Molecular and Thermodynamic Explanations of Ion Motion — page 1.01
In a solution, the solute is randomly distributed within the three-dimensional space
How solutes move within the space depends
upon the random walk of the molecules. To
understand this, it is crucial to first create a
framework to describe the quantitative movement of the molecules. We can think about the
molecules moving in various directions within
the medium.
The quantitative descriptor commonly used is
the flux (J), defined as the amount of molecules that pass through a specified area in a
specified time.
Concentration (c)
The flux, J, will depend upon the concentration of the substance. For the diagram of
solutes in solution within an x,y,z coordinate
system (above), we can determine the
concentration within small volume elements
in adjacent regions. The flux will depend
upon the gradient of solute concentration
between the two volume elements.
⎛ Δc ⎞
J ≈ −⎜ ⎟
⎝ Δx ⎠
Δc
J=
mol of solute
cm2 • sec
The –ve sign indicates that the flux
occurs from regions of higher to
lower solute concentration.
Δx
Distance (x)
Molecular and Thermodynamic Explanations of Ion Motion — page 1.02
Concentration (c)
More accurately, rather than approximating flux, J, as proportional to ∆c/∆x, it is proportional to a
point tangent to the concentration versus distance curve: That is, J is proportional to ∂c/∂x (the
partial derivative of concentration, c, with respect to the distance, x).
⎛ ∂c ⎞
J ≈ −⎜ ⎟
⎝ ∂x ⎠
Δc
Δx
Distance (x)
The units of the derivative, ∂c/∂x are (mol cm–3)/(cm), or mol cm–4. To arrive at the units of
flux, J (mol cm–2 sec–1), a coefficient with units of cm2 sec–1 must be included:
J = −D •
dc
cm2 mol
mol
⇒(
)( 4 ) ⇒(
)
sec cm
dx
sec• cm2
D is known as the Diffusion coefficient. The above equation is known as Fick's Law of Diffusion.
It follows a general form:
Flux = (Conductance to Flux)•(’Driving Force’).
Other physical relations follow a similar form. Electrical current, for example:
Current (I) = Conductance (g) • Voltage (V) [I=gV]
and water flow:
Flow (J) = Hydraulic Conductivity (L) • Pressure (P) [J=LP]
Summary: A formal description of molecular movement of molecules in solution relies
upon a framework to describe the quantity of molecules which pass across a region of
specified area during a defined period of time. To standardize units, a coefficient must be
introduced. In its final form, The equation, Fick's Law, is seen to be very similar to other
formal descriptions of flow, either current or mass flow of water.
So far, the description is phenomenological. A mechanistic explanation requires closer
examination of the movement of the molecule of interest.
Molecular and Thermodynamic Explanations of Ion Motion — page 1.03
Berg[1] uses an approach that follows a mechanistic explanation originally proposed by Einstein. It
starts with a one-dimensional case, with N(x) particles at x and N(x+δ) particles at x+δ along a line.
The symbol, δ, refers to a small distance away. How many particles will move across the boundary
from point x to point x+δ in a given
N(x)
N(x+δ) number of particles
time? If the probability for a particle
to move to the left is the same as the
probability to move to the right, then
at time t+τ, half the particles at x will
have moved to x+δ, and half the
x
x+δ
distance
particles at x+δ will have moved to x.
1/2N(x)
N(x+δ)
N(x)
1/2N(x)-1/2N(x+δ)
= -1/2[N(x+δ)-N(x)]
x
x+δ
1/2N(x+δ)
The net number of particles going from x to x+δ will be -1/2[N(x+δ) - N(x)], and the flux, J
(obtained by dividing by area and by time) will be:
Jx = −
[
]
1
N(x +δ )– N(x) / Aτ ,
2
δ2
multiplying by 2
δ
δA has units of volume, and
1δ 1
Jx = –
N(x +δ )– N(x)
2 δ 2 Aτ
1δ2 1
Jx = −
C(x +δ )− C(x)
2τ δ
re-arranging
re-arranging
1 δ 2 1 N(x +δ ) N(x)
Jx = −
−
2τ δ
δA
δA
1 δ C(x +δ )− C(x)
Jx = −
δ
2τ
2
[
[
[1]
N divided by volume is concentration
]
]
[
2
[
Berg, HC (1998) Random Walks in Biology. Princeton University Press. pp.17–21.
]
]
Molecular and Thermodynamic Explanations of Ion Motion — page 1.04
Continuing Berg’s approach[1]:
1δ2 1
Jx = −
C(x +δ )− C(x)
2τ δ
[
N(x)
N(x+δ)
number of particles
x
x+δ
distance
]
If we take the term
[C(x +δδ)– C(x) ]
to the limit δ → 0, then
C(x +δ )– C(x) ∂C
=
∂x
δ
therefore
Jx = –
1 δ 2 ∂C
2 τ ∂x
or (Einstein):
1 Δ2 ∂C
J =− • •
2 τ ∂x
These are the same form as Fick’s Law of
Diffusion (J = D • ∂C/∂x)
1 Δ2
is the Diffusion coefficient, D
2τ
cm2
with units of:
sec
where
1 Δ2
D= •
The molecular definition of the diffusion coefficient:
2 τ
can be recast to show that the average displacement, ∆,
is a function of the square root of time:
This prediction was used to verify
Δ = 2• D•τ
Einstein’s theory of Brownian
Motion. Since then, random walks have, in one form or another, permeated biophysical research.
