Passive

transport

across the

lipid bilayer

Membrane permeability to nonelectrolytes

Steps (any can be rate limiting)

1)

enter the membrane (potential barrier)

2)

diffusion through the bilayer core

3)

exit the membrane (potential barrier)

kT

E

a

e

P

P





0

E

a

correlates to the number of

H-bonds a permeant molecule
can form.

N is Avogadro's number and

f is the frictional coefficient

Nf

RT

D 























d

d

dx

dx

dC

D

dx

J

J

0

0

'

Diffusion of non-

electrolytes





mo

mi

C

C

D

J







Steady-state

flow

J = constant

Molecules in the

aqueous phase are

in equilibrium with

molecules in the

membrane phase.

dx

dC

D

J 



The chemical potential in the water phase (µ

w

) = the

chemical potential in the membrane (µ

m

):

m

o

m

m

w

o

w

w

C

RT

C

RT

ln

ln



















The concentration at the
surface of the membrane
(C

m

)

















RT

C

C

o

m

o

w

w

m





exp





o

i

o

m

o

w

C

C

RT

d

D

J

























exp

The

permeability

coefficient





























RT

d

D

P

o

m

o

w





exp



















RT

C

C

K

o

m

o

w

eq

m

p





exp

The membrane:water partition
coefficient (K

p

)

C

m

– concentration just inside

the hydrophobic core of the
bilayer,
C

aq

– concentration in the

aqueous solution.

d

DK

P

P







d

z

D

z

K

dz

P

0

)

(

)

(

1

For a non-

uniform

membrane

K(z) – the depth-dependent
partition cofficient from water
into the membrane
D(z) – the depth-dependent
diffusion coefficient in the
membrane.
„d” – the membrane thickness.

Permeability

dependence on

temperature





o

i

p

C

C

K

d

D

J







P in membranes is

strongly

correlated with K

p

for nonpolar

solvent

Molecule diffusion

across the aqueous

layers adjacent to

either surface of

the membrane.

Unstirred Layers

1 µm to 500 µm

thickness

.

For water soluble compounds – diffusion across the

unstirred layers will have relatively less effect.

It is most prominent for relatively nonpolar

compounds – the diffusion across the membrane
itself will be relatively fast.

P – a membrane

permeability coefficient

D – an aqueous

diffusion constant.

d

i

and d

o

– the

thicknesses of unstirred
layers.

C

i

and C

o

– the bulk concentrations of the

compound,

C

mi

and C

mo

– the concentrations at the surface of

the membrane.



The flow through the membrane is





mo

mi

m

C

C

P

J







The flow through the unstirred layers





mi

i

i

i

C

C

d

D

J









o

mo

o

o

C

C

d

D

J





Therefore

mo

mi

C

C

P

J





mi

i

i

C

C

D

Jd





o

mo

o

C

C

D

Jd





After summing:

o

i

o

i

C

C

D

d

D

d

P

J





















1

The effect of unstirred layers is to decrease the
permeability so the apparent permeability
coefficient (P

app

) is smaller than P:

D

d

D

d

P

P

o

i

app







1

1

At steady-state

J

J

J

J

o

i

m







It is entropic in nature

Osmosis

The osmotic pressure

difference can only arise if

there is a physical object,

the

semipermeable membrane

,

present to apply force to the

solute particles.

Semipermeable

membrane is a

thin, passive

partition

through which

solvent, but not

solute, can pass

and the fluid is

incompressible.

Osmotically active = solutes which

Osmotically active = solutes which

can’t

can’t

diffuse through the

diffuse through the

semipermeable membrane.

semipermeable membrane.

Hypotonic

solution

Solute

molecule

HYPOTONIC SOLUTION

Hypertoni

c solution

Selectively

permeable
membrane

HYPERTONIC

SOLUTION

Selectivel

y

permeabl

e

membran

e

NET FLOW OF WATER

Solute molecule with

cluster of water

molecules

Water

molecule

Osmotic pressure:

Osmotic pressure:

force required to

force required to

prevent

prevent

osmosis.

osmosis.

