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Membrane Bioinformatics SoSe 2009 Helms/Böckmann. Last Week:. Plasma Membrane: composition & function, membrane models. Fats & Fatty Acids: Different Motor Protein: F1-ATP Synthase pes of fatty acids, strange lipids, composition of membranes. Membrane Electrostatics. Today:.

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## Membrane Bioinformatics SoSe 2009 Helms/Böckmann

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**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.B104 (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 bilayer with defect S.J. Marrink et al. JACS123 (2001) 8638-8639**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**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**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**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):**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:**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**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**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**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**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**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: Water Phase Lipid Phase Equilibrium between both phases: In equilibrium: S.J. Marrink et al. JACS123 (2001) 8638-8639**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**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!

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