[1]
Berg, HC (1998) Random Walks in Biology. Princeton University Press. pp.17–21.
Molecular and Thermodynamic Explanations of Ion Motion – page 1.05
Typical values for diffusion coefficients are shown below for assorted simple
compounds in air (viscosity 1.813 • 10–5 N m–2 sec) and water (viscosity 1.0002
• 10–3 N m–2 sec).
Compound
MW
Hydrogen
Helium
Oxygen
Benzene
2
4
32
78
H+
K+
Ca2+
Cl–
H 2O 2
1
39
40
35
34
Diffusion Coefficient (m2 sec–1)
Air
Water
6.11•10–5
4.50•10–9
–5
6.24•10
6.28•10–9
–5
1.78•10
2.10•10–9
–6
9.60•10
1.02•10–9
9.31•10–9
1.96•10–9
0.40•10–9
2.03•10–9
1.30•10–9
Generally, diffusion coefficients depend upon the molecular weight (size) of the compound, but clearly additional factors can affect the value. For ions, hydration shells are a
mejor determinant of differences in the diffusion coefficient, since they can affect the
apparent size of the molecule dramatically.
Molecular and Thermodynamic Explanations of Ion Motion — page 1.06
To describe the movement of molecules through a membrane, we need to consider a more
complex framework. We still use the general form of the flux equation: J=D•dc/dx, but a
diffusion coefficient alone is insufficient.
co (outside concentration)
ci (inside
concentration)
Flux will depend upon the ability
of the particle to enter the membrane
(partitioning)
d (distance)
c
Partitioning, K p = c(membrane )
(aqueous )
so the flux is now described by:
Kp
[c
– cinside ]
d outside
K
where D p = P, the permeability coefficient with
d
cm2
units of sec , or cm•sec-1
cm
J=D
There is a classic literature on the permeability of membranes. Much of the original work
was done on giant freshwater algae. Historically, this research led to the proposal that cells
are bounded by a lipoidal membrane, because permeability matches closely the partitioning
of substances between olive oil and water
Molecular and Thermodynamic Explanations of Ion Motion – page 1.07
Typical values for permeability coefficients
Membrane permeabilities of selected solutes in Chara, Nitella, human erythrocytes, and
artificial membranes1.
Solute
molecular olive oil :
weight
water
partition
coefficient
water
18
1.3•10-4
formamide 45
1.1•10-6
ethanol
46
3.6•10-2
ethanediol 58
4.9•10-4
butyramide 87
1.1•10-6
glycerol
92
7.0•10-5
erythritol
122
3.0•10-5
Chara
Nitella
Human
Artificial
ceratophylla mucronata erythrocyte lipid
membrane
2.5•10-3
2.2•10-5
1.6•10-4
1.1•10-5
5.0•10-5
2.0•10-7
1.2•10-3
7.6•10-6
5.5•10-4
-5
1.4•10
3.2•10-9
2.5•10-3
1.1•10-6
2.1•10-3
2.9•10-5
1.1•10-6
1.6•10-7
6.7•10-9
2.2•10-3
1.0•10-4
8.8•10-5
5.4•10-6
1
compiled by Weiss TF 1996 Cellular Biophysics. Volume I: Transport. MIT Press.
Original citations are Collander R 1954 The permability of Nitella cells to nonelectrolytes. Physiol. Plant. 7: 420–445, and Stein WD 1990 Channels, Carriers and
Pumps. Academic Press.
formamide
ethanediol
glycerol
erythritol
butyramide
For comparison, permeability coefficients for ions are much lower. In an
artificial membrane:
Na+
10-11 to 10-14 cm/sec
Cl10-11
+
H /OH
10-4 to 10-8
In general, neutral solutes are relatively permeable, depending upon molecular
weight and their ability to partition into a hydrophobic environment. Charged
molecules are barely capable of partitioning into hydrophobic enviroments.
H+/OH- is a notable exception among charged molecules.
Molecular and Thermodynamic Explanations of Ion Motion – page 1.08
Permeability of Chara cells. The compounds, molecular weight, permeability, and oil/water partition
data are shown. From Collander, R. (1954) The permeability of Nitella cells to non-electroytes,
Physiol. Plant. 7:420-445.
