mol of substance J = sec • cm

To describe the movement of molecules, we need a framework. The one which is commonly used is flux (J):

Flux, J, depends upon the concentration of the substance. Or rather, the difference inconcentration between two adjacent regions.

More accurately, rather than equating flux, J c/ x, it is a point tangent to the concentration versus distance curve: J dc/dx (the derivative of concentration with respectto distance x:

Molecular and Thermodynamic Explanations of Ion Motion – page 1.01

J = mol of substance

sec • cm2

0

5

10

15

20

25

Con

cent

ratio

n (c

)

Distance (x)

c

x J

0

5

10

15

20

25

Con

cent

ratio

n (c

)

Distance (x)

dc

dx J

cx


J D •dc

dx(

cm2

sec)(

mol

cm4 ) (mol

sec•cm2 )

The units of the derivative, dc/dx 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:

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 is 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.


To understand how a concentration gradient results in flux requires a mechanistic description of the movement of the molecule of interest. Einstein presented a solution based upon a random walk.

Starting with a line (a one-dimensional case):

with N(x) particles at x and N(x+ ) particles at x+ . The symbol, , refers to a smalldistance away.

How many particles will move across the boundary from point x to point x+ 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+ , half the particles at x will have moved to x+ ,and half the particles at x+ will have moved to x.

The net number of particles going from x to x+ will be -1/2[N(x+ ) - N(x)]

N(x) N(x+ ) number of particles

x x+ distance

1/2N(x)

1/2N(x+ )

N(x) N(x+ )

x x+

1/2N(x)-1/2N(x+ )= -1/2[N(x+ )-N(x)]

This presentation is based upon a paper by Albert Einstein published in 1908 in Zeit. fur Elektrochemie 14: 235-239. An english translation is available in a Dover publication entitled "Investigations on the Theory of the Brownian Movement".


Jx

1

2[ N(x + )– N(x)]/ A ,

multiplying by2

2

Jx –12

2

2

1A

[ N(x + )– N(x)]A has units of volume

Jx

1

2

2 1 [ N(x )

A

N(x)

A]

N divided by volume is concentration

Jx

1

2

2 1 [C(x ) C(x)]re-arranging

Jx

12

2

[C(x ) C(x)]


Now, if we take the term

[C(x + )– C(x)]to the limit 0, then

C(x + )– C(x) Cx

therefore

Jx –1

2

2 C

x

where 1

2

2

with units of:cm2

sec

is the Diffusion coefficient, D

Flux depends upon random movements of particles. Concentration gradients result in net movement because there are more particles in localized regions.

The speed at which this occurs depends upon the Diffusion coefficient, whose units cm2•sec-1 imply the amount of space a particle will explore in a small unit of time, t. Therefore, the Diffusion coefficient is a property of the particle.

This is the same form as Fick’s Law ofDiffusion (J = D • dC/dx)


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

dP, the permeability coefficient with

units ofcm

, 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


Journal of the History of Biology 33: 71–111, 2000.©2000 Kluwer Academic Publishers. Printed in the Netherlands.

Diffusion Theory in Biology:A Relic of Mechanistic Materialism

PAUL S. AGUTTERFormerly of the Department of Biological SciencesNapier UniversityColinton RoadEdinburgh, EH10 5DT, U.K.

P. COLM MALONE129 Viceroy CloseBirmingham B5 7UY, U.K.

DENYS N. WHEATLEYCell PathologyUniversity of AberdeenMacRobert Building581 King StreetAberdeen AB24 5UA, U.K. (E-mail: [email protected])

