Question

This problem explores what happens to the resting steady state of a membrane, after the lon pumps are suddenly turned off, say by addition of a neurotoxin. Namely, sodium ions are far from equilibrium in the resting state. The concentration of sodium inside the cell is \(12 \mathrm{mM}\) while outside the cell it is \(145 \mathrm{mM}\); the membrane voltage Is \(V_{m c m}=-90 \mathrm{mV}\). This means that when pumps are turned off sodium will rush into the cell. (a) Using Ohm's law, find the current per unit area carried by sodium ions just after the pumps have been shut off. For the conductance of sodium channels per unit area take the measured value of \(0.13 \Omega^{-1} \mathrm{m}^{-2}\). Reexpress your answer as current per unit length along a giant axon, assuming a diameter of \(1 \mathrm{mm}\) (b) Find the charge per unit length of the axon contributed by all the sodium ions inside the axon. What would the corresponding quantity equal if the interior concentration of sodium matched the fixed exterior concentration? (c) Given the current of sodium computed in (a), make a rough estimate of how long will it take for the concentration of sodium inside the axon to reach the value outside the cell? You may assume that the current is constant in time for this estimate. (d) In the chapter we described a nerve impulse as a membrane depolarization event that lasts for about a millisecond. Compare with the time you just computed and comment on the size of the charge perturbation caused by a propagating action potential.

Short answer

To solve this particular problem, we first calculate the sodium ion current per length using Ohm's law and the given values in the problem. We then calculate the charge per unit length before and after the sodium concentration inside matches with the outside. Then we estimate time it would take for the sodium concentration inside the cell to equalize with the outside using the obtained currents and the change in charge. Finally, we calculate the charge perturbation during a neural impulse and comment on its comparison with the obtained charge changes.

Step-by-step solution

Calculate the sodium ion current

Firstly, we can calculate the current of the sodium ions using Ohm's law, which is given by \(I = GV\). Here, G is the conductance which is given as \(0.13 \, \Omega^{-1} \, m^{-2}\), and \(-90 \, mV\) is the voltage across the membrane at resting state. To get I in current per length on a giant axon, we need to multiply it with the cross-sectional area of the axon, which is \(\pi (d/2)^2\). Assuming a diameter of \(1 \, mm\), substituting the values into the equation and performing the calculation will give us the sodium ion current by area.

Calculate charge per unit length

The quantity of charge per unit length before and after the sodium concentration matches is calculated using the formula charge = concentration x volume x charge of a single ion. The volume here is the volume of the axon of unit length, i.e., it's the volume of a cylinder of unit height. Substitute the values into the formula and perform the calculation for both before and after the sodium concentration matches.

Estimate the time to equalize sodium concentration

Using the currents calculated in step 1, we can make a rough estimate of the time it takes for the sodium concentration inside the axon to achieve equilibrium with the outside concentration. We use the formula time = change in charge / current. Here, the change in charge is the difference between the two charges per unit length obtained in the previous step. Substitute the values into the formula and perform the calculation.

Comment on the size of the charge perturbation

In the final step, we compare this time with the duration of a nerve impulse event which is about a millisecond. To estimate the size of the charge perturbation caused by a propagating action potential, we use the equation charge = current x time. Substitute the values into the equation and perform the calculation. After obtaining the value, give a brief comment comparing it with the change in the sodium ion charges from step 2.

Key concepts

Neurotoxin Effect on Ion Pumps

Neurotoxins can have a profound effect on the function of nerve cells by targeting ion pumps, which are critical in maintaining the resting membrane potential. Ion pumps, like the sodium-potassium pump, actively transport ions across the neuronal membrane to maintain a high concentration of potassium ions inside the cell and a high concentration of sodium ions outside the cell.

When a neurotoxin disrupts these ion pumps, the carefully maintained ion gradients break down. Sodium ions (a^+a^+a^+)) start to rush into the cell due to both their concentration gradient and the negative electrical gradient inside the cell. This influx of sodium leads to depolarization, which, if unchecked, can result in cell damage or death. Such a disturbance can have dramatic effects on nerve function and can lead to symptoms ranging from muscle weakness to paralysis.

Ohm's Law in Biology

Ohm's Law is an important concept in biology, particularly in neurophysiology, as it helps us understand how electrical currents travel through neurons. In its basic form, it states that the current (a)) through a conductor between two points is directly proportional to the voltage (aV)) across the two points, and inversely proportional to the resistance (aR)) of the conductor: aa = afrac{V}{R}a.

Applied to a biological membrane, Ohm's Law allows us to calculate the ionic current passing through ion channels. The voltage in this case is the membrane potential, and the resistance is related to the conductance of the ion channels. This mathematical relationship becomes pivotal when examining the immediate effect after ion pumps are inactivated, as it provides a way to quantify the change in ionic current that contributes to changes in the membrane potential.

Sodium Ion Concentration Gradient

The sodium ion concentration gradient is key to both the resting membrane potential and the propagation of nerve impulses. Under normal conditions, there is a higher concentration of sodium ions (a{Na}^+a)) outside of a neuron than inside. This gradient is due to the action of the sodium-potassium pump, which moves three sodium ions out of the cell for every two potassium ions it brings in. The energy for this active transport comes from ATP.

At rest, the membrane is more permeable to potassium ions, which move out of the cell, leading to a negative membrane potential. When neurotoxins inhibit ion pumps, this gradient begins to diminish as sodium rushes into the cell. The change in the gradient alters the cell's membrane potential and can affect the cell's ability to fire an action potential, which is the basis of nerve impulse propagation.

Nerve Impulse Propagation

Nerve impulse propagation is the process by which neurons communicate with each other and with other types of cells. It involves the movement of ions across the neuronal membrane, resulting in a wave of depolarization called an action potential. This depolarization is a well-orchestrated sequence of events wherein sodium ions enter the cell, followed by the exit of potassium ions to repolarize the cell.

The speed with which this propagation occurs depends on a variety of factors including the axon diameter and the presence of myelin. Damage to ion pumps by neurotoxins can significantly alter the membrane potential, potentially preventing the nerve impulse from propagating efficiently. In such a case, the sodium influx might not stop at merely signaling another neuron but could lead to a prolonged inflow, disturbing the delicate balance necessary for normal nerve function.