Equilibrium Potentials II

Electrochemistry of the Nerve Cell

Key ideas for this unit:

• Balance of equilibrium potentials (the Donnan equilibrium)
• An important steady state: the resting potential of the nerve cell

Balancing Multiple Equilibrium Potentials: The Donnan Equilibrium

• Membranes are usually permeable to more than one ion.
• Is it still possible to satisfy the requirements for equilibrium?
• Yes, if the three rules - osmotic balance, bulk electroneutrality, and the balance of the concentration and electrical gradients - are all satisfied.
• Let's look at another problem to explore this more fully.

Figure 7: Solving the unknowns can generate a cell that is fully in equilibrium

• Note that in this problem, the membrane is permeable to chloride ($\text{Cl}^-$) and potassium ($\text{K}^+$) ions, as well as to water; it is impermeable to sodium ($\text{Na}^+$) ions (located on the outside of the cell, i.e., the right side of Figure 7) and to the intracellular proteins (located on the inside of the cell, i.e., the left side of Figure 7).
• Here is how to analyze the problem:
  • We have many unknowns on the left side, and many fewer on the right side, so we should probably focus on the right side first.
  • Recalling the requirement for bulk electroneutrality on the outside (the right side), calculate the concentration of chloride ions on the outside.
  • Given that information, it is possible to calculate the total number of particles on the outside (the right side).
  • On the left hand side, we have one equation for the total number of particles inside, which has to be equal to the concentration of particles on the outside; write down this equation.
  • We have a second equation that we can write down based on the requirement for bulk electroneutrality on the inside of the cell (the left side).
  • We have three unknowns, and only two equations so far. But we can get a third equation by the requirement of the balance of the concentration and electrical gradients.
  • Earlier, we defined the Nernst or equilibrium potential equation for chloride ions (Equation 5). We can also readily define the Nernst or equilibrium potential equation for potassium ions, recalling that the charge on the ion, $z$, is $+1$. So the form of the equation for potassium ions is $$E_{\text{K}^+} = 58\ \text{mV} \times \log_{10} \frac{\left[\text{K}^+\right]_\text{out}}{\left[\text{K}^+\right]_\text{in}}. \quad\text{(Equation 6)}$$
  • Now, if the membrane potential is in equilibrium, the equilibrium potential for the chloride ions and the equilibrium potential for the potassium ions must be equal to one another, since each much equal the membrane potential at equilibrium ($E_{\text{Cl}^-} = E_{\text{K}^+} = V_m$). We state this mathematically by setting Equation 5 equal to Equation 6: $$58\ \text{mV} \times \log_{10} \frac{\left[\text{Cl}^-\right]_\text{in}}{\left[\text{Cl}^-\right]_\text{out}} = 58\ \text{mV} \times \log_{10} \frac{\left[\text{K}^+\right]_\text{out}}{\left[\text{K}^+\right]_\text{in}}.$$
  • We can divide both sides of the equation by 58 mV, and raise both sides to the power of 10; the result will be $$\frac{\left[\text{Cl}^-\right]_\text{in}}{\left[\text{Cl}^-\right]_\text{out}} = \frac{\left[\text{K}^+\right]_\text{out}}{\left[\text{K}^+\right]_\text{in}}.$$
  • This can be re-written as $$\left[\text{Cl}^-\right]_\text{in}\ \left[\text{K}^+\right]_\text{in} = \left[\text{K}^+\right]_\text{out}\ \left[\text{Cl}^-\right]_\text{out}. \quad\text{(Equation 7)}$$
  • You now have three equations for the three unknowns. Please solve for the specific values given in Figure 7, including the value of the membrane potential.
• You can check your answers, after having done the calculation yourself, by clicking here. Please include your work and the results of your calculation in your notebook.
• Notice that neither the chloride nor potassium ions are distributed evenly between the two compartments at equilibrium.
  • This occurred because of the presence of charges impermeable to the membrane, the sodium ions.
  • This type of equilibrium is called the Donnan equilibrium.