Abstract. Diffusion theory explains in physical terms how materials move through a medium, e.g. water or a biological fluid. There are strong and widely acknowledged grounds for doubt-ing the applicability of this theory in biology, although it continues to be accepted almost uncritically and taught as a basis of both biology and medicine. Our principal aim is to explore how this situation arose and has been allowed to continue seemingly unchallenged for more than 150 years. The main shortcomings of diffusion theory will be briefly reviewed to show that the entrenchment of this theory in the corpus of biological knowledge needs to be explained, especially as there are equally valid historical grounds for presuming that bulk fluid movement powered by the energy of cell metabolism plays a prominent note in the transport of molecules in the living body. First, the theory’s evolution, notably from its origins in connec-tion with the mechanistic materialist philosophy of mid nineteenth century physiology, is discussed. Following this, the entrenchment of the theory in twentieth century biology is analyzed in relation to three situations: the mechanismof oxygen transport between air and mammalian tissues; the structure and function of cell membranes; and the nature of the inter-mediary metabolism, with its implicit presumptions about the intracellular organization and the movement of molecules within it. In our final section, we consider several historically based alternatives to diffusion theory, all of which have their precursors in nineteenth and twentieth century philosophy of science.

Keywords: diffusion theory, Fick, 19th century physiology, mechanistic materialism, oxygensecretion, metabolic organization

caveat emptor: It is important to note that

diffusion is a poor explanation for many

examples of molecular motion in biological

organisms. Simple models explain the movement of

many molecules, but not all. Nor have we acknowledged the crucial effect of turbulent

mass flow, a key aspect of molecular movement. The

non-universality of diffusive properties has led to

ideological attacks on biology.


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

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:

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

J mobility (cmsec) • activity(mole

cm3 ) • driving force (d dx)

J = u • c • ddx


From thermodynamics, the chemical potential is:

= RT(ln c) zF

so,

d RTd(ln c) zFd RT1c

dc d

d

dx of d

RT

c

dc

dxzF

d

dx

J u • c •[ RT

c

dc

dxzF

d

dx]

J (uRT )(dcdx

) zFucddx

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:

actually are: We need to integrate over the boundary conditions of the membrane.

R is the gas constantT is the temperature °Kc is the concentrationz is the valence (ionic charge)F is the Faraday constant

is the electrical potential

J (uRT )(dcdx

) zFucddx


o (outside potential)

i (insidepotential)

L (distance)

o - i

/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,

Inserting / x = /L:

J D(dc

) zFucL

re arranging to isolate the differentials:

-D

J + zFuc L

dc

we can integrate over the boundary conditions

co: concentration outside; ci: concentration inside

0 to L the width of the membrane

-DJ + zFuc L 0

L

c o

c i

which yields:

J P(zF

RT)[co ci • exp(zF

RT)]

[1 exp(zFRT)]

where P = D/L

This is called the Goldman constant field equation. Constant field because we

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:

J P(zF


RT)]

[1 exp(zFRT)]

0 0

0 co ci • exp(zFRT)

co

ci

exp(zFRT)

lnco

ci

zFRT

RT

zFln

co

ci

55log10

co

ci

(mV)


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

ci55log10

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 ( = 0).

J P(zF


RT)]

[1 exp(zFRT)]

For small potentials approaching zero:

exp(zFRT) 1 zF

RT

J PzF

RT[co ci exp(1 zF

RT)]

1 (1 zFRT)

J PzF

RT[co ci(1

zFRT)]

( zFRT)

J P[co ci(1zF

RT)] P[co ci ci(zF

RT)]

since = 0

J P[co ci ]

the equation for an uncharged solute.

If we set = 0, exp(0)=1, so we get 0/0, undefined. Therefore, we resort to a mathematical sleight of hands to solve for the case, = 0.

Ionic Mobility – page 2.01

Molecular motion can be described from either thermodynamic or mechanistic 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. Both of these descriptions were introduced by Einstein (among others) during the early 1900's. In essence, they describe the mobility of the molecule. But what is mobility? To address this we return to a one dimensional random walk and examine the effect of an applied force.

x+ – x x+ + distance

particle of mass m force acting in the x direction

The force Fx results in an acceleration in the +ve direction (x+ +), 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 + x or – x once every seconds.

m Fx


{+ x velocity

distance moved is: + = x + a 2/2 distance moved due to acceleration.


{- x velocity

distance moved is: – = – x + a 2/2 distance moved due to acceleration.

Note the introduction ofacceleration. 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, .

