Nernst. Today’s Topics. •Ion Transport Across Membranes (A Brief Primer) •The Generation of the Resting Membrane Potential. David S. Weiss Department of Physiology 5674325 weissd@uthscsa.edu. 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
5674325
weissd@uthscsa.edu
Transport Across Membranes
•Diffusion Through the Lipid Bilayer
•Carrier or ProteinMediated Transport
Generation of the
Resting Membrane Potential
Measurement of the Resting Potential
+

0 mV
70 mV
Diffusion
Fick’s Law of Diffusion
Net rate of diffusion
Concentration gradient
Diffusion coefficient
Area of the plane diffusing across
StokesEinstein 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 extracellular).
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]<=107 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 (EmEK)
INa=gNa (EmENa)
Ohm’s Law (V=IR) or
I=V/R or I=Vg
ICl=gCl (EmECl)
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 (EmEK)
INa=gNa (EmENa)
ICl=gCl (EmECl)
At steady state: IK+INa+ICl= 0
Therefore: gK (EmEK) +gNa (EmENa) +gCl (EmECl)= 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
Measurement of the Resting Potential
+

0 mV
70 mV
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]<=107 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
GoldmanHodgkinKatz Equation