Deuterium hydroxide
Ethyl acetate
Methyl acetate
sec.-Butanol
Methanol
n-Propanol
Ethanol
Paraldehyde
Urethane
iso-Propanol
Acetonylacetone
Diethylene glycol monobutyl ether
Dimethyl cyanamide
tert-Butanol
Glycerol diethyl ether
Ethoxyethanol
Methyl carbamate
Triethyl citrate
Methoxyethanol
Triacetin
Dimethylformamide
Triethylene glycol diacetate
Pyramidone
Diethylene glycol monoethyl ether
Caffeine
Cyanamide
Tetraethylene glycol dimethyl ether
Pinacol
Diacetin
Methylpentanediol
Antipyrene
iso-Valeramide
1,6-Hexanediol
n-Butyramide
Diethylene glycol monomethyl ether
19
88
74
74
32
60
46
132
89
60
144
162
70
74
148
90
75
276
76
218
73
234
231
134
194
42
222
118
176
118
188
101
118
87
120
25000
25000
9300
5700
7200
5500
12000
5200
3800
7500
2600
1900
1900
2300
1800
1600
2400
990
1100
705
661
655
406
357
292
285
229
209
191
192
182
177
139
134
0.0007
2.5
0.43
0.25
0.0078
0.13
0.032
1.9
0.074
0.047
0.081
0.12
0.073
0.23
0.11
0.019
0.025
0.5
0.0056
0.44
0.0049
0.033
0.26
0.006
0.033
0.0045
0.0056
0.071
0.024
0.032
0.023
0.0068
0.0095
0.0042
Trimethylcitrate
Proprionamide
Formamide
Acetamide
Polyethylene glycol monoethyl ether
Succinamide
Glycerol monoethyl ether
N,N-Diethyl urea
1,5-Pentanediol
Dipropylene glycol
Glycerol monochlorhydrin
1,3-Butanediol
2,3-Butanediol
1,2-Propanediol
N,N-Dimethyl urea
1,4-Butanediol
Ethylene glycol
Glycerol monomethyl ether
N,N-Dimethyl urea
1,3-Propanediol
Ethyl urea
Polyethylene glycol diacetate
Thiourea
Diethylene glycol
Methyl urea
Urea
Triethylene glycol
Polyethylene glycol diacetate
Tetraethylene glycol
Dicyanodiamide
Hexanetriol
Hexamethylene tetramine
Polyethylene glycol monoethyl ether
Glycerol
Pentaerythritol
234
73
45
59
200
99
120
116
104
134
110
90
90
76
88
90
62
106
88
76
88
380
76
106
74
60
150
480
194
84
134
140
400
92
136
121
79
76
66
66
54
40
39
34
31
30
24
21
17
15
14
12
12
12
10
6.6
6.3
3.6
3.8
3.2
1.3
1
0.8
0.71
0.46
0.42
0.39
0.15
0.032
0.002
0.047
0.0036
0.00076
0.00083
0.0049
0.0074
0.0076
0.0061
0.002
0.012
0.0043
0.0034
0.0017
0.0023
0.0021
0.00049
0.0026
0.0017
0.0012
0.00044
0.00015
0.00047
0.00021
0.00007
Permeability (cm/sec)(X107)
100000
10000
1000
100
10
1
0.1
0.01
0.001
0.00001
0.0001
0.001
0.01
0.1
1
10
100
Oil/water partition (left, squares) or Molecular weight (right, circles)
1000
Molecular and Thermodynamic Explanations of Ion Motion – page 1.09
For all cells, plant, animal, bacteria etc., charged solute concentrations vary widely
between the inside and outside environment:
Animals
intracellular
extracellular
K+
high
low
Na+
low
high
Ca2+
low
high
Em (milliVolts)
-60 mV
0 mV
high
low
varies
low
low
high
-180 mV
0 mV
Plants
intracellular
extracellular
The presence of a voltage difference (Em) affects ion movement, and therefore must be considered an
additional driving force affecting flux, J. Thus, concentration differences, per the membrane flux
equation, J = P•(ci-co) (one solution of the basic equation J=P•dc/dx), are insufficient. To include the
electrical potential, we need to consider a more complete description of the energy potential of the ion.
To do this, we use a concept called the chemical potential: μ, where flux, J, is proportional to the
chemical potential gradient ∂μ/∂x. That is, ∂μ/∂x replaces the concentration gradient, ∂c/∂x.
The equation
J = –mobility (cm sec ) • activity(mole cm3 ) • driving force (dμ dx )
arises from an approach that considers the mobility times concentration (units of moles/cm2 sec), where
the mobility is the velocity per unit force, and force is the chemical potential gradient (∂μ/∂x). The
activity is used rather than concentration in more rigorous derivations, to account for the non-ideal
nature of solutes in solution.
chemical potential
J = –u • c • dμ dx
mobility
concentration
This derivation is taken from Schultz, SG. 1980. Basic Principles of Membrane Transport.
Cambridge University Press. It is a 'classic' derivation, relying upon thermodynamics.
Molecular and Thermodynamic Explanations of Ion Motion – page 1.10
From thermodynamics, the chemical potential is:
μ = RT(ln c)+ zFΨ
so,
1
dμ = RTd(ln c)+ zFdΨ = RT dc + dΨ
c
RT dc
dΨ
d
of dμ =
+ zF
c dx
dx
dx
R is the gas constant
(8.314 J mol–1 K–1)
T is the temperature °K
c is the concentration
z is the valence (ionic charge)
F is the Faraday constant
(9.649 • 104 J mol–1 V–1)
ψ is the electrical potential
so,
J = −u • c •
[ RTc dcdx + zF dΨ
]
dx
J = −(uRT )(
dc
dΨ
)− zFuc
dx
dx
This is known as the Nernst-Planck equation. In this form, D, the Diffusion coefficient is uRT
(units: cm2 sec–1), often called the Einstein relation, which is an outcome of Einstein's mechanistic
molecular derivation. It describes the ability of the molecular ion to explore space on the basis of
its mobility, u, and its kinetic energy, RT (the mole form of kT, which defines the velocity of the
molecule).