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

then the average displacement,2

, is:

( xa 2

2) (- xa 2

2 )

2the x terms cancel out:

(a2

2) (a 2

2 )

2(a 2

2 ) (the average displacement).

The average velocity (recall = ) is:

a 2

2 a2 or, =

1

2

Fx

m•

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

=Fx

f where f 2m

To obtain a more meaningful description of frictional drag:

f 2m •

2

2

2

2

since 2 =2

2 then f

2m 2

2

22

m 2

but2

2D (the Diffusion coefficient), so

f2m 2

DFinally, substituting the kinetic equation, m 2 kT

fkT

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 • • r • where r is the radius of the sphere

and is the viscosity of the solution.

From the relationship =Fx

f , Fx f •

For an ion, the force is an electrical one: z • e •

where z is the valence, e is the electron charge and the potential.

So, z • e • 6 • • r • •

The ionic mobility is defined by the ionic velocity per volt of driving force.

u =z • e

6 • • r • 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:

0 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:0

MA M A– so solution conducticity: 0 0–0 at infinite

(salt) 0–

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

infinite dilution: 0

hydration Mobility

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

+ . . 36.3

4+ 1.48 . 7.52

Cs+ 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 IonicChannels of Excitable Membrane. Sinauer Associates.

Properties of Ions

The enthalpies of hydration hydrationshould 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)


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 ( 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 Section 5).

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 3.01

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 Dc

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

tD

2c

x2c

tD[ 2c

x2

2c

y2

2c

z 2 ]Jr(r ) D

c

rct

D1r 2

•r

•(r 2 •cr

)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( Dt)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) ]

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(1a

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 • • 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 • • D • a • C0

=1

1 + • 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

a

C = 0 C = C0 at x >> s

The electrical properties of cells include four electrical concepts: Voltage, Current,Resistance and Capacitance

Ohm's Law defines the relationship between Voltage, Current and Resistance:

V = I • Runits: Voltage Amperes Ohms

(V) (A) (Ω)(Coulombs/second)(charge transport)

Resistance(Ω)

Current(A)

Voltage(V){

or....... Electrical potential (voltage)in series witha resistancegenerates acurrent

Current passingthrough a resistancegeneratesan electrical potential(voltage)

For a cell:

-100 mV 0 mV

10 MΩ

V = I • RI = V/R10-8 A = 10-1 V/107 Ω(10 nA)

Amicroelectrode impaled into the cell is usedto measure the electrical potential across thecell plasma membrane

I

Resistive Networks – page 4.01

There is a mental short-hand for units used in Ohm's Law:

10-9 A = 10-3 V / 106 Ω 10-12 A = 10-3 V / 109 ΩnA mV MΩ pA mV GΩnanoAmps milliVolts megaOhms picoAmps milliVolts gigaOhms

(suitable for cellular impalements) (suitable for patch clamp)

Ohm's Law can be viewed as a current-voltage relation:

Current

Voltage

I = g • V

slope (g) = conductance (units: Siemens)where,

g = 1 / R

(conductance is the inverse of resistance)


If this line is linear,it is calledohmic, sinceit 'obeys' Ohm'sLaw. Most cellsdo not haveohmic I–V relations

In a biological system, resistances are often more complex. They may beconfigured in parallel or in series

Resistances in seriesare simpler

Resistances in parallelare more complex

For example, if R1 and R2 are both 5 MΩ, Rtot = 2.5 MΩ

R1

R2}Rtotal = R1 + R2

R1 R2

or Rtotal =R1 •R2(R1+ R2 )

Rtotal =1

( 1R1

+ 1R2)

(obtained by multiplying byR1/R1 and R2/R2)


Intuitively, if you have two resistances in series, current has to flow throughboth of them, so total resistance is additive. But if the resistances are in parallel,the current has two alternative pathways, so total resistance is lower thaneither pathway alone.

Is this important? Yes.

Take an epithelial tissue:

membrane resistance

cytoplasmicresistance

between cellresistance

gap junctionresistance

The resistive networkfor two cells................

OK, now extend this to a thousand cells.Now, apply a voltage, where is the current going to flow?


