1 Membrane Bioinformatics SoSe 2009 Helms/Böckmann.

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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

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Today:

Self-organization of membranes (self-assembly, stability of lipid bilayers, order parameters)

Elasticity of bilayers (theory, experiment, simulation)

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Aggregation in Simulation Studies:

S.J. Marrink et al. J.Phys.Chem.B 104 (2000) 12165-12173

Rate approx. 14ps101

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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 defectS.J. Marrink et al. JACS 123 (2001) 8638-8639

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Aggregation in Simulation Studies:Vesicle Aggregation in coarse-grained molecular dynamics:

Coarse-grained molecular dynamics:

Four atom types: polar, non-polar, apolar, chargedFour water molecules = 1 coarse grained polar atom50fs time step instead of 2fs for ‚conventional‘ all-atom molecular dynamics simulationsIncreased dynamics: effective speed increase ≈4Total speed-up: 43 1010 S.J. Marrink et al. JACS 125 (2004) 15233-15242

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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 micellesPore frequenciesEffect of detergent molecules... S.J. Marrink et al. JACS 125 (2004) 15233-15242

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Phase transition multi-lamellar to inverted hexagonal phase:

S.J. Marrink et al. Biophys.J. 87 (2005) 3894-3900

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Aggregation in Simulation Studies:Hexagonal phase:


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rhombohedral phase:


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Self-Organization of Membranes

Ebind : energy required to expose hydrophobic region of amphiphile to water

hydrophilic head

hydro-phobic tail cccln

hcR

cn : number of C-atoms

nm126.0lcc : average C-C bond length projected on chain

Area of hydrophobic chain: cchcc lRn2=A

Enthalpic change for exposure (energy required to create new water-hydrocarbon interface):

•lRn2=•A=E cchccbind

Free enthalpy change (free energy):

STESTHG bind

Define: for lipids aggregated in micelle or bilayer 0G

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Statistical Physics: Entropy of an Ideal Gas

Canonical partition function: r

rc eZ

r : energy of state r

Tk1

B

r

: sum over all possible states r of the gas

Free Energy F=E-TS: cZln1

F

Entropy S: cB ZlnEkS

c

r

rr

c

Zlne

Z1

EAverage energy E of the system:

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N1N1

p,...,p,r,...,rN3c pdpdrdrde

h

1!N

1Z N1N1

Partition function for a gas of undistinguishable particles:

N! different possibilities to arrange N identical atoms in the sum for the partition functionh3 phase space volume occupied by one state (normalization)

Energy of an ideal gas: N1

2N

21 r,,rV

m2p

m2p

E

kinetic energy potential energy

Rewrite the partition function as:space

N3momentumZZ

!N1

Z N

N1r,...,rV

space VrdrdeZ N1

2/1

m2p

momentumm2

h1

dpeh1

Z

2

with

No interaction between

particels (V=0)

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2/N3

N3N m2

h

1V

!N1

Z

Putting everything together:

cB ZlnEkS

TNkZln

E B23

We want to calculate the entropy of an ideal gas:

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h

mk2ln

23

Tln23

NV

lnNkS2

BB

Which can be rewritten as:

€

S = kB5

2− ln ρ ⋅

h

2πmkBT

⎡

⎣ ⎢

⎤

⎦ ⎥

3 ⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

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Entropy S per molecule of an ideal gas at number density ρ:

3

BB Tmk2

hln

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kS

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

STEG bind dependent only on density of lipids ρ

-low ρ : entropy dominates, solution phase is dominated-large ρ : Ebind favors condensed phase

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Cross-over between phases:

STE0G bind

•lRn2=•A=E cchccbind

3

BB Tmk2

hln

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kS

=

TBkbindE

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Bagg e

Tmk2h

-threshold for aggregation decreases as the binding energy of lipids increases

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1st case: single chain phospholipid with 10 carbons (400 Dalton)

pm1.5m101.5Tmk2

h 12

B

Length scale:

Surface tension: 22

nmmol

kJ3.0J/m05.0

(for short alkanes)

Effective radius of single chain: 0.2nm

20Tk

E

B

bind M3.0agg

2nd case: double chain phospholipid with 10 carbons per chain (570 Dalton)

pm3.4m103.4Tmk2

h 12

B

Length scale:

Effective radius of double chain: 0.3nm 30Tk

E

B

bind

M102 5agg

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single chain phospholipid with 10 carbons (400 Dalton) M3.0agg

double chain phospholipid with 10 carbons per chain (570 Dalton) M102 5agg

Experimental: M1010 23

Experimental: M1010 35

CMC = critical micelle concentration :

-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 lipidsExponential decrease with number of chain carbons:

cmc decreases faster for double chain PC

TBkcclhcRcn2

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Bagg e

Tmk2h

RnPC

RnRnPC

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Molecular Packing in Different Aggregate Shapes

Important quantities:

Area per lipid

Volume of single, satu-rated hydrocarbon chain:

hc

hchc l

a

33chc nm10n9.264.27

I. Spherical Micelle:

3

2

R34

volume

R4surface

Number of molecules (area a0, volume vhc):

0

hc

hc3

02

a3R

/R34

or a/R4

If equal:

Condition:

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la

lR

hc0

hc

hc

Spherical micelles are favored by large vaues for the area/lipid

2R

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II. Cylindrical Micelle:

R

t

tRvolume

Rt2surface2

Number of molecules in the section:

0

hc

hc2

0

a2R

/tR or a/Rt2

If equal:

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la31

lR

hc0

hc

hc

Condition for cylindrical micelles:

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III. Bilayer:

hc0

0

lavolume

asurface

Ideal bilayer: 1la hc0

hc

1la2

1

hc0

hc

Condition for bilayers:

Typical area/lipid: 50...70Å2

Typical chain length: 16 carbon atoms ≈ 20ÅVolume: 916Å3

Double chain phospholipids:

92.065.0la hc0

hc

double chain phospholipids preferentially form lipid bilayers!

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IV. Inverted Micelle:

volume > area x chain length(small headgroup area)

1la hc0

hc

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A thermodynamics view:

N)p,T(pVTSE)N,p,T(G

pVE)N,p,S(H

TSE)N,V,T(F

NpVTS)N,V,S(E

Energy

Free Energy

Enthalpy

Free Enthalpy / Gibbs Free Energy

Thermodynamic Potentials:

dNVdpSdT)pV(d)TS(ddEdG

dNVdpTdS)pV(ddEdH

dNpdVSdT)TS(ddEdF

dNpdVTdSdE

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

NV

,NS

eN)N,V,S(E

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A thermodynamics view:

Entropy S is maximal for the equilibrium state of a closed system: (second law of thermodynamics)

Vp, given for min,G

VT, given for min,F

VE, given for max,S

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

STG

TSH

STF

TSE

N,p

N,p

N,V

N,V

VpG

VpH

pVF

pVE

N,T

N,S

N,T

N,S

T,pNG

: chemical potential

µ minimal at equilibrium!

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Molecular Aggregation:

Two phases:

Lipid Phase Water Phase

ndGd ndGd

Equilibrium between both phases:

dnndnddG

mequilibriu 0dG

process sspontaneou 0dG

In equilibrium: S.J. Marrink et al. JACS 123 (2001) 8638-8639

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Chemical potential for ideal gas:

Ideal gas(*): p

RTNV

NRTpV

o

*o

o*o

o*

p

op

pp

lnRT

plnplnRTpdp

d

pdp

RTd

Inserting (*): clnRT

VRTN

lnRT

o

*o

c=molar concentration of an ideal gas

e)temperatur constant (at dpNV

NdG

d

pVTSEN1

NG

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clnRT

clnRT

o

o

Equilibrium concentrations of lipids in lipid and in water phase:

RT/GRT/oo eecc

: distribution coefficient

Equilibrium constant for the transfer of lipids from bilayer/micelle to water phase:

RNk ,ec*C ABTBk/

o

Empirical rule for one chain amphiphiles:

kJ/mol n311 c

Lyso-DPPC: M 102*C 5

DPPC: M 10*C 12

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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:

n

1

n

nK

1

A

AK

AnA

: equilibrium constant

Number of molecules per object:

1n

1n

n1

n1

Kx1

nKx1x

AAAnA

: x= A1

Model predicts a sharp transition at the critical micelle concentration!

1 Membrane Bioinformatics SoSe 2009 Helms/Böckmann.

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Transcript of 1 Membrane Bioinformatics SoSe 2009 Helms/Böckmann.