BIOE 2520

Electrical Properties of the Membrane

Reading: Chapter 11 of Alberts et al.

Stephen Smith, Ph.D.433 Biotech Center

shs46@pitt.edu

Permeability of Lipid membrane • Lipid bilayer is virtually impermeable for

charged ions and molecules

Permeability:Gas > small uncharged molecules> Large uncharged molecules>> ions > charged larger molecules

NaClNaCl NaClNaClH2O

Osmosis• Osmosis: Diffusion of water molecules down its

concentration gradient.

• If membrane is only permeable to H2O:

• Osmotic Pressure: ρgh = ΔnRT

Osmosis and Cell Volume

Diffusion across a lipid bilayer

• Measurement of diffusion across a bilayer:– Diffusion rate ∝ concentration

gradient;– Diffusion rate depends on

substance solubility in lipid, which is measured by partition coefficient (between water and lipid);

– For charged particles, diffusion rates also depend on electrical potential across the bilayer;

Intracellular and Extracellular Ionic Concentrations

[K+]i > [K+]o

[Na+]i < [Na+]o

[Cl-]i < [Cl-]o

[Ca2+]i << [Ca2+]o

Membrane Transport Proteins

ATPases

• ATPases transport molecules across membrane against their electrochemical gradient.

Intracellular Na+ and K+ concentrations are maintained by Na+/K+ ATPase

• Na/K ATPase: transports 3 Na+ out and 2 K+ into the cell by hydrolyzing an ATP to ADP and Pi;

• Na/K ATPase: tetramer – α2β2.

ATPases

Ion Channels

• Gated Ion Channel: opening and closing is determined by membrane potential (voltage-gated) or gating molecules (signal molecules);

• Non-gated ion Channels;– Determine the resting membrane potential.

Potassium Channel• Tetramer with identical subunits;• High throughput rate: can pass 108 K+ ions per second (near

diffusion limit):– No binding site for K+;

• High selectivity: K+ is 104-fold more permeable than Na+;• Diameters of K+ and Na+ ions: 0.133 nm and 0.095 nm, how is

the selectivity achieved?

K+ channel ion-select filter

Conservation of K+ channel SequenceThere are consensus sequences in K+ channels that are conserved from bacteria, plants, to mammals.

Gly-Tyr-Gly in pore region forms the filter.

Doyle et al. Science, 1998, 280: 69-77.

Membrane Potential

Thermodynamics of Ideal solutions• Under isothermal and isobaric conditions, a system

reaches equilibrium when its electrochemical potential is at a minimum;

• The electrochemical potential of an ideal solution is:

∑=i

iμμ

iiPToi cRTFz ln)( , ++= ψμμ

Where zi and ci are the electric charge and concentration of the ith substance, ψ is the electrical potential, F and Rare the Faraday and gas constants, (μo)T,P only depends on the property of the solvent.

Thermodynamics of Ideal solutions

• If the membrane is only permeable to substance i, then the system reaches equilibrium when electrochemical potentials of substance i equal on both sides: out

iin

i μμ =

in

out

Nain

out

Na

outin

outout

Nainin

Na

NaNa

FzRT

NaNa

FzRT

NaRTFzNaRTFz

][][lg10ln

][][ln

]ln[]ln[

+

+

+

+

++

⋅==−

+=+

ψψ

ψψFor i =Na+:

5910ln ≈F

RTmV,

Nernst Equation:in

outoutin

cc

zFRT ln=−ψψ

Nernst Potential

59][][ln −==− +

+

in

out

K

outin

KK

FzRTψψ

If the membrane is only permeable to K+, then

mV

Nernst Equilibrium potentials

If plasma membrane is only permeable to:

K+: 91)139

4lg(59 −≈=−= outinKE ψψ mV

Na+: 64)12145lg(59 ≈=−= outin

NaE ψψ mV

Cl+: 86)4

116lg(59 −≈−=−= outinClE ψψ mV

EK, ENa, Ecl : equilibrium potentials for K+, Na+, Cl-.

Membrane permeable to more then one ion

• If the membrane is permeable to K+, Na+, and Cl-, then an equilibrium can not be maintained!

