# Role of Gibbs-Donnan Equilibrium and Transmembrane Electrochemical Gradient in Defining the Plasma Potassium Concentration

## ABSTRACT

Potassium is the most abundant exchangeable cation in the body. Most potassium resides in the intracellular compartment (ICF),
and a small amount is in the interstitial fluid compartment (ISF) and plasma space. Both plasma and intracellular potassium ions are
in equilibrium with potassium ions in the interstitial fluid compartment (ISF). The distribution of potassium between the plasma and
ISF is determined by the electrochemical gradient imposed by the presence of negatively charged, impermeant albumin molecules
and is known as Gibbs-Donnan equilibrium. The potassium ions in the ISF are in turn in equilibrium with intracellular potassium ions
as defined by the transmembrane electrochemical gradient. Current analysis relating the plasma potassium concentration [K^{+}]_{p} with
the intracellular potassium concentration [K^{+}]_{ICF} implicitly assumes that the plasma water potassium concentration [K^{+}]_{pw} is equal to the interstitial fluid potassium concentration [KK^{+}]_{ISF}.This is inaccurate since the [K^{+}]_{pw} is greater than [K^{+}]_{ISF} due to Gibbs-Donnan equilibrium. Given that hypokalemia and hyperkalemia are disorders of potassium balance that reflect changes in the intracellular potassium store, a new equation is derived to define the quantitative interrelationship between the [K^{+}]_{p} and [KK^{+}]_{ICF} taking into account the three compartmental distribution of potassium in the body fluids.

## INTRODUCTION

Potassium is present in all body tissues, and it is necessary for normal cellular function due to its role in maintaining intracellular fluid
volume and transmembrane electrochemical gradient. Although disorders of potassium balance are clinically based on changes in the plasma
potassium concentration [K^{+}]_{p}, it is well known that changes in the total body potassium stores predominantly originate from the intracellular
compartment (ICF). Indeed, the plasma potassium concentration, [K^{+}]_{p}, ranges from 3.5 to 5 meq/L, whereas the intracellular potassium
concentration [K^{+}]_{ICF} ranges from 140 to 150 meq/L [1]. Currently, there is no formula that defines the quantitative interrelationship between
the [K+]p and [K^{+}]_{ICF}. In this article, a new equation is derived to quantify the interrelationship between [K^{+}]_{p} and [K^{+}]_{ICF} based on the three
compartmental distribution of potassium resulting from Gibbs-Donnan equilibrium and transmembrane electrochemical gradient.

#### A New Formula Defining the Quantitative Interrelationship Between the [K^{+}]_{p} and [K^{+}]_{ICF}

The distribution of potassium ions between the plasma and interstitial fluid (ISF) is defined by Gibbs-Donnan Equilibrium (2): Figure 1

FE_{G}+RTln([K^{+}]_{pw}/[K^{+}]_{ISF})=0 (Eq.1)

where F = Faraday’s constant

E_{G} = electrical potential imposed by negatively charged, impermeant plasma proteins

R = ideal gas constant

T = absolute temperature in Kelvin

[K^{+}]_{pw}= plasma water potassium concentration

[K^{+}]_{ISF}= interstitial fluid potassium concentration

In([K^{+}]_{pw}/[K^{+}]_{ISF}) = -FE_{G}/RT (Eq.1A)

K^{+}]_{pw} = [K^{+}]_{ISF} (e^{-FEG/RT}) (Eq.1B)

The transmembrane electrical gradient is defined by the Goldman-Hodgkin-Katz equation (3,4):

V_{m} = RT/F ln((P_{K}+[K^{+}]_{ISF}+P_{Na}+[Na^{+}]_{ISF}+P_{Cl}-[Cl^{-}]_{ICF})/(P_{K}+[K^{+}]_{ICF}+P_{Na}+[Na^{+}]_{ICF}+P_{Cl}-[Cl^{-}]_{ISF})) (Eq.2)

where V_{m} = potential difference across the cell membrane

F = Faraday’s constant

R = ideal gas constant

T = absolute temperature in Kelvin

P_{K}+ = permeability of the membrane to potassium

P_{Na}+ = permeability of the membrane to sodium

P_{Cl}-= permeability of the membrane to chloride

[K^{+}]_{ICF} = intracellular fluid potassium concentration

[K^{+}]_{ISF}= interstitial fluid potassium concentration

[Na^{+}]_{ICF}= intracellular fluid sodium concentration

[Na^{+}]_{ISF}= interstitial fluid sodium concentration

[Cl^{-}]_{ICF}= intracellular fluid chloride concentration

[Cl^{-}]_{ISF}= interstitial fluid chloride concentration

FV_{m}/RT= ln ((P_{K}+[K^{+}]_{ISF}+P_{Na}+[Na^{+}]_{ISF}+P_{Cl}-[Cl^{-}]_{ICF})/(P_{K}+[K^{+}]_{ICF}+P_{Na}+[Na^{+}]_{ICF}+P_{Cl}-[Cl^{-}]_{ISF})) (Eq. 2A)