In the form:
J = −(uRT )(
dc
dΨ
)− zFuc
dx
dx
the equation is not very useful, because we need to know what ∂c/∂x and ∂ψ/∂x actually are: We
need to integrate over the boundary conditions of the membrane.
Molecular and Thermodynamic Explanations of Ion Motion – page 1.11
The fundamental assumption made to solve the Nernst-Planck equation is that
the electrical potential across the membrane is linear. Then, the linear slope,
ψ/L describes the electrical gradient ∂ψ/∂x.
ψo (outside potential)
ψi (inside
potential)
Slope:
∂ψ /∂x = (ψo - ψi / ∆x)
= ψ/L.
L (distance)
Inserting ∂Ψ/∂x = Ψ/L:
Ψ
dc
J = −D( )− zFuc
L
dx
re− arranging to isolate the differentials:
-D
dc = dx
J + zFuc Ψ L
we can integrate over the boundary conditions
co: concentration outside; ci: concentration inside
0 to L the width of the membrane
ci
-D
∫ J + zFuc
co
J = −P(
L
Ψ
=
L
∫ dx
which yields:
0
zFΨ [co − c i • exp( zFΨ RT )]
)
where P = D/L
[1− exp( zFΨ RT )]
RT
This is called the Goldman
constant field equation.
Constant field because we
assume ∂ψ/∂x is constant. We
can now predict flux, J, as a
function of the difference in
concentration and electrical
potential.
Molecular and Thermodynamic Explanations of Ion Motion – page 1.12
The Goldman constant field equation is the starting point for two special cases.
Case One: flux is zero (J = 0).
J = −P(
zFΨ [co − c i • exp( zFΨ RT )]
)
[1− exp( zFΨ RT )]
RT
0
0
0= co − ci • exp( zFΨ RT )
co
= exp( zFΨ RT )
ci
c
ln o = zFΨ RT
ci
RT co
ln = Ψ
zF ci
This is known as the Nernst equation. It is useful for
identifying permeant ions. For practical use, the
Nernst equation can be simplified by using log10
rather than the natural logarithm. At room
temperature for a monovalent:
58log10
63.0
co
= Ψ (mV)
ci
RT/F values (log10 equation)
61.0
59.0
57.0
55.0
53.0
0
5
10
15
20
25
30
35
40
45
Temperature (ºC)
Molecular and Thermodynamic Explanations of Ion Motion – page 1.13
Case Study: Effect of K+ on the membrane potential, E of a cell Nernst Potential.
The membrane potential, E measured
by impaling a micropipette into the cell.
–
– +
– +
– +
–
– +
–
+
–
+
10 mM KCl
10 mM KCl
0
E2
-50
E1
-100
-100 mV
K+
Cl–
+ve charge
in
-50 mV
K+
Cl–
100 mM KCl
c 2o
E2 − E1 = 55log10
ci
c 2o
E2 − E1 = 55log10 o
c1
Negative or positive? If you
set-up the equations correctly,
the solution will show the
c1o polarity. But, it's easy to lose a
− 55log10
'minus' sign. There is a simple
In this case study,
c i intuitive test.
+
higher K outside should cause
positive charge movement
= + 55 mV inward: depolarization.
In some cells, both plant and animal, increasing KCl outside of the cell causes a depolarization of
the membrane potentials. If the cell is selectively permeant to K+, and not Cl–, the depolarization
can be explained using the Nernst potential for K+. If a hyperpolarization occurred, Cl–
permeation would be responsible.
Molecular and Thermodynamic Explanations of Ion Motion – page 1.14
The Goldman constant field equation is the starting point for two special cases.....
Case Two: the potential, E is zero (ψ = 0).
zFΨ [co − c i • exp( zFΨ RT )]
J = −P(
)
[1− exp( zFΨ RT )]
RT
For small potentials approaching zero:
exp(zFΨ RT )= 1+ zFΨ RT
If we set ψ = 0, exp(0)=1, so we get
0/0, undefined. Therefore, we resort
to a mathematical sleight of hands to
solve for the case, ψ = 0.
zFΨ c o − ci exp(1+ zFΨ RT )]
J = −P
[
1− (1+ zFΨ RT )
RT
zFΨ c o − ci (1+ zFΨ RT )]
J = −P
[
RT
( zFΨ RT )
J = −P[c o − ci (1+ zFΨ RT )] = − P[co − ci − c i ( zFΨ RT )]
since Ψ = 0
J = −P[c o − ci ]
the equation for an uncharged solute.
