molecular and thermodynamic explanations of ion motion ... · molecular and thermodynamic...

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 move-ment of the molecules. We can think about the molecules moving in various directions within the medium.

J =mol of solute

cm2 • sec

The quantitative descriptor commonly used is the flux (J), defined as the amount of mol-ecules that pass through a specified area in a specified time.

The flux, J, will depend upon the concentra-tion 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.

Con

cent

ratio

n (c

)

Distance (x)

ΔcΔx

J ≈ −Δc

Δx

⎛

⎝ ⎜

⎞

⎠ ⎟

The –ve sign indicates that the flux occurs from regions of higher to

lower solute concentration.

Molecular and Thermodynamic Explanations of Ion Motion — page 1.02C

once

ntra

tion

(c)

Distance (x)

ΔcΔx

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).

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

dx⇒(

cm2

sec)(

mol

cm4 ) ⇒(mol

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.

J ≈ −∂c

∂x

⎛

⎝ ⎜

⎞

⎠ ⎟


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 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 particles at x+δ will have moved to x.

[1]Berg, HC (1998) Random Walks in Biology. Princeton University Press. pp.17–21.

N(x) N(x+δ) number of particles

x x+δ distance

1/2N(x)

1/2N(x+δ)

N(x) N(x+δ)

x x+δ

1/2N(x)-1/2N(x+δ)= -1/2[N(x+δ)-N(x)]

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

2[ N(x +δ )– N(x)]/ Aτ ,

multiplying by δ 2

δ 2

Jx = –12

δ 2

δ 2

1Aτ

[ N(x +δ )– N(x)]

δA has units of volume, and

Jx = −1

2

δ 2

τ

1

δ[ N(x +δ )

δA−

N(x)

δA]

N divided by volume is concentration

Jx = −1

2

δ 2

τ

1

δ[C(x +δ )− C(x)]

re-arranging

Jx = −12

δ 2

τ[C(x +δ )− C(x)

δ]

re-arranging


N(x) N(x+δ) number of particles

x x+δ distance

If we take the term

[C(x +δ )– C(x)

δ]

to the limit δ → 0, then

C(x +δ )– C(x)δ

=∂Cx

therefore

Jx = –1

2

δ 2

τ

∂C

∂x

where 1

2

Δ2

τ

with units of:cm2

sec

∂

is the Diffusion coefficient, D

These are the same form as Fick’s Law ofDiffusion (J = D • ∂C/∂x)

Continuing Berg’s approach[1]:

J = −1

2•

Δ2

τ•

∂C

∂x

or (Einstein):

D =1

2•

Δ2

τ

Δ = 2 • D • τ

Jx = −1

2

δ 2

τ

1

δ[C(x +δ )− C(x)]

[1]Berg, HC (1998) Random Walks in Biology. Princeton University Press. pp.17–21.

The molecular definition of the diffusion coefficient:can be recast to show that the average displacement, ∆,is a function of the square root of time:This prediction was used to verifyEinstein’s theory of BrownianMotion. Since then, random walks have, in one form or another, permeated biophysical research.

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 Diffusion Coefficient (m2 sec–1) Air WaterHydrogen 2 6.11•10–5 4.50•10–9

Helium 4 6.24•10–5 6.28•10–9

Oxygen 32 1.78•10–5 2.10•10–9

Benzene 78 9.60•10–6 1.02•10–9

H+ 1 9.31•10–9

K+ 39 1.96•10–9

Ca2+ 40 0.40•10–9

Cl– 35 2.03•10–9

H2O2 34 1.30•10–9

Generally, diffusion coefficients depend upon the molecular weight (size) of the com-pound, 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.


co (outside concentration)

ci (insideconcentration)

d (distance)

Flux will depend upon the abilityof the particle to enter the membrane(partitioning)

Partitioning, Kp =c(membrane )

c(aqueous )

so the flux is now described by:

J = DKp

d[coutside – cinside ]

where DKp

d= P, the permeability coefficient with

units of cm

, or cm•sec-1

To describe the movement of molecules through a membrane, we need to consider a morecomplex framework. We still use the general form of the flux equation: J=D•dc/dx, but adiffusion coefficient alone is insufficient.

cm2

sec

There is a classic literature on the permeability of membranes. Much of the original workwas done on giant freshwater algae. Historically, this research led to the proposal that cellsare bounded by a lipoidal membrane, because permeability matches closely the partitioningof substances between olive oil and water


Typical values for permeability coefficients

Membrane permeabilities of selected solutes in Chara, Nitella, human erythrocytes, andartificial membranes1.

Solute molecularweight

olive oil :waterpartitioncoefficient

Characeratophylla

Nitellamucronata

Humanerythrocyte

Artificiallipidmembrane

water 18 1.3•10-4 2.5•10-3 1.2•10-3 2.5•10-3 2.2•10-3

formamide 45 1.1•10-6 2.2•10-5 7.6•10-6 1.1•10-6 1.0•10-4

ethanol 46 3.6•10-2 1.6•10-4 5.5•10-4 2.1•10-3

ethanediol 58 4.9•10-4 1.1•10-5 2.9•10-5 8.8•10-5

butyramide 87 1.1•10-6 5.0•10-5 1.4•10-5 1.1•10-6

glycerol 92 7.0•10-5 2.0•10-7 3.2•10-9 1.6•10-7 5.4•10-6

erythritol 122 3.0•10-5 6.7•10-9

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 non-electrolytes. Physiol. Plant. 7: 420–445, and Stein WD 1990 Channels, Carriers andPumps. Academic Press.

For comparison, permeability coefficients for ions are much lower. In an artificial membrane: Na+ 10-11 to 10-14 cm/sec Cl- 10-11

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.

formamide ethanediol butyramide

glycerol erythritol


Permeability (cm/sec)(X107)

Oil/water partition (left, squares) or Molecular weight (right, circles)

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.

0.001

0.01

0.1

1

10

100

1000

10000

100000

0.00001 0.0001 0.001 0.01 0.1 1 10 100 1000

Deuterium hydroxide 19 25000 0.0007Ethyl acetate 88 2.5Methyl acetate 74 25000 0.43sec.-Butanol 74 9300 0.25Methanol 32 5700 0.0078n-Propanol 60 7200 0.13Ethanol 46 5500 0.032Paraldehyde 132 12000 1.9Urethane 89 5200 0.074iso-Propanol 60 3800 0.047Acetonylacetone 144 7500 0.081Diethylene glycol monobutyl ether 162 2600 0.12Dimethyl cyanamide 70 1900 0.073tert-Butanol 74 1900 0.23Glycerol diethyl ether 148 2300 0.11Ethoxyethanol 90 1800 0.019Methyl carbamate 75 1600 0.025Triethyl citrate 276 2400 0.5Methoxyethanol 76 990 0.0056Triacetin 218 1100 0.44Dimethylformamide 73 705 0.0049Triethylene glycol diacetate 234 661 0.033Pyramidone 231 655 0.26Diethylene glycol monoethyl ether 134 406 0.006Caffeine 194 357 0.033Cyanamide 42 292 0.0045Tetraethylene glycol dimethyl ether 222 285 0.0056Pinacol 118 229Diacetin 176 209 0.071Methylpentanediol 118 191 0.024Antipyrene 188 192 0.032iso-Valeramide 101 182 0.0231,6-Hexanediol 118 177 0.0068n-Butyramide 87 139 0.0095Diethylene glycol monomethyl ether 120 134 0.0042

