3. Lipids, membrane mechanics (RD)

Reminder: self assembly

v/a0lc → 1Small head-group and bulky chainsFor the same a0 and lc, the

HC chain volume, v, is to be twice the volume of micelle forming surfactantwhere 1/3< v/a0lc<1/2 ⇒

two chainsTwo chains ⇒ 1) hydrophobicity ↑ ⇒ CMC ↓CMC (micelle-form.) = 10-2 ÷ 10-5 MCMC (bilayer-form.) = 10-6 ÷ 10-10 M

2) residence time, τR↑ :τR(micelles) ∼ 55 x 10-9 / 10-3 ∼ 10-4sτR(bilayers) ∼ 55 x 10-7 / 10-10 ∼ 104s

3) flip-flop times , τflτfl ∼ 102 - 105s (depends more on the head-group

rather than the chain)

a0

R ≤ lcv

Packing parameter (shape factor): v/a0lc

spherical micelles : v/a0lc<1/3

non-spherical micelles : 1/3 < v/a0lc<1/2

vesicles or bilayers : 1/2 < v/a0lc< 1

inverted structures : v/a0lc >1

oil

w

What makes the lipids form vesicles instead of sheets?

• Answer: edge tension, λ (units: J/m)• Experimental values:

for lecithin bilayers -- 4x10-11 J/mfor pure SOPC -- 9x10-12 J/m

Will the sphere always win?

Rv

2Rv

radius of disk = 2Rv in orderto conserve surface area

3. Lipids, membrane mechanics (RD)

Microcalorimeters:

Features:

heat is supplied at the same rate to the 2 cells (applied current to heaters)

solution cell generally absorbs more ⇒ ∆T

feedback loop supply ∆Q to the solution cell, to equalize T

calculate ∆(dH/dt) from difference in power to maintain the same T

Heat capacity ∆CP = ∆Q/∆T Def: heat required to change

the temperature of a substance with 1K, [cal/K]

heat flow is equivalent to enthalpy changes: (dQ/dt)P = dH/dt

∆(dH/dt) = (dH/dt)sample - (dH/dt)reference

CP

time or T

endothermic

Differential scanning calorimetry (DSC): measures heat uptake

Features:

directly measures a thermodynamic property

observes a different signal - just about every reaction has an enthalpic component

measurable quantities: ∆H, ∆S, ∆CP

Def: monitor heat adsorbed vs. temperature: “titrate” with heat

cell withbuffer

insulation

thermocouples

cell withsample

heaters

3. Lipids, membrane mechanics (RD)

DSC Isotherms

CP : shift in the baseline at the startingtransient

glass transitions: cause a baseline shift Def: a rapid change of the specific heat, the coefficient of thermal expansion, the free volume, the dielectric constant

no enthalpy associated with it ⇒ called a second order transition

∆CP = ∆Q/∆T

endothermic

exothermic

3. Lipids, membrane mechanics (RD)

Membrane Composition

• Proteins: 30-75% (usually 50%)• Lipid (25-70%), carbohydrate up to 10% (glycolipids and glycoproteins)• Compositions vary between species, and between cells & tissues of same species

Membrane Functions

• Define the external boundary & internal compartments of cells• Regulate molecular traffic into & out of cells and organelles.• biological energy conservation (ion gradients).• cell-cell communication (receptors, etc.)

Note: Biological membranes have asymmetric distributions of lipid & protein (external ↔ internal) – external PLA2 cleaves sn-2

3. Lipids, membrane mechanics (RD)

Molecular motions of lipids

1. Lateral Diffusion

• Diffusion coefficients range from 10-7 (fast) to 10-10 (slow) cm2/s.• orders of magnitude faster than diffusion across themembrane

• Problem: for D=10-8 cm2/s, what is the distance (in µm) a lipid can traverse in 4s?

