Membrane Bioinformatics

creatorprocessBiotechnology

Oct 2, 2013 (3 years and 8 months ago)

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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!