Trimethylcitrate 234 121 0.047Proprionamide 73 79 0.0036Formamide 45 76 0.00076Acetamide 59 66 0.00083Polyethylene glycol monoethyl ether 200 66Succinamide 99 54 0.0049Glycerol monoethyl ether 120 40 0.0074N,N-Diethyl urea 116 39 0.00761,5-Pentanediol 104 34 0.0061Dipropylene glycol 134 31 0.002Glycerol monochlorhydrin 110 30 0.0121,3-Butanediol 90 24 0.00432,3-Butanediol 90 21 0.00341,2-Propanediol 76 17 0.0017N,N-Dimethyl urea 88 15 0.00231,4-Butanediol 90 14 0.0021Ethylene glycol 62 12 0.00049Glycerol monomethyl ether 106 12 0.0026N,N-Dimethyl urea 88 121,3-Propanediol 76 10Ethyl urea 88 6.6 0.0017Polyethylene glycol diacetate 380 6.3Thiourea 76 3.6 0.0012Diethylene glycol 106 3.8Methyl urea 74 3.2 0.00044Urea 60 1.3 0.00015Triethylene glycol 150 1Polyethylene glycol diacetate 480 0.8Tetraethylene glycol 194 0.71Dicyanodiamide 84 0.46 0.00047Hexanetriol 134 0.42Hexamethylene tetramine 140 0.39 0.00021Polyethylene glycol monoethyl ether 400 0.15Glycerol 92 0.032 0.00007Pentaerythritol 136 0.002


For all cells, plant, animal, bacteria etc., charged solute concentrations vary widely between the inside and outside environment:

Animals K+ Na+ Ca2+ Em (milliVolts)

intracellular high low low -60 mV extracellular low high high 0 mV

Plants intracellular high varies low -180 mV extracellular low low high 0 mV

J= –mobility (cmsec) • activity(mole

cm3 ) • driving force (dμdx)

J = –u • c • dμdx

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.

This derivation is taken from Schultz, SG. 1980. Basic Principles of Membrane Transport. Cambridge University Press. It is a 'classic' derivation, relying upon thermodynamics.

The equation

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.

mobilityconcentration

chemical potential


From thermodynamics, the chemical potential is:

μ = RT(ln c)+ zFΨ

so,

dμ = RTd(ln c)+ zFdΨ = RT1c

dc + dΨ

d

dx of dμ =

RT

c

dc

dx+ zF

dΨ

dx

J = −u • c •[ RT

c

dc

dx+ zF

dΨ

dx]

J = −(uRT )(dcdx

)− zFucdΨdx

so,

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:

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.

R is the gas constant (8.314 J mol–1 K–1)T is the temperature °Kc is the concentrationz is the valence (ionic charge)F is the Faraday constant (9.649 • 104 J mol–1 V–1)ψ is the electrical potential

J = −(uRT )(dcdx

)− zFucdΨdx


ψo (outside potential)

ψi (insidepotential)

L (distance)

Slope: ∂ψ /∂x = (ψo - ψi / ∆x)

= ψ/L.

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.

Inserting ∂Ψ/∂x = Ψ/L:

J = −D(dc

dx)− zFuc

Ψ

L

re− arranging to isolate the differentials:

-D

J + zFuc ΨL

dc = dx

we can integrate over the boundary conditions

co: concentration outside; ci: concentration inside

0 to L the width of the membrane

-DJ + zFuc Ψ

L

= dx0

L

∫c o

c i

∫ which yields:

J = −P(zFΨ

RT)[co − ci • exp(zFΨ

RT)]

[1− exp(zFΨRT)]

where P = D/L

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.


The Goldman constant field equation is the starting point for two special cases.

Case One: flux is zero (J = 0).

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:

J = −P(zFΨ


RT)]

[1− exp(zFΨRT)]

0 0

0= co − ci • exp(zFΨRT)

co

ci

= exp(zFΨRT)

lnco

ci

= zFΨRT

RT

zFln

co

ci

= Ψ58log10

co

ci

= Ψ (mV)

53.0

55.0

57.0

59.0

61.0

63.0

0 5 10 15 20 25 30 35 40 45

Temperature (ºC)

RT/F values (log10

equation)


Case Study: Effect of K+ on the membrane potential, E of a cell - Nernst Potential.

