Post on 16-Jan-2016
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, AustraliaDuangkamon 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
En
erg
y
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() A
6 B
12
E f () fb
A
2
1
( b)4 1
( b)4
B
5
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
sin
2 cos
a
n
rI 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/