Easy way to

Easy way to

measure

measure

osmolality:

osmolality:

Each Osm (of any

Each Osm (of any

solute) lowers the

solute) lowers the

freezing point of

freezing point of

water by ~ 2

water by ~ 2

o

o

C

C

Chemical Potential of Water

w

w

w

w

PV

X

RT







ln

0





µ

w

0

– standard chemical

potential of water

X

w

– molar fraction of water

P – pressure

V

w

– molar volume of water

Solutes Decrease the
Chemical Potential of
Water

0

ln

1







w

w

X

X

Semipermeable

membrane

1

)

2

(

1

)

1

(







w

w

X

X

Addition of an impermeable
solute to one compartment
drives the system out of
equilibrium.

)

2

(

)

1

(

)

2

(

ln

)

1

(

ln

w

w

w

w

X

RT

X

RT











There is a net

water flow

from

compartment

(2) to

compartment

(1).

Osmotic Equilibrium

)

2

(

)

1

(

w

w







At the equilibrium the
chemical potential of any
species is the same at every
point in the system.

w

w

w

w

V

P

X

RT

V

P

X

RT

w

w

)

2

(

)

2

(

ln

)

2

(

)

1

(

)

1

(

ln

)

1

(

0

0















w

w

w

V

P

V

P

X

RT

)

2

(

)

1

(

)

1

(

ln





)

1

(

ln

w

w

X

RT

PV







1

)

2

(

1

)

1

(







w

w

X

X

1





s

w

X

X

Solute

molar

fraction

in

physiological (dilute) solutions
is much smaller than water
molar fraction.

1



s

X

s

s

w

X

X

X









)

1

ln(

ln

s

w

RTX

PV 



s

w

X

V

RT

P 







Osmotic pressure

s

w

X

V

RT

P 







w

s

w

tot

s

w

w

w

s

w

s

w

s

s

s

V

C

V

V

n

V

V

n

n

n

n

n

n

n

X













Solute concentration (~0.1M) in
physiological (dilute) solutions is
much

smaller

than

water

concentration (55M).

w

s

n

n 

s

w

s

w

RTC

V

C

V

RT

P











vant’Hoff’s law

(the osmotic

pressure)

The osmolarity of a solution is equal to
the molarity of the particles dissolved in
it

.

3.

10 mmoles/liter of CaCl

2

= ???

2.

10 mmoles/liter of NaCl = 20 mosmoles/liter.

1.

10 mmoles/liter of glucose = 10 mosmoles/liter.

In a simple solutions the effect is additive.

Osmotic machine

Osmotic Flow

Water flows from the solution with a
low osmotic pressure to the solution
with a high osmotic pressure

.

0







P

At equilibrium

s

C

RT

P











Reverse osmosis

Reverse Osmosis is

Reverse Osmosis is

Used for Water

Used for Water

Purification

Purification

Osmotic pressure creates a depletion force

between large molecules

The depletion interaction –
molecular crowding



Each of the large objects is surrounded by a

depletion zone of thickness equal to the radius a of the
small particles – the centers of the small particles
cannot enter this zone.



The depletion zone reduces the volume available to

the small particles – eliminating it would increase
their entropy and hence lower their free energy.

The depletion interaction is short

range (<2a)

It is a measure of

the probability of the molecule

crossing the membrane

.

σ – selectivity/reflection coefficient

The

osmotic

pressure

gRTC





The effective osmotic

pressure depends on the

reflection coefficient:

gRTC

ef













non-

selective

membran

e

semiperme

able

membrane















P

L

J

P

V

Bulk
flow

Important summary points about osmosis

1.

The steady-state volume of the cell is

determined by the concentrations of impermeant
ions.

2.

Permeant solutes redistribute according to the

rules of electrodiffusion, and hence affect only the
transient volume of the cell.

3.

The more permeant

the solute, the more
transient its effects on
volume.

Volume

regulation of

living animal

cells

Ti
me

C

h

a

n

g

e

o

f

c

e

ll

vo

lu

m

e

Δ
V

Swelling

(water

uptake)

Volume regulation

Ion transport,

release of isotonic

solution

Osmoconformers

– animals, like sea slugs, that allow

the osmolarity of their internal environment to change
with that of the external environment.

Response to shrinking

Osmoregulators

– animals that do not allow the

osmolarity of their internal environment to change.

