BILAYER MEMBRANESof

COARSE−GRAINED MODELS

Department of Biomedical Engineering

Ben Gurion University

andSIMULATIONS

ODED FARAGO

Outline

1. Introduction

1.1 Atomistic vs. macroscopic models

1.2 Intermediate scale models

2. Water−free computer model

3. Numerical results

4. Pores − simulations and theory

4.1 Elasticity models

4.2 Pores in fluctuating membranes

5. Summary

6. Simulations of more complex systems

hydrophilic head

hydrophobic tail

>1 m

5−10 nm

µ

κ∼ 10−12 erg

B ∼ 100 erg/cm2

* Created by self−assembly of amphiphilic molecules

Membranes − some (well known) facts:

* Quasi two−dimensional objects

* Soft (but strong) materials

− May fluctuate strongly in the normal direction (out−of−plane)

− Resist lateral (in plane) stretching or compression

* Statistical mechanics at reduced dimensionality

− Variety of phases [solid, fluid, gel (=hexatic?)]

− Large normal fluctuations

− 2D generalizations of linear polymer chains

* Soft Matter

− Surface tension, bending elasticity and the interplay between them.

* Nano−biotechnology

− Liposome based drug delivery

− Gene therapy systems

Why are membranes interesting ?

Simple model for biomembranes.

a. Understanding the relation between the physical properties of the

lipid matrix and its biological functions.

b. Developing computational models which will serve as platforms

for more complicated systems including mixtures of lipids and

membranes proteins.

Why are membranes interesting ?

gauche bonds, penetration of water. thickness, chain tilt, fraction of

area per lipid, bilayer

Dynamics: motion of a singlemolecule, reorganization of smallpatches, aggregation (micelles).

Structure:transitions, phase diagram (solid−gel−fluid), domain formation(mixtures), self−assembly, elasticity,lateral diffusion, flip−flops, pores.

thermal fluctuations, shape

Length and time scales in membranes

Molecular level Membrane level

microscopic propertiesAtomistic computer models

how the macroscopic properties emerge from the microscopic entities and theinteractions between them?

Coarse−grained models

macroscopic behaviorContinuum theories

Example: the theory relating thethe shape of the lipid molecules and

morphology of the aggregate.

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Monge gauge

surface tension saddle−splay moduluslocal curvatures

bending modulus spontaneous curvature

H =

Z

SdA

[

σ+12

κ(c1 + c2−2c0)2 +κGc1c2]

Phenomenological continuum models

(Landau expansion in small curvature)Helfrich Hamiltonian

h(x,y)

H '12

Z

[

σ(

~∇h)2

+κ(

∇2h)2

]

dxdy

H '12 ∑

~q

[

σq2 +κq4]

|h~q|2

〈|h~q|2〉 =kT

σq2 +κq4

c0 = 0

2πL

≤ |~q| ≤2πa

− Monge gauge

Note:

* We assume that * The Gaussian curvature contributes a constant term.

In Fourier space:

Helfrich Hamiltonian (cont.)

(flat reference surface)

Equipartition theorem:

Spectral Intensity

one lipid molecule).

nanoseconds.

* Computationally very expensive (we need >150 atoms to simulate

* Restricted to small patches of 50−200 lipids and time scales of a few

covalent bonds

bond angles

non−bonded interactions (LJ)

electrostatics

+ WATER MOLECULES (20−50 Molecules per lipid)

+ INTERACTIONS:

Atomistic computer models

[Shelley et al., J. Phys. Chem. B 105 ,4464 (2001)]Coarse−grained model of a phospholipid molecule (DMPC)

The major challenge: Finding effective potentials that reproduce microscopic features of the system.

Coarse−grained computational models I(coarse−description of specific systems)

108 ,7397 (1998)][R. Goetz and R. Lipowsky, J. Chem. Phys.

Coarse−grained computational models II(simplified models of amphiphilic molecules)

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+ WATER PARTICLES (10−30 Molecules per Lipid)

* Still computationally expensive (15−40 particles per "lipid")

* No specific lipid systems, only general properties

* Larger membranes, Larger time scales

SIMPLIFIED INTERACTIONS

Lennard−Jonesspringsmany−bodyno electrostatics!

Repulsion

Attraction

Attraction

* only Lennard−Jones pair interactions

* NO WATER !!!

* lipids are modeled as rigid trimers

Even more simplified model (Minimal ?)

