Action Potential III: Sodium Current, Patch Clamp, and Ion Channels

Introduction

• In the last unit, we analyzed the capacitative current, and the leak and potassium currents and conductances underlying the action potential.
• In this unit, we will complete our analysis of the currents contributing to the action potential by analyzing the sodium current and the sodium conductance.
• We will then go further down to the molecular level of individual ion channels, and discuss their behavior as discovered using the technique of patch clamp.

Video on the Analysis of the Sodium Current

Analysis of the Sodium Current in the Squid Giant Axon

Analysis of the Sodium Current

Figure 1: Sodium currents in response to depolarizing voltage steps from rest to 0 mV, 50 mV, and 100 mV. All other currents subtracted.

Figure 2: Sodium conductances in response to the same depolarizing voltage steps as in Figure 1.

Figure 3: Sodium conductances in response to two voltage steps from rest to 0 mV, 10 ms apart.

Figure 4: Sodium conductances in response to two voltage steps from rest to 0 mV, 20 ms apart.

Figure 5: Sodium conductances in response to two voltage steps from rest to 0 mV, 40 ms apart.

Figure 6: The m and h gates, and the sodium conductance.

• To analyze the sodium currents, we can assume that the capacitative and leak currents have been subtracted away, and that the potassium current has been blocked (e.g., with the drug TEA).
• Figure 1 shows the currents we record as we step the membrane from the resting potential to increasingly positive potentials.
  • The bottom voltage step, from the resting potential (about -61 mV) to 0 mV, generates a large downward deflection (the bottom trace in the top panel).
    • As we mentioned in the last unit, inward current recorded under voltage clamp is represented using negative values.
    • Thus, the downward deflection represents a negative, inward current, which is depolarizing.
    • This makes good sense; if the membrane is stepped to a depolarizing potential, the resulting potential across the membrane will increase the probability that the voltage-dependent sodium channels will open and allow sodium ions to flow along their concentration gradient into the cell, bringing positive charges into the cell and depolarizing the neuron.
    • Unlike the potassium current that we studied in the previous unit, after some time, the sodium current collapses back to zero. This suggests that the channel inactivates.
  • The results of the middle voltage step, to +50 mV, also make sense after some thought. The effects of this step correspond to the middle trace in the top of the figure.
    • There is a small inward current, but the overall current is very close to zero.
    • What is the reason for this? Recall that, for the squid giant axon, the Nernst or equilibrium potential for sodium is approximately +55 mV.
    • Thus, stepping the membrane to +50 mV is almost moving it to the equilibrium potential for the sodium ions. The driving force, $V_{m} - E_{\text{Na}^+} = +50\ \text{mV} - +55\ \text{mV} = -5\ \text{mV}$, will be very small, and thus the overall sodium current will be greatly reduced.
  • This analysis helps to explain the effect of the third and largest voltage step, to +100 mV. The current resulting from this voltage step is the top trace of the top panel of the figure.
    • The membrane has been stepped beyond the equilibrium potential for sodium ions; as a consequence, the electrical gradient now strongly opposes and is greater than the concentration gradient of sodium, so that the net flux of sodium ions becomes outward. As a consequence, the sodium current reverses direction, and becomes positive. This is also clear from a consideration of the driving force, which switches sign: $$V_{m} - E_{\text{Na}^+} = +100\ \text{mV} - +55\ \text{mV} = +45\ \text{mV}$$.
• It would be worthwhile to compare these results to those obtained when analyzing the potassium currents, in the previous unit. Since the Nernst or equilibrium potential for potassium ions is -75 mV, and all of the voltage steps were made positive to the resting potential, the driving force in response to each of the steps was large and positive, generating a large positive, outward (hyperpolarizing) current.
  • If potassium ion currents were not blocked, but all other ionic currents were blocked, to what potential would the membrane have to be stepped to eliminate net flux of potassium ions? Please write down your answer, and check your answer here.