Molecular and Thermodynamic Explanations of Ion Motion – page 1.15
So, molecular motion can be described from either mechanistic or thermodynamic perspectives. In
either case, diffusive flux is defined by two terms: the 'driving force' and a diffusion coefficient,
defined as cm2 sec–1, or uRT. To gain a clearer understanding of the molecular nature of flux when the
molecule is charged and an electrical force is present, we need to return to a one dimensional random
walk and examine the effect of an applied force. This gives insight into the nature of u, the mobility
particle of mass m
force acting in the x direction
m
x+δ–
Fx
x
Note the introduction of
acceleration. We are no longer
dealing with a particle moving at
some constant velocity, but one
that is subjected to accelerative
force (for example, a voltage
field) and responds accordingly
x+δ+
distance
The force Fx results in an acceleration in the +ve direction (x+δ+), a = Fx/m, where
the units of acceleration are cm sec-2. The particle moves to the right or to the left
with an initial velocity +νx or –νx once every τ seconds.
{
+νx velocity
distance moved is: δ+
= νxτ + aτ2/2
distance moved due to
acceleration.
x+δ–
x
x+δ+
distance
-νx velocity
{
distance moved is: δ–
= –νxτ + aτ2/2
distance moved due to
acceleration.
x+δ–
x
x+δ+
distance
Molecular and Thermodynamic Explanations of Ion Motion – page 1.16
If the probability that the particle goes left or right is the same,
δ +δ
then the average displacement, + − , is:
2
2
2
Average displacement
(ν xτ + aτ 2 )+ (-ν xτ + aτ 2 )
depends upon
acceleration, a and time, τ.
2
the ν xτ terms cancel out:
(aτ
2
2
aτ )
2 )+ (
2 = ( aτ 2 ) (the average displacement).
2
2
Average velocity equals average
δ
displacement per time interval
) is:
The average velocity (recall ν =
τ
2
aτ
2 = aτ or, ν = 1 Fx • τ
2
τ
2m
a frictional drag coefficient, f , is used to describe the resistance to movement:
ν=
Fx
f
where f = 2mτ
To obtain a more meaningful description of frictional drag:
f = 2mτ •
δ2
τ2
δ2
τ2
δ2
2mν 2 2τ
since ν = 2 then f = 2 = 2 mν 2
δ
τ
δ
τ
2
2τ
= D (the Diffusion coefficient), so
δ2
2mν 2
f=
D
Finally, substituting the kinetic equation, mν 2 = kT
This definition of the diffusion coefficient originates
kT
kT
Therefore: D =
f=
with Einstein and Smoluchowski, and is described in
detail by Berg HC. 1993 Random Wallks in Biology.
f
D
but
Princeton University Press.
Molecular and Thermodynamic Explanations of Ion Motion – page 1.17
An ion can be described as a sphere made up of the ion itself and a cloud of water molecules
surrounding the ion. In this case, the frictional drag coefficient is described by:
f = 6 • π • r •η where r is the radius of the sphere
and η is the viscosity of the solution.
F
From the relationship ν = x , Fx = f •ν
f
For an ion, the force is an electrical one: z • e •ψ
where z is the valence, e is the electron charge and ψ the potential.
So, z • e •ψ = 6 • π • r •η •ν
The ionic mobility is defined by the ionic velocity per volt of driving force.
u=
ν
z •e
=
with units of
ψ 6 • π • r •η
cm
sec
volts
cm
Ionic mobility can be converted directly to a measureable value, conductivity:
λ = z • F • u where z is the valence
and F is Faraday constant.
RT
The Diffusion coefficient: D =
•u
F
0
λ is the conductivity of an ionic species. In solution:
0
Note that
MA ↔ M + + A –
(salt) λ +
0
λ–
0
so solution conducticity: Λ0 = λ+0 + λ0– at infinite
dilution. Solution conductivity is concentration
dependent.
Molecular and Thermodynamic Explanations of Ion Motion – page 1.18
Conductivity
Λ
200
The graph shows the concentration
dependence of conductivity for an easily
dissociated salt (KCl) and a weakly
dissociated salt (acetic acid). Redrawn
from Castellan GW 1971 Physical
Chemistry. 2nd edition. Addison-Wesley.
KCl
100
acetic acid
A measure of the energetics of water ‘binding’
to the ion to create a hydration sphere is the
enthalpy of hydration
0
0
0.1
0.2
0.3
concentration
The enthalpies of hydration ∆H°hydration
should not be confused with enthalpies of
sovation (salt dissolvation: MA <---> M+ + A–
(aq)). It is the energy released when the ion
reacts with water: M+ <---> M+ (aq).
infinite dilution: Λ = Λ0
Properties of Ions
Ion
+
Tl
H+
NH4+
Cs+
Rb+
K+
Na+
Li+
Cl–
F–
Br–
I–
NO3–
Mg2+
Ca2+
Sr2+
Ba2+
.
.
.
.
Atomic
Radius
(Å)
1.44
.
.
1.48
.
1.69
1.48
1.33
0.95
0.6
1.81
1.36
1.95
2.16
2.9
0.65
0.99
1.13
1.35
∆H°hydration
Mobility
(kcal/mole)
10-4(cm/sec)/(V/cm)
7.74
Data are taken from
36.3
compilations by Bertl
7.52
Hille 1984 Ionic
Channels of Excitable
8.01
Membrane. Sinauer
8.06
Associates.
7.62
5.19
4.01
7.92
5.74
Palmgren (2001, Ann. Rev.
8.09
Pl. Physiol. Pl. Molec. Biol.
