The GHK equation, equilibrium potentials, and membrane potentials

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Table of Contents

• Introduction
• Lipid bilayers act as barriers to ions
• Ion channels allow ions to move across lipid bilayers
• Equilibrium potentials: concepts
• Equilibrium potentials: the Nernst equation and application
• Other factors: Na⁺ channels and the GHK equation
• Other factors: Na⁺/K⁺ pumps

Introduction

The electrophysiology of membranes is a fascinating subject and underlies our understanding of just about everything in neuroscience. Unfortunately, it is also one of the subjects in physiology and neuroscience that students struggle with mightily. Rather than overwhelm students with too much detailed information, here we try to focus on the bare fundamentals so that students can develop a knowledge base to understand more complex concepts if they so choose in future studies.

The GHK Simulator is meant to simulate a membrane patch recording rig connected to a chart recorder – this is a device that on one end has a very sharp glass microelectrode that is impaled onto the membrane of a cell and reads voltage across the membrane, and on the other end is attached to a machine that has a pen that moves up and down (the y-axis) as the microelectrode reads different voltages. The machine also has a roll of paper that scrolls continuously so that the length of the paper (the x-axis) represents time.

One way to use this website is to open the simulator in a separate window or tab, and play with it whilst browsing the content on this page. If you are more technically inclined or are simply curious and want to know more, you may wish to install Python and run the Python version of the GHK simulator on your personal computer – it’s prettier with more colors and more responsive. Also, it looks more like an oscilloscope (which doesn’t do anything functionally but it looks more cool).

For some background information, read on. This reading is not meant to be comprehensive – students should read their assigned texts first. The information below is meant to be condensed. We assume familiarity with basic cell biology and chemistry.

You of course can use the simulator in any way you choose; suggested activities are in orange boxes below.

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Lipid bilayers act as barriers to ions

Cell membranes are primarily made from two kinds of biological macromolecules: phospholipids and proteins. We first focus on phospholipids.

Phosopholipids form bilayers in an aqueous environment. Because the inner layer of the bilayer is strongly hydrophobic, the bilayer serves as an effective barrier to charged particles such as ions (e.g., Na⁺, K⁺, Cl^–, etc.). All cells exist in a “salty” aqueous environment – the intracellular cytoplasm, the extracellular fluids (such as lymph or blood), and even the environment where the organism lives are often liquid water that contains various salts. For our discussion, we focus on NaCl and KCl.

In water, these salts dissociate into ions. For instance, NaCl dissociates into Na⁺ and Cl^–, KCl dissociates into K⁺ and Cl^–, and so forth. Ions, of course, carry an electrical charge. In the electrical systems of household appliances and electronic devices, electrons flow through wires to carry information or to do work; in cells, ions flow across membranes to carry information or to do work.

While phospholipid bilayers are barriers to ions, water can pass through phospholipid bilayers via osmosis (effectively diffusion of water). Osmosis is physiologically important but we generally ignore the effects of osmosis when learning about electrophysiology. This is because: (1) osmosis is a lot slower than the kinds of electrical changes that occur in membranes; and (2) there are other cellular mechanisms that maintain osmotic balance that are beyond the scope of this discussion.

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Ion channels allow ions to move across lipid bilayers

Next, let’s discuss some of the proteins that are relevant to electrophysiology. Ion channels are a type of transmembrane protein that allow certain types of ions to passively diffuse across a lipid bilayer. Think of them as tiny little tunnels that span the thickness of the membrane. Different kinds of ion channels are generally selective for the kinds of ions they let through – Na⁺ channels only let Na⁺ ions through, K⁺ channels only let K⁺ ions, through, etc.

Most ion channels are gated – this means that the channel pore that spans the lipid bilayer is usually closed or inaccessible to ions unless the channel detects some kind of stimulus. The stimulus that opens up different channels depends on the particular channel, and is discussed further below. Some ion channels are leaky. In some cases, an ion channel is gated by something but some amount of ions can leak through. In other cases, an ion channel is intentionally leaky and basically does nothing but leak as its primary function. For the purposes of discussing the GHK equation, we care about one particular type of leaky channel called the K⁺ (read: potassium) leak channel. This channel is not gated and always leaks K⁺.