Solving an Equilibrium Problem

Solving an equilibrium problem

The Resting Potential of the Nerve Cell

We need to take only one more step to get to the resting potential.

• We've seen that the membrane can be permeable to multiple ions, and that it can nevertheless be in equilibrium, as long as the three rules stated above are obeyed (osmotic balance, electroneutrality, and a balance of the concentration and electrical gradients).
• What happens if the membrane is permeable to multiple ions, but the equilibrium or Nernst potentials of the ions to which it is permeable are very different?
• To explore this question, let's solve a still more complicated problem as an equilibrium problem, and then explore what happens if one additional ion becomes permeable whose equilibrium or Nernst potential is far from the equilibrium or Nernst potentials of the other ions that previously permeated the membrane.
• Part of the reason for adding more complexity with each of the problems we have examined, if you haven't guessed this already yourself, is to make the model cells more and more similar to real nerve cells, which have all of these complexities, and more.
• Let us solve the following problem, assuming that the membrane is in equilibrium:

Figure 8: Solving the unknowns can generate a cell that is fully in equilibrium until sodium ions become permeable

• You may want to try doing this yourself without reading any further; see if you can set it up and solve it yourself.
• Here's a way to approach this problem:
  • Since there is only one unknown on the right-hand side, it would probably be worth focusing on that side first.
  • Applying the rule of bulk electroneutrality makes it possible to solve for the concentration of chloride ions on the outside (the right-hand side).
  • Now you can add up the total concentration of particles on the outside, and this serves to constrain the concentration on the inside (the left-hand side).
  • In fact, you can write down an equation for the inside of the cell based on osmotic balance; you should do so.
  • You can write a second equation for the inside of the cell based on bulk electroneutrality. The one new wrinkle is that there are charged impermeable particles (designated as A), and they do not have a simple integral charge; they represent charged proteins within the cell. You will need to take this into account when you write the equation for electroneutrality.
  • Finally, the figure indicates that only the potassium and chloride ions are permeable to the membrane, at least initially. You can then use Equation 7 developed in the previous section to write down a third equation for the system.
  • Solve for the different concentrations, and for the equilibrium or Nernst potential across the membrane.
• To check your work, you can click here. Please include your work and the results of your calculation in your notebook.
• Now, what would happen if the membrane suddenly also became permeable to sodium ions?
  • How could that happen? As we will see a bit later in the course, ion channels can be gated, that is, their ability to permeate ions can be affected by other factors, such as the voltage across the membrane, or the presence of a chemical that binds to a part of the channel. In both of these situations, if the channel's configuration changes, it may now be able to permeate ions, when previously it was unable to do so.
  • The first step would be to calculate the equilibrium or Nernst potential for the sodium ions, based on the concentrations of sodium ions inside and outside of the cell. Given that sodium ions have a charge of $+1$, the relevant version of the Nernst equation to use is $$E_{\text{Na}^+} = 58\ \text{mV} \times \log_{10} \frac{\left[\text{Na}^+\right]_\text{out}}{\left[\text{Na}^+\right]_\text{in}}. \quad\text{(Equation 8)}$$
  • Use the values of internal and external sodium ions that you computed to find the equilibrium conditions for the cell shown in Figure 8.
  • Is this value similar to the equilibrium membrane potential that you computed for the membrane? To check your work, you can click here. Please include your work and the results of your calculation in your notebook.
• We now have a new situation: there is a "tug of war" between the equilibrium potentials, two of which are trying to move the membrane potential to a very negative value, and one which is trying to move it to a very positive potential.
  • What is the final resulting potential of this "tug of war"? It is the resting potential, which is a steady state, not an equilibrium, because it is maintained by net flows of different ions across the membrane.
  • It is possible to derive an equation to describe the resulting potential, using the following three assumptions:

(1) The flux of one ion across the membrane does not affect the flux of any other ion that may also be crossing the membrane;

(2) The membrane is in steady state, so that the flux is independent of location, and

(3) The electric field across the membrane is constant, and the partitioning of ions inside the cell and just outside the membrane is equal to the partitioning of ions outside the cell and on the inside part of the membrane.

