Question

A hypothetical neuron has the following properties: ENo +40mV: Ea -70 mV; Ex -85 mV. (5 pts) i. If the membrane were permeable only to potassium, what would be the resting membrane potential? What would happen to ENa if the external sodium concentration was lowered (with internal concentration of sodium remaining the same)? Explain your answer ii. ili. Use the constant field equation to calculate the RMP given a PNa:PK:PCI permeability ratio of 1 :0.04 0.45. Show your work.

Answer 1

In the question, it is given that a hypothetical neuron has the following equilibrium potential for Na⁺, Cl^- and K⁺ ions :

E_Na = +40mV, E_Cl = -70mV and E_K = -85mV..

IMPORTANT CONCEPTS - Equilibrium potential and resting membrane potential (RMP)

1. Equilibrium potential for a particular ion is defined as a function of the ratio of the intracellular and extracellular concentrations of the ion. The formula is given below for K⁺ ion, which is essentially the same as the Nernst equation.

$E_{eq,K^+} = \frac{RT}{zF} \ln \frac{[K^+]_{o}}{[K^+]_{i}} ,$ ----------------------- (eq. 1)

where E_eq,K⁺ is the equilibrium potential for potassium, measured in volts

R is the universal gas constant, equal to 8.314 joules·K⁻¹·mol⁻¹

T is the absolute temperature, measured in kelvins

z is the number of elementary charges of the ion in question involved in the reaction

F is the Faraday constant, equal to 96,485 coulombs·mol⁻¹ or J·V⁻¹·mol⁻¹

[K⁺]_o is the extracellular K⁺ concentration

[K⁺]_i is the intracellular K⁺ concentration

Since the equilibrium potentials for various ions are already given to us, we need not use the above equation. However, it is important to note that if the intracellular concentration of the ion is more than the extracellular concentration, the equilibrium potential will be negative, and vice versa. If both concentrations are equal, the equilibrium potential will be zero.

2. Resting membrane potential for a particular cell is defined as a function of the sum of equilibrium potentials of all ions that are permeable across the cell membrane. It is calculated using the constant field equation (also called as Goldman-Hodgkin-Katz equation), which is given below.

$E_{{m}}={\frac {RT}{F}}\ln {\left({\frac {P_{{Na^{+}}}[Na^{+}]_{{o}}+P_{{K^{+}}}[K^{+}]_{{o}}+P_{{Cl^{-}}}[Cl^{-}]_{{i}}}{P_{{Na^{+}}}[Na^{+}]_{{i}}+P_{{K^{+}}}[K^{+}]_{{i}}+P_{{Cl^{-}}}[Cl^{-}]_{{o}}}}\right)}$ ------------------ (eq. 2)

where P_Na+, P_K+ and P_Cl- are the the relative permeabilities of each of the ions.

Note that since only Na⁺, K⁺ and Cl^- ions play an important role in membrane potential, only these three ions are considered.

Answer i. In case of only one ion being permeable, the resting membrane potential is equal to the equilibrium potential for that ion. In this case, it is given that K⁺ is the only ion permeable. Therefore, the resting membrane potential will be equal to the equilibrium potential of K⁺ ion, i.e. -85mV.

Answer ii. In the above given equation (eq. 1), replace K⁺ with Na⁺. If the extracellular Na⁺ were to be increased, the equilibrium potential will increase, since the extracellular concentration value lies in the numerator of the equation.

Therefore, on increasing the external Na⁺ concentration, E_Na will become higher than +40mV.

Answer iii. The resting membrane potential can be considered as a summation of the equilibrium potentials of each of the permeable ions, multiplied by their permeability factor. In this case, the permeability factors for Na+, K+ and Cl- are given as 1.0, 0.04 and 0.45, respectively. Therefore,

resting membrane potential = (1.0)(+40) + (0.04)(-85) + (0.45)(-70)

= 40 - 3.4 - 31.5.

= +5.1mV.