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Equilibrium potentials: concepts

Molecules will always diffuse so that they will spread out evenly in space. Another way to look at this is that molecules will diffuse from an area of high concentration to an area of low concentration (we sometimes say this as “molecules diffuse down their concentration gradient”). This is a passive process and does not require any external input of energy (in fact, the gradient itself is the energy source). This is true for ions in solution as well. If I dissolve KCl crystals into pure water in a beaker, the crystals will initially sink to the bottom of my beaker. As the ions dissociate into the water, the concentration of K⁺ (and Cl^–) will initially be higher at the bottom of the beaker. But over time, they will spread out evenly throughout the beaker. You can speed this up by stirring the water.

There are other forces that can cause molecules to move around besides concentration gradients. With regards to ions, one other important motive force is electrical charge. A positively-charged cation (such as K⁺) will be attracted to negative charges, and a negatively-charged anion (such as Cl^–) will be attracted to positive charges.

Let’s imagine a vertical cell membrane where the concentration of KCl to the left of the membrane is 100 mM. Let’s also imagine that “left” here represents the cytoplasm, or the “inside” of the cell. We can write this as [KCl]_in = 100 mM. Similarly, let’s imagine that the right of the membrane represents the extracellular space (the “outside” of the cell) and has a KCl concentration of [KCl]_out = 5 mM. These concentrations are actually fairly similar to real-life animal cells. If this membrane contains no ion channels at all, then there is potential for KCl (or more precisely, there is potential for K⁺ and Cl^–) to move down its concentration gradient (from left to right) – but it doesn’t, because the lipid bilayer is a barrier to these ions and there are no ion channels for them to move through.

Now, let’s add K⁺ leak channels to this membrane. There is now a pathway for K⁺ ions (but not Cl^– ions) to move down its concentration gradient (from left to right). At what point does K⁺ movement equilibrate? At a first glance, you might imagine that it is when [K⁺]_out = [K⁺]_in. But each K⁺ ion that moves from left to right creates a net -1 electrical charge on the left side caused by a surplus of Cl^– ions remaining on the left side. This -1 electrical charge attracts K⁺ ions back towards the left side. The force generated by the negative electrical charge will actually attract all cations, but in our example K⁺ is the only cation in the solution.

Initially the rightward force generated by the K⁺ concentration gradient is much stronger than the leftward electrical force. But as more and more K⁺ ions move right, the negative electrical charge becomes larger and larger and the leftward force acting on K⁺ becomes stronger and stronger. At some point, these two forces reach equilibrium. When they do, we find that there has been an overall net movement of K⁺ from left to right. This means that at equilibrium, there will be a net difference in charge between the left and right sides of the membrane; the left side will be more negative than the right side (you can also say it the other way around: that the right side is more positive than the left). This kind of equilibrium is called a Nernst equilibrium. When a system reaches a Nernst equilibrium, the difference in charge across the membrane is usually expressed as a voltage (usually in millivolts). We call the voltage across a membrane a “membrane potential“, and we call the membrane potential at a Nernst equilibrium the “equilibrium potential“.

Important concept: the amount of net K⁺ movement needed to establish an equilibrium potential is actually very small! This means that we generally consider [K⁺]_out and [K⁺]_in to be changed so infinitesimally that for practical purposes they are unchanged. In other words, in our example after an equilibrium is reached, [K⁺]_out is very close to 5 mM and [K⁺]_in is very close to 100 mM. It doesn’t take a lot of K⁺ movement to generate a membrane potential in the tens of millivolts range.

Another important concept: measurable Nernst equilibriums don’t exist in live cells! That’s because there are all kinds of cations and anions in a live cell, whereas a Nernst equilibrium (which deals with single cation/anion pairs, such as K⁺ and Cl^–) only occurs either in a laboratory setting (the KCl in a beaker example above) or as a hypothetical “thought experiment”. See the section below for the practical reason we care about equilibrium potentials.

Use the GHK simulator to change the values of [K⁺]_in and [K⁺]_out (while keeping everything else the same) to see the effect it has on E_K. If you touch other controls by acceident you can click the “Reset” button to reset the simulator to default values.

