membrane & action potentials module part ii: a multi-ion membrane potential in vitro

Membrane & Action Potentials Module

Part II: A Multi-Ion Membrane Potential in vitro

Module Outline

• Part I. A Single-Ion Membrane Potential in vitro

• Part II. A Multi-Ion Membrane Potential in vitro

• Part III. The in vivo Membrane Potential

• Part IV. The Action Potential

II: A Multi-Ion Membrane Potential in vitro

• Welcome to Part II of the Membrane & Action Potentials Module!

• As you may have guessed from the title, in this section we will consider membrane potentials that involve more than one ion.

• The basic principles that you learned in Part I still hold true of course; however, there are a few important differences to consider.

Definitions

• Here’s that list of definitions again.

Current – the flow of charged particles (Amperes; Amps)

Potential – the separation of charge (Volts; V)

Capacitor – a non-conducting medium that allows electrostatic interaction between charges

on either side of it (Farads)

Same Instructions…

• While running this module is orders of magnitude simpler than say, syncing your Outlook Calendar w/ iRocket, some brief instructions might still be valuable.– On some of the following slides you will see the commands

“(click)” and “(next)”.– When you see “(click)” an animation is on the way. Just strike the

Down Arrow or Left-Click and enjoy!– The “(next)” command means that when you strike the Down

Arrow or Left-Click you’ll be moving on to the next page.– The color of the membrane indicates the ions to which it is

permeable. For example, a red membrane means that the membrane is permeable to red ions (potassium as you’ll soon see)

• Got it? Alrighty then…happy viewing!


So far we have only considered cases in which only one ion was involved in generating the membrane potential.

Let’s start with the potassium-only situation from Part I and see what happens if we add another ion to the mix

How ‘bout sodium (Na)?

(next)

Temperature 298K

Membrane Permeability PK = 1 PNa = 0 PCl = 0

Solution A 10mM KAc Solution B 10mM KAc

0mV

10mM K+

10mM Ac-

K+

Ac-

Ac-

Ac-

K+

K+

K+K+

K+

10mM K+

10mM Ac-

K+

Ac-

Ac-

Ac-

K+

A

Ac-Ac-K+

K+

B

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+

K+

Temperature 298K


Solution A 10mM KAc Solution B 10mM KAc

Temperature 298K


Solution A 10mM KAc 10mM NaAcSolution B 10mM KAc 10mM NaAc

Temperature 298K



10mM K+

20mM Ac-

10mM Na+

10mM K+

20mM Ac-

10mM Na+

10mM K+

10mM Ac-

10mM K+

10mM Ac-


Ok…let’s add some Sodium Acetate (NaAc) and make the membrane permeable to sodium (click)…

Whoops…I meant “…make the membrane permeable to sodium AND potassium.” (click)

And…(click)…

Hey! The voltmeter still read 0mV?

Why?...You got it! No gradients!

(next)

0mV

K+

Ac-

Ac-

Ac-

K+

K+

K+K+

K+

K+

Ac-

Ac-

K+

A

Ac-K+

K+

B

Ac-

Ac-

Ac-

K+

K+

K+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

10mM K+

20mM Ac-

10mM Na+

10mM K+

20mM Ac-

10mM Na+

Temperature 298K




Just like in Part I, without concentration gradients for either potassium ions or sodium ions the net potassium current will be zero (click) as will the net sodium current (click).

Alright then…let’s establish a gradient.

We’ll start with a potassium gradient.

(next)

0mV

K+

Ac-

Ac-

Ac-

K+

K+

K+K+

K+

Ac-

Ac-

K+

A

Ac-

K+

B

Ac-

Ac-

Ac-

K+

K+

K+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Temperature 298K




Ok…So now we have established a potassium gradient.

For the moment, we’ve also made the membrane impermeable.

What do you think will happen when we make it permeable to potassium and sodium again?