  • If potassium ion currents were not blocked, but all other ionic currents were blocked, and the membrane was stepped to -80 mV, what direction would the net flux of potassium ions follow (into or out of the nerve cell)? Please write down your answer, and check your answer here.
• Let's return to the consideration of the sodium current and conductance.
• Since we know the current and the voltage, we can solve for the sodium conductance using the equation $$g_{\text{Na}^+} = \frac{I_{\text{Na}^+}}{(V_{m} - E_{\text{Na}^+})}.$$
• Once this is done, the sodium conductances in response to the same voltage steps are shown in Figure 2.
  • Now that the driving force has been factored out, the sodium conductance steadily increases in magnitude as the voltage steps increase from 0 mV to 50 mV to 100 mV.
• Hodgkin and Huxley found that the rising phase could be fit using a gate that rapidly opened in response to depolarizing voltage, but that three such gates needed to become conducting for the ion channel to open and conduct current.
  • They designated this gate m, and the requirement that three gates open for current to flow implied that the probability of this occurring was $m \times m \times m = m^3$.
• What about the collapse of the sodium conductance, i.e., its inactivation?
  • To characterize this part of the sodium conductance, Hodgkin and Huxley first stepped the membrane to a depolarized potential (e.g., 0 mv), then, after variable periods of time, they again stepped the membrane to the same depolarized potential.
    • When they waited only 10 ms, the current (and thus the conductance) were greatly reduced, as shown in Figure 3.
    • When they waited 20 ms, the current in response to the second voltage pulse (and thus the conductance) were larger, though still reduced, as shown in Figure 4.
    • When they waited 40 ms, the current in response to the second voltage pulse (and thus the conductance) were almost as large as they had been initially, as shown in Figure 5.
  • These data allowed them to calculate the rate at which the inactivation gate, which they called h, changed in response to voltage.
  • Unlike either the n or m gates, which open in respond to depolarizing voltage (and whose probability of being open thus rises from 0 to 1 in response to depolarizing membrane potentials), they assumed that the h gate closed in response to depolarizing membrane potential, i.e., its probably of being open fell from 1 to 0 over time.
• Thus, their data suggested that the best fit for the sodium conductance would be based on the probability of three m gates being open, and the h gate also being opened, or $m^3h$.
• As shown in Figure 6, this yielded an overall sodium conductance that looked reasonably close to that measured from their data.
• You now have enough information to answer the following questions:
• Question 1. Please use the current clamp simulation of the action potential to answer this question. Run the simulation with the default values.
  • Explain the changes you see in the m, n and h gates, and in the sodium and potassium conductances. In particular, please focus on the values of the gates at the onset of the two depolarizing pulses (i.e., please measure the gates at $t=10\ \text{ms}$ and at $t=24\ \text{ms}$).
  • Now shorten the inter-stimulus interval to 5.6 ms. Again, focus on the values of the gates at the onset of the second depolarizing pulse (i.e., please measure the gates at $t=19.6\ \text{ms}$). Explain the changes you see in the m, n and h gates at the onset of the second pulse.
  • What happens if you lower the inter-stimulus interval to 5.5 ms? Again, focus on the values of the gates at the onset of the second depolarizing pulse (i.e., please measure the gates at $t=19.5\ \text{ms}$). Explain in terms of the gates and the conductances. In particular, focus on the maximum or minimum value of the m and h gates achieved before the second depolarizing pulse. Please explain the differences you observe.
• Question 2. Press the Reset button. Reduce the Number of pulses to 1, and reduce the Fast transient sodium conductance from 120 to 12. This is similar to applying a drug such as tetrodotoxin (TTX) which blocks sodium channels. Run the simulation.
  • What happens to the sodium gates (m and h)? What happens to the sodium conductance? What happens to the sodium current? What does this do to the action potential?
  • Now increase the Stimulus current first pulse to 25 nA. What do you observe? Is this an action potential?
  • If you use the same 25 nA current, but now change the sodium conductance to 0, what do you observe? Is this an action potential? Does this affect your answer to the previous question? Explain.

