1
Membrane Bioinformatics
SoSe
2009
Helms/Böckmann
2
Last Week:
Plasma Membrane:
composition & function,
membrane models
Fats & Fatty Acids:
Different
Motor Protein: F
1

ATP
Synthase
pes of fatty acids, strange
lipids,
composition of membranes
Membrane Electrostatics
3
Today:
Self

organization of membranes (self

assembly,
stability of lipid bilayers, order parameters)
Elasticity of bilayers (theory, experiment, simulation)
4
Aggregation in Simulation Studies:
S.J. Marrink et al. J.Phys.Chem.B
104
(
2000
)
12165

12173
Rate approx.
5
Aggregation in Simulation Studies:
Fast initial aggregation of lipids,
separation into lipid and aqueous domains
(
200
ps)
Formation of bilayer

like phase with
defects (≈
5
ns)
Defect lifetime ≈
20
ns
bilayer with defect
S.J. Marrink et al. JACS
123
(
2001
)
8638

8639
6
Aggregation in Simulation Studies:
Vesicle Aggregation in coarse

grained molecular dynamics:
Coarse

grained molecular dynamics:
Four atom types: polar, non

polar, apolar, charged
Four water molecules =
1
coarse grained polar atom
50
fs time step instead of
2
fs for ‚conventional‘ all

atom molecular dynamics simulations
Increased dynamics: effective speed increase ≈
4
Total speed

up:
S.J. Marrink et al. JACS
125
(
2004
)
15233

15242
7
Aggregation in Simulation Studies:
Vesicle Aggregation in coarse

grained molecular dynamics:
What we can learn from simulation studies about aggregation (future):
Aggregation rates, dependency on temperature, pressure, ...
Ab initio
lipid distribution for mixed lipid systems, mixed micelles
Pore frequencies
Effect of detergent molecules
...
S.J. Marrink et al. JACS
125
(
2004
)
15233

15242
8
Aggregation in Simulation Studies:
Phase transition multi

lamellar to inverted hexagonal phase:
S.J. Marrink et al. Biophys.J.
87
(
2005
)
3894

3900
9
Aggregation in Simulation Studies:
Hexagonal phase:
S.J. Marrink et al. Biophys.J.
87
(
2005
)
3894

3900
10
Aggregation in Simulation Studies:
rhombohedral phase:
S.J. Marrink et al. Biophys.J.
87
(
2005
)
3894

3900
11
Self

Organization of Membranes
E
bind
: energy required to expose hydrophobic region of amphiphile to water
hydrophilic
head
hydro

phobic tail
: number of C

atoms
: average C

C bond length projected on chain
Area of hydrophobic chain:
Enthalpic change for exposure (energy required to create new
water

hydrocarbon interface):
Free enthalpy change (free energy):
Define:
for lipids aggregated in micelle or bilayer
12
Statistical Physics: Entropy of an Ideal Gas
Canonical partition function:
: energy of state r
: sum over all possible states r of the gas
Free Energy F=E

TS:
Entropy S:
Average energy E of the system:
13
Statistical Physics: Entropy of an Ideal Gas
Partition function for a gas of undistinguishable particles:
N! different possibilities to arrange N identical atoms
in the sum for the partition function
h
3
phase space volume occupied by one state
(normalization)
Energy of an ideal gas:
kinetic energy
potential energy
Rewrite the partition function as:
with
No interaction
between
particels (V=
0
)
14
Statistical Physics: Entropy of an Ideal Gas
Putting everything together:
We want to calculate the entropy of an ideal gas:
Which can be rewritten as:
15
Self

Organization of Membranes
Entropy S per molecule of an ideal gas at number density
ρ
:
Assumption
1
:
lipids in solution sufficiently dilute behaviour of lipids as ideal gas
Assumption
2
:
entropy of bulk water unchanged

γ
includes changes in entropy of close water molecules upon ordering
dependent only on density of lipids
ρ

low
ρ
: entropy dominates, solution phase is dominated

large
ρ
: E
bind
favors condensed phase
16
Self

Organization of Membranes
Cross

over between phases:
=

threshold for aggregation
decreases
as the binding
energy of lipids
increases
17
Self