Let’s find out…

(next)

100mM K+

110mM Ac-

10mM Na+

10mM K+

20mM Ac-

10mM Na+

0mV

K+

Ac-

Ac-

Ac-

K+

K+

K+K+

K+

K+

Ac-

Ac-

K+

A

Ac-K+

K+

B

Ac-

Ac-

Ac-

K+

K+

K+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+

K+K+

K+

K+

Ac-

Ac-

Ac-

Ac-

Ac- Ac-

K+Temperature 298K




Just to warn you, this might get a little complex, so although the following processes happen simultaneously, we’ll take them stepwise. (click)

Things begin very similarly to the “potassium-only” situation.

With the membrane permeable to potassium (and sodium) and a potassium gradient present, there will be a net flux of potassium from A to B.

(click)

(next)

100mM K+

110mM Ac-

10mM Na+

10mM K+

20mM Ac-

10mM Na+

0mV

K+

Ac-

Ac-

Ac-

K+

K+

K+K+

K+

Ac-

Ac-

K+

A

Ac-

K+

B

Ac-

Ac-

Ac-

K+

K+

K+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+

K+K+

K+

K+

Ac-

Ac-

Ac-

Ac-

Ac- Ac-

----------

++++++++++

K+

Temperature 298K




This flux will tend to make side A negative, thus creating an opposing electrical “pull” on potassium, which would increase until the system reached equilibrium (click).

But wait!

That same negative potential on Side A will also exert a “pull” on Sodium ions, which will cause a net flux of Sodium ions from Side B to Side A (click)!

(next)

100mM K+

110mM Ac-

10mM Na+

10mM K+

20mM Ac-

10mM Na+

K+

Ac-

Ac-

Ac-

K+

K+

K+

K+

K+

Ac-

Ac-

K+

A

Ac-

K+

B

Ac-

Ac-

Ac-

K+

K+

K+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+

K+K+

K+

K+

Ac-

Ac-

Ac-

Ac-

Ac- Ac-

K+ Temperature 298K




As you might expect, that net flow of positively charged sodium ions from Side B to Side A will tend to make Side A more positive than Side B.

Although this doesn’t eliminate the potential established by potassium completely, this sodium current will “cancel out” some of the potassium-generated membrane potential.

(click)

(next)

100mM K+

110mM Ac-

10mM Na+

10mM K+

20mM Ac-

10mM Na+

Ac-

Ac-

Ac-

K+

K+

K+

K+

Ac-

Ac-

K+

A

Ac-

K+

B

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+

K+

K+

K+

Ac-

Ac-

Ac-

Ac-

Ac- Ac-

-

-

-

-

-

+

+

+

+

+

----------

++++++++++

K+

K+ K+

K+


Eventually, a new ‘steady-state’ membrane potential will be reached.

But what will that new potential be?

Hmm…instead of going right to the answer let’s start by making an educated guess based on what we’ve already learned.

First, let’s use the Nernst equation to figure out the potassium equilibrium potential.

Why? You’ll see…just trust me for now.

(next)

RT 100

zF 10lnEK = = -59mV

K+

100mM K+

110mM Ac-

10mM Na+

10mM K+

20mM Ac-

10mM Na+

Ac-

Ac-

Ac-

K+

K+

K+

K+

Ac-

Ac-

K+

A

Ac-

K+

B

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+

K+

K+

K+

Ac-

Ac-

Ac-

Ac-

Ac- Ac-

-

-

-

-

-

+

+

+

+

+K+

K+ K+

K+


And what about sodium? Can we use the Nernst equation for sodium even though there’s no gradient?

We touched on this briefly in Part I, but it turns out that you can!

Recall that the Nernst equation calculates the membrane potential required to balance the “force” generated by the chemical gradient.

There’s zero chemical gradient, so therefore 0mV is the potential required to balance the chemical “force”.

