Spontaneous ordering of semiflexible polymers on nanotubes and nanospheres

Simcha Srebnik

Chemical Engineering

Technion

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

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

0

20

-1000

100200

300 -500

0

500

-100

-50

0

50

100

150

200

-400

0-200

00

-700

0

-600

0

-500

0

-400

0

-300

0

-200

0

-100

000

500

1000

1500

2000

2500

3000

3500

4000

4500

lR 2212 10 lR 2212 106

lR 4212 10

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

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

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

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

lii

iii

ii

iiii

,,

,,1

es

es

uuuu

resesrr

i – 1

i

si + 1

l

e

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

11

Carbon Nanotubes

• First reported by IIjima in 1991 (“microtubules”)– Nature 354 (1991) 56-58. – Over 5000 citations!

Examples of helical wrapping

12

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

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)

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

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

-200

2040

-40-20020400

20

40

60

80

100

1,expmin kTUUp oldnew

Recipe: adsorption and frustration.

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

• The total potential energy of a given polymer configuration is given by:

helix multiple

i ijijLJ

nfrustratio

B

adsorptioni

icyltot rUUrUU

)()()(

-50

0

50

-50

0

500

20

40

60

80

100

-50

0

50

-50

0

500

20

40

60

80

100

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

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50

fads

0

1

2

3

4

5

0 2 4 6 8

Nt

R

0

1

2

3

4

5

0 10 20 30 40 50

Nt

1.62

3

4

5

1.2

1

lp

lp

Effect of concentration

Nc=2 Nc=3

Nc=5 Nc=8

N=100, R=3, k=50

0

20

40

60

80

100

120

0 2 4 6 8

% a

ds k=0k=5k=10k=50k=100

0

1

2

3

4

0 2 4 6 8Nc

Nt

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

40

60

80

0 20 40 60

adsorption

helix

• 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

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

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.

Polymer wrapping of a sphere

N=1000 monomers confined to a sphere with radius =10s

0

5

10

15

20

25

30

35

40

0 5 10 15 20 25

l p,m

in

Ref. 10

VIM

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

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

Acknowledgement

• Liora Levi• Yevgeny moskovitz• Hely Oizerovich• Inna Gorevitz• Iliya Kusner

• ISF• Rubin Scientific and Medical Research Fund