As you know from our discussions on membrane potential in living cells, our cells rely on the creation of an ionic gradient for the membrane potential. If you'll remember, an ionic gradient is basically two gradients combined - the chemical and electrical gradients "combine" together to make the ionic gradient. As first year medical students who haven't fully experienced physiology yet (after all, you've only been here a couple of weeks!) you may not realize this, but no self-respecting physiologist is happy unless s/he can apply an equation to something (some say this is the result of deep-seated ... oh never mind...). Luckily for us, Nernst derived an equation that allows us to determine at what point the two forces (chemical and electrical gradients) balance each other - in other words, at what point we have an ionic equilibrium. In the next paragraph we'll talk about the two equations (one is an easier derivative that works for us in biology) that Nernst derived and then we'll talk about what it means.

 

The Nernst equation

As originally described by Nernst, the equilbrium potential for any ion is:

E =
RT

ln
[X] extracellular
zF
[X] intracellular

 

where:
E = The Nernst Equilibrium Potential
R= Ideal Gas Constant
T = Temperature (in Kelvin)
z = the charge of the ion (valence)
F = Faraday's number
ln = natural log (based on e)
[x] = concentration of the ion
 

 

Luckily for us in biology, many of the variables are constants (e.g. Temperature is assumed to be body temperature and we will assume that to be 37 C) AND we can substitute the log (base 10) for the natural log (e) - so this equation collapses to something far less intimidtating:

E =
z(61.5) x
log
[X] extracellular
[X] intracellular
 

Yes, you'll still need a calculator, but at least you know where the buttons to push are!

 

Nernst made certain assumptions that one must be aware of when you use this equation. Those assumption are:

  • This equation can only be solved for one ion at a time (e.g. either sodium or potassium, but not both)
  • The membrane must be completely permeable to that ion.
  • The ion must be at equilbrium

Basically, unless these three conditions are met, the membrane potential will NOT be the same as the Nernst equilibrium potential. Note that the fact that not all of these conditions are met for most ions most of the time doesn't stop us from using the equation - it just means that the Vm will not be the same as what we calculate! There is one more thing as well - if we want to solve for a negative ion (like chloride), we use the absolute value of the valence (1 rather than -1), but invert the intracellular and extracellular equations (so intracellular is now in the numerator (top) of the equation). Because we only solve for one ion at a time, the "E" in the equation will have a subscript denoting what ion you solved for. Therefore, we will talk about the Nernst potential for sodium, etc...


Solving the Nernst Equation

We are now going to take a few minutes to solve the Nernst equation for sodium so you can see how to do it... Although there are conditions where I do myself like to use the Nernst equation (we'll talk about those below), it is not likely that you'll have to do this as a physician (Neurologists, Neurosurgeons, and cardiologists - Ignore that last sentence, particularly if you decide to do a little research!). However, seeing the Nernst potentials for each ion is useful to the rest of our discussion. Take a deep breath, calm yourself, find your calculator, and let's do some math...

The Nernst potential for Sodium (ENa)

First - we'll collect the numbers we need:

ENa =
z(61.5) x
log
[X] extracellular
[X] intracellular

 

where:
ENa = The Nernst potential for sodium
z = the charge of the ion (valence) = +1
[x]i = intracellular concentration = 9.2 mEq
[x]e = extracellular concentration = 130 mEq

 

Substituting these into our equation, we now have:

 
ENa =
1(61.5) x
log
130 mEq

9.2 mEq

 

 

This then collapses to:

 
ENa =
61.5 x
log
14.13043...

Note that I have truncated the decimal, but all calculations from here were done with the full number, so you may get a slightly different answer than shown below if you use this number.

Calculating the log leads to:

 
ENa =
61.5 x
1.1501555249613

 

 

All that's left is to finish out the (by now) simple multiplication:

 
ENa =
+ 70.73 mV

 

 

So the Nernst Equilibrium potential for sodium is +70.73 mV...


So What?

Of course, after doing all that work, one wonders what the point of it was. There are two ways to look at what the number we got (they are the same, you may just find one of them conceptually easier to grasp).

  • If we stick an electrode into our cell at a time when the membrane is completely permeable to sodium and the sodium has come to equilibrium, the membrane potential will be +70.73 mV. (Note: remember that only very small amounts of sodium need to move to change the membrane potential - the concentration doesn't actually change).
  • At 70.73 mV, the electrical gradient acting on sodium is exactly balanced by the chemical gradient described by the intracellular and extracellular values I gave you. If we change the membrane potential to a more polarized potential (> 70.73 mV), the electrical gradient is greater then the chemical gradient and sodium will move out of the cell if given the chance , while if the Vm is closer to zero (less polarized, < 70.73 mV) the chemical gradient is stronger and sodium moves into the cell (as we normally see).