• • We will not show the derivation, but the result is known as the constant field equation, or the Goldman-Hodgkin-Katz (GHK) equation, assuming that the major ions to which the membrane is permeable are sodium, potassium and chloride: $$V_{m} = \frac{RT}{\mathcal{F}} \ln \frac{p_{\text{Na}^+} \left[\text{Na}^+\right]_\text{out} + p_{\text{K}^+} \left[\text{K}^+\right]_\text{out} + p_{\text{Cl}^-} \left[\text{Cl}^-\right]_\text{in}} {p_{\text{Na}^+} \left[\text{Na}^+\right]_\text{in} + p_{\text{K}^+} \left[\text{K}^+\right]_\text{in} + p_{\text{Cl}^-} \left[\text{Cl}^-\right]_\text{out}}. \quad\text{(Equation 9)}$$
    • Note that the permeabilities of the channels in the membrane to sodium, potassium or chloride ions are represented by $p_{\text{Na}^+}$, $\ p_{\text{K}^+}$, and $\ p_{\text{Cl}^-}$, respectively.
    • Note also that if the permeabilities to any two ions are set to zero, the equation will reduce to the Nernst potential equation for the third ion.
    • Once again, at about $19^\circ\ \text{C}$ (a few degrees below room temperature), the natural logarithm can be converted to a logarithm in base 10, and the initial constants can all be simplified to $58\ \text{mV}$. Note that the valence, $z$, is applied to each ratio independently, which is why the inner and outer chloride concentrations are reversed in the numerator and denominator relative to those of the sodium and potassium ions.
• Using the values computed for the model cell shown in Figure 8, estimate the resting potential of the model cell. Assume that the nerve cell is equally permeable to sodium, potassium, and chloride ions, i.e., $p_{\text{Na}^+} = p_{\text{K}^+} = p_{\text{Cl}^-} = 1$.
• Check your answer by clicking here. Please include your work and the results of your calculation in your notebook.
• Using the constant field or Goldman-Hodgkin-Katz equation, if you are given the concentrations of ions on either side of the membrane and their relative permeabilities, you can predict the resulting steady-state resting potential of a nerve cell.

From Equilibrium to Steady State: How the Resting Potential Differs from an Equilibrium Potential

A first step towards understanding the difference between an equilibrium potential and the resting potential

A Realistic Resting Potential: Different Permeabilities and Ion Concentrations, and the Need for an Ion Pump

A more realistic analysis of the resting potential and its underlying fluxes

The Effect of Current Injection on the Fluxes Underlying the Resting Potential

How the underlying ionic fluxes are changed by current injection

A Qualitative Description of the Constant Field (Goldman Hodgkin Katz) Equation

From the individual Nernst Potentials to the Resting Potential in terms of concentrations and ion permeabilities

Problem 1: The Nernst Potential and the Constant Field Equation

You now have enough information to work on Problem Set 2, Problem 1. Please do so now.

Problem 2: Permeabilities and the Membrane Potential

You now have enough information to work on Problem Set 2, Problem 2. Please do so now.

