Page 34 - Fluid, Electrolyte, and Acid-Base Disorders in Small Animal Practice
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Applied Physiology of Body Fluids in Dogs and Cats 23
APPENDIX At equilibrium, G ¼ 0, and solving the equation for
C
E m yields:
The cell membrane is composed of a hydrophobic lipid
bilayer with embedded protein molecules that play struc- RT ½c I
þ
tural and functional roles. This configuration allows the E m ¼ ln ð2Þ
zF ½c o
þ
cell membrane to act as an electrical capacitor that stores
energy. Some of the embedded proteins act as hydrophilic þ
At 37 C and with a monovalent ion (e.g., K ), the term
pores in the membrane. One embedded functional pro- RT/zF ¼ 26.67 mV. Converting to the base 10 loga-
þ
þ
tein is Na ,K -ATPase, which pumps sodium out of
rithm and specifying potassium as the cation:
and potassium into the cell in an Na/K ratio of 3:2. In
this model, the cell membrane acts as a capacitor; the
þ
½K
hydrophilic protein pores provide resistance; and the E m ¼ 26:67 ð2:303Þ log 10 þ I
þ
Na ,K -ATPase provides energy. ½K O
þ
þ
The intracellular concentration of potassium (140 E m ¼ 61 log ½K I ð3Þ
þ
mEq/L) is much higher than its extracellular concentra- 10 ½K O
tion (4 mEq/L). Consequently, potassium diffuses out of
the cell down its concentration gradient. However, the The Nernst equation is valid only when there is no net
cell membrane is impermeable to most intracellular current flow.
anions (e.g., proteins and organic phosphates). A net The Goldman-Hodgkin-Katz constant-field equation
negative charge develops inside the cell as potassium ions is a modification of the Nernst equation used to calculate
diffuse out of the cell, and a net positive charge the membrane potential based on the membrane perme-
accumulates outside the cell. As a result, a potential differ- ability ratio for sodium and potassium (P Na /P K ). This
ence is generated across the cell membrane. The principal equation allows determination of the individual ionic
extracellular cation is sodium, which enters the cell rela- contributions to E m by summing the individual
tively slowly down its concentration and electrical concentrations and permeability effects:
gradients, because the cell membrane is much less perme-
þ
þ
able to sodium than to potassium. Diffusion of potassium P K ½K þ P Na ½Na I
I
E m ¼ 61 log ð4Þ
from the cell continues until the ECF acquires sufficient 10 P K ½K þ P Na ½Na
þ
þ
O o
positive charge to prevent further diffusion of potassium
ions out of the cell. where P Na and P K are the membrane permeabilities for
The ratio of intracellular and extracellular concen- sodium and potassium.
þ
þ
trations of potassium ([K ] I /[K ] O ) is the major deter- A term r is included in the constant-field equation to
minant of the resting cell membrane potential take into account the effect of the electrogenic Na ,
þ
difference. This potential difference is demonstrated by K -ATPase pump under steady-state conditions. This
þ
the Nernst equation, which is derived from the general term is usually assigned the Na/K transport ratio of the
C
equation for free-energy change (G ): Na ,K -ATPase (r ¼ 3/2 ¼ 1.5). If the membrane
þ
þ
permeability of potassium is assigned a value of 1.0 and
þ the cell membrane is known to be 100 times more
½c
C
DG ¼ RT ln I þ zFE m ð1Þ permeable to potassium than to sodium,
þ
½c
o
þ þ
where R is the gas constant (8.314 J/K/mol), T is E m ¼ 61 log 10 rP K ½K þ P Na ½Na I
I
þ
þ
þ
the absolute temperature in K ( C þ 273), [c ] I is the rP K ½K þ P Na ½Na o
O
concentration of cation inside the cell, [c ] O is the con-
þ
þ þ
centration of cation outside the cell, z is the valence, F 1:5P K ½K þ 0:01P Na ½Na I
I
E m ¼ 61 log 10 ð5Þ
þ
is the Faraday constant (96,484 C/Eq), and E m is the 1:5P K ½K þ 0:01P Na ½Na o
þ
O
membrane potential in volts.
The first term on the right side of this equation Any ion that is not actively transported across the
represents the osmotic work required to transport membrane cannot contribute to the membrane potential,
1 mol of particles across the membrane against a concen- and the transmembrane distribution of such an ion must
þ
þ
tration gradient of [c ] I /[c ] O , and the second term follow the resting potential. Chloride is not considered in
represents the electrical work required to transport the the Goldman-Hodgkin-Katz equation, because chloride
same number of particles across the membrane against is usually passively distributed across the cell membrane
an electrical gradient. according to the prevailing E m .
