School of Mathematical Sciences

Life Impact The University of Adelaide

Instability of Instability of CC6060 fullerene fullerene interacting with lipid bilayerinteracting with lipid bilayer

Nanomechanics Group, Nanomechanics Group, School of Mathematical Sciences, School of Mathematical Sciences, The University of Adelaide,The University of Adelaide,Adelaide, SA 5005, AustraliaAdelaide, SA 5005, Australia

Duangkamon Baowan, Barry J. Cox and James M. Hill

5th-9th February, 2012International Conference on Nanoscience and Nanotechnology,Perth, Australia

Lipid bilayer

Understanding how nanoparticles of different shape interact with cell membranes is

important in drug and gene delivery. Yang & Ma (Nature Nanotechnology 2010) give

computer simulation results for the translocation of nanoparticles of elipsoidal shape

across a lipid bilayer. Here we give an analytical model for the instability of a fullerene

passing through a circular hole in a lipid bilayer of assumed variable radius b. This might

mimic a patient receiving mild heat treatment, such as from ultra-violet light, causing skin

nanopores to change in size. The model predicts that a fullerene placed on the skin surface

is likely to relocate within the skin. We determine the minimum energy configuration for the C60 fullerene Z, measured from the fullerene centre to the upper bilayer surface, and

initially for increasing b follows a perfect circle. As the hole radius increases beyond a

critical value (b=6.81 Å) the fullerene relocates inside the layer until the radius acquires

the value b≤17.96 Å, and for hole radii beyond that value the fullerene is attracted to the

mid-plane layer and remains there. Results for spherical gold nanoparticles are included.

Lipid bilayer

A lipid bilayer is very thin as compared to its lateral dimensions, and despite being only a

few nanometers thick, the bilayer comprises several distinct chemical regions through its

cross-section. These regions and their interactions with an aqueous environment have been

characterized using x-ray reflectometry, neutron scattering and nuclear magnetic resonance

techniques. The first region on either side of the bilayer is the hydrophilic head group which

is typically around 8-9Å thick. The hydrophobic core of the bilayer is typically 30-40Å thick,

but this value varies with chain length and chemistry. Moreover, the core thickness varies

significantly with temperature, and particularly near a phase transition.

Lipid bilayer

In this presentation, we utilise the 6-12 Lennard-Jones potential function and the

continuous approximation in order to determine the interaction energy between a lipid and a C60 fullerene. We assume that the atoms are uniformly distributed over the entire surface

of the molecules and that the molecular interaction energy can be obtained from surface or

volume integrals over the molecules. We first determine the equilibrium spacing of a bilayer without a C60 fullerence moving through an assumed circular hole in the bilayer. In

the following slide, the 6-12 Lennard-Jones potential function and the continuous approximation are presented. For the inter-spacing for lipid bilayer without the C60

fullerene, we describe the model formulation and give numerical results for the lipid bilayer without the C60 fullerene. On assuming a circular hole in the lipid bilayer, the

energy behaviour for a C60 fullerene penetrating through the hole is determined, and we

discuss the overall behaviour.

Interaction energy between non-bonded molecules

• The non-bonded interaction energy is obtained by summing the interaction potential energies for each atomic pair:

• In the continuous model, the interaction energy is obtained assuming constant surface atomic densities over each molecule:

where n1 and n2 are the mean atomic surface densities for each molecule, and r is the distance between two typical surface elements dS1 and dS2 on two non-bonded molecules.

i j

ijrvE )(

2121 )( dSdSrnnE

Lennard-Jones potential energy

Combined interaction energy

Ener

gy

6/A r

12/B r

12 6

4E rr r

• Mathematician who held a chair of Theoretical Physics at Bristol University (1925-32)

• Proposed Lennard-Jones potential (1931)

(October 27, 1894 – November 1, 1954)

“Father of modern computational chemistry”

Lennard-Jones sphere-point interaction

E() A6

B12

E f () fb

A2

1( b)4

1( b)4

B5

1( b)10

1( b)10

Discrete & continuous models

• Discrete model takes each atom as the centre of a

spherically symmetric electron distribution.• Continuous model assumes a uniform atomic density

over the entire surface.

“The continuous model may be closer to reality than a discrete set of Lennard- Jones centres.”

Girifalco, Hodak & Lee, Physical Reviews B (2000).

Modelling lipid bilayer

Head group modelled as a flat plane. Tail group modelled as a rectangular box.

Inter-spacing for lipid bilayer without C60 fullerene

We first determine the inter-spacing between the two layers, by modelling the molecular interaction energy for the lipid bilayer as consisting:1.Interaction energy between two head groups,2.Interaction energy between head and tail groups,3.Interaction energy between two tail groups.

Numerical results

Energy profile for lipid bilayer without C60 fullerene where δ is the perpendicular distance between the two layers and l is the tail length which is assumed to be in the range 15 – 20 Å.

We find that the interspacing δ is 3.36 Å, a small value that is: •Ten times smaller than the hydrophobic core thickness,•Three times smaller than the hydrophilic core thickness.

Energy behaviour for C60 penetrating lipid bi-layer hole

The atomic interaction energy between a lipid bilayer and a spherical fullerene is assumed to comprise:1.Energy for two head groups and a C60,

2.Energy for two tail groups and a C60.

Lipid bilayer is assumed to be an

infinite plane consisting of two head groups and two tail groups

and with a spacing δ = 3.36. Å.

Numerical results

Energy profiles for a C60 fullerene interacting with holes of radius b=0,1,2, …, 10 Å as a function of the perpendicular distance Z with tail length l assumed to be 15 Å.

The centre of the C60 is located at the origin Z = 0, when b0=6.8102 Å.

Numerical results

Relation between minimum energy location Zmin and hole radius b.

For b ≤ 6.81 Å, the fullerene behaves like a hard sphere at rest in the hole.

For 6.81< b ≤17.96 Å, the fullerene penetrates through the bilayer.

For b > 17.96 Å, fullerene is attracted to mid-plane layer and remains there.

Note: 6.81 = 3.55 + 3.26 10.87 = 3.55 + 7.32

22 2min 6.81Z b

min

1

7.33

4.31tanh 10.87 / 7.32

Z

b

Penetration of gold nanoparticle through bilayer

Spherical gold nanoparticle modelled as dense spherical molecule and interaction evaluated as spherical volume integral.

2

2 20 0

sin2 cos

a

nrI d drd

r Z r

Volume integral for sphere and point at a distance Z apart.

System set as previously.

Numerical results for gold

•Consider three spherical gold nanoparticles with a =10, 15 and 20 Å.

•Penetration behaviour is similar to fullerene witha surface instability connecting exterior and interior regions of bilayer.

Relation between minimum energy location Zmin and hole radius b for threespherical gold nanoparticles of radii a = 10, 15, 20 Å.

Summary

Modelling is employed to determine molecular interaction energy and the structural dimensions of a lipid bilayer.

6–12 Lennard-Jones potential and the continuous approach are employed to determine the equilibrium spacing between two layers of the lipid, and it is found to be 3.36 Å.

On assuming a central circular hole in the lipid bilayer, the penetration behaviour of a C60 fullerene is determined.

As the hole radius increases, there exists instability at the critical radius b = 6.81 Å and for b > 6.81 Å, the fullerene penetrates through the bilayer.

18

Acknowledgement

• All colleagues in the Nanomechanics Group• Australian Research Council

http://www.maths.adelaide.edu.au/nanomechanics/

Thank you!http://www.maths.adelaide.edu.au/nanomechanics/http://www.maths.adelaide.edu.au/nanomechanics/