1. Membrane Bioinformatics SoSe 2009 Helms/Böckmann

2. Last Week: Plasma Membrane: composition & function, membrane models
Fats & Fatty Acids:
Different Motor Protein: F1-ATP Synthasepes 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:
Rate approx.
S.J. Marrink et al. J.Phys.Chem.B 104 (2000)

5. Aggregation in Simulation Studies:
Fast initial aggregation of lipids, separation into lipid and aqueous domains (200ps)
Formation of bilayer-like phase with defects (≈5ns)
Defect lifetime ≈20ns
S.J. Marrink et al. JACS 123 (2001)
bilayer with defect

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
50fs time step instead of 2fs for ‚conventional‘ all-atom molecular dynamics simulations
Increased dynamics: effective speed increase ≈4
Total speed-up:
S.J. Marrink et al. JACS 125 (2004)

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)

8. Aggregation in Simulation Studies:
Phase transition multi-lamellar to inverted hexagonal phase:
S.J. Marrink et al. Biophys.J. 87 (2005)

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

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

11. Self-Organization of Membranes
Ebind : energy required to expose hydrophobic region of amphiphile to water
hydrophilic head
: number of C-atoms
: average C-C bond length projected on chain
Area of hydrophobic chain:
hydro-phobic tail
Define: for lipids aggregated in micelle or bilayer
Enthalpic change for exposure (energy required to create new water-hydrocarbon interface):
Free enthalpy change (free energy):

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. No interaction between particels (V=0)
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
h3 phase space volume occupied by one state (normalization)
Energy of an ideal gas:
No interaction between particels (V=0)
kinetic energy
potential energy
Rewrite the partition function as:
with

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
Assumption 1: lipids in solution sufficiently dilute behaviour of lipids as ideal gas
Entropy S per molecule of an ideal gas at number density ρ:
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 : Ebind 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
1st case:
single chain phospholipid with 10 carbons (400 Dalton)
Length scale:
Surface tension:
(for short alkanes)
Effective radius of single chain: 0.2nm
2nd case:
double chain phospholipid with 10 carbons per chain (570 Dalton)
Length scale:
Effective radius of double chain: 0.3nm

18. Self-Organization of Membranes
CMC = critical micelle concentration :
single chain phospholipid with 10 carbons (400 Dalton)
Experimental:
double chain phospholipid with 10 carbons per chain (570 Dalton)
Experimental:
cmc strongly depends also on the hydrophilic headgroup
computed numbers are very sensitive to the geometric properties (e.g. radius)
RnPC
Single chain lipids uniformly higher cmc than double chain lipids
Exponential decrease with number of chain carbons:
cmc decreases faster for double chain PC
RnRnPC

19. Molecular Packing in Different Aggregate Shapes
Area per lipid
Volume of single, satu-rated hydrocarbon chain:
Important quantities:
I. Spherical Micelle: 2R
Number of molecules (area a0, volume vhc):
If equal:
Condition:
Spherical micelles are favored by large vaues for the area/lipid

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:
Double chain phospholipids:
Typical area/lipid: Å2
Typical chain length: 16 carbon atoms ≈ 20Å
Volume: 916Å3
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:
Thermodynamic Potentials:
Energy
Free Energy
Enthalpy
Free Enthalpy / Gibbs Free Energy
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. (second law of thermodynamics)
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)

26. Molecular Aggregation:
Chemical potential for ideal gas: ( ) e)
temperatur
constant
(at dp N V dG d pV TS E 1 G = m + -
Ideal gas(*):
Inserting (*):
c=molar concentration of an ideal gas

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 An
Rest of lipids is isolated: concentration A1
Equilibrium:
: equilibrium constant
Number of molecules per object:
: x= A1
Model predicts a sharp transition at the critical micelle concentration!