Principles of the model:

OF, J. Chem. Phys.119 , 596 (2003)

1 1.5 2 2.5r

-2

-1

0

1

2

3

4

5

U(r

)

1−1

2−23−3

1−2

2−3 1−3

Repulsion

Attraction

Attraction1

2

3

Problem: Water (via the hydrophobic interactions) confines the lipids to

the membrane

Incorrect solution: strong attraction between lipids (solid membrane)

Correct solution: shallow potentials − smooth energy landscape

non−additive pair potentials − mimics hydrophobicity

Phase diagram

solid membrane fluid membrane

1000 Molecules

Small projected area (high density) Large projected area (low density)

self−diffusion coefficient

0 50000 1e+05 1.5e+05 2e+05t [MC time units]

0

5

10

15

20

25

30

∆ r´

2 [

σ2]

low area-density

high area-density

D≡ limt→∞

∆r′(t)2

4t≡ lim

t→∞

14Nt

N

∑i=1

[(~ri(t)−~rCM(t))− (~ri(0)−~rCM(0))]2

Fluid membranesin−plane diffusion

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

|q|2 [1/ a

2]

0

5

10

15

20

25

1/(

a2<

|hq|2

>)

[1

/ a4

]

〈|hq|2〉 ∝kT

σq2 +κq4κ ∼ 40kT

Fluid membranes

out−of−plane fluctuations

spectral intensity

The bending modulus

... and allows the trans−bilayer diffusion (flip−flops)of lipids

pore formation reduces the elastic energy ...

Pores and flip−flops

> 0< 0

− stretching− compression

As − tensionless (Schulman) area

σ = KA(A − As)

As

AKA

Rpore

Epo

re

Epore (R) = λ 2πR − σ πR2

Rmax =γ

σ

Emax =πγ2

σ

λ ∼ 10−11 N

σrapture ≤ 10−1 N

m

Pores − mechanism for reducing the membrane area

Surface tension

− area compressibility− membrane area

Pore nucleation energy (Litster, 1975)

Epore (A, Apore) = KA(A − Apore − As)

2

2As− KA

(A − As)2

2As+ λLpore

λ

pore a

rea

Rpore

E por

e

λ = λ′ −→ ∆E ∼λ4/3

K1/3A

(As)1/3

Pore energy:

Pore formation − finite size dependence

First order transition (circular pore)

− size dependence

− metastable pores

λ′ ∝ KA(A − As)3/2

As

A−Asλ > λ′

λ < λ′

λ = λ′

λ > λ′

2. membrane fluctuations around equlibrium height

1. pore height fluctuations (equilibrium profile)

3. fluctuations of pore shape

Pore formation − role of thermal fluctuations

Small fluctuations of a membrane with quasi−spherical hole

For small fluctuations (harmonic approximation)

these three contributions decouple!

122 , 044901 (2005).OF and CD Santangelo, J. Chem. Phys.

r0

Ffluct ≡ F (r0) − F (0)

Ffluct ≈−πσr20 +kBT2

∑m,n

ln

[

σ(λm,n1

)2 +κ(λm,n1

)4

σ(λm,n1,(0))

2 +κ(λm,n1,(0))

4

L2− r20

L2

]

elastic energy (smaller projected area)

entropy (change in the spectrum of thermal undulations)

* Membrane fluctuations and the surface tension

Jm(λm,n1 r0)Ym(λm,n1

L)− Jm(λm,n1 L)Ym(λm,n1

r0) = 0 Jm(λm,n1,(0)L) = 0r0 = 0

ξ =√

κ/σFfluct = −πr

2

0σ +kBTr

20

αl20

2 − α −

l0

πξ

2

ln

πξ

l0

2

+ 1

≡ −πr20(σ − ∆σ) ≡ −πr2

0σeff (α ∼ 1.7)

Eigenvalue equation:

Ffluct ≡ F (r0) − F (0)

∆F ' 2πr0

λ +bkBT

πl0

ln

bd2dBλ

kBT l0

− 2

≡ 2πr0(λ − ∆λ) ≡ 2πr0λeff

Hole fluctuations and the line tension

entropy (thermal fluctuations)line tension energy

r0

Fpore (R) = λeff 2πR − σeff πR2

λeff < λ ; σeff < σ

R pore

Fp

ore

R pore

Fpore

R pore

Fpore

Entopically induced pores

λeff > 0 ; σeff > 0

λeff < 0 ; σeff < 0

λeff > 0 ; σeff < 0

Req =λeff

σeff

2. Computer simulations reveal a variety of phenomena

commonly observed in real bilayer systems.

3. Pore formation is a mechanism for reducing the membrane area.

The energy barrier for this process increases with the size of the

system.

1. We present a simple and efficient computer model of

bilayer membranes. The absence of water from the simulation

cell greatly reduces the computational effort.

making their effective values smaller than their bare ones.

Depending on the sign of the effective coefficients, the opening

of a pore may be:

4. Thermal fluctuations renormalize the surface and line tension,

A) thermodynamically unfavorable

B) a thermally activated process

C) occur spontaneously

Summary

More complex Systems − DNA−lipid complexes

* Form spontaneously

* The new generation of gene−therapy vectors

* Their phase behavior is dominated by electrostatic,

bending, and stretching energies.

Thanks to ...

Fyl Pincus

Christian D. Santangelo

Cyrus R. Safinya and his group

Niels G. Jensen