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Equilibrium potentials: the Nernst equation and application

In a real cell, K⁺ leak channels are a real thing, and K⁺ ions attempting to reach their Nernst equilibrium indeed contributes a great deal to the resting membrane potential of a typical cell. However, there are multiple types of anions and cations present on either side of the membrane, and they all collectively affect the voltage across the membrane. Therefore, we don’t usually think of the equilibrium potential of any given ion (such as K⁺) as something that physically exists or as something that can be measured. Rather, we think of it as something we can calculate if we know the values of [K⁺]_out and [K⁺]_in, and interpret the equilibrium potential as “where K⁺ movement wants to bring the membrane potential to“. The equilibrium potential E_X for ion X can be calculated using the Nernst equation:

• R is the gas constant: R = 8.314 joule/K/mol
• T is temperature (in Kelvin)
• z is the charge of the ion (K⁺ = +1, Cl^– = -1, etc.)
• F is Faraday’s constant: F = 9.6485 x 10⁴ coloumb/mol (or joule/volt/mol)

Here is an example. Let’s say that you use a microelectrode and chart recorder to measure the membrane potential (V_m) of a resting cell, and come up with the value V_m = -62 mV (the membrane potential of a resting cell is also called the resting potential). Using analytical chemistry, you measure the ionic concentrations of K⁺ inside and outside the cell as [K⁺]_in = 100 mM and [K]_out = 5 mM. Using the Nernst equation, and assuming the temperature is 37 ^oC, you can calculate the equilibrium potential for K⁺ as E_K = -80 mV.

We interpret this as follows: given these K⁺ concentrations, K⁺ is contributing (i.e., “pushing” it towards -80 mV) to the membrane potential of the cell but other (yet to be determined) factors are causing the membrane potential to be more positive than E_K. Furthermore, given the calculated value of E_K, there is a relatively weak driving force (defined as the difference between V_m and E_K; V_K = V_m – E_K = -62 -(-80) = +18 mV) on K⁺ ions such that K+ ions are being pushed outwards, but other (yet to be determined) factors are maintaining the status quo of K⁺ ionic concentrations. (Positive values for driving force mean outward force, and negative values mean inward force.)

Use the GHK simulator to change the values of [K⁺]_in and [K⁺]_out to see the effect it has on V_m.

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Other factors: Na⁺ channels and the GHK equation

So what are these yet-to-be-determined other factors? Let’s first consider Na⁺ channels. After K⁺ leak channels, Na⁺ channels make the next-most contribution to the membrane potential of typical resting cells. Here we discuss Na⁺ channels in a collective sense, even though there are lots of different kinds of Na⁺ channels with different properties. Unlike K⁺ leak channels, Na⁺ channels are collectively gated but leaky. What they are gated by is not very important for our current discussion (although this is briefly discussed below), as long you understand that they are gated – that is, the channel pore is closed unless some specific gating stimulus is detected by the channel protein.

Use the GHK simulator to change the values of [Na⁺]_in and [Na⁺]_out to see the effect it has on E_Na and V_m. Do you notice anything different about how changing Na⁺ concentrations changes V_m vs. how changing K⁺ concentrations changing V_m? Which ion has a greater influence on V_m?

The “influence” of different channels on membrane potential can come in two different ways: there might be different amounts of a particular channel present, or a channel can be more (or less) permeable to ions. For instance, a leak channel will be very permeable, whereas a gated channel that is just a little leaky will be less permeable. We use an aggregate measure called relative permeability (P_X, where X is some ion) to account for both these reasons. P_X is a unit-less value and it also a relative value. This means that considering P_K has no meaning unless you consider it relative to P_Na (another way to say this is that the absolute values of P_K and P_Na are meaningless, but the ratio P_K/P_Na is what’s important; if you think about this a little you will see this is true from the GHK equation). Using relative permeability, we can write an equation that takes into account multiple ions to calculate and/or predict the membrane potential V_m. This is the famous Goldman-Hodgkin-Katz (GHK) equation:

You can also add in other ions to the equation, such as Cl^–, Mg²⁺, Ca²⁺, etc. For now to keep things simple we only consider Na⁺ and K⁺, which are the most important. Note how mathematically similar it is to the Nernst equation above. However, there are some key differences:

• The two equations calculate different things. The Nernst equation calculates the equilibrium potential of one type of ion; the GHK equation calculates the membrane potential, taking into account the concentrations of ions you include in the equation (in our example, Na⁺ and K⁺).
• The two equations as written here have a slight difference in the coefficient in front of the log function. In the Nernst equation there is the term z, which is the charge of the ion in question. In the version of the GHK equation above, both Na⁺ and K⁺ have a charge of +1 so we omit z.
• We don’t include P_X in the Nernst equation. P_X is a relative value, so when considering just one ion, it makes no sense to include it (and even if we did, it would just cancel itself out from the numerator and denominator).