Thus, the equilibrium potential for sodium is:

(next)

RT 10

zF 10lnENa = = 0mV

K+

100mM K+

110mM Ac-

10mM Na+

10mM K+

20mM Ac-

10mM Na+

Ac-

Ac-

Ac-

K+

K+

K+

K+

Ac-

Ac-

K+

A

Ac-

K+

B

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+

K+

K+

K+

Ac-

Ac-

Ac-

Ac-

Ac- Ac-

-

-

-

-

-

+

+

+

+

+K+

K+ K+

K+

K+

100mM K+

110mM Ac-

10mM Na+

10mM K+

20mM Ac-

10mM Na+

Ac-

Ac-

Ac-

K+

K+

K+

K+

Ac-

Ac-

K+

A

Ac-

K+

B

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+

K+

K+

K+

Ac-

Ac-

Ac-

Ac-

Ac- Ac-

-

-

-

-

-

+

+

+

+

+K+

K+ K+

K+


Ok then, back to our educated guess…

We know now that the potassium “wants” to make the membrane potential -59mV.

We also know that the sodium “wants” to make the membrane potential 0mV.

So, it’s logical to assume that the sum of the influence of both ions on the membrane potential will leave us somewhere between -59mV and 0mV.

(click)

Under these conditions, the potential across the membrane reaches -44mV.

(next)

-44mV


Now…you might be thinking to yourself that maybe we could have predicted this all along, just like we did in Part I…

Well, you know what? You’re right!

Just like that smart guy Nernst put together an equation that lets us calculate single-ion equilibrium potentials, three smart guys developed a way to calculate membrane potentials that involve several ions.

Their names were Goldman, Hodgkin, and Katz, and their equation is called the Goldman-Hodgkin-Katz (GHK) equation.

Looks pretty complicated huh?Have no fear though, we’ll be going through this one too…

(next)

RT PIx[Ix]OUT + PIy[Iy]OUT + PIz[Iz]OUT + etc…

F PIx[Ix]IN + PIy[Iy]IN + PIz[Iz]IN + etc…lnVm =

K+

100mM K+

110mM Ac-

10mM Na+

10mM K+

20mM Ac-

10mM Na+

Ac-

Ac-

Ac-

K+

K+

K+

K+

Ac-

Ac-

K+

A

Ac-

K+

B

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+

K+

K+

K+

Ac-

Ac-

Ac-

Ac-

Ac- Ac-

-

-

-

-

-

+

+

+

+

+K+

K+ K+

K+

-44mV


If you’ve had a chance to take a close look, you might have recognized that this equation is quite similar to the Nernst equation.

In fact, with the exception of “Vm”, which stands for “membrane potential” and the fact that the “z” has been dropped (more on this later), everything up to the “ln” is identical to the Nernst equation.

In fact, as you’ll see, the GHK equation is basically giving us a weighted average of equilibrium potentials (i.e. Nernst equations).

Ok then…let’s have a look at the rest of the equation (i.e. the stuff after the “ln”).

(next)



K+

100mM K+

110mM Ac-

10mM Na+

10mM K+

20mM Ac-

10mM Na+

Ac-

Ac-

Ac-

K+

K+

K+

K+

Ac-

Ac-

K+

A

Ac-

K+

B

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+

K+

K+

K+

Ac-

Ac-

Ac-

Ac-

Ac- Ac-

-

-

-

-

-

+

+

+

+

+K+

K+ K+

K+

-44mV


Beginning with the numerator…

PIx[Ix]OUT + PIy[Iy]OUT + PIz[Iz]OUT + etc…

Put into words, this means, “the relative permeability of ion x (PIx), times the concentration of ion x on the outside of the membrane ([Ix]OUT), plus the relative permeability of ion y (PIy), times the concentration of ion y on the outside of the membrane ([Iy]OUT), plus the relative concentration of ion z (PIz)…and so on and so forth.

That means that the denominator…

PIx[Ix]IN + PIy[Iy]IN + PIz[Iz]IN + etc…

means all that same stuff, except that instead of considering ion concentrations on the outside of the membrane, we’re considering the ionic concentrations on the inside of the membrane.