The activation energy (E

a

) required for water

diffusion in an entirely aqueous environment –

5

kcal/mol

.

The activation energy (E

a

) required for water

diffusion through the lipid bilayer –

10-20

kcal/mol.

always passive; bidirectional;

osmosis-driven

Water Transport Across Cell Membrane

Diffusion through lipid
bilayers

slow, but enough for many
purposes

Channel-
mediated

Fast adjustment of water

concentration is necessary (RBC,
brain, lung).

Large volumes of water

needed

to

be

transported

(kidneys).

Aquaporins in the Kidney

It filters and eliminates toxic substances
from the blood.

• To maintain water
balance, > 99% of
water is reabsorbed
before it leaves the
kidney as urine.

•

Adult

human

kidneys filter >150 l
of blood each day.

This is achieved by the filtration of blood in

nephrons, which have important functions in the

reabsorption of water, active solute transport and

acid–base balance.

Aquaporin

Bacterial

Water

Channel

Aquaporin-1

Cryo-electron
microscopy

maps

of

water channel proteins
(viewed

from

cytoplasmic side).

Red blood

cell water

channel

AQP1

The lens

fiber water

channel

MIP or

AQP0

The

bacterial

water

channel

AqpZ

The AQP1 tetramer

Membrane permeability to ions

The energy needed to move an ion into

the membrane lipid phase is nearly 100

kT.































w

hc

B

r

q

E







1

1

8

0

2

Image forces

reduce ΔE

B

by

10 - 15%

but

cations

times

anions

times

molecules

neutral

P

P

P







1000

-

20

10

8

Due to the

internal

membrane

potential ~

+240 mV

(dipole

potentiaol).

Born energy concept makes no difference between „-” and „+”

The ion transport

across membrane

depends on:

Free energy

profile

Electric

potential

Concentration

profile

The movement of ions.

Diffusion

dx

dc

D

J





Electrophoresis

dx

d

uc

J







dx

d

uc

dx

dc

D

J









zF

uRT

D 

Nerst-Einstein relation

JzF

I 

Faraday’s constant

The current across the
membrane

dx

d

c

RT

DzF

zF

dx

dc

zFD

I























dx

d

c

RT

zF

dx

dc

zFD

I



Nerst-Plank relationship

the Nernst Equation

Integrating
across the
membrane

i

o

o

i

c

c

zF

RT

]

[

]

[

ln









At equilibrium

0



I

dx

d

dx

dc

c

zF

RT







1

(C/cm

2

) · surface area/(V · Faraday constant) =

3.7 ·10

-18

moles

This is only 1 out of ~ 30,000 K

+

in the cell.

How much K

+

would flow out to establish a V

m

of -60

mV in an idealized cubic cell with edge dimension of
10µm and a membrane permeable only to K

+

?

Equivalent cirquit

m

CV

Q 

When the voltage across the

capacitance changes a

current will flow.

dt

t

dV

C

I

m

C

)

(



The battery

– the resting

potential

V

rest

.

The resistance –

dissipative current across

the membrane.

Capacitance of biological

membranes ≈ 1 µF/cm

2

.

The value of V

m

is independent of

the concentration or voltage profile

within the membrane!

Comments

Nerst equation gives the value of

membrane potential



nerst

at which

the ion is in steady-state

equilibrium.

At this value of



nerst

, the

electrostatic energy per mole

(zF



m

) is exactly counterbalanced

by the chemical energy per mole

(RTln(c

i

/c

o

)).











 



RT

zF

c

c

nerst

o

i



exp























































dx

d

RT

zFC

dx

dC

D

zF

C

RT

dx

d

Nf

C

J

o







ln

Permeation of electrolytes













RT

zF

RT

zF

e

dx

d

RT

zFC

dx

dC

Ce

dx

d

/

/



















exp(zF/RT)

can

be

an

integrating factor

dx

Ce

d

D

Je

RT

zF

RT

zF



























zF

C

RT







ln

0

Electrochemical potential

If J is constant across the
membrane:





RT

zF

mo

RT

zF

mi

d

RT

zF

mo

mi

e

C

e

C

D

dx

e

J

/

/

0

/















C

mi

and C

mo

are related to the concentrations C

i

and

C

o

in the bulk aqueous phases by the

membrane:water partition coefficient (K

p

) and by

the surface potential.













