Membrane Bioinformatics SoSe 2009 Helms/Böckmann

1. Membrane BioinformaticsSoSe 2009Helms/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.B104 (2000) 12165-12173

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 bilayer with defect S.J. Marrink et al. JACS123 (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 • 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. JACS125 (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. JACS125 (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. Free enthalpy change (free energy): 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):

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 h3 phase space volume occupied by one state (normalization) Energy of an ideal gas: No interaction between particels (V=0) potential energy kinetic 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. 2nd case: double chain phospholipid with 10 carbons per chain (570 Dalton) Length scale: Effective radius of double chain: 0.3nm 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

18. Experimental: Experimental: Self-Organization of Membranes CMC = critical micelle concentration : single chain phospholipid with 10 carbons (400 Dalton) double chain phospholipid with 10 carbons per chain (570 Dalton) • 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: 50...70Å2 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. 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: Water Phase Lipid Phase Equilibrium between both phases: In equilibrium: S.J. Marrink et al. JACS123 (2001) 8638-8639

26. G 1 ( ) m = = - + E TS pV N N dG V m = = d dp (at constant temperatur e) N N Molecular Aggregation: Chemical potential for ideal gas: 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!