Thermodynamic equilibrium potentials (Nernst, Donnan).

Diffusion potential, Goldman-Hodgkin-Katz equation.

Passive electrical properties of biological membranes

anion (–)cation (+)

The cell membrane acts as a capacitor:-lipid bilayer high electrical resistance(good insulator, ~103- 109 Ωcm)-aqueous solutions on both sides(conductive material)

Capacitance of biological membranes is expressed normalized to unit surface area (normally 0.4-1.0 μF/cm2)

Potential difference across of biological membranes

anion (–)cation (+)

•An electrical potential difference between the two sides of the membrane ~70 mV means ~105 V/cm electric field (E=U/d). This is an enormous electric field!

• relatively small charge separation produces a large potential difference across the membrane:

Q=CU = 1.0 μF/cm2 ×70 mV = 7×10−8 C/cm2

1 M ion has a charge of ~96,500 C, thus 7×10−8 C/cm2 corresponds to

~10-12 mole ion/ cm2 which is negligible as compared to the ion content of living cells.

7 nm

r=5 μm

Appendix to slide #3

Vcell =4 r3 π/3= 5.233 ×10−13 L[K+] = 150 mmol/L

that is 7.85 ×10−14 mole K+ ions inside

Acell= 4 r2 π = 3.14 ×10−6 cm2

7.85 ×10−14 mole K+/3.14 ×10−6 cm2 =2.5 ×10−8 mole ion/ cm2 maximum ion density.

For 70 mV potential difference: 10-12

mole ion/ cm2 ion density is required only!! (Ten thousand times less !!)

Biological membranes (e.g. plasma membrane) separate compartments having remarkably different ionic composition

Ion [extracellular] mM [intracellular] mM

Na+ 140 10-20K+ 5 140-150Ca2+ 1-2 10−4

Comparison of resting membrane potential (Em) in different cell types:

Neuron –70 mVSkeletal muscle – 80 mVCardiac muscle – 80 mVRed blood cell – 11 mVT lymphocyte – 50 mV

Ψi

Ψo

PD = Em= Ψi –Ψo

selective permeabilityof the membrane

unequal distribution of ions in the extra- and intracellular space

membrane potential

transmembrane ion fluxes

for steady (trans)membrane potential zero net charge flux is required

thermodynamic equilibrium potential•Donnan potential•Equilibrium potential,

Nernst equation

zero net flux for eachpermeable ion:

non-zero net flux of permeable ions but sum of the charge fluxes equals to zero

K+

K+

K+

Na+

•Goldman-Hodgkin-Katzvoltage equation

Thermodynamic equilibrium potential:

model for the derivation of the Nernst equation

The electrochemical potential of the permeable ions (K+) :

At equilibrium the electrochemical potential for the permeable ion should be equal in the two compartments:

zFψRTlncμμ KK0K ++=

0)ψzF(ψ]RTln[K]RTln[KΔμ IIIIII

K =−+−= ++

Ψ+= zFΔ[K][K]RTlnΔμ II

I

K mV59][K][Kln

zFRTΔΨ II

I

−=−= +

+

K+: 0.1 M K+: 0.01 M

Cl–: 0.01 M

side I. side II.

Cl– : 0.1 M

— +

T=22 oC

Summary of equilibrium potential (Nernst equation):

1, This potential difference results from diffusion of permeable ions( diffusion potential), no metabolic energy is required.

2, This potential difference is generated if the membrane is selectivelypermeable to one ion species only (in our case K+, ΔΨ=EK)

3, At equilibrium potential there is no net flux of K+.

The system is at thermodynamic equilibrium, diffusion is balanced by the electric potential difference, forward and backward fluxes are equal. Electroneutrality is preserved in the bulk solutions, there is a small charge separation directly across the membrane to account for ΔΨ=EK. Excess of K+ and Cl- in side II and side I, respectively, is practically unmeasurable.

Biological significance I.

1, Given that the plasma membrane is permeable to one ion species and concentration gradient exists for that ion membrane potential can be calculated form the Nernst equation.

Ion [x]i [x]o Ex

K+ 140 5 − 89 mVNa+ 10 140 +70 mVCl− 5 100 −80 mV

2, If the transmembrane potential equals Ex the transmembrane flux of ion species x =0.

o

i

x ][x][xln

zFRTE +

+

−=

(T= 37 oC)

Biological significance II.

In contrast in living cells:

1, the membrane is permeable to several ion species the charge separation generated by one ion species is destroyed by others, the diffusion potential is no longer stationary.

2, biological membranes are not at thermodynamic equilibrium, net flow of ions occurs at resting potential.

3, Nernst equation predicts that change in the membrane potential (e.g. action potential) occurs only if the extra- and intracellular concentrations of the permeable ion change dramatically. This is against homeostasis.

1, For real biological membranes the most significant contribution to the membrane potential can be calculated from the Goldman-Hodgkin-Katz voltage equation, generally quoted as net diffusion potential (Ediff).

2, Major difference as compared to the Nernst equation is is that Ediffis NOT a thermodynamic equilibrium potential.