e^{FVm/RT}= ((P_{K}+[K^{+}]_{ISF}+P_{Na}+[Na^{+}]_{ISF}+P_{Cl}-[Cl^{-}]_{ICF})/(P_{K}+[K^{+}]_{ICF}+P_{Na}+[Na^{+}]_{ICF}+P_{Cl}-[Cl^{-}]_{ISF})) (Eq. 2B)

[K^{+}]^{ISF}=([eFVm/RT(P^{K}+[K^{+}]^{ICF}+P^{Na}+[Na^{+}]^{ICF}+P^{Cl}-[Cl^{-}]^{ISF})-P^{Na}+[Na^{+}]^{ISF} -P^{Cl}-[Cl^{-}]^{ICF}]/P^{K}+) (Eq.2C)

Incorporating Eq. 2C into Eq. 1B:

∴[K^{+}]_{pw}=([e^{FVm/RT} (P_{K}+[K^{+}]_{ICF}+P_{Na}+[Na^{+}]_{ICF}+P_{Cl}-[Cl^{-}]_{ISF})-P_{Na}+[Na^{+}]_{ISF}-P_{Cl}-[Cl^{-}]ICF]/ PK^{+}) (e^{-FEG)/RT}) (Eq.3)

Since the plasma water potassium concentration [K^{+}]_{pw} is related to the plasma potassium concentration

[K^{+}]_{p} by the mass concentration of water, (pH_{2}O) (5):

[K^{+}]_{pw} (pH_{2}O)=[K^{+}]_{p} (Eq.4)

∴[K^{+}]_{p}=([e^{FVm/RT}(P_{K}+[K^{+}]_{ICF}+P_{Na}+[Na^{+}]_{ICF}+P_{Cl}-[Cl^{-}]ISF)-P_{Na}+[Na^{+]ISF-PCl-[Cl-]ICF]/PK+) (e-FEG/RT) (pH2
}

## DISCUSSION

Although disorders of potassium balance are clinically diagnosed by changes in the plasma potassium concentration, it is well appreciated that changes in the plasma potassium concentration reflect alterations in the total body potassium content that originate predominantly from the intracellular potassium compartment [1]. Currently, the quantitative interrelationship between the plasma potassium concentration and intracellular potassium concentration is not known. In this article, a new mathematical equation is derived to quantify the interrelationship between the plasma potassium concentration and intracellular potassium concentration since changes in the plasma potassium concentration ultimately reflects alterations in the intracellular potassium concentration.

It is well known that the plasma water potassium concentration
[K^{+}]_{pw} is greater than the interstitial fluid potassium concentration
[K^{+}]_{ISF} due to Gibbs-Donnan equilibrium (6,7). The presence of
negatively charged, impermeant plasma proteins will tend to attract
potassium ions from the interstitial fluid into the plasma space,
thereby resulting in a higher [K^{+}]_{pw}. At Gibbs-Donnan equilibrium,
the freely diffusible potassium ions will distribute in such a
manner that its chemical gradient is equal in magnitude to the
electrical gradient imposed by the negatively charged, impermeant
plasma proteins as defined by Eq. 1. As a result, the potassium
concentration in the interstitial fluid is typically 0.95 times the
potassium concentration in the plasma [6]. Potassium ions in
the interstitial fluid are in turn in equilibrium with intracellular
potassium ions as defined by the transmembrane electrochemical
gradient [8]. Potassium is present at higher concentration inside
the cell than outside, whereas sodium and chloride are present at
higher concentration outside the cell [8]. Due to its concentration
gradient, potassium will diffuse across the cell membrane from the
intracellular compartment to the interstitial fluid compartment.
As potassium leaves the cell, an excess of positive charge builds
up on the exterior of the cell membrane, and an excess of negative
charge builds up on the interior of the cell. Consequently, the
interior of the cell becomes negative relative to the exterior,
generating a difference in electrical potential across the membrane.
Since like charges repel and unlike charges attract each other, the
electronegative cell interior and electropositive cell exterior will
in turn oppose the diffusive movement of potassium down its
concentration gradient. When the electrical potential difference
across the cell membrane is equal to the chemical force driving
potassium out of the cell, there is no net movement of potassium
in either direction and electrochemical equilibrium is attained.
The electrical potential difference across the cell membrane that
exactly opposes the chemical gradient for an ion is known as the
equilibrium potential. The magnitude of this equilibrium potential
is therefore a function of the concentration gradient for the ion. For a
cell where there is only one permeant ion, the equilibrium potential
for that ion will be equal to the resting membrane potential of
the cell. However, the resting membrane potential is slightly less
negative than the potassium equilibrium potential because other
types of ions also contribute to the resting membrane potential. In
addition to potassium, the cells are also permeable to sodium and chloride. In particular, permeability to sodium is the main reason
for the difference in the resting membrane potential from the
potassium equilibrium potential. Since the sodium concentration is
much higher outside of a cell than inside, sodium will diffuse down
its chemical gradient into the cell, making the cell interior more
positive relative to the outside. Therefore, the sodium equilibrium
potential that exactly opposes the sodium concentration gradient
will be positive. If sodium were to be the only permeant ion, the
resting membrane potential will be positive. Since both sodium and
potassium are able to cross the membrane, the resting membrane
potential will be in between the sodium equilibrium potential and
potassium equilibrium potential. Given that the resting membrane
is much more permeable to potassium than to sodium, the resting
membrane potential is closer to the potassium equilibrium
potential than to the sodium equilibrium potential.