(next)

K+

100mM K+

110mM Ac-

10mM Na+

10mM K+

20mM Ac-

10mM Na+

Ac-

Ac-

Ac-

K+

K+

K+

K+

Ac-

Ac-

K+

A

Ac-

K+

B

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+

K+

K+

K+

Ac-

Ac-

Ac-

Ac-

Ac- Ac-

-

-

-

-

-

+

+

+

+

+K+

K+ K+

K+

-44mV


That’s all there is to it, but before we start putting it to use, here are three notes about using the GHK equation.

1) In the part after the “ln”, we can use as many or as few terms (i.e. PI[I]’s) as we need. For example, if we only have two ions in solution or if the membrane were only permeable to two ions, then we only have to use use two terms. Thus, the equation would become

RT PIx[Ix]OUT + PIy[Iy]OUT

F PIx[Ix]IN + PIy[Iy]IN

lnVm =

2) For negatively charged ions we put the concentration of that ion on the inside of the membrane in the numerator and the concentration of that ion on the outside of the membrane in the denominator.

3) The GHK equation (as written) cannot be used for multivalent ions (i.e. no Ca2+ or Mg2+).

(next)

K+

100mM K+

110mM Ac-

10mM Na+

10mM K+

20mM Ac-

10mM Na+

Ac-

Ac-

Ac-

K+

K+

K+

K+

Ac-

Ac-

K+

A

Ac-

K+

B

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+

K+

K+

K+

Ac-

Ac-

Ac-

Ac-

Ac- Ac-

-

-

-

-

-

+

+

+

+

+K+

K+ K+

K+

-44mV


Alrighty then, now that we know the rules let’s put that GHK equation to work!





lnVm =

Since the membrane is permeable to only potassium and sodium we only have to consider those two ions. Therefore, the equation reduces down to:

(next)K+

100mM K+

110mM Ac-

10mM Na+

10mM K+

20mM Ac-

10mM Na+

Ac-

Ac-

Ac-

K+

K+

K+

K+

Ac-

Ac-

K+

A

Ac-

K+

B

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+

K+

K+

K+

Ac-

Ac-

Ac-

Ac-

Ac- Ac-

-

-

-

-

-

+

+

+

+

+K+

K+ K+

K+

-44mV


And let’s change the subscripts and variables too so that the equation is specific for our situation.



lnVm =

RT PK[K+]B + PNa[Na+]B

F PK[K+]A + PNa[Na+]A

lnVm =

Great!

Now all we have to do is plug in the numbers and let our calculators do the work for us.

(next)

K+

100mM K+

110mM Ac-

10mM Na+

10mM K+

20mM Ac-

10mM Na+

Ac-

Ac-

Ac-

K+

K+

K+

K+

Ac-

Ac-

K+

A

Ac-

K+

B

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+

K+

K+

K+

Ac-

Ac-

Ac-

Ac-

Ac- Ac-

-

-

-

-

-

+

+

+

+

+K+

K+ K+

K+

-44mV


Plugging in the constants & the numbers from the box below gives us…

Temperature 298K



(8.314)(298) (1)(10) + (1)(10)

(96500) (1)(100) + (1)(10)lnVm =

RT PK[K+]B + PNa[Na+]B

F PK[K+]A + PNa[Na+]A

lnVm =

= (approx.) -44mV

Hey what do ya know! -44mV!

(next)K+

100mM K+

110mM Ac-

10mM Na+

10mM K+

20mM Ac-

10mM Na+

Ac-

Ac-

Ac-

K+

K+

K+

K+

Ac-

Ac-

K+

A

Ac-

K+

B

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+

K+

K+

K+

Ac-

Ac-

Ac-

Ac-

Ac- Ac-

-

-

-

-

-

+

+

+

+

+K+

K+ K+

K+

-44mV

Temperature 298K



Temperature 298K



K+

150mM K+

170mM Ac-

20mM Na+

5mM K+

145mM Ac-

140mM Na+

K+

Ac-

Ac-

Ac-

K+

K+

Ac-

K+

A

Ac-

B

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+K+

K+

K+

Ac-

Ac- Ac-

0mV


Ok…so now that we’ve looked at what happens when we start with a potassium gradient, let’s see what happens when we have both potassium & sodium gradients across the membrane just like in a real cell!