Ion Channel Currents and Macroscopic Currents

• Although the work of Hodgkin and Huxley was very influential, their hypothesis that ion channels were responsible for ion transport through the membrane was not definitively proven by their work.
• For several decades after their work, researchers attempted to find indirect evidence supporting the idea of ion channels, using analysis of noise fluctuations in the membrane potential, or studying the very tiny gating currents that should flow across the membrane, based on the Hodgkin and Huxley model of ion channels.
• Because of the difficulty of obtaining good space clamp, and because of the very small size of many mammalian and primate cells, much of the work on membrane biophysics continued to focus on large nerve cells, many found in molluscan invertebrates.

Figure 7: Schematic diagram of patch clamp.

• A major breakthrough occurred in the late 1970s and early 1980s with the invention of patch clamping by Neher and Sakmann. A schematic of the technique is shown in Figure 7.
  • A relatively blunt micro electrode whose tip has been made extremely smooth by fire polishing is lowered onto the cell of interest.
  • Suction is applied to the electrode, inducing the membrane of the nerve cell to adhere very tightly to the electrode.
  • The tight adherence causes a very high resistance seal (on the order of gigaohms) to form between the electrode and the membrane.
  • The very tight seal over a very small patch of membrane acts to greatly amplify any currents that may pass through single channels within the patch.
• There are a number of different ways patch clamp can be used.
  • The configuration shown in Figure 7 is referred to as the cell-attached configuration, and is ideal for studying an ion channel when one wishes the cytoplasmic side of the channel to remain exposed to all the chemicals within the cell. One can control solutions in the electrode, and thus modify the extracellular medium this way, but not the intracellular medium.
  • To control both the inside and outside solutions, one can pull the electrode back, tearing the small patch of membrane away from the cell, leaving the small patch attached to the tip of the electrode. This is referred to as the inside-out configuration, since the cytoplasmic side of the channel is now exposed to the larger solution.
  • If suction is increased so much that the patch of membrane is broken, the inside of the electrode and the cytoplasm of the cell become continuous, and this is referred to as the whole-cell configuration. For many small cells, this provides sufficient current through the large tip of the electrode that good space clamp can be maintained, and thus it becomes possible to voltage clamp the entire cell.
  • Finally, in the whole cell configuration, if the electrode is pulled gently away from the cell, the membrane adhering to the outside of the electrode will pinch off, and this forms the outside-out configuration, since the membrane facing the solution was originally on the outside of the nerve cell, and the membrane facing the electrode solution was on the inside of the nerve cell.
• Because patch clamping can be applied to all regions of a nerve cell (soma, axon and dendrites), regional distributions of ion channels can be determined. The ability to patch very small (7 micron) neurons in vertebrates, as well as large molluscan nerve cell bodies, has provided a vast amount of data about the biophysics of a much wider range of ion channels than those that were originally studied by Hodgkin and Huxley.
• What is remarkable is that, despite all of these advances, many of the inferences that Hodgkin and Huxley drew based on their studies of the squid giant axon turned out to be substantially correct.
  • For example, their model of how the potassium ion channel was gated by voltage appears to fairly accurately correspond to the underlying structure of potassium ion channels, whose genes have been identified, and whose structures have been fairly fully characterized. Four charged "gates" do have to translocate for the channel to conduct ions.

Figure 8: Gates, channel currents, and macroscopic currents.

• What is the relationship between the single channel currents and the overall current that we have been analyzing until now?
  • We can understand the relationship by looking at Figure 8, which shows the output of a model of the gates, individual channels, and populations of ion channels.
    • The top of the figure shows four traces in green; these are the configuration of the individual n gates in the potassium channel.
    • The large trace, below this, in blue, shows when the channel becomes conducting. Note that it is initially off, even though some of the gates are on, because all four must be on for the channel to conduct. Notice that it also stays on only part of the time, because as soon as one gate goes into the nonconducting state, the channel as a whole turns off.
    • The trace beneath this, the smooth green line, shows the predicted macroscopic potassium current that should be observed as the membrane is stepped to this particular potential. The blue line and the green line are radically different.
    • However, beneath this are 20 sweeps of the simulation averaged (the ragged blue line) superimposed on the green line. Although in twenty sweeps there are still some discrepancies, the average current is starting to approach the predicted current. This is the equivalent of summing up the current through 20 channels.
  • Thus, once again, like the formation of the Nernst or equilibrium potential, when we look at the molecular level, we see that processes are random and highly variable. However, when we sum these processes up over large populations, they begin to look much smoother and more predictable.

