Membrane potentials

Thermodynamic treatment:

Nernst-Planck flux equation

Nernst equation Constant field equation

Equivalent electrical circuit:

Parallel conductance model

Active transport: pumps, carriers

Electrochemical potential

ui is the electrochemical potential (contribution of one mole of the ith constituent to free energy and is the drive for flux across the membrane)

Solute transport occurs from region of higher to lower electrochemical potential

For membrane transport: ui = ui

o + RTlnCi + ziFV + …

ziFV is the contribution of the electrical potential ie the work required to bring a mole of ions with charge z from zero potential to potential V (F = 9.65 x 104 coulombs/mole)

uio is the standard electrochemical potential at C= 1M,

V=0

Equilibrium

At equilibrium the electrochemical potential is the same on each side of membrane;

uio + RTlnC1 + ziFV1 = ui

o + RTlnC2 + ziFV2

V2-V1 = E = RT ln C1

ziF C2

Nernst potential

Z = valenceR = gas constant: 8.314 (volts x coulombs/KO x mol)T = absolute temperatureF = Faraday constant: 96,500 coulombs/mol

Nernst Equation: how it works

IN = 100 mM KCl OUT = 10 mM KCl

= K+

= Cl-

-

+

-

+

--

-

+```

(-)(+)

• A membrane separates solutions of 100 mM and 10 mM KCl:

• The membrane is permeable to K+, but not to Cl-

• K+ ions diffuse across the membrane from left to right

• The movement of K+ makes the inside negative with respect to outside - this membrane potential prevents further net movement of K+

• Equilibrium is quickly established; the magnitude of is computed from the Nernst equation

mvK

K

z in

out 60)1(*60100

10log*60

][

][log

60=−===

For a real cell

Area of cell membrane = 1000 µm2 ()Radius = 8.92 µmVolume = 2974 x 10-15 liters (4/3r3)

140 mM K+

5 mM Na+

145 mM Cl-

5 mM K+

140 mM Na+

145 mM Cl-

K+

Vm = 58.17 log [5]/[140] = -87 mV

Questions: 1) How many K+ ions did the cell lose? 2) Did the K+ concentration inside the cell change?

How many K+ ions did the cell lose?

Use Q= CV

Know Vm = -87 mV

Calculate C

A = 1000 µm2 = 10-5 cm2 (1 µm = 10-4 cm)

Cm = 1 µF/cm2 x A (cm2) = 10-6 F/cm2 x 10-5 cm2 = 10-11 F = 10 pF

Q = 0.087 V x 10-11 F = 8.7 x 10-13 coulombs

Using Faraday’s constant: 8.7 x 10-13 C / 96,500 C/mol = 8.7 x 10-18 moles

Using Avogadro’s #: 8.7 x 10-18 moles x 6 x 1023 molecules/mole =

5 x 106 K+ ions

A single K channel ~2 pA current passes 107 ions/sec

Did the K+ concentration inside the cell change?

140 mM K+ = 0.14 moles/liter

0.14 moles/l x 6 x 1023 molecules/mole

= 0.84 x 1023 molecules/liter

Cell volume is 2974 x 10-15 liters

0.84 x 1023 molecules/liters x 2974 x 10-15 liters

= 2.3 x 1011 K+ ions in the cell

So losing 5 x 106 ions is not much

Ki does not change (in some cases Ko can!)

Ion equilibrium potentials in some cells

Ions are not in equilibrium across membranes

Membrane potentials with more than one ion

In Out

100 mM KCl 10 mM KCl

10 mM NaCl 100 mM NaCl

• If membrane is permeable only to K+ - what is ?

-

-

-

-

+

+

+

+

-

-

-

-

+

+

+

+

mvEV Km 60100

10log60 −===

• If membrane is permeable only to Na+?

• What if the membrane is 10 times more permeable to K+ than to Na+?

mvVm 2.42−=

Vm in this case is somewhat more positive than Ek

mvEV Nam 6010

100log60 ===

-

-

-

-

+

+

+

+

can’t use Nernst equation in this situation – ions are not in equilibrium.

Ions are not in equilibriumNeed to calculate ion fluxes at steady-state

Nernst-Planck Flux Equation

J = L dµ/dxFlux (J) proportional to gradient of electrochemical potential

J = L d [RTlnC + zFV] dx

= L [RT dC + zF dV] C dx dx

For a neutral diffusing substance (z=0), flux given by Fick eqn

J = -D dC/dx = L RT dC C dx

so L = -DC/RT

Substituting and rearranging:

J = -D dC + C zF dV dx RT dx

Nernst-Planck Equation

V

d

µo

V(x) Constant field approximation:

dV = Vdx d

Integrate Nernst-Planck eqn from 0 to d across membrane:

J = -zF D V RT d

C(d)ezFV/RT - C(0)

ezFV/RT - 1

The concentration of an ion just inside the membrane is related toIts concentration in the bulk solution by a partition coefficient:

C (membrane) = k

C (bulk solution)

Permeability coefficient P= Dk/d

J = -zFV P RT

C(1)ezFV/RT - C(2)

ezFV/RT - 1

J -mol/cm2sD= cm2/sP= cm/s

What is the permeability constant?

