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Nernst. Today’s Topics. •Ion Transport Across Membranes (A Brief Primer) •The Generation of the Resting Membrane Potential. David S. Weiss Department of Physiology 567-4325 [email protected] Transport Across Membranes •Diffusion Through the Lipid Bilayer

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Nernst

Today’s Topics

•Ion Transport Across Membranes (A Brief Primer)

•The Generation of the Resting Membrane Potential

David S. Weiss

Department of Physiology

567-4325

[email protected]


Transport Across Membranes

•Diffusion Through the Lipid Bilayer

•Carrier- or Protein-Mediated Transport


Generation of the

Resting Membrane Potential




Fick’s Law of Diffusion

Net rate of diffusion

Concentration gradient

Diffusion coefficient

Area of the plane diffusing across

Stokes-Einstein Relation

Kinetic energy

Viscosity

Six

3.14

Molecular radius


Einstein Relationship

Diffusion Distance (µM)

1

10

100

1000 (1mm)

10,000 (1cm)

Time Required

0.5 msec

50 msec

5 sec

8.3 min

14 hrs


A

B

Electrochemical Potential

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

[x]A

µ(x) = µA(x) - µB(x) = RTln + zF(EA - EB)

[x]B


A

B

[x]A

[x]B

µA(x) = µo(x)+ RTln[x]A + zFEA

µB(x) = µo(x)+ RTln[x]B + zFEB

[x]A

µ(x) = µA(x) - µB(x) = RTln + zF(EA - EB)

[x]B

At equilibrium: µ = 0

RTln + zF(EA - EB) = 0


A

B

[x]A

[x]B

-RT

RT

[x]A

[x]B

[x]B

[x]A

zF

zF

At equilibrium: µ = 0

RTln + zF(EA - EB) = 0

Rearranging gives:

EA - EB = ln = ln

[x]B

RT

Nernst Equation

Ex = ln

[x]A

zF


[x]B

RT

Nernst Equation

Ex = ln

[x]A

zF

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+

RT

[x]B

EK+ = ln

[x]A

zF

Sample Problem

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

A

B

60

60

[x]B

[0.01]B

EK+ = log10

EK+ = log10

[x]A

z

z

[0.1]A

EK+ = -60 mV

Put -60 mV on side A with respect to side B and there will be no net

movement of K+ ions.


0 mV

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.


0.1 M NaCl

0.01 M NaCl

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?


+ 60 mV

-

-

-

+

-

+

+

+

+

-

+

+

-

+

-

-

0.1 M NaCl

0.01 M NaCl

A

B

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


Important Point

0 mV

0.1 M NaCl

0.01 M NaCl

A

B

Impermeable membrane

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


Ion Concentrations in a Typical Mammalian Cell*

Out

[Na]=145 mM

[K]=4 mM

[K]=155 mM

In

[Na]=12 mM

[Cl]=4.2 mM

[Ca]<=10-7 mM

[Cl]=123 mM

[Ca]=1.5 mM

*actual values may vary


Equivalent Circuit of the Membrane

Extracellular

RNa

RCl

RK

Cm

ENa

+67

ECl

EK

-90

-98

Intracellular


Equivalent Circuit of the Membrane

Extracellular

RNa

RCl

RK

Cm

ENa

+67

ECl

EK

-90

-98

Intracellular

Derivation of the chord conductance equation:

}

IK=gK (Em-EK)

INa=gNa (Em-ENa)

Ohm’s Law (V=IR) or

I=V/R or I=Vg

ICl=gCl (Em-ECl)

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


Extracellular

RNa

RCl

RK

Cm

ENa

+67

EK

ECl

-98

-90

Intracellular

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

gCl

gK+gNa+gCl

gNa

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.