Patch Clamp Analysis of Ion Channels

• You now have enough information to answer several questions using a simulation of patch clamp to analyze ion channels. Once again, you may wish to open the simulation on a separate web page. As always, feel free to experiment a bit with the simulation before starting on the questions.

• Question 3: Reset the simulation to its defaults, and then set both the number of activation and inactivation gates to 0. This represents a "leak" channel. Set the channel reversal potential to -75 mV (e.g., equal to the Nernst or equilibrium potential for potassium ions in the squid giant axon) and channel conductance to 10 pS.
  • Recall the ion conductance equation derived in Unit II (Passive Membrane Properties); use this to write down an equation to predict the current based on the conductance, reversal potential, and membrane potential. Use this equation to predict the current during both a holding voltage of -100 mV and a subsequent step up to 0 mV. Please make sure to write down your predictions. Test your prediction using the simulation.
• Question 4: Change the number of activation gates to 1. While simple, there are several types of channels for which this single-gate model is a good fit (e.g., M current potassium channels).
  • Step from -100 mV to 0 mV. Run the simulation several times and observe the resulting currents and gates. Let $m_{1}$ be a variable that is 0 when the gate is closed, and 1 when the gate is open. Using $m_{1}$, rewrite your equation from Question 3 so that it takes into account the state of the gate. To help figure this out, what should the value of the current be if the gate is closed? If the gate is open so that it fully conducts, based on the answer to Question 3, what should the current be? Please make sure to write down the equation. How does this compare with the results in Question 3, especially when the channel is at the holding voltage of -100 mV? Use this to explain what happens if the gate closes in the middle of a trial.
• Question 5: Change the number of activation gates to 4. This four-gate model is similar to what Hodgkin and Huxley proposed for voltage-gated potassium channels.
  • As in the previous question, step from -100 mV to 0 mV. Run the simulation and observe the results. What has to be true of the gates for current to flow? Let $n_{1}$, $n_{2}$, $n_{3}$, $n_{4}$ each represent the state of one of the activation gates. Rewrite your equation from the previous question to handle the four activation gate case.
  • Now, instead of thinking about the current through a single channel, consider the average channel current in a large collection of channels. The variables $n_{1}$, $n_{2}$, $n_{3}$, $n_{4}$ now represent probabilities that can vary continuously between 0 and 1. How does the equation you wrote down a moment ago change if $n_{1} = n_{2} = n_{3} = n_{4}$? For simplicity, you can use the symbol $n$ to represent the probability of any one of the gates being opened or closed. Please write the modified form of the equation assuming this is true.
• Question 6: Reset the simulation to its defaults. Notice that there are now 3 activation gates and 1 inactivation gate. Set the Channel Reversal Potential to +55mV, so that the channel now resembles a sodium channel in the squid giant axon.
  • Using the equation from Question 3, the new channel reversal potential, and its open conductance of 10 pS, predict the maximum current through the gate when it is stepped to +100 mV. Please write down your prediction, and then check it using the simulation. Run the simulation a few times and study the results.
  • What is the relationship between the current through a single channel and the activation and inactivation gates? To answer this, let $m_{1}$, $m_{2}$, $m_{3}$ be the state of each of the activation gates, and $h$ the state of the inactivation gate (1 if open, 0 if closed). Write an equation to predict the current based on the membrane potential, reversal potential, channel conductance, and the state of the gates.
  • Now, think again about the average channel current in a large collection of channels, and treat $m_{1}$, $m_{2}$, $m_{3}$, $h$ as continuously varying probabilities. How does your equation change if $m_{1} = m_{2} = m_{3}$? For simplicity, you can use the symbol $m$ to represent the probability of any one of the activation gates being opened or closed. Please write the modified form of the equation assuming this is true.