Poly(styrene) 5 15Poly(acrylamide) 6.5 23Cellulose diacetate 26 230Poly(para-benzamide) 200 7000DNA (in double helix) 300 13000Poly(benzyl-l-glutamate) (α-helix) 500 30000 lp ~ 0.6
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Modeling ‘ideal’ semiflexible chains
• Current computer resources limit our simluations to chains with ~102 monomers. – Develop model for analyzing conformational behavior
of very long chains.– Limited to non-interacting systems.
laa
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Polymer adsorption on curved manifolds
• Noncovalent functionalization of nanotubes using polymer wrapping– Dispersion of CNTs in aqueous or organic media– Mechanical reinforcement– Fluorescent labeling– Sensors and biosensors (conjugated
polymers/biopolymers)
• Polymer in or on spheres– DNA packaging in viruses, vesicles, or cells– Protein encapsulation– Colloidal and micellar suspensions
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11
Carbon Nanotubes
• First reported by IIjima in 1991 (“microtubules”)– Nature 354 (1991) 56-58. – Over 5000 citations!
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Examples of helical wrapping
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B. McCarthy, J. N. Coleman. J. Phys. Chem. B, 2002, 2210
PmPV coating
HupR protein on MWNTs
Balavoine and Shultz. Angew. Chem., 1999, 1912
Zheng et al., Nature materials, 2 (2003)338.
DNA
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Forces leading to helical wrapping
• Molecular modeling suggests that ssDNA can bind to carbon nanotubes through -stacking, resulting in helical wrapping. (Zheng et al., Nature Materials 2 (2003) 338).
• Alignment of backbone aromatic rings was also thought to determine interactions between CNTs and polymers (Zaiser and coworkers, J Phys Chem B 109 (2005) 10009; Coleman and coworkers, J Phys Chem B 106 (2002) 2210-2216).
– Note: all molecular modeling studies based their conclusions regarding polymers on short oligomers
• Shinkai and coworkers used TEM and AFM to confirm periodic helical structure of polysaccharides adsorbed on CNTs. Argue that helical pattern is observed because of their strong helix-forming nature. (JACS 127 (2005) 5875-5884)
• ‘General phenomenon’ argued by Baskaran et al. from studies on various polymers. (Chem Mater 17(2005)3389)
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Smalley’s postulate
• Monolayer wrapping results from a thermodynamic drive to eliminate the hydrophobic interface between the tubes and their aqueous medium.
• Random adsorption is not likely to result in sufficient coverage; single tight coil would introduce significant bond-angle strain in the polymer backbone;
• multiple helices are the likely configuration.
Smalley and coworkers, Chem Phys Lett 342 (2001) 265
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Simplest MC simulation
• Dilute semiflexible polymer solution• Impenetrable infinite cylinder• Periodic boundaries• LJ interactions• MC moves
– Reptation– Kink-jump– Pivot
• Metropolis acceptance
– 106 equilibration moves– Averages every 103 for
additional 107 iterations -40
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2040
-40-20020400
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1,expmin kTUUp oldnew
Recipe: adsorption and frustration.
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Potential of nanotube
• Surface-averaged Lennard-Jones potential between the CNT and monomers:
2
02/52/11 16
3512
638outer
inner
R
Rcyl xx
ddU
where
cos)(2)( 22 RDRDx
LJ
R
R
z
cyl UdzddrrUouter
inner
2
0 0
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• The total potential energy of a given polymer configuration is given by:
helix multiple
i ijijLJ
nfrustratio
B
adsorptioni
icyltot rUUrUU
)()()(
-50
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-50
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-50
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R=2, N=100
k=50k=0
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8
fads
R
0
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0 10 20 30 40 50
fads
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5
0 2 4 6 8
Nt
R
0
1
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0 10 20 30 40 50
Nt
1.62
3
4
5
1.2
1
lp
lp
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Effect of concentration
Nc=2 Nc=3
Nc=5 Nc=8
N=100, R=3, k=50
0
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0 2 4 6 8
% a
ds k=0k=5k=10k=50k=100
0
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0 2 4 6 8Nc
Nt
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Transitions
2
2
1( ) ( , )3
N
iG m g m i
N
1, , , ,1
1 2, ,1
1/( 1) (cos cos )(cos cos )( , )
1/( 1) (cos cos )
N mi j i j i j m i jj
Ni j i jj
N mg m i
N
( ) exp( / ) cos(2 / )G m m m P
0
20
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0 20 40 60
adsorption
helix
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• Helical pitch depends on NT radius and chain flexibility
Helical pitch
0
10
20
30
40
50
0.1 1 10 100
av,
degr
ees
R/l
lp
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What drives helical polymer wrapping?
• Hydrophobic drive?– Monolayer adsorption also achieved with weak interactions
between monomers and tube for semiflexible chains– Not sufficient to induce helicity
• Helical polymers?– Too stringent, semiflexible polymers sufficient
• Helicity of nanotube (-stacking)– Geometry (tube radius) and chain flexibility provide strong
drive for helical wrapping
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VIM on sphere
i–1
i
i+1
es
A
O
B
10-2 10-1 100 101 102 103 104
10-4
10-3
10-2
10-1
100
101
102
103
104
<R2 >
/l p2
L/lp
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 2 4 6 8 10
<R2 >
/l p2
L/lp
5s
10s
2s
Position of bead i+1 is determined from a point along the path of a great circle connecting monomer i and the intersection of line OA with the surface of the sphere.
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Polymer wrapping of a sphere
N=1000 monomers confined to a sphere with radius =10s
0
5
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0 5 10 15 20 25
l p,m
in
Ref. 10
VIM
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Conclusions
• weak surface interactions are sufficient to overcome low entropy barrier of semiflexible chains and lead to monolayer adsorption
• helix is a stable ‘universal’ state for polymers determined solely by surface curvature (NT and sphere) and polymer bending energy.
• geometry determines helical pitch at intermediate radii for semiflexible chains
• multiple helices form due to vdW interactions between monomers which are sufficient to overcome (small) translational entropy of adsorbed chains
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Conclusions (2)
• Available computational resources limit our simulations to relatively short chains
– The semiflexible chain can be effectively modeled through a summation of energy and entropy ‘vectors’ that determine the growth or position of a monomer based solely on the two previous monomers
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Acknowledgement
• Liora Levi• Yevgeny moskovitz• Hely Oizerovich• Inna Gorevitz• Iliya Kusner
• ISF• Rubin Scientific and Medical Research Fund