Organization of Membranes
1
st case:
single chain phospholipid with
10
carbons (
400
Dalton)
Length scale:
Surface tension:
(for short alkanes)
Effective radius of single chain:
0.2
nm
2
nd case:
double chain phospholipid with
10
carbons per chain (
570
Dalton)
Length scale:
Effective radius of double chain:
0.3
nm
18
Self

Organization of Membranes
single chain phospholipid with
10
carbons (
400
Dalton)
double chain phospholipid with
10
carbons per chain (
570
Dalton)
Experimental:
Experimental:
CMC
=
c
ritical
m
icelle
c
oncentration :

cmc strongly depends also on the hydrophilic headgroup

computed numbers are very sensitive to the geometric properties (e.g. radius)
Single chain lipids uniformly higher cmc than double chain lipids
Exponential decrease with number of chain carbons:
cmc decreases faster for double chain PC
R
n
PC
R
n
R
n
PC
19
Molecular Packing in Different Aggregate Shapes
Important quantities:
Area per lipid
Volume of single, satu

rated hydrocarbon chain:
I. Spherical Micelle:
Number of molecules (area a
0
,
volume v
hc
):
If equal:
Condition:
Spherical micelles are favored by
large vaues for the area/lipid
2
R
20
Molecular Packing in Different Aggregate Shapes
II. Cylindrical Micelle:
R
t
Number of molecules in the section:
If equal:
Condition for cylindrical micelles:
21
Molecular Packing in Different Aggregate Shapes
III. Bilayer:
Ideal bilayer:
Condition for bilayers:
Typical area/lipid:
50
...
70
Å
2
Typical chain length:
16
carbon atoms ≈
20
Å
Volume:
916
Å
3
Double chain phospholipids:
double chain phospholipids preferentially form lipid bilayers!
22
Molecular Packing in Different Aggregate Shapes
IV. Inverted Micelle:
volume > area x chain length
(small headgroup area)
23
Molecular Packing in Different Aggregate Shapes
A thermodynamics view:
Energy
Free Energy
Enthalpy
Free Enthalpy / Gibbs Free Energy
Thermodynamic Potentials:
total differentials:
The potentials are all extensive quantities, i.e.:
Thermodynamic potentials are
state variables
, i.e. they depend unambiguously on the
state variables T,p,N,V,S
24
A thermodynamics view:
Entropy S is maximal for the equilibrium state of a closed system:
(second law of thermodynamics)
Often the Free Enthalpy or the Gibbs Free Energy G is referred to as the
Free Energy of a system
Thermodynamic Forces: derivatives of the
thermodynamic potentials
: chemical potential
µ minimal at equilibrium!
25
Molecular Aggregation:
Two phases:
Lipid Phase
Water Phase
Equilibrium between both phases:
In equilibrium:
S.J. Marrink et al. JACS
123
(
2001
)
8638

8639
26
Molecular Aggregation:
Chemical potential for ideal gas:
Ideal gas(*):
Inserting (*):
c=molar concentration of
an ideal gas
e)
temperatur
constant
(at
dp
N
V
N
dG
d
pV
TS
E
N
1
N
G
27
Molecular Aggregation:
Equilibrium concentrations
of lipids in lipid and in
water phase:
: distribution coefficient
Equilibrium constant for the transfer of lipids from bilayer/micelle to water phase:
Empirical rule for one chain amphiphiles:
Lyso

DPPC:
DPPC:
28
Molecular Aggregation:
Cooperativity in Aggregation:
Micelles usually have a specific size (narrow distribution), between
20
and
60
molecules
Assume:
Every micelle is n

mer:
concentration A
n
Rest of lipids is isolated:
concentration A
1
Equilibrium:
: equilibrium constant
Number of molecules per object:
: x= A
1
Model predicts a sharp transition at the critical micelle concentration!
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