Today’s Topics

Today’s Topics

David S. WeissDepartment of Physiology

[email protected]

•Ion Transport Across Membranes (A Brief Primer)

•The Generation of the Resting Membrane Potential

Nernst

Transport Across Membranes

•Diffusion Through the Lipid Bilayer

•Carrier- or Protein-Mediated Transport

Generation of the

Resting Membrane Potential

Measurement of the Resting Potential

+- 0 mV

-70 mV

Diffusion

Fick’s Law of Diffusion

J = -DAdcdx

Net rate of diffusion

Diffusion coefficient Area of the plane diffusing across

Concentration gradient

D =kT

6r

Kinetic energy

Viscosity

Molecular radiusSix

3.14

Stokes-Einstein Relation

Einstein Relationship

(x2 ) 2Dt

Diffusion Distance (µM)1

10100

1000 (1mm)10,000 (1cm)

Time Required0.5 msec50 msec

5 sec8.3 min14 hrs

Electrochemical Potential

A B

µ = µo + RTlnC + zFE

A B µ = µo + RTlnC + zFE

We are interested in the difference in the electrochemical potential between the two sides (i.e., intra- and extra-cellular).

µA(x) = µo(x)+ RTln[x]A + zFEA

µB(x) = µo(x)+ RTln[x]B + zFEB

µ(x) = µA(x) - µB(x) = RTln + zF(EA - EB)[x]A

[x]B

µA(x) = µo(x)+ RTln[x]A + zFEA

µB(x) = µo(x)+ RTln[x]B + zFEB

µ(x) = µA(x) - µB(x) = RTln + zF(EA - EB)[x]A

[x]B

A B

At equilibrium: µ = 0

RTln + zF(EA - EB) = 0[x]A

[x]B

A B

At equilibrium: µ = 0

RTln + zF(EA - EB) = 0[x]A

[x]B

Rearranging gives:

EA - EB = ln = ln -RTzF

[x]A

[x]B

RTzF

[x]B

[x]A

Ex = ln [x]B

[x]A

RTzF

Nernst Equation

Ex = ln [x]B

[x]A

RTzF

Nernst Equation

This equation determines the voltage difference that must be imposed between side A and side B to prevent the movement of ions due to the chemical force.

-or-

This equation determines the voltage at which the electrical and chemical forces are balanced; that is, there is no net movement of ions.

0.1 M K+ 0.01 M K+ Calculate the potential difference required to oppose the movement of K+ ions.

Sample Problem

60 z

EK+ = log10 [x]B

[x]A

EK+ = ln [x]B

[x]A

RT

zF

60 z

EK+ = log10 [0.01]B

[0.1]A

EK+ = -60 mVPut -60 mV on side A with respect to side B and there will be no net movement of K+ ions.

A B

A More Realistic Situation:

0.1 M NaCl 0.01 M NaCl

A B

Membrane impermeable to anions, permeable to cations.

At time 0, the membrane is made permeable to Na+ only.

0 mV


A B

Membrane impermeable to anions,

permeable to cations.

If we apply -90 mV (Side A with respect to Side B) before the membrane is made permeable to sodium,what will happen?


A B

++++

++++

----

----

+ 60 mV

Membrane impermeable to anions,

permeable to cations.

The +60 mV, or the Nernst potential, is also called the reversal potential (Erev). Also sometimes called the equilibrium potential.

Nernst potential = reversal potential = equilibrium potential


A B

0 mV

Impermeable membrane

Important Point

If ions cannot move, then no potential difference will be created!

Ion Concentrations in a Typical Mammalian Cell*

[Na]=12 mM

[Na]=145 mM

[K]=155 mM

[K]=4 mM

[Ca]=1.5 mM

[Ca]<=10-7 mM

[Cl]=123 mM

[Cl]=4.2 mM

In

Out

Ion [X]out [X] in Ratio E X (mV)Na 145 12 12 +67K 4 155 0.026 -98Ca 1.5 <10-7 <15000 >=128Cl 123 4.2 30 -90

*actual values may vary

Equivalent Circuit of the Membrane

Cm

Extracellular

Intracellular

RNa RCl RK

+67 -98-90ENa ECl EK

Equivalent Circuit of the Membrane

Cm

Extracellular

Intracellular

RNa RCl RK

+67 -98-90ENa ECl EK

Derivation of the chord conductance equation:IK=gK (Em-EK)INa=gNa (Em-ENa)ICl=gCl (Em-ECl)

These equations calculate the current flowing across the membrane for each ion.