There are some simple rules which depend upon the relative magnitude of thevarious resistances.

In series, if one resistance is much larger than the other:

the total resistance is approximately equal to the larger resistance:

In parallel, total resistance is approximately equal to the smaller resistance:

R1>> R2Rtotal ≈

R1>> R2 Rtotal =R1 •R2(R1+ R2 )

≈R1 •R2R1

≈ R2

R1 = R2 Rtotal =R2

(2•R)=R2

Thus, relative approximations of current flow can be made.

Formally, Thevenin's equivalent circuit theorem states that any two terminalnetwork of resistors and voltage sources is equivalent to a single resistor in serieswith a single voltage source. So the epithelial network:

Can be simplified to:

There is a problem.......

For an epithelial tissue, the simplified pathway can be measured. But if one ismeasuring the voltage difference and resistance across the tissue, it is not possibleto dissect out the relative contributions of each current pathway.

R1


R1 < R2

R1

d1

R2

d2

r1

r2

To normalize resistance, one uses resistivity , or specific resistance:

Material Resistivity (units: cm)

Glass 1012

•1015

dH2O 25•10

6

0.1 N KCl 90

cell membrane 1•109


Resistance in biological systems is a relatively simple concept. Although we focus on electrical resistance, resistive analysis is also used to research water and gas flow through complex tissues. In the context of electrophysiology, the resistance you measure in a solution will vary with the distance between the two probes of the measuring device. And resistance of a cell will vary with the size of the cell.

Increased distance, d is analogous to more resistors in series.

Increased radius r is analogous to more resistors in parallel. Thus, the larger the cell, the lower the resistance

(tap water has much lower resistivity thandH

2O due to ions in the water)

Voltage Dividing

Suppose you have a voltage source in series with a number of resistors:

So,

V1 = (R1 / Rtotal ) • Vtotal

The multiple resistors act to 'divide' the voltage, hence the term voltage divider.

Is this important? Yes.

R1

R2

R3

Vtotal

V1

V2

V3

Itotal

There is a total current flowthrough the resistive networkof Itotal which must pass throughall three resistors. The voltage acrossR1, R2, or R3, would be:

V1 =Itotal • R1

V2 =Itotal • R2

V3 =Itotal • R3

and,Vtotal = V1 + V2 + V3


Here are some examples where voltage dividing is an experimental problem.

Velectrode = (Relectrode / Rmembrane + Relectrode) • Vmembrane

If Relectrode ≈Rmembrane, then Velectrode ≈ 1/2 • Vmembrane.

If Relectrode >> Rmembrane, then Velectrode ≈Vmembrane.

Rshunt

Relectrode which includesthe resistance of the instrumentation.

Imembrane

}Vtotal

What you want is to measure the voltage across the membrane, Vmembrane.The equivalent circuit:

Rmembrane

Vmembrane

RmembraneRshunt

Vmembrane

Relectrode

Velectrode

Velectrode(what you measure).

Rshunt complicates resistance measurements and can attenuate measurements of Vmembrane.It is most significant in small cells. Ignoring it for now:

ground


In real life, the electrode is connected to an electrometer, so that Relectrode includesnot only the electrode resistance, but also the electrometer resistance, a specificationcommonly called the input impedance, usually about 1011 Ω.In small cells, Rmembrane may approach 1011Ω, errors (significant underestimationof Vmembrane) may result.

Current Sources and Resistance

Suppose you have a battery and some resistors, and want to inject a specifiedcurrent into the cell, usually to measure Rmembrane.

Iinjected = Vapplied / RtotalIinjected = Vapplied / (Rmembrane+ Relectrode)If Relectrode >> Rmembrane, then the magnitude of thecurrent will be independent of Rmembrane.

Iinjected

Rmembrane

Vapplied

Relectrode

RelectrodeRmembrane

VmembraneVapplied


Capacitance

Capacitance is the final elementary electrical property we need to consider.

Capacitance is defined by the relation:

C = Q / V, or Q = C • Vunits: Farads Coulombs Volts

Intuitively, capacitance can be considered the ability or capacity to hold charge.