And while we’re at it, let’s start using potassium & sodium concentrations that approximate those of a typical excitable cell (e.g. neuron, myocyte, etc.).

(click)

(next)K+

100mM K+

110mM Ac-

10mM Na+

10mM K+

20mM Ac-

10mM Na+

Ac-

Ac-

Ac-

K+

K+

K+

K+

Ac-

Ac-

K+

A

Ac-

K+

B

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+

K+

K+

K+

Ac-

Ac-

Ac-

Ac-

Ac- Ac-

-

-

-

-

-

+

+

+

+

+K+

K+ K+

K+

-44mV


Notice that the potassium ion concentration is high on Side A (150mM) and low on Side B (5mM) and the sodium ion concentration is low on Side A (20mM) and high on Side B (140mM).

Thus, Side A is similar to the inside of a cell and Side B is similar to the outside of a cell. Hey, ya know what? Let’s go ahead and label them that way. (click)

Also, notice that the membrane is permeable to neither potassium nor sodium right now.

(next) Temperature 298K



K+

150mM K+

170mM Ac-

20mM Na+

5mM K+

145mM Ac-

140mM Na+

K+

Ac-

Ac-

Ac-

K+

K+

Ac-

K+

Ac-

A B

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+K+

K+

K+

Ac-

Ac- Ac-

0mV

IN OUT

Temperature 298K


INSIDE 150mM KAc 20mM NaAcOUTSIDE 5mM KAc 140mM NaAc

Temperature 298K



IN OUT


Before opening things up to both ions though, let’s see what happens when the membrane is permeable to one or the other.

As always, we’ll start with potassium. (click)

What did we get? - 87mV!

As you know by now, we could have predicted this using the Nernst equation.

(next)

Temperature 298K



K+

150mM K+

170mM Ac-

20mM Na+

5mM K+

145mM Ac-

140mM Na+

K+

0mV

Ac-

Ac-

Ac-

K+

K+

Ac-

K+

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+K+

K+

K+

Ac-

Ac- Ac-

-87mV

----------

++++++++++


And…if you’ve had a chance to consider the GHK equation carefully, you might have also figured out that we could have used it to predict the new potential as well.

That becomes obvious when you recognize that under these conditions, the GHK equation reduces down to the Nernst.

The GHK equation is:

150mM K+

170mM Ac-

20mM Na+

5mM K+

145mM Ac-

140mM Na+

IN OUT

K+

K+

Ac-

Ac-

Ac-

K+

K+

Ac-

K+

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+K+

K+

K+

Ac-

Ac- Ac-

-87mV



And the Nernst equation is/becomes:

See (click), they’re identical!

(next)

which becomes…

RT PIx[Ix]OUT

F PIx[Ix]IN

lnVm = =RT (1)[K+]OUT

F (1)[K+]IN

ln

=RT [Ix]OUT

zF [Ix]IN

lnEI = RT [K+]OUT

(1)F [K+]IN

ln

----------

++++++++++


Ok, let’s start over (click) and see what happens when we make the membrane permeable to sodium only.

Actually, we don’t need to do that…do we? We can just predict what it will be with the Nernst equation, right?

The sodium equilibrium potential is +50mV!




Temperature 298K



IN OUT

K+

150mM K+

170mM Ac-

20mM Na+

5mM K+

145mM Ac-

140mM Na+

K+

Ac-

Ac-

Ac-

K+

K+

Ac-

K+

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+K+

K+

K+

Ac-

Ac- Ac-

0mV

150mM K+

170mM Ac-

20mM Na+

5mM K+

145mM Ac-

140mM Na+

IN OUT

K+

K+

Ac-

Ac-

Ac-

K+

K+

Ac-

K+

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+K+

K+

K+

Ac-

Ac- Ac-

-87mV

----------

++++++++++

RT [Na+]OUT

zF [Na+]IN

lnEI = (8.314)(298) 140

(1)(96500) 20ln =

= (approx.) + 50mV


That makes sense right?