3, Net ion fluxes flow across the membrane, only the sum of the charges moving is zero.

4, Derivation of the GHK voltage equation is difficult, kinetic equations for ion fluxes has to be determined.

Diffusion potential (Ediff), Goldman-Hodgkin-Katz equation

Intracellular space

Extracellular space

dψ

[c]o

[c]i

cpΔdxdcDJdiff −=−=

dxdψ

RTFzcD

dxdψzucJE −=−=

⎟⎠⎞

⎜⎝⎛ ψ

+−=+=dxdc

RTFz

dxdcDJJJ Ediff

⎟⎠⎞

⎜⎝⎛ ψ

+−==dxdc

RTFz

dxdczFDJFzI

l

⎟⎠⎞

⎜⎝⎛ ψ

+−==dxdc

RTFz

dxdczFDJFzI

This equation can be solved over the membrane thickness, l, and results in an equation describing the current density as a function of

•Permeability coefficient of ion s (ps)•Valence of ion s (zs)•Intra- and extracellular concentrations of ion s (si and so)•Potential difference (E)

Now take into consideration three ion species (K+,Na+, and Cl−) and solve for IK + INa + Icl = 0. The result is the GHK voltage equation:

iCloNaoK

oCliNaiKm [Cl]p[Na]p[K]p

[Cl]p[Na]p[K]pln

FzRTEdψ

++

++−==

-140

-120

-100

-80

-60

-40

-20

01 10 100

[K]i=150 mM[Na]i=15 mM[K]0+[Na]0=154 mM

0.001

0.01

0.050.10.2PNa/PK

[K]o (mM)E

m (

mV

)

oK

Nao

iK

Nai

m

[Na]PP[K]

[Na]PP[K]

lnzFRTE

+

+−=

Take home message: Ediff is determined by the ion concentrations and the relative permeability of the membrane to the ions.

Thermodynamic equilibrium potential: Donnan potential

t=0

Na+: 100 mM Na+:100 mM

Cl–: 100 mM

side II. side I.

A– : 100 mM

t=∞

Na+: 133 mM Na+:67 mM

Cl–: 67 mM

side II. side I.

A– : 100 mMCl–: 33 mM

— +

The electrochemical potential of permeable ions:

At equilibrium the electrochemical potential for each ion species should be equal in the two compartments:

ψ++μ=μ zFclnRT ii

i 0

0=ψ−ψ+−=μΔ ++ )(zF]Naln[RT]Naln[RT IIIIII

Na

0=ψΔ+=μΔ zF]Na[]Na[lnRT I

IINa I

II

]Na[]Na[ln

zFRT

+

+−=ΔΨ II

I

]Cl[]Cl[ln

zFRT

−

−−=ΔΨ

Na+: 133 mM Na+:67 mM

Cl–: 67 mM

side II. side I.

A– : 100 mMCl–: 33 mM

— +

Similarly, for Cl−:

What determines the magnitude of the Donnan potential?

1,

2, The principle of electroneutrality must be obeyed for the bulk solution:

•[Na+]I = [Cl–]I = c•[Na+]II = [Cl–]II +[A–]II

IIIIIIII

I

I

II]Cl[]Na[]Cl[]Na[isthatr

]Cl[]Cl[

]Na[]Na[ −+−+

−

−

+

+===

c2[A]c4[A]

r][Na][Na 22

I

II ++==+

+

Combining 1, and 2, (and some math) gives:

lnrzFRT

][Na][Naln

zFRTΔΨ I

II

−=−= +

+

A = anion conc.

1, Start from the principleof electroneutrality:

[Na+]I=100 mM–x[Na+]II=100 mM+x[Cl–]I=100 mM–x[Cl–]II= x

Thus, at equilibrium: Total amount of ions:[Na+]I= 67 mM II: 100 mM A–

[Na+]II=133 mM 133 mM Na+

[Cl–]I=67 mM 33 mM Cl–

[Cl–]II= 33 mMI: 67 mM Na+

67 mM Cl–

2, Use [Na+]II [Cl–]II= [Na+]I [Cl–]I and solve for x

How to find out the equilibrium concentration of ions?

(100 + x) x = (100-x) (100-x)

x = 33.3

mV][][ln

zFRT

]Na[]Na[ln

zFRT

I

II

I

II18

67133

−=−=−=ΔΨ +

+

Important issues of Donnan equilibrium:

•It is a consequence of non permeable anions in the cell (side II)

where A is the concentration of a protein anion A- and c is the ion concentration in side I after reaching equilibrium.

• Develops in the absence of metabolic energy and active transport processes.

•The PD generated is in the order of -20 mV.•The PD is a result of a very small charge separation, side II has a small excess of Cl– while side I has a small excess of Na+. •Ions are at thermodynamic equilibrium, no net flux occurs.•Permeable ions have the same permeation properties.•generates a huge osmotic imbalance which is not tolerated in living

cells (the concentration of ions inside is much larger than outside).

cAcA

rNaNa

I

II

2][4][

][][ 22 ++

==+

+