More formally, a capacitor of C farads with V volts across its terminals has+ve Q coulombs of charge on one plate, and –ve Q coulombs of charge on theother plate.

Q+ Q–

Voltage

dQdt=ddtC • V= I = C

dVdt

Now, taking the derivative with respect to time:

(coulombs per second:the same as current, I)

(the capacitance,C is constant)

That is, the current, I is proportional to the rate of change in voltage, dV/dt.Voltage must be changing to cause current to pass across the capacitor.


Analogous to resistance, capacitors sum differently when they are configuredin parallel or in series:�

In parallel:

Ctotal = C1 + C2

In series:

C1

C1

C1 C1Ctotal =

11C1+ 1C2

Ctotal =C1 • C2C2 +C2


Suppose we have a circuit:

Now,

Integrating:

which looks like:

RQ+

Q–C

At time = 0, we close theswitch and discharge thecapacitor charge, Q.

CdV

dt= I = -

V

R

Negative, because it is a discharge. V/R follows from Ohm's Law: I=V/R

V= V(0) • e- t

RC

V (0) (Vtime=0)

37%

t = R • C time

Voltage


CdV

dt

V

R1

VdV

1

RCdt

1V

dV1

RCdt

ln(V )t

RCA

V (t) etRC • eA

At time t = 0, etRC 1

so, eA V (0), thus

V (t) V (0) • etRC

Alternatively, capacitor charging:

Now,

R

C

At time = 0, we close theswitch and charge thecapacitor.

time

ViV

Vi

t = RC

63%

Is this important? Yes. Cell membranes have capacitance. In addition, electronicinstrumentation and electrodes have capacitance. These affect the timedependence of voltage and current measurements in a variety of ways.

I = C dVdt= (Vi - V)

RintegratingV=Vi(1- e

- tRC)


We now have an introductory understanding of the four electrical propertiesof cells: Voltage, Current, Resistance, and Capacitance.

Amodel of a cell:

EK

gK

ENa

gNa

ECl

gCl

Na, K,or Hpump

The ions which are transported across the membrane include K+, Na+, and Cl–.The ionic conductance (recall g = 1/R) for each of these ions is variable. A majoruser of cellular metabolism is the ion current pump: using ATP to transport K+and Na+, or H+ across the membrane.

We now need to consider the tools that allow us to measure the electricalproperties of the cell.

Ecell

RcellRinput Vmeasure

ground

cell electrometerRinput is the input resistance(sometimes called the inputimpedance). Duringmeasurements, the electrical

network dictates that:

Vmeasure =Rinput

Rinput + Rcell

Ecell

Therefore, accuracy depends upon Rinput >> Rcell. Since Rcell is often about 20 MΩ,Rinput must be very high to measure Ecell accurately.


The electrical properties of cells are a consequence of ion fluxes and steady stateion concentrations within and outside the cell. Therefore, current is not due toelectron flow, but to ion flow:

I = z • F • J

And voltage when the current is zero (I = 0)(the reversal potential):

Amperes Valence Faraday Fluxunits: (Coulombs/sec) (net charge on ion) (96,480 coulombs/mole) (moles/sec)

E= z •RTF• ln[

cico]

Gas Constant, Temperature Ratio of IonPotential Valence and Faraday Concentrations

units: (voltage) (net charge on ion) (25.3 mV at 25º C)

The equation holds true if one and only one ion is permeant through the cellmembrane.


If the membrane is permeant to more than one ion, the equivalent circuit is:

EK

RK

ENa

RNa

ECl

RCl

For simplicity: gNa = 1/RNa, etc.

Then:

Etotal =gKg∑ •EK +

gNag∑ •ENa +

gClg∑ •ECl

If any conductance changes, gK, gNa, or gCl, then so doesVtotal.Is this important? Yes.

