Problem 94 - College Physics

Question

If the actual conductance of an axon's ion channel is 10 \(\mathrm{pS}\) and the peak current during a squid axon's action potential (the electrical disturbance that propagates down the axon is 5 \(\mathrm{mA} / \mathrm{cm}^{2}\) what is the density of channels in the membrane? Assume that the voltage across the membrane is 50 \(\mathrm{mV}\). A. 1\(/ \mathrm{m}^{2}\) B. 100\(/ \mathrm{cm}^{2}\) C. 1\(/ \mathrm{cm}^{2}\) D. 100\(/ \mu \mathrm{m}^{2}\)

Step-by-Step Solution

Answer

D. 100/ \mu \text{m}^2

Step 1: Understand the Basics

The conductance of a single ion channel is given as 10 \( \text{pS} \) (picosiemens), and we need to find the density of these channels given the current density is 5 \( \text{mA/cm}^{2} \) and the membrane voltage is 50 \( \text{mV} \). Conductance is related to current by \( I = G \cdot V \), where \( I \) is the current, \( G \) is the conductance, and \( V \) is the voltage.

Step 2: Convert Units

Firstly, convert the given conductance from picosiemens to siemens: \( 10 \text{ pS} = 10 \times 10^{-12} \text{ S} \). Next, convert the voltage from millivolts to volts: \( 50 \text{ mV} = 0.050 \text{ V} \).

Step 3: Calculate the Current Through a Single Channel

Use the formula \( I = G \cdot V \) to find the current through a single channel: \( I = (10 \times 10^{-12} \text{ S}) \times (0.050 \text{ V}) = 5 \times 10^{-13} \text{ A} \).

Step 4: Determine Number of Channels Required for Given Current Density

The peak current density provided is 5 \( \text{mA/cm}^{2} \), which is equivalent to \( 5 \times 10^{-3} \text{ A/cm}^{2} \). To find the number of channels, divide the total current density by the current through a single channel: \( \frac{5 \times 10^{-3} \text{ A/cm}^{2}}{5 \times 10^{-13} \text{ A}} = 10^{10} \text{ channels/cm}^{2} \).

Step 5: Convert to Desired Units

The answer choices require the density in different units. \( 10^{10} \text{ channels/cm}^{2} \) is equivalent to \( 100 / \mu \text{m}^2 \), which matches one of the given options.

Key Concepts

Ion Channel Density

Ion channel density is a measure of how many ion channels are present per unit area of a membrane. Ion channels are crucial for transmitting electrical signals in nerve cells.
Each ion channel allows ions to pass through the membrane, affecting the voltage and, consequently, the signal transmission. To determine ion channel density, you divide the total number of channels by the membrane area. In this exercise, you're given the conductance per channel and need to find out how many such channels are there per area unit, assuming you know the current density and membrane voltage.
Understanding this concept is essential for comprehending how signals are transmitted along the axon.

Current Density Calculation

Current density refers to the amount of electrical current flowing through a unit area of the membrane. This measure is crucial for understanding the efficiency of signal transmission in neurons.
You can calculate current density by dividing the total current by the area through which the current flows.In the original problem, the current density is given as 5 \( \text{mA/cm}^2 \). This means that for every square centimeter of membrane, 5 milliamperes of current flow through.
By knowing the current density, and the conductance and voltage of single ion channels, you can calculate how many such channels must exist per unit area to support the observed current.

Membrane Voltage

Membrane voltage refers to the electrical potential across the neuron's membrane. This voltage difference arises from the uneven distribution of ions between the inside and outside of the cell.
It plays a vital role in the conduction of electrical signals along the axon.In neurons, a change in membrane voltage can trigger the opening or closing of ion channels, which is essential in action potentials—the rapid rises and falls in voltage that transmit signals.
In this case, the membrane voltage is given as 50 \( \text{mV} \) (millivolts), which is typical during an action potential peak. Knowing this value helps in calculating the current through a single channel using the formula \( I = G \cdot V \).

Conductance Units Conversion

Conductance measures the ease with which electric current passes through a conductor or, in this case, an ion channel. It's typically measured in siemens (S).
However, conductance values in biological systems are often very small and hence expressed in picosiemens (pS).To perform calculations, you need to convert these units into a standard form. For example, converting 10 pS to siemens involves multiplying by \( 10^{-12} \).
Similarly, the membrane voltage needs conversion from millivolts to volts, where 50 mV is equivalent to 0.050 V. These conversions are crucial for applying formulas accurately, as seen when calculating the current through a channel.