The default values of the GHK simulator are those of a typical cell at rest. From playing with the GHK simulator you can see that based on default values, K⁺ has a much greater influence than Na⁺ at changing the membrane potential, even though the driving force for Na⁺ is much stronger than that for K⁺ (V_Na = V_m – E_Na = -62 – 61 = 123 mV; the positive value tells us it is an inward force). This is because P_K is much greater than P_Na. Biologically, we can interpret this to mean that either: (1) There are far more K⁺ channels than there are Na⁺ channels; and/or (2) K⁺ channels are more permeable than Na⁺ channels. Whether (1) is true or not depends on what kind of cell you are looking at. In most neurons there are Na⁺ leak channels (i.e., Na⁺ channels whose purpose is to leak) that help regulate the resting membrane potential and are present at low levels relative to K⁺ leak channels. But (2) is almost universally true. As mentioned above, this is because most Na⁺ channels are gated. Thus, at rest K⁺ has a large influence on the membrane potential while Na⁺ has only a small influence.

Neurons (you’re here because you’re interested in neurobiology, right?) usually become “activated” when they detect an appropriate stimulus. The gating mechanisms of Na⁺ channels usually open up in response to that stimulus. The activating stimulus depends on the type of neuron and specific Na⁺ channel, but can include:

• changes in membrane potential: the Na⁺ channels that are activated in response to changes in membrane potential are called voltage-gated Na⁺ channels
• neurotransmitters released by other neurons: the Na⁺ channels that are activated in response to changes in membrane potential are called ligand-gated Na⁺ channels
• changes in physical force exerted upon the neuron: the Na⁺ channels that are activated in response to changes in membrane potential are called mechanically-gated Na⁺ channels

You will probably learn about these kinds of channels in your cell biology or neurobiology courses.

Use the GHK Simulator to emulate the effect of gated Na⁺ channels opening and closing by moving the slider for P_Na left and right. What effect does this have on the membrane potential?

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Other factors: Na⁺/K⁺ pumps

If there are K⁺ leak channels and Na⁺ leak channels on the cell membrane, then over time you would expect that more and more K⁺ would leak out of the cell and more and more Na⁺ would leak into the cell. Yet, in resting live cells ionic concentrations are very stable. The reason is not because the concentrations are at an equilibrium. Rather, the stable ionic concentrations of a resting cell are the result of a steady state that is maintained by a very important protein, the Na⁺/K⁺ pump.

This transmembrane protein is known by several other names: the Na⁺/K⁺ exchanger (or counter-exchanger), the Na⁺/K⁺ transporter (or antiporter), and/or the Na⁺/K⁺ ATPase. It uses the energy derived from ATP hydrolysis to pump Na⁺ out of the cell while pumping K⁺ into the cell at the same time. You will notice that the default Na⁺ and K⁺ concentrations inside and outside of the cell ([Na⁺]_out = 150 mM, [Na⁺]_in = 15 mM, [K⁺]_out = 5 mM, [K⁺]_in = 100 mM) are congruent with the direction of pumping of this protein. One convenient way of thinking of the purpose and function of this protein is this: Na⁺ is used to “activate” cells via gated Na⁺ channels as described above. However, Na+ leaks into the cell even when it’s not desired. Therefore, this protein is constantly pumping out Na⁺ as an act of cellular “maintenance”. In many ways, it acts like a bilge pump on an oceangoing ship to keep out seawater that splashes aboard from waves.

Some final trivia that you may or may not find interesting. Most textbooks will tell you that the the Na⁺/K⁺ pump transports 3 Na⁺ ions outward for every 2 K⁺ ions it transports inwards; this is true but also not super relevant in terms of understanding its basic purpose of keeping Na⁺ out of the cell. What I believe is a more interesting piece of trivia is that an estimated 20%-40% of the brain’s energy consumption is thought to be dedicated to running this pump. Your brain is figuratively speaking a leaky ship that is constantly bailing out “water”, except that water is actually sodium ions.

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