2. Transbilayer movement (flip-flop)

• Very slow process (energetically unfavorable to move headgroup into hydrophobic interior), unless aided by a translocase• For PC t1/2= several days• Flip-flop of some PL (PA and PG) and FA can be induced by a transmembrane pH gradient• Cholesterol will flip-flop by passive diffusion• Proteins not at all

Membrane permeability – next lecture

3. Lipids, membrane mechanics (RD)

Membrane mechanicsTopics overview:

1) Classical modes of deformation 2) Membrane elasticity

(essential for lipid-protein interactions, behavior of mixtures, lyotropic phase transitions e.g. bilayer -> inverted hexagonal)

stretching elasticitymembrane curvature, bending elasticitylysismembrane elasticity measurements

micropipettes, fluctuation analysis, measurements on red blood cellseffect of cholesterol

3) Shear viscosity of membranes(essential for mobility and activity of membrane proteins, membrane diffusion)

Classical modes of deformation

stretching elasticity [dyn/cm]

dilatational surf.viscosity [sp]

bending stiffness [erg]

bending surf. viscosity [sp]

shear elastic modulus [dyn/cm]

shear surface viscosity [sp]

intermonolayer friction [dyn.s/cm3]

Stretching BendingShear

bilayer membrane

Intermonolayer slip

, - essential for elasticity of membranesin fluid state

, , , - to be considered when dealing witherythrocytes, membranes frompolymerized lipids

3. Lipids, membrane mechanics (RD)

Stretching elasticity of membranes

( )2

21 AAKgext δ=Elastic energy density:

Isotropic lateral extension:A

A+δA

τ

τ

τ

τ

K – area compressibility modulus

Methods for measuring K :- micropipette technique - photon correlation spectroscopy, cryoelectron microscopy

and DLS of small vesicles subjected to osmotic stress- NMR and X-ray diffraction of strongly dehydrated

multibilayer arrays

AAK δ=τMean tension:

Cells under tension: Pressure can come from osmotic effects:

∆π=RTΣCs

Or from mechanical effect such as microtubilipolymerisation during cell division

Increase suction pressure, P

best resolution (better than 0.1% for δA); can test reversibility, instantaneous tension area dilation at rupture

LRR

RAv

pp ∆

−π≅δ 125µm

∆L

2Rp

2Rvvp

p

RRPR22 −

=τ

0

10

20

0.00 0.02 0.04AAδ

τ, dyn/cm

Example:(polymer membrane)

Typical values for lipids: K = 100 ÷ 200 dyn/cm

For cholesterol containing membranes: e.g. SOPC:cholesterol=1:1 → K ≈ 850 dyn/cmerythrocyte lipid extract (50% cholesterol)

→ 750 dyn/cm

For sheets (5nm thick) ofrubber: 100 dyn/cmpolyethylene: 5000 dyn/cmsteel: 107dyn/cm

Evans, Rawvicz, PRL, 64, 2094 (1990)Helfrich, Servus, Nuovo Cimento, D3, 137 (1984)

Micropipette technique:

Red blood cell in a pipette at different pressures

3. Lipids, membrane mechanics (RD)

Bending elasticity of membranes

Bending moment:

−+∆κ= 0

11 CRR

Myx

2

02

2

2

2

21

−

∂∂+

∂∂= C

yu

xugbend κElastic energy density:

M

M

M

M

κ - bending elastic modulus

1) Bending a bilayer with free ends (freely sliding monolayers)= splay of a single smectic layer: κ = Κ11h

h

2

2

2

2 1;1yu

Rxu

R yx ∂∂=

∂∂= - principal radii of curvature C0 – spontaneous curvature:

zΠ(z)

Example for creating intrinsic bending moment

κ=Π= ∫−

00

2/

2/0 ;)( MCdzzzM

d

d

2) Bending closed bilayer i.e. vesicle → additional constraints:fixed volume Vconstant average area ⟨A⟩fixed monolayer area difference ∆A

compression

expansion

∫

+≈∆

211

yx RRdAA h

“bilayer coupling constraint”

global bending modulus, κ‘(similar to κ)

Basic concepts for calculating shapes and shape transitions

of vesicles and cells

Gaussian curvature:

- constant for closed vesicles ⇒ ignored

Important for lyotropic phase transitions and for torus-shaped vesicles, high genus

vesicles

Mutz, Bensimon PRA,43,4525 (1991)

yxG RR

C 12 =

3. Lipids, membrane mechanics (RD)

3. Lipids, membrane mechanics (RD)

Methods for measuring κ :- micropipette technique (most versatile, difficult)- electric field deformation technique (cells, vesicles)- cell poking- fluctuation spectroscopy (measures forces in the femto-Newton range)- new: with differential confocal miccroscopy

Micropipette technique: aspiration at low tensions (entropic regime)

Bending elasticity of membranes (cont.)

τπκ

=δ ln8

TkAA B

-3

0

3

0.00 0.02 0.04AAδ

ln(τ)

Red-blood cell: 12 kBT (?enigma?)

Typical value for κ ≈ 20 kBT

Fluctuation spectroscopy:

ϕ

R(ϕ)

( ) ( ) ( )

πϕ+πϕ+=ϕ ∑∑

nn

nn nbnaRR sincos10

Fourier expansion of the deviations from spherical shape with equivalent radius R0

Döbereiner, et al, PRE, 55, 4458 (1997)

0 200 400 600-0.1

0.0

0.1

0.2

0.3

0.4

a2 - elipticity a3 - assymetry

an

time, sec

κπ=τ

8TkK B

xCrossover tension:

τx ∼ 1dyn/cm

Membrane lysis:Rupture tension (τlysis ) < 10 dyn/cm

(cholesterol increases τlysis)Maximum area expansion: between 2 and 5%

Klysis ∝τ

2n

B aTk

∝κ

3. Lipids, membrane mechanics (RD)

Vesicle at the bottom

Deformation < 1µm

Differential confocal microscopy (Lee, Ling, Wang 2001)

Intercorrelation of the elastic parameters:

From theory of elasticity (for 2d shells) → 2−≈κ hK

2121053 −×÷=κ cmK - stays constant

Effects of various factors on the elastic properties:Cholesterol increases both K and κSmall molecules e.g. peptides, short bipolar lipid (“bola”)

decrease the bending stiffness (possibly due to creating variation in h)

d

L

Undulation forces: membrane close to a surface

For d < L the long wavelength undulations freeze out → interaction potential:

2)(d

dκ

≈ TkV B

Reflection interference contrast microscopy →Radler, et al. PRE, 51,4526 (1995)

3. Lipids, membrane mechanics (RD)

Shear surface viscosity

Energy and force representation of Hook’s law:

( )221 22 −λ−λµ= −

shg

λ - lateral extension ratioconstant area

( )tss ∂λ∂η=τ ln2

Shear surface viscosity measurement:

diffusion of molecular probes (unreliable for frozen bilayers; large scattering of the data) e.g. FRAP

tether formation experiments (gives upper bound)

optical dynamometry

Viscous dissipation in the bilayer - difficult to measure: deformation response is dominated by dissipation in water

tethervesicle

Fluorescence photobleaching recovery (FRP or FRAP)

2D diffusion in cell membranes or in concentrated solutions

Fluorophores: green fluorescent protein (GFP)fluorescent label bound to a protein, nucleic acid, lipid

time

spot bleached by a laserRecovered fluorescence fraction:

ω – Gaussian beam waist

3. Lipids, membrane mechanics (RD)

1. ‘Falling-ball viscosimetry’:measure friction coefficient calculate viscosity

g

vesiclelatexparticle

free bulk bulk + membrane

η

ηS

bRπη=ζ 612.0

93.26−

ηηη+πη=ζ

b

ssb R

R

Membrane shear surface viscosity contribution to friction:

Typical value for ηs :

3 x 10-6 dyn.s/cm (SOPC)

units: [bulk viscosity] x [thickness]~ 5 nm~ 150 cp

(water: 1 cp)

Optical dynamometry

2. Brownian motion

kBT

ζTBkD =

3. Optical trap dynamometry

FRPradiation pressure

force

xxkRP&ζ−=0