Ion Pumps Maintain or Even Alter the Resting Potential

• Since the resting potential involves unbalanced flows of ions across the membrane—sodium ions into the nerve cell and potassium ions out—the differences in concentration of these ions will run down over long times.
• To prevent this, nerve cells invest a great deal of energy in ion pumps that can move sodium ions out of the cell and potassium ions in. Generally, the pumps consist of proteins embedded in nerve cell membranes that have sites capable of binding to ions. They use the energy currency of the body—adenosine tri-phosphate (ATP)—to activate themselves and transfer ions bound on the outside to the inside of the cell, and vice versa. For example, there is a pump called sodium-potassium ATPase that breaks down ATP so it can transfer sodium ions out of the cell and potassium ions into the cell.
• Thus, because the resting potential requires a continuous input of energy to maintain the concentration gradients that drive it, the resting potential is a steady state, much like the temperature in an air-conditioned room.
• In large volume nerve cells, the number of ions that flow into or out of the cell to maintain either the resting potential or (as we will learn shortly) the action potential are very small relative to the very large number of ions in the bulk solution, so many cells can continue to generate action potentials for many hours even after the pump has been deprived of energy or blocked.
• One compound that can block ion pumps is ouabain.
• For some kinds of pumps, the number of ions brought into the cell is not equal to the number of ions that the pump removes from the cell. For example, the sodium-potassium ATPase binds three intracellular sodium ions and two extracellular potassium ions; as a consequence, it removes more positive charge from inside the cell than it brings in. Effectively, this is like injecting a small amount of negative charge into the cell. Such a pump is electrogenic, that is, it makes the cell's resting potential slightly more negative than it would be if it were not operating.
• As we explained above, any treatment that induces the cell's potential to become more negative is hyperpolarization, and any treatment that induces the cell's potential to become less negative is depolarization.
• Thus, another way to describe the sodium-potassium ATPase is to say that it directly contributes to the resting potential by making it slightly more hyperpolarized than it would be if the pump were not operating. Stopping the pump will lead to a slight, slow depolarization (though this is usually not enough to cause the cell to fire action potentials).

An Electrogenic Ion Pump and the Resting Potential

How an electrogenic ion pump can affect the resting potential, and what happens if the pump is poisoned

A Real Nerve Cell: The Squid Giant Axon

To build up the complexity of the actual nerve cell in steps, we created a series of increasingly realistic model cells and analyzed them carefully. What about a real nerve cell?

• One particular nerve cell that has been extensively studied is that of the squid. The animal uses the nerve to generate very fast contractions of its mantle so that it can propel water in one direction; this rapidly propels the animal in the opposite direction, out of harm's way.
• As we will learn in a later unit, one way to speed up the propagation of electrical signals through a nerve is for it to have a large diameter, and the squid giant axon has a very large diameter. This makes it an ideal preparation to study, and we will learn more about it when we discuss action potentials.
• The concentrations of ions within the squid giant axon, and in the medium that usually bathes it in vivo, have been measured. We can use our principles to analyze these results and see if they are consistent with the rules we have stated.
  • Inside the squid giant axon, the potassium ion concentration is 400 mM, the sodium ion concentration is 50 mM, the chloride concentration is 52 mM, and the concentration of impermeable negatively-charged particles is 385 mM.
  • Outside of the squid giant axon, in the medium that ordinarily bathes it, the potassium ion concentration is 20 mM, the sodium ion concentration is 440 mM, the chloride concentration is 560 mM.
  • Here's a chance to use what you have learned so far:
    • Is the squid giant axon in osmotic balance? If not, would it shrink or swell? Please do the calculation, and check your answer here. Please include your work and the results of your calculation in your notebook.
    • Do the particles on either side of the squid giant axon satisfy electroneutrality? Please do the calculation, and check your answer here. Please include your work and the results of your calculation in your notebook.
    • Finally, what are the Nernst or equilibrium potentials of the three permeable ions (sodium, potassium, and chloride)? Based on calculating these potentials, is the squid giant axon in equilibrium, or in a steady state? Please do the calculation, and check your answer here. Please include your work and the results of your calculation in your notebook.
    • What happens if you perturb the membrane potential, say by injecting a steady hyper polarizing current into the axon? If you do, you will move the membrane closer to the equilibrium potential of the potassium ions, and so fewer will flow out of the neuron; at the same time, you will move the membrane further from the equilibrium potential of the sodium ions, and so more will flow into the neuron. As a consequence, if you stop injecting the hyper polarizing current, the greater influx of sodium ion and reduced efflux of potassium ions will cause the membrane to depolarize back to the potential it had originally; so the resting potential is a stable potential, even if it is created by a steady state of ion fluxes. In general, a stable potential is one to which the neuron will return if it is slightly perturbed away from that potential.
• Controlling the resting potential controls the overall excitability of a nerve cell, and, as we will see, many inputs from other neurons, or even from within the neuron, can alter the resting potential, and change the pattern of activity of the nerve cell.

Problem 3: Using the Constant Field Equation

You now have enough information to work on Problem Set 2, Problem 3. Please do so now.