Steady State Membrane Potential• Ionic flux is proportional to its driving

forces;• Driving forces across a membrane:

– Concentration gradient: ,– Voltage Vm;

• Ionic flux:• At steady state, Vm is constant (no net charge

movement across the membrane),– Net current is zero:

ini

outi

i cc

zFRTE ln=

)( imii EVgj −=

∑ =i

ij 0

Steady State Membrane Potential

• Ionic flux:– Can be modeled with an equivalent electric

circuit;

)( imii EVgj −=

ji

Vm- - - -

+ + + ++ + + +- - - -

Cout

Cin

Ei

gi

Vm

Steady State Membrane Potential

• Multiple ionic currents: )( imii EVgj −=

Steady State Membrane Potential

• At steady state, net current is zero:

or

– Membrane potential:

∑ =i

ij 0

∑ =−i

imi EVg 0)(

∑∑

=

ii

iii

m g

EgV

(Chord conductance equation)

An Example

• gK= 10 gNa

91)139

4lg(59 −≈=KE

64)12145lg(59 ≈=NaE

mV

EEgg

EgEggg

EgEgV NaK

NaNa

KNaKNa

NaK

NaNaKKm

9.7611

64)91(1011

1010

10

−=+−×

=

+=

++

=++

=

Animal Cell membrane potential is predominately determined by non-gated K+ channels

• The membrane potentials of animal cells are proximately -70 mV.

• Membrane potential will shift if the conductance (permeability) of one or more ions change– If more K+ channels open, potential will be more negative

(hyperpolarization);– If more Na+ channels open or some K+ channels are blocked, potential

will be less negative (more positive) (depolarization).

ClNaK

ClClNaNaKKoutin

gggEgEgEg

++++

=−ψψ

Measurement of cell membrane potential

Microelectrode method:

Gibbs-Donnan Equilibrium

• If membrane is permeable to Na and Cl, the two side will eventually have equal [Na+] and [Cl-] at equilibrium.

• No electrical potential exits at the interface.

200 mMNaCl

100 mMNaCl

150 mMNaCl

150 mMNaCl

Gibbs-Donnan Equilibrium

• If one side contain large non-diffusible (impermeable) ions (X-, in our example), an unsymmetrical equilibrium (Gibbs-Donnan equilibrium) can be established:

• An equilibrium electrical potential (Gibbs-Donnan equilibrium potential) is present at the interface, and often an osmotic gradient exits.

100NaX

100NaCl

67 Na+

67 Cl-

133 Na+

33 Cl-

100 X-

18 mV

+++++++

-------

2211 ][][][][ −+−+ = ClNaClNa

1

2

2

1

][][

][][

−

−

+

+

=ClCl

NaNa

or

Patch Clamps: Measurement of single channel current

Patch Clamps

Patch Clamps

Na+/Ca2+ antiporter

• Many animal cells utilize plasma Na+/Ca2+

antiporter to maintain low intracellular [Ca2+];– Exchange 3 extracellular Na+ for 1 intracellular Ca2+;– [Ca2+]in < 0.2 μM; [Ca2+]out ~ 2 mM;

outininout CaNaCaNa ++++ +⎯→⎯+ 22 33

Cellular pH• In animals:

– extracellular pHo ≈ 7.4, intracellular pHi ≈ 7.2;• If unregulated and [H+] is at equilibrium, then

)(59][][lg10ln outin

in

out pHpHHH

FRT

−=⋅=Δ +

+

ψ mV

2.659

=Δ

+=ψ

outin pHpH 70−=Δψ mV

Cellular pH• Cells regulate intracellular pH:

– With Na+/H+, Cl-/HCO3-, and Na+HCO3

- /Cl-

antiporters (exchanger);

Transepithelial Transport• Epithelium: a single layer of cells that lines the

inside of internal organs such as intestine, stomach, kidney, etc.

Uptake of glucose from intestinal lumen

Transepithelial Transport

• Acidfication of the stomach lumen by parietal cells

Water Movement Across Plasma Membrane

• Osmosis: water flows in the direction that minimizes difference of solute concentrations (electrochemical potential)..

Water Movement Across Plasma Membrane

• Cell swells in hypotonic (low osmolarity) solution;

• Cell shrinks in hypertonic (high osmolarity) solution;

• Water permeability in cell membrane is often 10 times higher than pure lipid bilayer:– presence of water channels.

• Cells can regulate its volume in response to external disturbance by activation of ion channels, contransporters, and ATPases.