At these concentrations, the net flow of sodium ions would be from “OUT” to “IN”, thereby making the inside of the membrane more positive.

Furthermore, there is only a 7-fold sodium concentration gradient, whereas there is a 30-fold potassium concentration gradient.

Thus, we should expect that the magnitude of the electrical potential required to balance the sodium gradient will be smaller than what it took to balance the potassium gradient. That is, less than 87mV…

Well then, + 50mV make perfect sense!

Let’s skip the “sodium only” experiment then and move straight to making the membrane permeable to both potassium & sodium.

(next)

IN OUT

K+

150mM K+

170mM Ac-

20mM Na+

5mM K+

145mM Ac-

140mM Na+

K+

Ac-

Ac-

Ac-

K+

K+

Ac-

K+

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+K+

K+

K+

Ac-

Ac- Ac-

0mV


Actually, while we’re doing this prediction thing, instead of doing the experiment, let’s use the GHK equation to figure out what the potential will be when the membrane is equally permeable to potassium & sodium.

= (approx.) -4mV!




IN OUT

K+

150mM K+

170mM Ac-

20mM Na+

5mM K+

145mM Ac-

140mM Na+

K+

Ac-

Ac-

Ac-

K+

K+

Ac-

K+

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+K+

K+

K+

Ac-

Ac- Ac-

0mV

(8.314)(298) (1)(5) + (1)(140)

(96500) (1)(150) + (1)(20)lnVm =


Great!...Well…actually, that’s a little weird…

-4mV is between -87mV and +50 like we would expect, but it’s is nowhere near the potential we see across the membrane of most cells (i.e. around -65mV to -85mV)!

The concentrations that we used are just like those found in a real cell, right?

So what’s wrong?

(next)

Temperature 298K



IN OUT

K+

150mM K+

170mM Ac-

20mM Na+

5mM K+

145mM Ac-

140mM Na+

K+

Ac-

Ac-

Ac-

K+

K+

Ac-

K+

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+K+

K+

K+

Ac-

Ac- Ac-

0mV

IN OUT

K+

150mM K+

170mM Ac-

20mM Na+

5mM K+

145mM Ac-

140mM Na+

K+

Ac-

Ac-

Ac-

K+

K+

Ac-

K+

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+K+

K+

K+

Ac-

Ac- Ac-

0mV


Maybe we just calculated something wrong.

I know, let’s actually do the experiment and see what we get. (click)

Hmm…it really is -4mV!

“What’s going on?”, you ask.

Go on to the next slide find out.

(next)

Temperature 298K



Temperature 298K



-

-

-

+

+

+

-4mV


Wouldn’t you know it…It’s a permeability issue again!

Remember that at rest, most cells are far more permeable to potassium than they are to sodium.

What effect do you think this property has on the resting membrane potential?

(next)

Temperature 298K



IN OUT

K+

150mM K+

170mM Ac-

20mM Na+

5mM K+

145mM Ac-

140mM Na+

K+

Ac-

Ac-

Ac-

K+

K+

Ac-

K+

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+K+

K+

K+

Ac-

Ac- Ac-

-

-

-

+

+

+

-4mV


If you guessed that it makes the membrane potential closer to the potassium equilibrium potential, you’re absolutely right!

To most of us though that probably doesn’t make much sense right now, so we’re going take a moment to think about it a little…

…keeping it simple of course!

(next)

Temperature 298K



IN OUT

K+

150mM K+

170mM Ac-

20mM Na+

5mM K+

145mM Ac-

140mM Na+

K+

Ac-

Ac-

Ac-

K+

K+

Ac-

K+

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+K+

K+

K+

Ac-

Ac- Ac-

-

-

-

+

+

+

-4mV


Recall that the flow of ions across the membrane generates the potential.