Therefore, the transmembrane sodium and potassium concentration gradients are key determinants of the membrane potential and are maintained by the activity of the Na+-K+ ATPase, which actively transports sodium and potassium against their electrochemical gradients.

Similarly, the chloride concentration gradient also contributes
to the resting membrane potential. Consequently, in determining
the resting membrane potential across the cell membrane in which
Na+,K+ and Cl- are the major contributors to the membrane potential,
the Goldman-Hodgkin-Katz equation was derived to account for the
selectivity of the membrane’s permeability to sodium, potassium
and chloride ions [3,4]. Currently, it is implicitly assumed that the
interrelationship between the plasma potassium concentration
[K^{+}]_{p} and intracellular potassium concentration [K^{+}]_{ICF} is defined
by the Goldman-Hodgkin-Katz equation by inaccurately assuming
that the plasma water potassium concentration is equal to the
interstitial fluid potassium concentration [K^{+}]_{ISF}. However, due to
Gibbs-Donnan equilibrium, the potassium concentration in the
interstitial fluid is typically 0.95 times the potassium concentration
in the plasma [6].

Given that hypokalemia and hyperkalemia are disorders
of potassium balance which predominantly reflect changes
in the intracellular potassium store, a new formula, Eq. 5, is
derived to define the quantitative interrelationship between
the plasma potassium concentration [K^{+}]_{p} and intracellular
potassium concentration [K^{+}]_{ICF}
. Eq. 5 is derived based on the
three compartmental distribution of potassium among the
plasma, interstitial fluid and intracellular fluid compartments as
defined by Gibbs-Donnan equilibrium (Eq. 1) and transmembrane
electrochemical gradient (as defined as the Goldman-Hodgkin-Katz
equation, Eq.2). Eq. 5 also incorporates the mass concentration
of water in its derivation to account for the fact that the plasma
is normally composed of 93% plasma water and that the plasma
potassium concentration is 0.93 times the plasma water potassium
concentration: [K^{+}]_{pw} (0.93) = [K^{+}]_{p} (9).

## CLINICAL IMPLICATIONS

It is well known that changes in the serum sodium
concentration are sensed by osmoreceptors in the hypothalamus, whereas changes in the serum calcium level are sensed by calciumsensing
receptors in the parathyroid cell, kidneys and other tissues
[10,11]. How are changes in the serum potassium concentration
sensed by the cells? Changes in potassium balance will initially
result in changes in the serum potassium concentration. Changes
in the serum potassium concentration will in turn lead to a
subsequent change in interstitial fluid potassium concentration
due to reestablishment of Gibbs-Donnan equilibrium. The change
in interstitial fluid potassium concentration will then alter
the transmembrane electrochemical gradient, resulting in the
transcellular shift of potassium across the cells and a subsequent
change in the resting membrane potential. Therefore, fluctuations
in the serum potassium concentration are sensed by the cells by an
alteration in the membrane potential. Indeed, the cell membrane
hyperpolarizes or depolarizes in response to a decrease or increase
in serum potassium concentration respectively [12]. It is also known
that hyperpolarization or depolarization of the cell membrane
disrupts normal electrical excitability and can potentially lead to
life-threatening cardiac arrhythmias (12). Alterations in the serum
potassium concentration are therefore sensed by the cells via
changes in the [K^{+}]_{ICF} (intracellular fluid potassium concentration)
and V_{m} (membrane potential) as reflected by these terms in Eq. 5.

