# The Resting Membrane Potential

## The Resting potential

The resting potential of a cell is the membrane potential that would be maintained if there were no action potentials, post-synaptic potentials, or other active changes in the membrane potential. In most cells the resting potential has a negative value, which by convention means that there is excess negative charge inside compared to outside. The resting potential is mostly determined by the concentrations of the ions in the fluids on both sides of the cell membrane and the ion transport proteins that are in the cell membrane. How the concentrations of ions and the membrane transport proteins influence the value of the resting potential is outlined below.

How is the resting membrane potential produced and maintained?

• The Donnan Effect
• Membrane Selectivity (Difference in permeabilities of ions)
• Active Transport (Na/K ATPase)

The Donnan Effect- Simplified the donnan effect can be described as large impermeant negatively charged intracellular molecules attracting positively charged ions such as Na+, K+ and repelling negative ones such Cl-

Membrane selectivity- Membrane selectivity is the difference in permeabilities between different ions for example the plasma membrane is far more permeable to potassium than sodium.

Active transport- Active transport is the mediated process of moving particles across biological membrane against a concentration gradient. If the process uses chemical energy, such as from ATP, it is termed primary active transport. Secondary active transport involves the use of an electrochemical gradient.

The three factors above all responsible for the resting membrane potential and also maintaining it at a steady voltage.

## The Nernst Equation

The Nernst equation is a mathematical equation which can be applied physiologically to cells to produce Equilibrium potentials (The transmembrane potential difference required to exactly balance a given concentration gradient) for certain ions, the Nerst equation is named after the German physical chemist who first formulated it, Walther Nernst.

The equation is as follows

$W=RTln{\frac {[X]compartment1}{[X]compartment2}}$

Where-

• W = Chemical work
• R = Gas Constant
• T = Absolute temperature(oK)

Electrical working in the opposite direction $EFz$

• E = The potential deifference across the membrane
• F = Faradays Constant (96500 coulombs/mole)
• z = Vaelncy of ion

At equlibrium the two above equations can be expressed together as

$EFz=RTln{\frac {[X]_{1}}{[X]_{2}}}$

If the equation is then solved for $E$

$E_{i}={\frac {RT}{Fz}}ln{\frac {[X]_{1}}{[X]_{2}}}$

This is the Nernst equation as it is used in physiology to calcualte an equilibrium potential generated by a given ion gradient. The term ${\frac {RT}{Fz}}ln$  can be replaced by a temperature-dependent conversion factor:

At 20oC $E_{i}=58\log _{10}\left({\frac {[X]_{1}}{[X]_{2}}}\right)$

At 37oC (body temperature) $E_{i}=61\log _{10}\left({\frac {[X]_{1}}{[X]_{2}}}\right)$

The Nernst equation doesn't precisely calcualte the membrane potential because it is rarely at one equlibrium potential. Another equation was formulated allowing us to produce more accurate values for the membrane potential, The Goldman-Hodgkin-Katz equation.

## The Goldman-Hodgkin-Katz Equation

The Goldman-Hodgkin-Katz (GHK) voltage equation is used in cell membrane physiology to determine the potential across a cell's membrane (Em) taking into account all of the ions that are permeant through that membrane. The discoverers of this are David E. Goldman of Columbia University, and the English Nobel laureates Alan Lloyd Hodgkin and Bernard Katz.

GHK equation

$E_{m}={\frac {RT}{F}}\ln {\left({\frac {P_{Na^{+}}[Na^{+}]_{\mathrm {out} }+P_{K^{+}}[K^{+}]_{\mathrm {out} }+P_{Cl^{-}}[Cl^{-}]_{\mathrm {in} }}{P_{Na^{+}}[Na^{+}]_{\mathrm {in} }+P_{K^{+}}[K^{+}]_{\mathrm {in} }+P_{Cl^{-}}[Cl^{-}]_{\mathrm {out} }}}\right)}$

Again just like the Nersnt equation the first term can be simplified by a temperature dependent conversion factor producing this at 20oC

$E_{m}=58\log _{10}{\left({\frac {P_{Na^{+}}[Na^{+}]_{\mathrm {out} }+P_{K^{+}}[K^{+}]_{\mathrm {out} }+P_{Cl^{-}}[Cl^{-}]_{\mathrm {in} }}{P_{Na^{+}}[Na^{+}]_{\mathrm {in} }+P_{K^{+}}[K^{+}]_{\mathrm {in} }+P_{Cl^{-}}[Cl^{-}]_{\mathrm {out} }}}\right)}$

The Cl term of the equation is inverted so that z (valency) can be omitted from the first term. In many physiological applications the Cl term can be omitted altogether to yield the equation

$E_{m}=58\log _{10}{\left({\frac {P_{Na^{+}}[Na^{+}]_{\mathrm {out} }+P_{K^{+}}[K^{+}]_{\mathrm {out} }}{P_{Na^{+}}[Na^{+}]_{\mathrm {in} }+P_{K^{+}}[K^{+}]_{\mathrm {in} }}}\right)}$

Cl can be left out of the equation because in the steady state Cl does not contribute to the setting of E_m, Cl is said to be passively permeable and rather than the Cl gradient setting the Em, Em sets the Cl gradient. In other words Ecl- is in equilibrium with Em, $E_{Cl}=E_{m}$  this confers membrane stability.

The single most imporatnt factor in setting Em is the potassium gradient