Question

Nerve impulses are electrical "signals" that pass through neurons in the body. The electrical potential is created by the differences in the concentration of \(\mathrm{Na}^{+}\) and \(\mathrm{K}^{+}\) ions across the nerve cell membrane. We can think about this potential as being caused by a concentration gradient, similar to what we see in a concentration cell (keep in mind that this is a very simple explanation of how nerves work; there is much more involved in the true biologic process). A typical nerve cell has a resting potential of about \(-70 \mathrm{mV}\). Let's assume that this resting potential is due only to the \(\mathrm{K}^{+}\) ion concentration difference. In nerve cells, the \(\mathrm{K}^{+}\) concentration inside the cell is larger than the \(\mathrm{K}^{+}\) concentration outside the cell. Calculate the \(\mathrm{K}^{+}\) ion concentration ratio necessary to produce a resting potential of \(-70 . \mathrm{mV}\). $$\frac{\left[\mathrm{K}^{+}\right]_{\text {inside }}}{\left[\mathrm{K}^{+}\right]_{\text {outside }}}=?$$

Short answer

The required potassium ion concentration ratio to produce a resting potential of -70 mV is approximately: \(\frac{[\mathrm{K}^{+}]_\text{inside}}{[\mathrm{K}^{+}]_\text{outside}} \approx 1.35\) Thus, the potassium ion concentration inside the nerve cell must be about 1.35 times the concentration outside the cell.

Step-by-step solution

Convert mV to V

First, we need to convert the resting potential from mV to V. To do this, divide the given value by 1000: \(-70 mV = -70 \times 10^{-3} V = -0.07 V\)

Substitute the known values into the Nernst equation

Now, we substitute the known values into the Nernst equation: E=-0.07 V, R=8.314 J/mol*K, T=298 K, z=1, F=96,485 C/mol \(-0.07 = \frac{8.314 \times 298}{1\times 96,485} \ln \frac{[\mathrm{K}^{+}]_\text{outside}}{[\mathrm{K}^{+}]_\text{inside}}\)

Solve for the ion concentration ratio

Now, we solve for the ratio of potassium ion concentrations: \(\ln \frac{[\mathrm{K}^{+}]_\text{outside}}{[\mathrm{K}^{+}]_\text{inside}} = \frac{-0.07 \times 1 \times 96,485}{8.314 \times 298}\) Calculate the value on the right side of the equation: \(\frac{-0.07 \times 1 \times 96,485}{8.314 \times 298} \approx -0.299\) Now, take the exponent of both sides to remove the natural logarithm: \(\frac{[\mathrm{K}^{+}]_\text{outside}}{[\mathrm{K}^{+}]_\text{inside}} \approx e^{-0.299}\) Calculate the exponent: \(e^{-0.299} \approx 0.74\)

Write the final answer

The required potassium ion concentration ratio to produce a resting potential of -70 mV is approximately: \(\frac{[\mathrm{K}^{+}]_\text{inside}}{[\mathrm{K}^{+}]_\text{outside}} \approx \frac{1}{0.74} = 1.35\) Thus, the potassium ion concentration inside the nerve cell must be about 1.35 times the concentration outside the cell.

Key concepts

Electrical Potential

Nerve impulses originate from a difference in electrical potential across the cell membrane of neurons. This potential is crucial because it allows the nerve cells to transmit signals efficiently. Electrical potential is essentially the ability to do work due to a separation of charges. In our case, this involves positive and negative ions located on opposite sides of the nerve cell membrane.
For neurons, the inside typically has a net negative charge compared to the outside. This difference is maintained by ion pumps and channels that regulate ion movement across the membrane. The most important ions involved are sodium (\( \text{Na}^+ \)) and potassium (\( \text{K}^+ \)).

• When these ions move across the membrane, they change the charge balance, affecting the potential.
• This differential is what creates the capacity for nerve impulse transmission.

The electrical potential can be measured in volts (V) or millivolts (mV), with nerve cells having a typical resting potential of around -70 mV. This means the inside of the neuron is negatively charged compared to the outside.

Concentration Gradient

A concentration gradient refers to the gradual change in the concentration of solutes in a solution as a function of distance through a solution. In nerve cells, this principle is crucial for creating the conditions necessary for transmitting nerve impulses. Typically, cells maintain a higher concentration of \( \text{K}^+ \) ions inside the cell compared to outside, while the concentration of \( \text{Na}^+ \) ions is higher outside the cell.
This difference creates a chemical gradient which drives the movement of ions through channels in the neural membrane. The goal is to reach equilibrium, where concentrations are the same on both sides, but due to selective permeability of the membrane and the action of ion pumps, this equilibrium is never reached. This dynamic state is essential in:

• Generating electrical signals across the nerve cell membrane.
• Maintaining the resting potential of the cell, which is necessary for nerve function.
• The interplay between concentration gradients and membrane permeability is fundamenal for nerve impulses.

Nernst Equation

The Nernst equation is a mathematical formula that relates the concentration of ions inside and outside a cell to the electrical potential across the cell membrane. This equation allows us to calculate the potential created by a single type of ion. It's essential in understanding how different ion combinations result in the resting potential of nerve cells.
The Nernst equation can be written as: \[ E = \frac{RT}{zF} \ln \left( \frac{[\text{ion}]_{\text{outside}}}{[\text{ion}]_{\text{inside}}} \right) \]Where:

• \(E\) is the voltage (potential difference across the membrane).
• \(R\) is the universal gas constant.
• \(T\) is the temperature in Kelvin.
• \(z\) is the charge number of the ion.
• \(F\) is the Faraday constant.
• \(\ln\) denotes the natural logarithm.

This equation helps us predict the direction in which ions will move and how this movement will affect the overall cell membrane potential.

Potassium Ion Concentration

Potassium ions \( (\text{K}^+) \) play an essential role in maintaining the resting potential of nerve cells. Because the concentration of \( \text{K}^+ \) is higher inside the cell than outside, it tends to move out of the cell. This movement is facilitated by potassium channels in the membrane, which are highly selective for \( \text{K}^+ \) ions.
In forming the resting potential, the cell membrane's permeability to \( \text{K}^+ \) is higher than for \( \text{Na}^+ \). This permeability difference is why \( \text{K}^+ \) is more influential in determining the resting potential and the ratio of ion concentrations is vital.

• Higher intracellular \( \text{K}^+ \) concentration compared to extracellular.
• The Nernst equation helps determine the necessary ion concentration ratio for a given potential.
• This ion gradient is crucial for nerve signaling and muscle contractions.

Resting Potential

The resting potential is the baseline electrical charge difference across the neuron's membrane when it is not actively transmitting a signal. This potential is typically around -70 mV, indicating that the inside of the neuron is more negative than the outside.
Resting potential is essential for preparing the neuron to fire action potentials or nerve impulses.

• It is maintained by the sodium-potassium pump, which moves \( \text{Na}^+ \) out of the cell and \( \text{K}^+ \) into the cell.
• This activity leads to a buildup of positive charge outside the cell.
• The membrane's permeability to specific ions, primarily \( \text{K}^+ \), fine-tunes the resting potential.

The resting potential thus sets the stage for the rapid changes of potential that occur when a neuron sends an impulse.