Question

Remembering the effect of cyanide (described in chapter 5), explain how you might determine the extent to which the \(\mathrm{Na}^{+} / \mathrm{K}^{+}\)pumps contribute to the resting membrane potential. Using a measurement of the resting membrane potential as your guide, how could you experimentally determine the relative permeability of the plasma membeane to \(\mathrm{Na}^{+}\)and \(\mathrm{K}^{+}\)?

Short answer

To determine the extent to which the Na+/K+ pumps contribute to the resting membrane potential, perform an experiment with a control group and a cyanide-treated group, and measure the resting membrane potential for both groups. Calculate the difference in resting membrane potential between the two groups, as this difference can be attributed to the contribution of the Na+/K+ pump. To determine the relative permeability of the plasma membrane to Na+ and K+ ions, use the Goldman-Hodgkin-Katz equation with the resting membrane potential measurement from Step 3 as the Vm value, and the concentrations of K+ and Na+ ions from the literature or experimental measurements. By solving the equation for PK/PNa, we can quantitatively determine the relative permeability of the plasma membrane to Na+ and K+ ions.

Step-by-step solution

Understand the process of the Na+/K+ pump and role of cyanide

The Na+/K+ pump is a protein complex that helps maintain the resting membrane potential. It does this by pumping three Na+ ions out of the cell and two K+ ions into the cell. ATP provides the energy for these pumps, and cyanide is known to inhibit the production of ATP by disrupting cellular respiration.

Experiment design to determine the Na+/K+ pump's contribution to the resting membrane potential

In order to determine the contribution of the Na+/K+ pumps to the resting membrane potential, we can perform an experiment with two groups of cells: a control group and a group treated with cyanide. We can then measure the resting membrane potential of both groups and compare the difference. The control group will have normal Na+/K+ pump function, whereas the cyanide-treated group will have reduced or inhibited Na+/K+ pump function due to ATP depletion.

Measure the resting membrane potential for both groups

Using a microelectrode, measure the resting membrane potential for both the control group and the cyanide-treated group. This allows you to establish the normal resting membrane potential for the cellular system of interest and the effects of cyanide.

Determine the contribution of the Na+/K+ pump to the resting membrane potential

Calculate the difference in resting membrane potential between the control group and the cyanide-treated group. This difference can be attributed to the contribution of the Na+/K+ pump to the resting membrane potential. The larger the difference, the more significant the contribution of the Na+/K+ pump to the resting membrane potential.

Determination of relative permeability of the plasma membrane to Na+ and K+ ions

To determine the relative permeability of the plasma membrane to Na+ and K+ ions, use the Goldman-Hodgkin-Katz (GHK) equation: \[\mathrm{V_{m}} = \frac{RT}{F} \ln \frac{P_{K}[\mathrm{K_{o}}^{+}] + P_{\mathrm{Na}}[\mathrm{Na_{o}}^{+}]}{P_{K}[\mathrm{K_{i}}^{+}] + P_{\mathrm{Na}}[\mathrm{Na_{i}}^{+]}}\] Here, Vm is the membrane potential, R is the gas constant, T is the temperature, F is the Faraday's constant, PK and PNa are the permeabilities of K+ and Na+ ions, and the subscripts "o" and "i" represent concentrations of ions outside and inside the cells. We can use the GHK equation to determine the relative permeabilities when the resting membrane potential measurements made in Step 3 are used as the membrane potential (Vm) value. The concentrations of K+ and Na+ ions can be obtained from the literature or experimental measurements, and we can experimentally vary the conditions and measure the new resting membrane potential to compute the permeabilities. By solving the equation for PK/PNa, we can quantitatively determine the relative permeability of the plasma membrane to Na+ and K+ ions.

Key concepts

Understanding the Na+/K+ Pump and Its Impact on Cells

The Na+/K+ pump, a crucial membrane protein, serves as a guardian of cellular homeostasis, diligently maintaining the delicate balance of sodium (Na+) and potassium (K+) ions across the membrane. Imagine the pump as a tiny gatekeeper, expending energy in the form of ATP to send three Na+ ions out and welcome two K+ ions in, which creates an essential electrochemical gradient.

• The cycle of the Na+/K+ pump sets up a steep concentration gradient, with high Na+ outside the cell and high K+ inside. This gradient is fundamental to various cellular functions, including nerve impulse transmission and muscle contraction.
• When cyanide enters the scene, it disrupts ATP production, effectively tying the hands of our diligent pump and preventing it from doing its job, which can lead to a change in the resting membrane potential.

So, to grasp the pump's contribution, one might compare the resting potential in normal cells with cells that have been 'poisoned' by cyanide, observing the shift in potential when the pump's activity is compromised.

Membrane Permeability: A Gateway Influencing Cell Function

Let's now turn our attention to membrane permeability. If the Na+/K+ pump is the gatekeeper, membrane permeability represents the ease with which gates allow passengers (ions) to pass through. Different membranes have unique preferences, some favoring positive ions like Na+, others permitting K+ ions to slip through more easily.

• The relative permeability of these ions can have far-reaching effects on a cell's state; think communication in neurons or heartbeats in cardiac cells.
• By playing with these permeability settings (like modifying the ratio of Na+/K+ passage), scientists can learn how cells will respond and consequently understand the vital statistics for the body's electrical system.

Using the changes in the resting membrane potential as a reference, one can deduce the relative permeability by observing how the potential fluctuates under various controlled experimental conditions.

Goldman-Hodgkin-Katz Equation: The Cellular Accounting System

The Goldman-Hodgkin-Katz (GHK) equation serves as the meticulous accountant of cellular ionic economics, mathematically balancing the checkbooks of ion permeability and concentrations.

Breaking Down the GHK Equation

The formula revolves around Vm, the membrane potential, and extends a cordial invitation to the concentrations of K+ and Na+ ions, inside (i) and outside (o) the cell, as well as their respective permeability values, PK and PNa.

• The resulting calculation gives us the net membrane potential, answering the big question: 'How does the cell's unique mix of ion passage preferences affect its voltage at rest?'
• By taking experimental measurements, plugging in those numbers, and observing the outcomes, researchers can adjust the ionic permeability values to fit the cells' measured resting potential, effectively reverse-engineering the equation to understand cellular behavior.

This equation enables the scientific dissection of complex ionic contributions and their profound impact on crucial functions like heartbeat regulation and signal transmission in the nervous system.

Neuronal Electrochemical Gradient: The Power Source of Nerve Signals

Focusing on the neuronal electrochemical gradient, imagine it as the battery that powers the elaborate network of nervous communication highways. This gradient, a sort of voltage across the cell's membrane, is crucial for generating and transmitting nerve impulses.

• Integral to this process is the complementary forces of concentration gradients (difference in ion concentrations) and electrical gradients (difference in charge across the membrane).
• Neuronal communication relies on the interplay between these gradients, with the Na+/K+ pump setting the stage by creating high Na+ outside and high K+ inside the cell, and membrane permeability dictating the rate at which these ions travel.

The dance between these gradients facilitates the complex electrical signals that enable thoughts, sensations, and movements—a testament to the intricate balance and power hidden within cells.