Click here to activate an animation that illustrates this (this animation is narrated, so have some way of listening ready to go).

(I inadvertently used a slightly different set numbers for the calculation of the Nernst Potential - leading to a difference of about 1 mV. Don't panic! None of the important facts is altered by this. The animation program just isn't very amenable to editing after the fact! Dr.K.)


The Nernst Potential and resting membrane potential (Vm)

It is fairly common to "want" the Nernst equilbrium potential to resemble the resting Vm of a cell - and as you can quite clearly see, ENa clearly does not! The reason for this is that, at rest, the cell does not meet the criteria for the Nernst potential for sodium - if you will recall, we said that the Nernst Potential is calculated based on the assumption that the cell is fully permeable to the ion in question. Thinking back to lecture, the permability of a neuron/muscle cell at rest to sodium is virtually zero. So we do NOT meet the criteria for being at the Nernst potential for sodium!

One of the ions that has considerable permeability at rest is potassium. If we solve for the Nernst Potential for potassium (EK ), we see something interesting:

EK =
z(61.5) x
log
[X] extracellular
[X] intracellular

 

where:
EK = The Nernst potential for potassium
z = the charge of the ion (valence) = +1
[x]i = intracellular concentration = 150 mEq
[x]e = extracellular concentration = 3.5 mEq

 

Substituting these into our equation, we now have:

 
EK =
1(61.5) x
log
3.5 mEq

150 mEq

 

 

 

 
EK =
61.5 x
log
0.0233333...

 

 

Calculating the log leads to:

 
EK =
61.5 x
-1.6320232147...

 

 

And finally, simple multiplication gives us :

 
EK =
- 100.3694277 mV

 

 

Because the cell membrane is much more permeable to potassium than sodium at rest, the resting membrane potential is much closer to EK than it is to ENa . The reason that the resting membrane potential and the EK are not identical is that the membrane is not completely permeable to potassium. One other ion that has substantial resting membrane permability is Chloride... Take a few minutes and figure out the ECl. Click here to check your work!


When might you use the Nernst Potential

Despite the fact that I told you in class you will never have to solve the Nernst equation as a physician, there are two places you might want to:

  1. To decide what happens when ion concentrations change: Although intracellular concentrations of the ions don't change commonly, the extracellular concentration can and does change in certain clinical situations (fairly common ones, actually). These changes can change the resting membrane potential and have serious effects on the patient (particularly on the cardiac muscle). The most reliable way of deciding what will happen to the membrane potential is to figure out what happens to the Nernst Potential at the new concentrations. For example, hyperkalemia (high extracellular potassium) is a fairly common clinical occurrence with some nasty cardiovascular effects. To decide what happens to excitable tissue, just plug the "new" extracellular value into the equation (use the same intracellular value as before). Let's say that the new extracellular potassium level is 7 mEq.. Plug that in and see what happens! Click here to see this example worked out. You do have to keep in mind the permeability issue - since excitable tissue is permeable to potassium at rest, the cell's resting membrane potential will change with changes in the extracellular potassium levels (either move closer to threshold and become too excitable or hyperpolarize and become less excitable - neither one is a great option if we're talking cardiac tissue!). Just another note: keep in mind that the Nernst Potential is describing what happens to the cell in a very isolated situation - there are channels that we have talked about yet whose behavior will change and so the net effect on the cell is not entirely what is predicted by the Nernst Potential.
  2. In research, to identify the ion that is flowing through a channel: An electrode has no problem detecting ion flow through a channel - after all, that is current flow, something that an electrode is designed to detect. It can even tell us which direction the flow is going (into or out of the cell), also very useful. What the electrode can't do simultaneously is tell us what ion is moving - after all, all ions with positive charges will look the same to the electrode. However, there is enough difference in the Nernst potential for the different ions that we can use the Nernst potential to help us figure out which ion is moving through the channel. Let's say we have a channel that is allowing a positive ion to enter the cell - but we don't know if that ion is sodium or calcium. Well, we know that the ENa is +69 mV, while ECa is much different. If we bring the cell's membrane potential to + 70 mV and see the current flow reverse (start to leave the cell) we have circumstantial evidence that the ion moving through this channel is sodium. In the scientific literature, you will see this referred to as the reversal potential - we're not using the name Nernst potential because we have only circumstantial (not direct) evidence that the ion is sodium.