Passive
transport
across the
lipid bilayer
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.