7.96
52:817–845) lists ionic radii
7.41
of selected dehydrated
2.75
cations (but without direct
3.08
citation) as follows: H3O+
(1.15 Å), Na+ (1.12), K+
3.08
(1.44,
Ca2+ (1.06)
3.3
.
-72
-79.2
-85.8
-104.6
-131.2
-82
-114
-79
-65
.
-476
-397
-362
-328
Molecular and Thermodynamic Explanations of Ion Motion – page 1.19
Ion mobility versus atomic radius
Mobility ([m sec-1]/[Volt m-1]))
10
+
8
K
Rb +
Cs
Tl+
+
NH4
6
F
Na
+
The following graphs
explore the relations
between ionic size,
mobility, and energies of
hydration (∆H°hydration,
an indirect measure of
the degree to which the
ion is hydrated by
surrounding water
molecules, effectively
increasing the apparent
radius of the ion).
Br– I–
Cl–
NO3–
–
+
+
4
Li
2+
Ca2+ Sr
2
Ba
2+
Mg2+
1.0
1.5
2.0
2.5
Atomic Radius (Angstroms)
3.0
Enthalpy (hydration) (kcal mole-1)
Hydration enthalpy versus atomic radius
-50
+
Li
K+
F–
Na+
Cs+
Rb+
–
Cl
Br–
I–
-140
-230
Ba2+
-320
Sr2+
2+
-410
-500
Ca
Mn2+
Mg2+
1.0
1.5
2.0
2.5
Atomic Radius (Angstroms)
3.0
Molecular and Thermodynamic Explanations of Ion Motion – page 1.20
Mobility versus hydration enthalpy
Mobility ([m sec-1]/[Volt m-1]))
10
Cl– Rb+Cs+
–
Br
I–
K+
8
F–
6
Na+
4
Mg
2
-500
Li+
2+
2+
2+
Ca
-410
Sr2+ Ba
-320
-230
-140
Enthalpy (hydration) (kcal mole-1)
-50
It should be clear that the relation between ionic size, mobility, and energies of hydration is
complex. What is not shown on the graphs is the predicted mobility of the ions based upon the
Einstein formalism. But a simple examination of the very distinct behaviour of the the divalent
ions, and a subset of monvalents indicates that no general theory can suffice to explain mobility.
Of great significance to electrophysiologists is the relation between the physical chemical properties of
ions and the well-known selectivity of ion channels, which can distinguish between very similar ions, such
as potassium and sodium (see Page 1.25).
Conductance (relative to sodium)
Gramicidin is one example of a very simple proteinaceous pore structure
which traverses the membrane and exhibits ion selectivity. Relative cation
selectivity is shown versus ionic mobilities below.
15
H+
12
9
6
Li+ Cs+ Rb+
NH4+
+
+
+ Tl
Na
K
3
0
0
5
10
15
20
25
Mobility
30
35
40
Fick's Equations and Diffusion to Capture – page 1.21
In our initial description of ion
motion, we presented a simple one
dimensional analysis that
identified flux J, as a function of
the concentration gradient.
J = −D
∂c
∂x
However, a concentration gradient that is time-invariant is unlikely. In most cases, the
concentration gradient will change with time.
In one dimension
In three dimensions
Radial flux if the
2
geometry
is
∂c
∂c
2
2
2
∂
∂
∂c
∂
c
c
c
spherically symmetric
=D 2
∂t
∂x
∂t
=D
[ ∂x
2
+
∂y
2
+
∂z
2
]
These are geometric variants of Fick's Second Law of
Diffusion.
∂c
∂r
∂c
1 ∂
∂c
= D 2 • •(r 2 • )
∂t
r ∂r
∂r
J r(r ) = − D
In biological systems, it is common for molecules to be supplied from one source and be
removed at another location. This occurs during uptake of molecules from the
extracellular medium. The example shown below is a calcium gradient in growing hyphal
cells. Tip-localized calcium diffuses away from the growing tip and is sequestered.
Fick's Equations and Diffusion to Capture – page 1.22
Calcium diffusion results
in a gentler gradient over
time, as indicated. The
actual data is shown. The
predictions are based on
a steady supply of calcium at the tip and
sequestration behind the
tip:
[Ca 2+ ] =
Cytosolic [Ca2+] (nM)
Actual data is shown as well as the time dependence of the calcium gradient.
450
390
32 sec
330
8
270
0.5 sec
210
2
150
0
5
2
M
e(− x /4 Dt ) + [Ca 2+ ]basal
1/2
2(πDt)
10
15
20
Distance from the Tip (μm)
where M is the initial concentration (the best fit value was 4 μM), D is the diffusion
coefficient, [Ca2+]basal is the sub-apical [Ca2+] (the best fit value was 175 nM), and t is
time. The Ca2+ gradient was initially fit to obtain an estimate of the diffusion coefficient
(5.6 μm2 sec-1) using a 4 second time interval, when the hyphae would have grown
about 1.2 μm. Within the time frame 0.5 to 32 sec, diffusion causes a gentler gradient
compared to other cytological features of growing hyphae. In aqueous solutions, the
diffusion coefficient for Ca2+ is about 775 μm2 sec-1 in dilute CaCl2. Intracellular Ca2+
diffusion coefficients are 2-15 μm2 sec-1.
