I. Nernst Potential.

Ions move, both passively and actively, across cell membranes. In real cells many different ions are constantly flowing across the membrane driven by both diffusive and active (ATP-using) processes. To begin to understand the consequences of this movement, let’s first consider the dynamics of a single ion – potassium (K+).

Imagine that inside a cell is comprised of dissolved KCl (that is K⁺ and Cl^- ions) and that the membrane is permeable only to K⁺ions. Imagine that the cell is placed in water. Draw this scenario.

1. Describe the movement of the K⁺ ions and the Cl-^- ions.

2. Why would the net movement of K⁺ ions that you described in part a) ever stop? When the net movement does stop, would there be an equal number of K⁺ ions on both sides of the membrane? Why or why not?

3. Define the Nernst Potential in terms of the competing effects of chemical and electrical gradients.

II. Potential Across an Ion Channel

The flow of ions across membranes allows us to think about cells as analogous to electric circuits. Using our understanding of the Nernst potential we can predict how ions should flow across cell membranes.

Consider the ion channel shown at right, and suppose that the Nernst potential established by the channel is such that the outside of the membrane is at a higher electric potential than the inside.

1. In which direction would each of the following ions flow across such a channel? Na⁺, Cl^-, Ca²⁺

Now let’s represent the ion channel by the following circuit:

2. What does each of the circuit elements (the battery and the resistor) represent biologically?

3. Ion channels are sometimes described as having “variable resistance.” What does that mean? Biologically speaking, what would it mean for the channel resistance to change?

4. Let’s say that the circuit drawn represents a Cl-^- ion channel and the “top” of the battery (the long line) is outside the cell (so that the “bottom” of the battery, the short line, is inside the cell). Would the current in the circuit flow clockwise or counterclockwise? Would the Cl-^- ions flow into or out of the cell?

III. Multiple Channels: the Goldman Equation

Recall that because of the potential difference across the membrane, we can think of a cell membrane as analogous to a capacitor. To understand where this potential difference comes from, we can think about a cell membrane with multiple channels, each permeable to a different ion. We can represent such a membrane with a circuit like this:

The conductance gis the inverse of resistance (high resistance means low conductance and vice versa). The equilibrium potential difference associated with a particular ion is given as E_ion, as shown next to the batteries in the circuit. This E_ionis the potential difference across the cell membrane that would exist if the cell was only permeable to that particular ion, and if there were no current.

1. The resting potential difference (the potential difference for most cells, including nerve cells when they are not firing) across the cell membrane due to all the channels combined, V_m,is equivalent to the potential difference across the capacitor C_min the circuit. If the cell is only permeable to K⁺, what is the relationship between V_mand E_K+?

Now let’s think what the resting potential difference V_mwould be if the membrane is permeable to all three ions to different degrees. In other words: given the equilibrium potentials E_ionand conductances g_ionfor each ion, what is the overall resting membrane potential difference V_m? We could use the circuit above to figure it out(it’s good circuit practice… so give it a try if you have extra time!).

The result (after some renaming of variables to express conductances in terms of permeabilities, as is usually done in biology) is called the Goldman Equation:

2. Why does it make sense that the extracellular concentration of positive ions is on the top but the extracellular concentration of negative ions is on the bottom?

3. What is the role of the permeabilities in changing the value of the resting potential? What is the role of temperature?

4. Given this equation, how can the cell change its membrane potential? (E.g. during an action potential the cell membrane becomes depolarized and then hypoerpolarized.) What is one way that the cell could make this happen?