}Ohm’s Law (V=IR) orI=V/R or I=Vg

Cm

Extracellular

Intracellular

RNa RCl RK

+67 -98-90ENa EClEK

Derivation of the chord conductance equation:

IK=gK (Em-EK)INa=gNa (Em-ENa)ICl=gCl (Em-ECl)

At steady state: IK + INa + ICl = 0

Therefore: gK (Em-EK) + gNa (Em-ENa) + gCl (Em-ECl) = 0

Solve for Em: Em = EK+ ENa + ECl gK

gK+gNa+gCl

gNa

gK+gNa+gCl

gCl

gK+gNa+gCl

This is the chord conductance equation. It allows one to calculate the membrane potential given the relative conductances of the ions. It is simply a weighted average.

Cm

Extracellular

Intracellular

RNa RCl RK

+67 -98-90ENa EClEK

gK

gK+gNa+gCl

gNa

gK+gNa+gCl

gCl

gK+gNa+gCl

Em = EK+ ENa + ECl

Significance

+67 mV

-90 mV

-98 mV

gNa gK, gCl>>

Cm

Extracellular

Intracellular

RNa RCl RK

+67 -98-90ENa EClEK

gK

gK+gNa+gCl

gNa

gK+gNa+gCl

gCl

gK+gNa+gCl

Em = EK+ ENa + ECl

Significance

+67 mV

-90 mV

-98 mV

gNa gK, gCl>>

gK gNa, gCl>>

Cm

Extracellular

Intracellular

RNa RCl RK

+67 -98-90ENa EClEK

gK

gK+gNa+gCl

gNa

gK+gNa+gCl

gCl

gK+gNa+gCl

Em = EK+ ENa + ECl

Significance

+67 mV

-90 mV

-98 mV

gNa gK, gCl>>

gK gNa, gCl>>

gNa = gK

Cm

Extracellular

Intracellular

RNa RCl RK

+67 -98-90ENa EClEK

gK

gK+gNa+gCl

gNa

gK+gNa+gCl

gCl

gK+gNa+gCl

Em = EK+ ENa + ECl

Significance

+67 mV

-90 mV

-98 mV

gNa gK, gCl>>

gK gNa, gCl

gNa = gK

gK gNa, gCl>> >>

0 mV

Cm

Extracellular

Intracellular

RNa RCl RK

+67 -98-90ENa EClEK

gK

gK+gNa+gCl

gNa

gK+gNa+gCl

gCl

gK+gNa+gCl

Em = EK+ ENa + ECl

Significance

+67 mV

-90 mV

-98 mV

gNa gK, gCl>>

gK gNa, gCl

gNa = gK

gK gNa, gCl>> >>

0 mV

[K]i = [K]o

Measurement of the Resting Potential

+- 0 mV

-70 mV

So, what is the source of theResting Membrane Potential?

Cm

Extracellular

Intracellular

RNa RCl RK

+67 -98-90ENa EClEK

gK

gK+gNa+gCl

gNa

gK+gNa+gCl

gCl

gK+gNa+gCl

Em = EK+ ENa + ECl

Significance

+67 mV

-90 mV

-98 mV

gNa gK, gCl>>

gK gNa, gCl

gNa = gK

gK gNa, gCl>> >>

0 mV

Ion Concentrations in a Typical Mammalian Cell*

[Na]=12 mM

[Na]=145 mM

[K]=155 mM

[K]=4 mM

[Ca]=1.5 mM

[Ca]<=10-7 mM

[Cl]=123 mM

[Cl]=4.2 mM

In

Out

Ion [X]out [X] in Ratio E X (mV)Na 145 12 12 +67K 4 155 0.026 -98Ca 1.5 <10-7 <15000 >=128Cl 123 4.2 30 -90

*actual values may vary

Cm

Extracellular

Intracellular

RNa RCl RK

+67 -98-90ENa EClEK

gK

gK+gNa+gCl

gNa

gK+gNa+gCl

gCl

gK+gNa+gCl

Em = EK+ ENa + ECl

Significance

+67 mV

-90 mV

-98 mV

gNa gK, gCl>>

gK gNa, gCl

gNa = gK

gK gNa, gCl>> >>

0 mVgCl gNa, gK>>

Cm

Extracellular

Intracellular

RNa RCl RK

+67 -98-90ENa EClEK

gK

gK+gNa+gCl

gNa

gK+gNa+gCl

gCl

gK+gNa+gCl

Em = EK+ ENa + ECl

Em = lnRTF

PK[K]o + PNa[Na]o + PCl[Cl]i

PK[K]i + PNa[Na]i + PCl[Cl]o

One Last Equation

Goldman-Hodgkin-Katz Equation

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