196 GROUP I AND II METALS IN BIOLOGICAL SYSTEMS
The second term will be positive for a cation (K + , for instance) and negative
for an anion (Cl − ). To understand how this might be applied to any cell, con-
sider a simple “ cell ” that contains potassium ions, chloride ions, and protein
inside the cell with concentrations described in Figure 5.2 : 50 mM K + , 25 mM
Cl − , and 25 mM protein (that cannot diffuse through the membrane). For the
purposes of this example, we will consider that the cell is bathed in a 50 mM
KCl isotonic solution — that is, the concentration of K + and Cl − outside the cell
is 50 mM. For this cell the following equations are established:
ΔG
RT
F
zRT
F
= z[]
[]
+==
[]
[]
−
+
+−
ln ln −K
K
Cl
Clin
outin
outψψ (^0) (5.4)
−
[]
[]
⎡
⎣⎢
⎤
⎦⎥
−=
[]
[]
−
+
+−
−RT
F
zRT
F
ln ln zK
K
Cl
Clin
outin
outψψ(5.5)ln lnK
K
Cl
Clout
inin
out+
+−
−[]
[]
=
[]
[]
(5.6)
and therefore
(^) [][ ]KCl K Cl+− + −in in=[][ ]out out (5.7)
It is also true that:
(^) []KK++in+[]out= 100 (5.8)
(^) []Cl−−in+[]Cl out= 75 (5.9)
(^) []KClProt+−in=[]in+[]−. (5.10)
Using these equations, one fi nds that the equilibrium concentrations of K + and
Cl − and protein inside and outside the cell are as shown in Figure 5.2. If one
then uses the equilibrium concentrations of [K + ] in and [K + ] out to calculate the
potential generated across the membrane, one fi nds
ψ=
[]
[]
=≈−
+
+RT
F
ln. logK
K
out mV
in0 059
43
57
(^7) (5.11)
Figure 5.2 Ion concentrations inside and outside a hypothetical cell.
[K+] = 50 mM [K+] = 50 mM
[Cl-] = 50 mM
inside cell outside cell
[Cl-] = 25 mM
[Prot-] = 25 mM
[K+] = 57 mM [K+] = 43 mM
[Cl-] = 43 mM
inside cell outside cell
[Cl-] = 32 mM
[Prot-] = 25 mM
Before Donnan Equilibrium After Donnan Equilibrium