RT

zF

K

C

C

mi

i

p

i

mi





exp













RT

zF

K

C

C

mo

o

p

o

mo





exp





RT

zF

o

RT

zF

i

p

d

RT

zF

o

i

e

C

e

C

DK

dx

e

J

/

/

0

/



















RT

zF

i

d

RT

zF

p

m

e

C

C

dx

e

d

d

DK

J

/

0

0

/









































RT

zF

C

C

PQ

J

m

i



exp

0



m

is the membrane potential

Q

P

At

constant

field.

)

0

(

)

(









 Ex

x

If



(0) =

0

Ed

d

m







)

(





1

)

/

exp(

/





RT

zF

RT

zF

Q

m

m





1)

If the flow J is zero

we have the

Nernst

equation

:











 



RT

zF

C

C

m

o

i



exp

)

(

i

o

C

C

P

J





2)

If 

m

= 0, Q = 1 we

have the Fick's Law:

3)

If there is no

concentration gradient
(C

i

/C

o

= 1):

RT

PCzF

J

m







Since I = -JFz, the
equation is equivalent
to Ohm's Law:

m

g

I





The
conductance.

RT

F

PCz

g

2







RT

zF

i

m

m

m

e

C

C

RT

zF

RT

zF

P

J

/

0

1

)

/

exp(

/

























A final expression for the flow

The concentration equalization can be
circumvented when:

1)

The transported substances may be bound

by a macromolecule inside the cell, e.g., O

2

binding by hemoglobin.





















zF

C

C

RT

G

1

2

ln

2)

There are a membrane

potential which influences the
distribution of ions.

3)

There are thermodynamically favorable

process which are coupled to transport – active
transport.

Equilibria of weak acids and weak bases

At neutral pH, weak acids and weak bases are

predominantly in their charged forms (A

-

and BH

+

).

The charged species do not permeate across the

membrane’s hydrophobic barrier.

The charged species are in equilibrium with

uncharged species that will permeate the
membrane.

The uncharged species (B) will reach the

equilibrium (B

o

= B

i

).

Unprotonated species will
be in equilibrium with the
protonated form:

i

i

i

o

o

BH

H

B

BH

H

B

K

]

[

]

[

]

[

]

[

]

[

]

[

0













Since [B]

o

= [B]

i

o

i

o

i

H

H

BH

BH

]

[

]

[

]

[

]

[











For a weak

base









BH

H

B

For a weak

acid.

i

o

o

i

H

H

A

A

]

[

]

[

]

[

]

[











A

H

A









Evidence for protein-mediated

transport

Permeability Coefficients of Natural and

Synthetic Membranes to D-Glucose and D-

Mannitol at 25

o

C

Passive transport of polar molecule

through a transport protein - no

energy used.

This includes water, sugars, amino acids, ions.

 Like enzymes

- bind and transport substrate

molecules, ONE at a time.

Properties of facilitated

transport

 Passive

– down concentration gradient -

energy-independent.

 Specific


Dependent
on
temperature



A rate of solute movement across the

membrane is

saturable

.

 Can be inhibited

 Fast –

the flow may approach diffusion limit

e.g. 10

7

ions/sec.

Amino acid residues of the

transporter interact with

"dehydrated" solute



Molecule must shed

their water of hydration

before they can cross

the membrane

Forming hydrophilic

passageway or package

through membrane



Reduce energy

barrier

n

i

is the stoichiometry for transport

of substrate i.

















































i

i

i

m

i

out

in

i

out

i

in

i

i

i

i

F

z

C

C

RTin

n

n

n

G









They are hydrophobic compounds which can

complex an ion and carry it across a lipid

bilayer.

Classification of ionophores

neutral ionophores

(e.g. Valinomycin)

carboxylic ionophores

(e.g.

Nigericin)

protonophores

Ionophores

Small agents produced by microorganisms to kill

other microorganisms

Two basic types: mobile

carriers & channels

 Pores are not affected by temperature.

