Simulations of the Nernst potential

Bioelectricity: Electric model of the cell membrane

Simulations of the Nernst potential

As a reminder we once more write down the Nernst equation.

The Nernst equation for the equilibrium potential The equilibrium potential of an ion is given by \[E_\mathrm{ion}= \frac{RT}{zF}\cdot \ln\left(\frac{C_{\mathrm{e}}}{C_{\mathrm{i}}}\right) = -\frac{RT}{zF}\cdot \ln\left(\frac{C_{\mathrm{i}}}{C_{\mathrm{e}}}\right)\] This electric potential represents the resting membrane potential in a model of the cell with a single ion species and a single ion channel type, and the value depends on the absolute temperature \(T\), the valence \(z\) (the number of unit charges per ion), and the ion concentrations inside (\(C_{\mathrm{i}}\)) and outside (\(C_{\mathrm{e}}\)) the cell near the membrane. \(R\) is the gas constant, and \(F\) is Faraday's constant (equal to the charge of one mole of monovalent ions). At a temperature of \(29.2{}^{\circ}\mathrm{C}\) you can also compute the Nernst potential in millivolt through the equation \[E_\mathrm{ion}= \frac{60}{z} \log_{10}\left(\frac{C_{\mathrm{e}}}{C_{\mathrm{i}}}\right)\]

Use the simulation below to practice with the Nernst potential and to answer the following questions. First, think about all answers and check them later in the explanation.

Click on the link Simulation of Nernst potentials if you want to use the simulation in a separate window.

Exploration 1: sodium ions
Click on the checkbox labelled Na: the default values for \(\mathrm{Na}^{+}\) concentration inside and outside the cell are then \([\mathrm{Na}^{+}]_\mathrm{i}=50\,\text{mM}\) and \([\mathrm{Na}^{+}]_\mathrm{e}=430\,\text{mM}\).

Increase the \(\mathrm{Na}^{+}\) concentration inside the cell (e.g. double it) and explore the impact on the Nernst potential \(E_\mathrm{Na}\).
Come up with a different way to achieve exactly the same Nernst potential \(E_\mathrm{Na}\) from the first situation as in the first task.
How must you set the sodium concentrations inside and outside the cell to make the Nernst potential \(E_\mathrm{Na}\) equal to 0 mV?
Find out how much the Nernst potential \(E_\mathrm{Na}\) changes with each 10-fold change in the concentration gradient.
Increase the temperature and explore the impact on the Nernst potential \(E_\mathrm{Na}\).

Answers

When the \(\mathrm{Na}^{+}\) concentration inside the cell increases and the quotient \(\frac{[\mathrm{Na}^{+}]_\mathrm{e}}{[\mathrm{Na}^{+}]_\mathrm{i}}\) decreases, it follows from the Nernst equation that the Nernst potential decreases. You see this happen in the simulation.
Most relevant in the Nernst equation is the quotient \(\frac{[\mathrm{Na}^{+}]_\mathrm{e}}{[\mathrm{Na}^{+}]_\mathrm{i}}\). So halving the \(\mathrm{Na}^{+}\) concentration outside the cell twice has the same impact as doubling the concentration inside.
When the \(\mathrm{Na}^{+}\) concentration inside and outside the cell are equalr, then the Nernst potential is equal to 0 mV.
The Nernst equation at a given temperature is equal to \[E_\mathrm{Na}=60\,\log_{10}\left(\frac{[\mathrm{Na}^{+}]_\mathrm{e}}{[\mathrm{Na}^{+}]_\mathrm{i}}\right)\] From the calculation rules for the base 10 logarithm follows that with each multiplication factor of 10 for the concentration ratio outside÷inside, the Nernst potential increases with 60 mV.
The Nernst potential depends linearly on the temperature. When the temperature increases, the Nernst potential will increase in absolute value.

Exploration 2: potassium ions
Press the reset button and then click on the checkbox labelled K: the default values for \(\mathrm{K}^{+}\) concentration inside and outside the cell are then \([\mathrm{K}^{+}]_\mathrm{i}=400\,\text{mM}\) and \([\mathrm{K}^{+}]_\mathrm{e}=20\,\text{mM}\).

