Goldmans ekvation Goldmans ekvation för
N
{\displaystyle N}
M
{\displaystyle M}
E
m
=
R
T
F
ln
(
∑
i
N
P
M
i
+
[
M
i
+
]
o
u
t
+
∑
j
M
P
A
j
−
[
A
j
−
]
i
n
∑
i
N
P
M
i
+
[
M
i
+
]
i
n
+
∑
j
M
P
A
j
−
[
A
j
−
]
o
u
t
)
{\displaystyle E_{m}={\frac {RT}{F}}\ln {\left({\frac {\sum _{i}^{N}P_{M_{i}^{+}}[M_{i}^{+}]_{\mathrm {out} }+\sum _{j}^{M}P_{A_{j}^{-}}[A_{j}^{-}]_{\mathrm {in} }}{\sum _{i}^{N}P_{M_{i}^{+}}[M_{i}^{+}]_{\mathrm {in} }+\sum _{j}^{M}P_{A_{j}^{-}}[A_{j}^{-}]_{\mathrm {out} }}}\right)}}
Detta resulterar i följande om vi utgår från ett membran som separerar två
K
x
N
a
1
−
x
C
l
{\displaystyle \mathrm {K} _{x}\mathrm {Na} _{1-x}\mathrm {Cl} }
E
m
,
K
x
N
a
1
−
x
C
l
=
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F
ln
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P
N
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[
N
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o
u
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K
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[
K
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o
u
t
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P
C
l
−
[
C
l
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i
n
P
N
a
+
[
N
a
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i
n
+
P
K
+
[
K
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i
n
+
P
C
l
−
[
C
l
−
]
o
u
t
)
{\displaystyle E_{m,\mathrm {K} _{x}\mathrm {Na} _{1-x}\mathrm {Cl} }={\frac {RT}{F}}\ln {\left({\frac {P_{Na^{+}}[Na^{+}]_{\mathrm {out} }+P_{K^{+}}[K^{+}]_{\mathrm {out} }+P_{Cl^{-}}[Cl^{-}]_{\mathrm {in} }}{P_{Na^{+}}[Na^{+}]_{\mathrm {in} }+P_{K^{+}}[K^{+}]_{\mathrm {in} }+P_{Cl^{-}}[Cl^{-}]_{\mathrm {out} }}}\right)}}
Ekvationen liknar Nernsts ekvation men har en term för varje jon. Nernsts ekvation kan ses som ett specialfall av Goldmans ekvation då enbart en jontyp finns:
E
m
,
N
a
=
R
T
F
ln
(
P
N
a
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[
N
a
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o
u
t
P
N
a
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[
N
a
+
]
i
n
)
=
R
T
F
ln
(
[
N
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o
u
t
[
N
a
+
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i
n
)
{\displaystyle E_{m,Na}={\frac {RT}{F}}\ln {\left({\frac {P_{Na^{+}}[Na^{+}]_{\mathrm {out} }}{P_{Na^{+}}[Na^{+}]_{\mathrm {in} }}}\right)}={\frac {RT}{F}}\ln {\left({\frac {[Na^{+}]_{\mathrm {out} }}{[Na^{+}]_{\mathrm {in} }}}\right)}}
E
m
{\displaystyle E_{m}}
volt , ekvivalent med joule /coulomb )
P
i
o
n
{\displaystyle P_{\mathrm {ion} }}
m /s )
[
i
o
n
]
o
u
t
{\displaystyle [ion]_{\mathrm {out} }}
mol /m 3 )
[
i
o
n
]
i
n
{\displaystyle [ion]_{\mathrm {in} }}
R
{\displaystyle R}
ideala gaskonstanten (joule per kelvin per mol)
T
{\displaystyle T}
F
{\displaystyle F}
Faradays konstant (coulomb/mol)Första termen innan parentesen kan avrundas till 61,5mV (37°C kroppstemperatur).
E
X
=
61
,
5
m
V
log
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[
X
+
]
o
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t
[
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n
)
=
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61
,
5
m
V
log
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t
[
X
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n
)
{\displaystyle E_{X}=61,5\ \mathrm {mV} \log {\left({\frac {[X^{+}]_{\mathrm {out} }}{[X^{+}]_{\mathrm {in} }}}\right)}=-61,5\ \mathrm {mV} \log {\left({\frac {[X^{-}]_{\mathrm {out} }}{[X^{-}]_{\mathrm {in} }}}\right)}}
Obs! Jonladdning avgör tecknet för membranpotentialens bidrag.
Ekvationen tillåter uträkning av membranpotential i de fall permeabiliteten hos jonerna är kända.
![{\displaystyle E_{m}={\frac {RT}{F}}\ln {\left({\frac {\sum _{i}^{N}P_{M_{i}^{+}}[M_{i}^{+}]_{\mathrm {out} }+\sum _{j}^{M}P_{A_{j}^{-}}[A_{j}^{-}]_{\mathrm {in} }}{\sum _{i}^{N}P_{M_{i}^{+}}[M_{i}^{+}]_{\mathrm {in} }+\sum _{j}^{M}P_{A_{j}^{-}}[A_{j}^{-}]_{\mathrm {out} }}}\right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4168067dcf31b41af9beddcc7e9a3beba72aab8c)
![{\displaystyle E_{m,\mathrm {K} _{x}\mathrm {Na} _{1-x}\mathrm {Cl} }={\frac {RT}{F}}\ln {\left({\frac {P_{Na^{+}}[Na^{+}]_{\mathrm {out} }+P_{K^{+}}[K^{+}]_{\mathrm {out} }+P_{Cl^{-}}[Cl^{-}]_{\mathrm {in} }}{P_{Na^{+}}[Na^{+}]_{\mathrm {in} }+P_{K^{+}}[K^{+}]_{\mathrm {in} }+P_{Cl^{-}}[Cl^{-}]_{\mathrm {out} }}}\right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2504511122b4215bf6fa6aef42441b3da96cec09)
![{\displaystyle E_{m,Na}={\frac {RT}{F}}\ln {\left({\frac {P_{Na^{+}}[Na^{+}]_{\mathrm {out} }}{P_{Na^{+}}[Na^{+}]_{\mathrm {in} }}}\right)}={\frac {RT}{F}}\ln {\left({\frac {[Na^{+}]_{\mathrm {out} }}{[Na^{+}]_{\mathrm {in} }}}\right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91cbc49fa386a36adc6d698bbeb5a5c761a54d2a)
![{\displaystyle [ion]_{\mathrm {out} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5d2e7fe4863a9c797f60607f91baeab4391607b)
![{\displaystyle [ion]_{\mathrm {in} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23ebffa4ae9cdb7523b8a4dd8e1752a6380b22e3)
![{\displaystyle E_{X}=61,5\ \mathrm {mV} \log {\left({\frac {[X^{+}]_{\mathrm {out} }}{[X^{+}]_{\mathrm {in} }}}\right)}=-61,5\ \mathrm {mV} \log {\left({\frac {[X^{-}]_{\mathrm {out} }}{[X^{-}]_{\mathrm {in} }}}\right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec5b86368dcb113f3d24ec9bb8937333bce58bc2)