 Carriers depend on the fluidity
of the membrane, so transport
rates are highly sensitive to
temperature, especially near the
phase transition of the membrane
lipids

At the
equilibrium



















in

out

m

H

H

F

RT

]

[

]

[

ln



Carbonylcyanide m-chlorophenyl

hydrazone (CCCP)

Protonophores

2,4-Dinitrophenol
(DNP)

m

out

in

F

H

H

RT

G

























]

[

]

[

ln

Both DNP and CCCP have a dissociable

proton (weak acids) and are hydrophobic.

The pH gradient provides a proton

motive force that is used to convert ADP

to ATP.

With

protonophores,

electron transport

proceeds (NADH is

oxidized, O

2

is

reduced) but no

ATP is synthesized.

The mitochondria

are

uncoupled

.

Uncouplers dissipate

the proton gradient.

Valinomycin

– neutral

ionophore

Relatively slow rate of K

+

transfer, 10

3

K

+

/sec per

molecule

Binds K

+

in

central cavity by
C=O
coordination,
shields charge.

The

valinomycin

surronds the

potassium ion with a

hydrophobic surface which

allows the ion to cross the

membrane.

It depends on the

membrane potential.

It creates a membrane

potential

by

transporting
capacitative charge.

It

crosses

the

membrane either with
or without a bound ion.

K

+

The selectivity of valinomycin for K

+

K

+

binds tightly, but affinities for Na

+

and Li

+

are

about a 10 000 -fold lower.

Factor 1

: Ionic radius (K

+

> Na

+

> Li

+

)

The smaller Na

+

ion cannot simultaneously interact

with all 6 oxygen atoms within valinomycin (Na

+

=

0.95 Å, K

+

= 1.32 Å).

Factor 2

: desolvation energy: water molecules

surrounding the ion must be stripped off before it
binds to the carrier:

Li

+

(aq)  Li

+

+ nH

2

O ΔG = 410 kJ/mol

Na

+

(aq) Na

+

+ nH

2

O

ΔG = 300 kJ/mol

K

+

(aq)  K

+

+ nH

2

O

ΔG = 230 kJ/mol

It "costs more" energetically to desolvate Na

+

and Li

+

than K

+

It is a K

+

/H

+

exchanger.

Nigericin does not carry a net charge across the

membrane

m

out

in

m

out

in

F

K

K

RT

F

H

H

RT

G















































]

[

]

[

ln

]

[

]

[

ln

G = 0

Nigericin will reach

equilibrium when the

[H

+

] and [K

+

] gradients

are proportional.

out

in

out

in

K

K

H

H

]

[

]

[

]

[

]

[











It has linear structure

with a carboxyl group on
one end and hydroxyls on
the other.

The carboxylic ionophores -Nigericin

It cyclize by head-to-tail

hydrogen bonding and will
cross the membrane with
the carboxyl group either
protonated or complexed to
an ion.

Alamethicin – A

Weakly Selective

Channel



Multi-conductance level channels,



Rapid switching between conductance
levels,



Weakly cation selective (ca. 4:1
cations:anions)

Unusual wide helix in membrane - 6.3

residues per turn with a central hole - 4 Å
diameter (a 

6:3

helix, NOT an -helix)

In a in lipid bilayer is an end-to-end

helical dimer

Gramicid

in

15 aa hydrophobic peptide

HCO-Val-Gly-Ala-D-Leu-Ala-D-Val-Val-D-

Val-Trp-D-Leu-Trp-D-Leu-Trp-D-Leu-Trp-

NHCH

2

CH

2

OH

The pore is  28 Å long and 4

Å in diameter when a dimer
forms.

The pore is lined by

backbone amide groups and
permits the transmembrane
flux

of

small

monovalent

cations (Na

+

, K

+

, H

+

).

Gramicidin pore

Channels constantly assemble

and dissociate (lifetime ~1 sec)

At high [gramicidin] overall

transport

rate

depends

on

[gramicidin]

2

.

The rate of K

+

transfer is 10

7

K

+

/sec

per

channel.

Single-channel

current trace

obtained with the

gA in a (DPhPC)/n-

decane bilayer.

cu

rr

en

t

time

ion flow

through one
channel

Characterization of

gramicidin channel.

Current transition

amplitude histogram.

Normalize

d survivor

plot

The average

lifetime

(1410 ms).

N(t) is the

number of

channels with

lifetime longer

than time t.