Membrane Bioinformatics SoSe 2009 Helms/Böckmann

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

Today: Self-organization of membranes (self-assembly, stability of lipid bilayers, order parameters) Elasticity of bilayers (theory, experiment, simulation)

Aggregation in Simulation Studies: Rate approx. S.J. Marrink et al. J.Phys.Chem.B 104 (2000) 12165-12173

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) 8638-8639 bilayer with defect

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) 15233-15242

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

Aggregation in Simulation Studies: Phase transition multi-lamellar to inverted hexagonal phase: S.J. Marrink et al. Biophys.J. 87 (2005) 3894-3900

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

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

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):

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:

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

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:

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

Self-Organization of Membranes Cross-over between phases: = -threshold for aggregation decreases as the binding energy of lipids increases

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

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

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

Molecular Packing in Different Aggregate Shapes II. Cylindrical Micelle: R t Number of molecules in the section: If equal: Condition for cylindrical micelles:

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!

Molecular Packing in Different Aggregate Shapes IV. Inverted Micelle: volume > area x chain length (small headgroup area)

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

(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!

Molecular Aggregation: Two phases: Lipid Phase Water Phase Equilibrium between both phases: In equilibrium: S.J. Marrink et al. JACS 123 (2001) 8638-8639

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

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:

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!