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.

The membrane potential, E measuredby impaling a micropipette into the cell.

–

+

––––

–

–

–

+++++

10 mM KCl

0

-50

-100E1

E2

100 mM KCl

10 mM KCl

-100 mV K+

Cl–

K+

-50 mV Cl–

+ve chargein E2 − E1 = 55log10

c o2

ci− 55log10

c o1

ci

E2 − E1 = 55log10

c o2

co1

= + 55 mV

Negative or positive? If you set-up the equations correctly, the solution will show the polarity. But, it's easy to lose a 'minus' sign. There is a simple intuitive test. In this case study, higher K+ outside should cause positive charge movement inward: depolarization.


The Goldman constant field equation is the starting point for two special cases.....

Case Two: the potential, E is zero (ψ = 0).

J = −P(zFΨ


RT)]

[1− exp(zFΨRT)]

For small potentials approaching zero:

exp(zFΨRT)= 1+ zFΨ

RT

J = −PzFΨ

RT[co − ci exp(1+ zFΨ

RT)]

1− (1+ zFΨRT)

J = −PzFΨ

RT[co − ci(1+ zFΨ

RT)]

( zFΨRT)

J = −P[co − ci(1+ zFΨRT)] = −P[co − ci − ci(

zFΨRT)]

since Ψ = 0

J = −P[co − ci ]

the equation for an uncharged solute.

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.


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

x+δ– x x+δ+ distance

particle of mass m force acting in the x direction

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.

m Fx


{+νx velocity

distance moved is: δ+ = νxτ + aτ2/2 distance moved due to acceleration.


{-νx velocity

distance moved is: δ– = –νxτ + aτ2/2 distance moved due to acceleration.

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


Average displacement depends upon

acceleration, a and time, τ.

If the probability that the particle goes left or right is the same,

then the average displacement, δ+ +δ−

2 , is:

(ν xτ + aτ 2

2)+ (-νxτ + aτ 2

2 )

2the νxτ terms cancel out:

(aτ 2

2)+ (aτ 2

2 )

2= (aτ 2

2 ) (the average displacement).

The average velocity (recall ν =δτ

) is:

aτ 2

2τ

= aτ2 or, ν =

1

2

Fx

m• τ

a frictional drag coefficient, f , is used to describe the resistance to movement:

ν =Fx

f where f = 2m

τ

To obtain a more meaningful description of frictional drag:

f = 2mτ •

δ 2

τ 2

δ 2

τ 2

since ν 2 =δ 2

τ 2 then f =

2mν 2

δ 2

τ

=2τ

δ 2mν 2

but 2τδ 2

= D (the Diffusion coefficient), so

f =2mν 2

DFinally, substituting the kinetic equation, mν 2 = kT

f =kT

D Therefore: D =

kT

f

This definition of the diffusion coefficient originates with Einstein and Smoluchowski, and is described in detail by Berg HC. 1993 Random Wallks in Biology.

Princeton University Press.

Average velocity equals average displacement per time interval


f = 6 • π • r •η where r is the radius of the sphere

and η is the viscosity of the solution.

From the relationship ν =Fx

f , Fx = f •ν

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 =νψ

=z • e

6 • π • r •η with units of

cmsec

voltscm

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:

Ionic mobility can be converted directly to a measureable value, conductivity:

λ0 = z • F • u where z is the valence

and F is Faraday constant.

The Diffusion coefficient: D =RTF

• u

Note that is the conductivity of an ionic species. In solution:λ0

MA ↔ M+ + A– so solution conducticity: Λ0 = λ+0 + λ–

0 at infinite

(salt) λ +0 λ –

0 dilution. Solution conductivity is concentration

dependent.


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.