Action potentials: Changes in conductances over time, yield changes in Voltage:

0

2

4

6

8

10

0

1

2

3

4

5

6

7

8gK

gNa

Vtotal

Time (seconds)

Vtotal


Under circumstances where the conductance changes as a consequence of voltagechanges, the current-voltage relation is no longer ohmic (That is, Ohm's Law nolonger applies). The effect of multiple conductances under ohmic conditions:

g 2•g 3•g

0.0

0.2

0.4

0.6

0.8

1.0

-6

-4

-2

0

2

4

6

8

10

E1

g

E2

g

-12-10-8-6-4-2024681012

g

2•g

3•gCurrent (I)

Voltage

-15

-10

-5

0

5

10

15

20 Current (I)

Voltage

E1

E1

OR

Conductance (g(E2))

Voltage

E1 E2

Current (I)

Voltage

E2

E1

E1

g(E1)

E2

g(E2)

OR

1

2

1 2

Now suppose g changes with voltage. As it changes,the reversal potential, where current is zero will changefrom E1 to E2. But in addition, the current can exhibitan 'N' shape, a classic property of action potentials.

1 2


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 5.01

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)

JI

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.

R •lA

100 ( • cm)•2.5 •10 7 (cm)

•(2.5 •10 7 (cm))2 12.73 •109

Conductance1

12.73 •109 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

Root Hair Case Study – page 01

Membrane Ion Transport Case Study:The Root Hair

During growth and development of the plant root, root hairs often play a key role in nodulation, and water and ion uptake. Root hairs are not an obligatory cell type on the root surface. However, when present, they have highly active respiration , and are known to preferentially express a number of transport proteins, such as the plasma membrane H+ ATPase, ammonium and nitrate transporters, and a phosphate transporter. In fact, root hair length increases when phosphorus levels are low. The

preferential expression of ion transporters and morphological responses to nutrient conditions both suggest that root hairs have special and unique functions in ion transport. In addition, they are an ideal system for examining ion transport mechanisms in situ because they have anatomical and cytological characteristics that simplify the micromanipulation, impalement, and micro-injection necessary to characterize ion transport. They are readily accessible and microscopic imaging is straightforward because of the lack of interference by surrounding tissue. This even results in an ability to manipulate specific intracellular targets, useful not only in studies of ion transport, but also cellular signal transduction in general.

Developing root hairs on a root, from the website of Dr. Moira Galway, Department of Biology, St. Francis Xavier University, Antigonish, Nova Scotia, Canada:

http://www.stfx.ca/people/mgalway/galwaymain.html

Rshunt

Relectrode which includes

the resistance of the instrumentation.}Vtotal

Rmembrane

VmembraneVelectrode

ground


Measurements of the electrical properties of the root hair cell require the impalement of a micropipette into the cell. The voltage between the inside and outside of the cell is measured by an electrometer, a specialised voltmeter connected to the micropipette by a silver/silver-chloride wire.

Ohm's Law defines the relationshipbetween Voltage, Current and Resistance:

V = I • Runits: Voltage Amperes Ohms

(Coulombs/second) (charge transport)

Resistance Current(A)

Voltage(V){

or.......Electrical potential(voltage)in series witha resistancegenerates a current

Current passingthrough aresistancegeneratesan electrical potential(voltage)

I


Example of electrical measurements of a cell. In this case, a fern gemetophyte cell impaled by an undergrdauate honours student, Harpreet Atwal. The impalement is with a double barrel micropipette, so that current can be injected into the cell, independent of the voltage measurement

Voltage (milliVolts)

Current (nanoAmps)

Cell WallPotential

SlowSealing

Impalement

Out ofCell

Over the years, most measurements of the electrical properties of higher plant cells relied upon impalement with one micropipette. This effectively limited the types of measurements to only one: voltage, because measurements of resistance with a single microelectrode are complicated by the fact that the micropipette itself can have a resistance (10-20 Mohm) similar to typical cell resistances. The best method to measure the full range of a higher plant cell's electrical properties is to use two microelectrodes. This may require the impalement of multiple micropipettes into a single cell, or double-barrel micropipettes. With two microelectrodes, one may be dedicated to voltage measurements, while the other microelectrode is used to inject current into the cell. The deflection of the voltage (measured with the first microelectrode) caused by the current injection (through the second microelectrode) can be used to calculate the membrane resistance.