Comes from flux equation: J = -DA dc/dx

J = flux (moles/second)A= areadc/dx = concentration gradient

x

Co

Ci

big

small

dc/dx ~ k (Co-Ci)/ x

So J = -DA k (Co-Ci)/ x

or J = -Dk/x A (Co-Ci)

= -P A (Co-Ci)

Permeability constant(can be measured)

k Is partition coefficient

For membrane permeable to Na+, K+, and Cl-:

INa+ = FzJNa+

IK+ = FzJK+

Icl-= FzJCl-

Itotal = INa+ + IK+ + Icl-

J = mol/cm2s F = 9.65 coul/molI = current density (coul/cm2 x sec-1)

When Itotal = 0

Vo = RT F

PNaNao + PKKo + PClCli

PNaNai + PKKi + PClClo

Vo = zero current potentialGoldman-Hodgkin-Katz equation

J = -zFV P RT

C(1)ezFV/RT - C(2)

ezFV/RT - 1

Rectification when C(1) < or > C(2)

J=0When C(1)ezFV/RT = C(2) or V=RTlnC(2)/C(1) zF

V∞, JzP C(1)FV RT

V-∞, J-z P C(2)FV RT

Limiting cases:

Fluxes depend on ion concentrations

What is Rectification?

Constant field: concentration asymmetry

Permeation mechanism: asymmetric barrier to ion entry or block

Gating

-100 -50

0.5

1000 mV

1.0G / Gmax

50

p

-100 -50 50 100

-15

-10

-5

5

10

15nA

I = N p i

K

Na

Resting Potential using the Goldman-Hodgkin-Katz eqn

iNaiK

oNaoKm NaPKP

NaPKPV

][][

][][log60

++

=

[Na]o = 145 mM [Na]i = ~12 mM

[K]o = 4 [K]i = 140 mM

K+

Na+

•In real cells there are many more open K+ channels than Na+ channels at rest

• Vm is somewhat more positive than EK

EK = -92.6 mv

ENa= +64.9 mv

Vm = -82 mv

• The Goldman-Hodgkin-Katz equation (constant field equation)

Where PK and PNa are the membrane permeability to K and Na, respectively

-

-

-

-

-

-

-

-

- - - -

The GHK Equation - continued

K

Na

P

P=α

• an equivalent formulation of the GHK equation is the following:

ii

oom NaK

NaKV

][][

][][log60

αα

++

= where

• The GHK equation is often expanded to include terms for Cl-

oCliNaiK

iCloNaoKm ClPNaPKP

ClPNaPKPV

][][][

][][][log60

++++

=

Note that internal [Cl] is in the numerator, external in the denominator

K+-Dependence of Resting Potential

• membrane potentials measured using microelectrodes

• depolarization – Vm becomes more positive (less negative) than resting potential • hyperpolarization – Vm becomes more negative than resting potential

• increasing the external [K+] depolarizes the membrane

⎟⎟⎠

⎞⎜⎜⎝

⎛

++

⋅= ++

++

ii

OOm NaK

NaK

F

RTV

][][

][][ln

α

α

Dependence of the resting membrane potential on [K+]o and on the PNa/PK ratio. The blue line shows the case with no Na+ permeability (i.e., PNa/PK = 0). Orange curves describe the Vm predicted by the GHK equation for α> 0. The deviation of these curves from linearity is greater at low [K+]o, where the Na+ contribution is relatively larger.

Application to resting and active membrane in squid axon(Hodgkin & Katz, 1949)

PNa PK

Na+K+

PNa PK

Na+ K+

Neurons can have different PNa/PK

EK

ENa

EK

ENa

K+o increases K permeability

Ko = 4 mM

PNa PK

Ko = 10 mM

PNa/PK = 1 PNa/PK = 0.5

Na+ K+

Vm is halfway between Vna and VK

Vm closer to VK

PNa

PK

External K+ can modify K conductance of resting membrane

K+

K+

K+

K+

K+K+

K+ K+

K+

K+

K+

PNa

PK

Na+ K+

Hyperkalemia in a neuron with a large resting K conductance

PNa

PK

K+

EK

ENa

EK*

hyperkalemia

PNa PK

Na+K+

EK

ENa

Hyperkalemia in a pacemaker cell

PNa

PK

Na+K+

hyperkalemia

Mg2+

Ko = 4 mM

K+

K+

K+K+

Mg2+

Ko = 10 mM

K+

K+

K+ K+

K+

K+

Removal of Mg2+ block accounts for effects of Ko on PK

Inward K fluxclears Mg fromchannel

K+

K+K+

K+

K+

K+

K+K+

Resting potential

Active ion transport in nerve

24Na

24Na

(Hodgkin & Keynes, 1955)

A case of too much K+

A 35 year old male with history of bipolar disorder, taking no medications,presented to the emergency department complaining of nausea, vomiting, lethargy, and abdominal pain 5 hours after ingesting an unknownnumber of digoxin tablets He was found partly conscious by a friend whocalled for emergency medical help. Paramedics found him to be agitatedwith a pulse of 30. Upon arrival to the ED, vital signs were blood pressure 162/87,pulse of 30 beats per minute, respiratory rate of 22 per minute and temperature of 97 oF. An ECG showed second degree AV block. The patient was intubated and digoxin antibody administered. The next day his digoxin level was reducedfrom 2.49 ng/ml to 0.69 ng/ml and he was transferred out of the ICU.

How does digoxin work?Can the GHK eqn explain the electrical effects of digoxin?