200

100

0

0 0.1 0.2 0.3

KCl

acetic acid

Con

duct

ivity

concentration

Λ

infinite dilution: Λ = Λ0

Ion Atomic ∆H°hydration

Mobility Radius (Å) (kcal/mole) 10-4(cm/sec)/(V/cm)Tl+ 1.44 . 7.74H+ . . 36.3NH4

+ 1.48 . 7.52Cs+ 1.69 -72 8.01Rb+ 1.48 -79.2 8.06K+ 1.33 -85.8 7.62Na+ 0.95 -104.6 5.19Li+ 0.6 -131.2 4.01Cl– 1.81 -82 7.92F– 1.36 -114 5.74Br– 1.95 -79 8.09I– . 2.16 -65 7.96NO3

– 2.9 . 7.41Mg2+ 0.65 -476 2.75Ca2+ . 0.99 -397 3.08Sr2+ . 1.13 -362 3.08Ba2+ . 1.35 -328 3.3

Data are taken from compilations by Bertl

Hille 1984 Ionic Channels of Excitable

Membrane. Sinauer Associates.

Properties of Ions

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).

Palmgren (2001, Ann. Rev. Pl. Physiol. Pl. Molec. Biol. 52:817–845) lists ionic radii of selected dehydratedcations (but without direct citation) as follows: H

3O+

(1.15 Å), Na+ (1.12), K+ (1.44, Ca2+ (1.06)

A measure of the energetics of water ‘binding’ to the ion to create a hydration sphere is the

enthalpy of hydration


Br– I–

Cl–Cs+

Rb+K+

F–Na+Li+

Ba2+

Sr2+

Ca2+

Mn2+

Mg2+

Ca2+ Sr2+

Mg2+

Ba2+Li+

Na+F–

I–Br–

Cl–Cs+Rb+

K+

NH4+

Tl+

NO3–

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).

Ion mobility versus atomic radius

Hydration enthalpy versus atomic radius

Atomic Radius (Angstroms)

Atomic Radius (Angstroms)

1.0 1.5 2.0 2.5 3.0

1.0 1.5 2.0 2.5 3.0

Ent

halp

y (h

ydra

tion)

(kc

al m

ole-1

)M

obili

ty (

[m s

ec-1

]/[V

olt m

-1])

)

2

4

6

8

10

-50

-320

-230

-140

-410

-500


Br–

I–

Cl– Cs+Rb+

K+

F–

Na+

Li+Ba2+Sr2+

Ca2+Mg2+

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).

Mobility versus hydration enthalpy

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.

0 5 10 15 20 25 30 35 400

3

6

9

12

15

Mobility

Con

duct

ance

(re

lativ

e to

sod

ium

)

H+

Tl+

Li+

Na+

Cs+

K+

NH4+

Rb+

Enthalpy (hydration) (kcal mole-1)

Mob

ility

([m

sec

-1]/

[Vol

t m-1

]))

4

6

8

10

-50-320 -230 -140-410-5002

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 geometry is spherically symmetric∂c

∂t= D

∂2c

∂x2∂c

∂t= D[ ∂2c

∂x2 +∂2c

∂y2 +∂2c

∂z 2 ]Jr(r ) = −D

∂c

∂r∂c∂t

= D1r 2

•∂

∂r•(r 2 •

∂c∂r

)These are geometric variants of Fick's Second Law of Diffusion.

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.


0.5 sec2

832 sec

Cyt

osol

ic [

Ca2+

] (n

M)

Distance from the Tip (μm)

[Ca2+] =M

2(πDt)1/2 e(−x 2 /4 Dt ) + [Ca2+]basal

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 cal-cium at the tip and sequestration behind the tip:

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.

Actual data is shown as well as the time dependence of the calcium gradient.

450

390

330

270

210

1500 5 10 15 20 25


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:

Thus, r = 6 • D • t1010

109

108

107

106

105

104

103

102

101

10-6 10-5 10-4 10-3 10-2 10-1 100 101 102

Dif

fusi

on T

ime

(sec

onds

)

Diffusion Distance (meters)

1 century

1 year

1 day

1 hour

1 minute

1 month O2 moleculea (1.80 • 10–9 m2 sec–1)

Hemoglobina (7.00 • 10–11 m2 sec–1)

Diffusion works best at small distances.

aBrouwer 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.