Increase the \(\mathrm{K}^{+}\) concentration inside the cell and explore the impact on the Nernst potential \(E_\mathrm{K}\).
Come up with a different way to achieve exactly the same Nernst potential \(E_\mathrm{K}\) from the first situation as in the first task.
How must you set the sodium concentrations inside and outside the cell to make the Nernst potential \(E_\mathrm{K}\) equal to 0 mV?
Find out how much the Nernst potential \(E_\mathrm{K}\) changes with each 10-fold change in the concentration gradient.

Answers

When the \(\mathrm{K}^{+}\) concentration inside the cell increases and the quotient \(\frac{[\mathrm{K}^{+}]_\mathrm{e}}{[\mathrm{K}^{+}]_\mathrm{i}}\) decreases, it follows from the Nernst equation that the Nernst potential decreases. You see this happen in the simulation.
Most relevant in the Nernst equation is the quotient \(\frac{[\mathrm{K}^{+}]_\mathrm{e}}{[\mathrm{K}^{+}]_\mathrm{i}}\). So halving the \(\mathrm{K}^{+}\) concentration outside the cell twice has the same impact as doubling the concentration inside.
When the \(\mathrm{K}^{+}\) concentration inside and outside the cell are equal, then the Nernst potential is equal to 0 mV.
The Nernst equation at the given temperature is equal to \[E_\mathrm{K}=60\,\log_{10}\left(\frac{[\mathrm{K}^{+}]_\mathrm{e}}{[\mathrm{K}^{+}]_\mathrm{i}}\right)\] From the calculation rules for the base 10 logarithm follows that with each multiplication factor of 10 for the concentration ratio outside÷inside, the Nernst potential increases with 60 mV.

Exploration 3: chloride ions
Press the reset button and then click on the checkbox labelled Cl: the default values for the \(\mathrm{Cl}^{-}\) concentration inside and outside the cell are then \([\mathrm{Cl}^{-}]_\mathrm{i}=30\,\text{mM}\) and \([\mathrm{Cl}^{+}]_\mathrm{e}=450\,\text{mM}\).

Increase the \(\mathrm{Cl}^{-}\) concentration inside the cell and explore the impact on the Nernst potential \(E_\mathrm{Cl}\).
Come up with a different way to achieve exactly the same Nernst potential \(E_\mathrm{Cl}\) from the first situation as in the first task.
How must you set the sodium concentrations inside and outside the cell to make the Nernst potential \(E_\mathrm{Cl}\) equal to 0 mV?
Find out how much the Nernst potential \(E_\mathrm{Cl}\) changes with each 10-fold change in the concentration gradient.
Lower the temperature and explore the impact on the Nernst potential \(E_\mathrm{Cl}\).

Answers

When the \(\mathrm{Cl}^{-}\) concentration inside the cell increases and the quotient \(\frac{[\mathrm{Cl}^{-}]_\mathrm{e}}{[\mathrm{Cl}^{-}]_\mathrm{i}}\) decreases, it follows from the Nernst equation that the Nernst potential decreases (mind the minus sign because the valence of the ion species). You see this happen in the simulation.
Most relevant in the Nernst equation is the quotient \(\frac{[\mathrm{Cl}^{-}]_\mathrm{e}}{[\mathrm{Cl}^{-}]_\mathrm{i}}\). So halving the \(\mathrm{Cl}^{-}\) concentration outside the cell twice has the same impact as doubling the concentration inside.
When the \(\mathrm{Cl}^{-}\) concentration inside and outside the cell are equal, then the Nernst potential is equal to 0 mV.
The Nernst equation at the given temperature is equal to \[E_\mathrm{Cl}=-60\,\log_{10}\left(\frac{[\mathrm{Cl}^{-}]_\mathrm{e}}{[\mathrm{Cl}^{-}]_\mathrm{i}}\right)\] From the calculation rules for the base 10 logarithm follows that with each multiplication factor of 10 for the concentration ratio outside÷inside, the Nernst potential decreases with 60 mV.
The Nernst potential depends linearly on the temperature. When the temperature decreases, the Nernst potential will decrease in absolute value.