Dark Treatment


For root hairs (and other cells), the electrical properties consist of a voltage difference (the membrane potential) and resistance in parallel with the membrane capacitance. Capacitance is the ability of the cell to hold charge: Q=CV, where Q is the net charge (in coulombs), C is the capacitance (in farads), and V is the voltage difference. Typical values for these three components are -190 mV voltage difference, 20-40 Mohm resistance, and ca 1 nanofarad capacitance. The voltage differences and resistances are related through Ohm's law: V (voltage) = I (current) R (resistance). In root hairs, the plasma membrane H+ ATPase generates a H+ current that is the major contributor to the voltage difference between the cytoplasm and the outside of the cell, based upon inhibition of the pump with vanadate.


Nernst PotentialDifferences in the concentration of an ion inside and outside the cellcause the formation of an electrical potential:

where z is the valence of the ion, F is the Faraday constant (96,490 Jmol-1 V-1), R is the gas constant (8.314 J mol-1 K-1), and T is thetemperature (K)[1]. For room temperature, we can simplify:

[1] Nobel PS (1991) Physicochemical and Environmental Plant Physiology. Academic Press, San Diego.

ERT

zF• ln

c in

c out

K+K++

E (mV) 55 • log

The Nernst potential is used to quantify the contribution of potassium to the transmembrane voltage of the root hair plasma membrane

c in

c out


H+

ATP ADP + Pi

active pump

2H+

symport

Cl- K+

K+

channels

plasma membraneproton ATPase

K+

Cl-

Some of the transporters known to exist in root hairs are shown below, along with electrical measurements of the plasma membrane and vacuolar membranes.

Voltage Difference: -170 mVMembrane Resistance: 5.3 MegaOhmMembranbe Capacitance: 1 nanoFarad

Voltage Difference: +22 mVMembrane Resistance: 1.6 MegaOhmMembranbe Capacitance: 1 nanoFarad

The activity of the proton pump generates a proton motive force, made up of the transmembrane voltage and pH difference.

plasma membrane

vacuolar membrane

RT

F• ln

aoutH

ainH

The proton motive force provides the energy for uptake of other ions. The proton/chloride symport uses both components


During electrical measurements, the root hair is growing. Growth represents an expansion in the cytoplasmic and vacuolar volumes. The ionic composition of both compartments must be maintained during growth. In part because growth is caused by a high internal hydrostatic pressure, that causes the expension of the apex of the root hair, somewhat like blowing up a balloon.

plasma membrane cell wallvacuolar membrane20 m

The growth rate is about one micron per minute

The volume increase can be described as the continuous addition of a cylindrical element to the root hair. The volume increase is the Area • Length,

2•h. For a cell of radius 4 μm and growth rate of 1 μm min-1, the volume increase is 50 μm3 min-1, or 50 femtoliters min-1. To maintain a cell osmolarity of about 500 mOsmol requires an ion influx of 25 fmol min-1: (~500 mMole liter-1)(50•10-15 liters min-1)


The data examines the mechanisms that transport the necessary ions into the cell to maintain the cell's osmolarity during cell expansion. Treatment with a potassium channel inhibitor, TEA-Cl, causes the transmembrane voltage to become more negative and inhibits root tip growth, indicative of a requirement for potassium uptake during cell expansion. Cyanide inhibits respiration. This depletes cellular ATP, so that the proton pump cannot function, resulting in a large depolarization, indicative of the contribution of the proton pump to the transmembrane potential.

H+

ATP ADP + Pi

K+

channels

+

Normally, the potassium ion influx causes a slightly more positive transmembrane potential. When the channels are inhibited, the positive ions no longer depolarize the potential, which becomes more negative. The magnitude of the inward potassium current, about 0.6 nA, is more than sufficient to maintain the hydrostatic pressure during growth:

+ve charge in +ve charge out

–

JI

zF

0.6 •10 9(coulombs/sec)

1 • 96,480 (coulombs/mole)• 60 (sec/min) 373 femtomol/min

TEA inhibition reveals the potassium current

Documents

mol of substance J = sec • cm