< x(t) >=1

N[xi(t −1) ± δ]

i=1

N

∑ = 0

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):

< x 2(t) >= 2Dt

< r2(t) >= 6Dt


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.

For the specified boundary conditions, the solutions are:

C(r ) = C0(1−a

r) and, Jr (r )= −DC0

a

r 2

The molecules are absorbed at a rate equal

to the sphere area times the inward flux (Jr (a)):

I = 4 • π • D • a • C0

Diffusion to a spherical absorber

C = 0 C = C0 at x >> s

sThe molecules are absorbed at a rate of: I = 4 • D • s • C0

Diffusion to a disk absorber

Diffusion to a cell covered with N absorbing disks.

I

4 • π • D • a • C0

=1

1 + π • aN • s

The molecules areabsorbed at a rate less thanfor a spherical absorber

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

N absorbing disks

I/I0

π•a/s

a

C = 0 C = C0 at x >> s

Gramicidin is a very simple protein-aceous pore structure which traverses the membrane and exhibits ion con-ductance. Relative cation selectivity is shown versus ionic mobilities.

0 5 10 15 20 25 30 35 400

3

6

9

12

15

Mobility

Con

duct

ance

(re

lativ

e to

sod

ium

)

H+

Tl+

Li+

Na+

Cs+

K+

NH4+

Rb+

Channel Function and Structure – page 1.25

5Å 25Å

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.

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

channelTechniqueBLM

-

voltage clamp

fusion

trans cis

membrane (BLM) technique. In this method, the gramicidin channel is incorporated into a lipid membrane seperating two compartments. Ion concentrations in the compartments can be controlled. The channel activity is monitored using a current to voltage converter, an electronic design also used to measure ion channels in the patch clamp technique. This is how ion conductances were measured (below).

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.


M+

M+

M+

M+

M+Passage of ions through the pore is measured as current. Current occurs in step-like transitions, due to opening and closing of the channel.

3 picoAmpere at 100 mV: 30 pico Siemen conductance

Current can be converted into flux:

I = z • F • Jamperes(coulombsper second)

valence

Faradayconstant(96,490 coulombs/mole)

Flux(mole/second)

J =I

zF=

10−9 (coulombs/mole)

+1 • 96,490 (coulombs/mole)= 1.036 •10−14 (moles/sec)

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 •lA

= 100 (Ω • cm)•2.5 •10−7 (cm)

π •(2.5 •10−7 (cm))2 =12.73 •109 Ω

Conductance=1

12.73 •109 Ω= 78.6 picoSiemens

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.


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.

CI- Malate2- NO3- I- Br- F-

(n=3) (n=18) (n=12) (n=4) (n=9) (n=4)PX/PCl 1±0.04 0.24±0.19 20.9±11.2 0.98±0.16 2.4±1.5 1.26±0.4IPeak(pA) -231±183 -77±58 -747±378 -146±112 -791±340 -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

5 6 7 8 90

5

10

15

20

25

NO3-

Br-

F-

I-CI-

PX/PCl

Mobility (cm/sec)/(V/cm)

The selectivity of a chloride channel from plants is compared to the mobility of the anion.

5 6 7 8 9

-800.0

-712.5

-625.0

-537.5

-450.0

-362.5

-275.0

-187.5

-100.0

F- NO3-

Br-

I-

CI-

The conductance through a chloride channel from plants is compared to the mobility of the anion.

IPeak(pA)

Mobility (cm/sec)/(V/cm)

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.

Chloride Channels (permeability properties of a CI- channel from a higher plant guard cell)


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.

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.

Cl–

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. Cl-

ions are shown as red spheres. Dashed lines highlight the pore entryways.


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

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