25
Fick's Equations and Diffusion to Capture – page 1.23
To determine how far a particle can travel by diffusion, we can determine the average
displacement. For a particle that can move in a positive or negative direction, there is a
problem, that the average displacement will be zero:
N
< x(t) >=
1
∑[x i (t −1) ± δ] = 0
N i=1
< x 2 (t) >= 2Dt
Instead, the root mean square is used, which yields the result:
for one dimension, or, for three dimensions (summing the x, y,
and z coordinates):
Diffusion works best at
small distances.
r = 6•D •t
Thus,
< r 2 (t) >= 6Dt
1010
Diffusion Time (seconds)
109
1 century
Hemoglobina
(7.00 • 10–11 m2 sec–1)
108
107
106
105
1 year
1 month
O2 moleculea
(1.80 • 10–9 m2 sec–1)
1 day
104
1 hour
10
3
102
10
1 minute
1
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
Diffusion Distance (meters)
a
Brouwer ST, L Hoof, F Kreuzer (1997) Diffusion coefficients of oxygen and hemoglobin measured by
facilitated oxygen diffusion through hemoglobin solutions. Biochim Biophys Acta. 1338:127–136.
Fick's Equations and Diffusion to Capture – page 1.24
Solutions to diffusive flux equations vary. The following example is presented in
Berg HC 1983 Random Walks in Biology. It models the situation for cell, which will
have a finite number of transporters at the plasma membrane to take up a molecule
from the extracellular environment. The modelling starts with diffusion to a cell,
examines diffusion to an absorbing disk, then puts the two together.
Diffusion to a spherical absorber
C=0
C = C0 at x >> s
For the specified boundary conditions,the solutions are:
a
a
C(r ) = C0 (1− ) and, Jr (r )= − DC0 2
r
r
The molecules are absorbed at a rate equal
to the sphere area times the inward flux (J r (a)):
a
I = 4 • π • D • a • C0
Diffusion to a disk absorber
The molecules are absorbed
at a rate of: I = 4 • D • s • C0
s
C=0
C = C0 at x >> s
Diffusion to a cell covered
with N absorbing disks.
1.0
I/I0
0.8
0.6
The molecules are
absorbed at a rate less than
for a spherical absorber
0.4
0.2
0.0
0
1
I
=
4 • π • D • a • C0 1 + π • a
N •s
1
π•a/s
2
3
N absorbing disks
4
5
Channel Function and Structure – page 1.25
5Å
Gramicidin is an example of a
very simple ion channel. It is
formed from two helical
cylinders, which may intertwine
as shown, or join at the ends to
create a 5Å pore through the
membrane.
25Å
Gramicidin was originally isolated from a soil bacteria, Bacillus brevis. It is anti-bacterial,
especially against Gram-positive bacteria. Lysis does not occur. It's toxicity depends upon the
phospholipid make-up of the membrane, phosphatidylethanolamine and phosphatidylserine
inhibit bactericidal activity. Early experiments on its efficacy as an antibiotic were promising,
but in fact it is toxic when applied systemically, and is only used therapeutically as a topical
application. The activity of the small peptide is measured commonly with the bilayer lipid
BLM
membrane (BLM) technique. In this method, the
channel
Technique
gramicidin channel is incorporated into a lipid
membrane seperating two compartments. Ion
concentrations in the compartments can be controlled.
voltage clamp
The channel activity is monitored using a current to
voltage converter, an electronic design also used to
fusion
measure ion channels in the patch clamp technique.
This is how ion conductances were measured (below).
cis
Dubos R, 1939 Studies on a
bactericidal agent extracted
from a soil bacteria. J. Exp.
Med. 70:1–17.
Hunter Jr. FE, Schwartz LS.
1967. Gramicidins. in
Gottleib & Shae, eds.
Antibiotics. Vol. I
Springer-Verlag.
Conductance (relative to sodium)
trans
Gramicidin is a very simple proteinaceous pore structure which traverses
the membrane and exhibits ion conductance. Relative cation selectivity is
shown versus ionic mobilities.
15
12
H+
9
6
+
+
Cs+ Rb +
NH4
+
Na+ K+ Tl
Li
3
0
0
5
10
15
20
Mobility
25
30
35
40
Channel Function and Structure – page 1.26
Passage of ions through the pore is measured as current.
Current occurs in step-like transitions, due to opening
and closing of the channel.
M+
M+
M+
M+
3 picoAmpere at 100 mV:
30 pico Siemen conductance
M+
Current can be converted into flux:
amperes
(coulombs
per second)
valence
Flux
(mole/second)
I=z•F•J
J=
Faraday
constant
(96,490 coulombs/mole)
I
10−9 (coulombs/mole)
=
= 1.036 •10−14 (moles/sec)
zF +1 • 96,490 (coulombs/mole)
1.036 •10−14 (moles/sec) • 6.023 •1023 (molecules/mole) = 6.24 •106 (molecules/sec)
We can test the experimentally measured flux with expected flux through a pore
having the dimensions of the gramicidin dimer: 5Å diameter by 25Å length.