Based on Eq. 5, hyperpolarization of the cell membrane induced by hypokalemia will in turn lead to re-establishment of the electrochemical gradient for sodium, resulting in increased transcellular sodium entry. Indeed, a deficit of intracellular potassium may result in the entry of sodium ions into the cell as supported by the finding of an increase in non extracellular sodium and an increased ratio of non extracellular sodium to exchangeable sodium in patients with diuretic-induced hyponatremia [13]. Diuretic-induced hyponatremia in these patients is thought to be due to hypokalemia-induced hyponatremia due to the transcellular shift of sodium ions into the cell [13]. According to Eq. 5, the chloride concentration gradient also contributes to the resting membrane potential. Evidence for the contribution of the chloride concentration gradient to the resting membrane potential is demonstrated in an experimental mouse model of thiazide-induced hypocalciuria [14]. It has been demonstrated on immortalized mouse distal convoluted tubule cells that thiazide diuretic inhibits cellular chloride entry mediated by apical membrane NaCl cotransport, resulting in a decrease in the intracellular chloride concentration. Since intracellular chloride is above its electrochemical equilibrium, the chloride equilibrium potenial required to maintain the transmembrane chloride concentration gradient will be more negativce due to the decrease in the intracellular chloride concentration. Consequently, the decrease in the intracellular chloride concentration hyperpolarizes the distal convoluted tubule cells towards their potassium equilibrium potential, and this membrane hyperpolarization in turn stimulates calcium entry by the apical membrane, dihydropyridine-sensitive calcium channels [14].

## CONCLUSION

In conclusion, current analysis implicitly assumes that the
interrelationship between the plasma potassium concentration
[K^{+}]_{p} and intracellular potassium concentration [K^{+}]_{ICF} is defined by
the Goldman-Hodgkin-Katz equation by inaccurately assuming that
the [K^{+}]_{pw} is equal to the interstitial fluid potassium concentration
[K^{+}]_{ISF}. In actuality, changes in the [K^{+}]_{p} will lead to an initial
alteration in the [K^{+}]_{ISF} due to re-establishment of Gibbs-Donnan
equilibrium. Changes in the [K^{+}]_{ISF} will in turn result in changes
in the [K^{+}]_{ICF} due to re-establishment of the transmembrane electrochemical potential. In addition, changes in the [K+]ICF
may consequently lead to the transcellular shift of sodium due
to perturbation in the transmembrane sodium electrochemical
gradient. The quantitative interrelationships of all these factors
are defined by Eq.5. Specifically, the transcellular entry of sodium
ions in hypokalemia-induced hyponatremia induced by a deficit of
intracellular potassium can be mathematically accounted for by
this new formula. Lastly and most importantly, this new formula
mathematically accounts for the three compartmental distribution
of potassium in the body fluids and the role of Gibbs-Donnan
equilibrium and transmembrane electrochemical gradient in
defining the plasma potassium concentration.

## DECLARATIONS

#### Data Availability

The data (derivation of a new formula defining the quantitative
interrelationship between [K^{+}]_{p} and [K^{+}]_{ICF}) used to support the
findings of this study are included within the article.

#### Author Contributions

M.K.N. conception and design of research; M.K.N., M-K.N. and DS.N. analyzed data; M.K.N. drafted manuscript; M.K.N., M-K.N. and D-S.N. edited, revised and approved final version of manuscript; M-K.N. and D-S.N. prepared figure.

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#### Article Type

Research Article

#### Publication history

Received Date: April 09, 2022

Published: September 06, 2022

#### Address for correspondence

Minhtri K Nguyen, David Geffen School of Medicine at UCLA, 911 Broxton Ave, Suite 202D, Los Angeles, California

#### Copyright

©2022 Open Access Journal of Biomedical Science, All rights reserved. No part of this content may be reproduced or transmitted in any form or by any means as per the standard guidelines of fair use. Open Access Journal of Biomedical Science is licensed under a Creative Commons Attribution 4.0 International License

#### How to cite this article

Minhtri KN, Minh-Kevin N, Dai-Scott N. Role of Gibbs-Donnan Equilibrium and Transmembrane Electrochemical Gradient in Defining the Plasma Potassium Concentration. 2022- 4(5) OAJBS.ID.000483.

**Figure 1:** Three-Compartmental Distribution of Potassium- The distribution of potassium between the plasma and
interstitial fluid compartments is defined by Gibbs-Donnan equilibrium, whereas the distribution of potassium
between the interstitial fluid and intracellular fluid compartments is defined by the transmembrane electrochemical
gradient. Although disorders of potassium balance are clinically diagnosed by changes in the plasma potassium
concentration, it is well appreciated that changes in the plasma potassium concentration reflect alterations in the
total body potassium store that originate predominantly from the intracellular potassium compartment. Therefore,
changes in the plasma potassium concentration are simply a reflection of changes in the intracellular potassium
concentration.