Extracellular

RNa

RCl

RK

Cm

ENa

+67

EK

ECl

-98

-90

Intracellular

>

>

Significance

gK

gK+gNa+gCl

gCl

gK+gNa+gCl

gNa

gK+gNa+gCl

Em =EK+ENa+ECl

gNa gK, gCl

+67 mV

-90 mV

-98 mV


Extracellular

RNa

RCl

RK

Cm

ENa

+67

EK

ECl

-98

-90

Intracellular

>

>

Significance

gK

gK+gNa+gCl

gCl

gK+gNa+gCl

gNa

gK+gNa+gCl

Em =EK+ENa+ECl

gNa gK, gCl

+67 mV

-90 mV

-98 mV


>

>

>

>

Significance

Extracellular

RNa

RCl

RK

Cm

ENa

+67

EK

ECl

-98

-90

Intracellular

gK

gK+gNa+gCl

gCl

gK+gNa+gCl

gNa

gK+gNa+gCl

Em =EK+ENa+ECl

gNa gK, gCl

+67 mV

-90 mV

-98 mV

gK gNa, gCl


>

>

>

>

Significance

Extracellular

RNa

RCl

RK

Cm

ENa

+67

EK

ECl

-98

-90

Intracellular

gK

gK+gNa+gCl

gCl

gK+gNa+gCl

gNa

gK+gNa+gCl

Em =EK+ENa+ECl

gNa gK, gCl

+67 mV

-90 mV

-98 mV

gK gNa, gCl


>

>

>

>

Significance

Extracellular

RNa

RCl

RK

Cm

ENa

+67

EK

ECl

-98

-90

Intracellular

gK

gK+gNa+gCl

gCl

gK+gNa+gCl

gNa

gK+gNa+gCl

Em =EK+ENa+ECl

gNa gK, gCl

+67 mV

gNa = gK

-90 mV

-98 mV

gK gNa, gCl


>

>

>

>

Significance

Extracellular

RNa

RCl

RK

Cm

ENa

+67

EK

ECl

-98

-90

Intracellular

gK

gK+gNa+gCl

gCl

gK+gNa+gCl

gNa

gK+gNa+gCl

Em =EK+ENa+ECl

gNa gK, gCl

+67 mV

gNa = gK

-90 mV

-98 mV

gK gNa, gCl


>

>

>

>

>

>

Significance

Extracellular

RNa

RCl

RK

Cm

ENa

+67

EK

ECl

-98

-90

Intracellular

gK

gK+gNa+gCl

gCl

gK+gNa+gCl

gNa

gK+gNa+gCl

Em =EK+ ENa+ECl

gNa gK, gCl

+67 mV

0 mV

gNa = gK

-90 mV

-98 mV

gK gNa, gCl

gK gNa, gCl


>

>

>

>

>

>

Significance

Extracellular

RNa

RCl

RK

Cm

ENa

+67

EK

ECl

-98

-90

Intracellular

gK

gK+gNa+gCl

gCl

gK+gNa+gCl

gNa

gK+gNa+gCl

Em =EK+ ENa+ECl

gNa gK, gCl

[K]i= [K]o

+67 mV

0 mV

gNa = gK

-90 mV

-98 mV

gK gNa, gCl

gK gNa, gCl



So, what is the source of the

Resting Membrane Potential?


>

>

>

>

>

>

Significance

Extracellular

RNa

RCl

RK

Cm

ENa

+67

EK

ECl

-98

-90

Intracellular

gK

gK+gNa+gCl

gCl

gK+gNa+gCl

gNa

gK+gNa+gCl

Em =EK+ ENa+ECl

gNa gK, gCl

+67 mV

0 mV

gNa = gK

-90 mV

-98 mV

gK gNa, gCl

gK gNa, gCl


Ion Concentrations in a Typical Mammalian Cell*

Out

[Na]=145 mM

[K]=4 mM

[K]=155 mM

In

[Na]=12 mM

[Cl]=4.2 mM

[Ca]<=10-7 mM

[Cl]=123 mM

[Ca]=1.5 mM

*actual values may vary


>

>

>

>

>

>

>

>

Extracellular

Significance

RNa

RCl

RK

Cm

ENa

+67

EK

ECl

-98

-90

Intracellular

gK

gK+gNa+gCl

gCl

gK+gNa+gCl

gNa

gK+gNa+gCl

Em =EK+ ENa+ECl

gNa gK, gCl

+67 mV

0 mV

gNa = gK

gCl gNa, gK

-90 mV

-98 mV

gK gNa, gCl

gK gNa, gCl


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

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

RT

F

Em = ln

Extracellular

One Last Equation

RNa

RCl

RK

Cm

ENa

+67

EK

ECl

-98

-90

Intracellular

gK

gK+gNa+gCl

gCl

gK+gNa+gCl

gNa

gK+gNa+gCl

Em =EK+ ENa+ECl

Goldman-Hodgkin-Katz Equation


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