Resistance = resistivity (100 Ω • cm for 120 mM salt) • length/area
R= ρ •
l
2.5 •10−7 (cm)
9
= 100 (Ω • cm) •
−7
2 = 12.73 •10 Ω
A
π •(2.5 •10 (cm))
Conductance=
1
= 78.6 picoSiemens
9
12.73 •10 Ω
The calculated value (79 pS) is higher than the experimental value (30 pS). Some
possible reasons include inaccuracies in pore and length measurements, limitations due
to diffusion from the external medium, and a lower resistivity within the pore, due to
steric hindrance.
Channel Function and Structure – page 1.27
Chloride Channels (permeability properties of a CI- channel from a higher plant guard cell)
Permeability ratios and peak current magnitudes for malate, nitrate, and halides over chloride.
Current reversal potentials for the anions shown were recorded under bi-ionic conditions in
the whole-cell patch-clamp configuration. Averaged values of reversal potentials were used to
calculate the permeability of these anions relative to chloride.
PX/PCl
IPeak(pA)
CI(n=3)
1±0.04
-231±183
Malate2(n=18)
0.24±0.19
-77±58
NO3(n=12)
20.9±11.2
-747±378
I(n=4)
0.98±0.16
-146±112
Br(n=9)
2.4±1.5
-791±340
F(n=4)
1.26±0.4
-771±361
Schmidt C, Schroeder JI (1994) Anion selectivity of slow anion channels in the plasma membrane of guard
cells. Large nitrate permeability. Plant Physiol. 106: 383–391
Mobility (cm/sec)/(V/cm)
5
25
6
7
8
9
-100.0
PX/PCl
-
NO3-
I
-187.5
CI-
20
-275.0
-362.5
15
-450.0
10
-537.5
-625.0
5
F0
5
Br
CI6
7
-
-712.5
I8
Mobility (cm/sec)/(V/cm)
The selectivity of a chloride channel
from plants is compared to the
mobility of the anion.
9
F-
NO3-
Br-
-800.0
IPeak(pA)
The conductance through a chloride
channel from plants is compared to
the mobility of the anion.
Notice the lack of correspondence between ionic mobility and either selectivity or
conductance. Analogous to the situation with a much simpler ion channel, gramicidin,
there is a complexity associated with the function of the ion channel which cannot be
explained by a simple comparison to molecular properties. In this context, the
determination of the structure of a chloride channel using x-ray crystallography was a real
breakthrough.
Channel Function and Structure – page 1.28
Cl–
Left: Ribbon representation of the StClC dimer from the extracellular side. The two subunits
are shaded differently. A Cl- ion in the selectivity filter is shown by arrows. Right: View from
within the membrane with the extracellular solution above. The channel is rotated by 90° about
the x- and y-axes relative to a. The black line (35Å) indicates the approximate thickness of the
membrane. From: R Dutzler, EB Campbell, M Cadene, BT Chait, R MacKinnon. 2002. X-ray
structure of a ClC chloride channel at 3.0 Å reveals the molecular basis of anion selectivity.
Nature 415: 287–294.
Channel Function and Structure – page 1.29
Cl–
Structure of the StClC selectivity filter. Left: Helix dipoles (end charges) point towards the selectivity filter. The
a-helices are shown as cylinders. The amino (positive, blue) and carboxy (negative, red) ends of a-helices D, F
and N are shown. The selectivity filter residues are shown as red cords surrounding a Cl- ion (red sphere). The
view is from 208 below the membrane plane; the dimer interface is to the right, and the extracellular solution
above. Part of a-helix J has been removed for clarity (grey line). Right: Stereo view of the Cl- ion-binding site.
Distances (,3.6 Å) to the Cl- ion (red sphere) are shown for polar (white dashed lines) and hydrophobic (green
dashed lines) contacts. A hydrogen bond between Ser 107 and the amide nitrogen of Ile 109 is shown (white
dashed line). From: R Dutzler, EB Campbell, M Cadene, BT Chait, R MacKinnon. 2002. X-ray structure of a
ClC chloride channel at 3.0 Å reveals the molecular basis of anion selectivity. Nature 415: 287–294.
The positive dipoles of amino and
hydroxyl groups create a
coordinated web of weak bonds that
bind the chloride ion.
Surface electrostatic potential on the ClC dimer
in 150mM electrolyte. The channel is sliced in
half to show the pore entryways (but not the full
extent of their depth) on the extracellular
(above) and intracellular (below) sides of the
membrane. Isocontour surfaces of -12 mV (red
mesh) and +12 mV (blue mesh) are shown. Clions are shown as red spheres. Dashed lines
highlight the pore entryways.
Channel Function and Structure – page 1.30
In the potassium channel, the potassium ion enters a large
vestibule. The selectivity filter is negatively charged. Its size
indicates that the potassium ion must shed its water molecules.
The negative oxygens effectively replace the water molecules.
Dehydration is highly energetic. Yet the entire process is probably
not: Entry of one ion would occur in tandem with the exit of
another ion.
DA Doyle, J Morais Cabral, RA Pfuetzner, A Kuo,
J M Gulbis, S L Cohen, BT Chait & R MacKinnon
(1998) The structure of the potassium channel:
molecular basis of K+ conduction and selectivity.
Science 280: 69-77