Also recall that the membrane’s permeability to an ion determines how freely that ion will be able to flow.

Well…then ions to which the membrane is most permeable will have the largest influence on the membrane potential, right?

That’s basically what’s going on in real cells.

That is, while sodium and potassium both influence the membrane potential, the membrane is about 100-fold more permeable to potassium.

Thus, potassium’s influence on the potential is greater than that of sodium, and therefore the resting membrane potential is much closer to potassium’s equilibrium potential than it is to sodium’s

(next)IN OUT

K+

150mM K+

170mM Ac-

20mM Na+

5mM K+

145mM Ac-

140mM Na+

K+

Ac-

Ac-

Ac-

K+

K+

Ac-

K+

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+K+

K+

K+

Ac-

Ac- Ac-

-

-

-

+

+

+

-4mV

Temperature 298K



IN OUT

K+

150mM K+

170mM Ac-

20mM Na+

5mM K+

145mM Ac-

140mM Na+

K+

Ac-

Ac-

Ac-

K+

K+

Ac-

K+

Ac-

Ac-

Ac-

K+

K+

Na+

Na+

Na+

Na+

Na+Na+

Na+

Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+K+

K+

K+

Ac-

Ac- Ac-

-

-

-

+

+

+

-4mV

IN OUT

K+

150mM K+

170mM Ac-

20mM Na+

5mM K+

145mM Ac-

140mM Na+

Ac-

Ac-

Ac-

Ac-

K+

Ac-

Ac-

Ac-

K+

Na+

Na+ Na+

Na+Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+K+

K+

K+

Ac-

Ac- Ac-

Temperature 298K




So let’s go ahead and make our membrane 100-fold more permeable to potassium too!

But…As always, we’ll start over first (click)

And now we’ll set the potassium permeability to 1 and the sodium permeability to 0.01 (a 100-fold difference)

(click)

So…under these conditions the membrane potential is -81mV.


Membrane Permeability PK = 1 PNa = 0.01 PCl = 0


K+

K+

K+

K+Na+

Na+

Na+

0mV

----------

++++++++++

-81mV

Note – In the previous animation, the decrease in sodium permeability was represented by making the sodium ions move more slowly as they crossed the membrane. Of course this is not what happens in reality. Sodium ions do not actually move any more slowly. They are simply less able to cross the membrane, so per-unit-time fewer ions cross.

(next

IN OUT

K+

150mM K+

170mM Ac-

20mM Na+

5mM K+

145mM Ac-

140mM Na+

Ac-

Ac-

Ac-

Ac-

K+

Ac-

Ac-

Ac-

K+

Na+

Na+ Na+

Na+Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+K+

K+

K+

Ac-

Ac- Ac-

K+

K+

K+

K+

Na+

Na+

Na+

----------

++++++++++

-81mV -81mV is between ENa(+50mV) and EK(-87mV), but much closer to EK…

Just like we expected!

And let’s check the GHK equation to make sure that it gives us the same thing.

=(approx.) -81mV !

Yup! It sure does!

(next)


(8.314)(298) (1)(5) + (0.01)(140)

(96500) (1)(150) + (0.01)(20)lnVm =

IN OUT

K+

150mM K+

170mM Ac-

20mM Na+

5mM K+

145mM Ac-

140mM Na+

Ac-

Ac-

Ac-

Ac-

K+

Ac-

Ac-

Ac-

K+

Na+

Na+ Na+

Na+Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+K+

K+

K+

Ac-

Ac- Ac-

K+

K+

K+

K+

Na+

Na+

Na+

----------

++++++++++

-81mV Alright! Congratulations! You’ve survived Part II.

Just like last time though, I’m sure there are still a few questions that you’d like answered.

Unfortunately though, I can’t read your mind.

So…here are a couple of questions that often come up and the answers to them.

1) Most real cells are also permeable to Cl-, so where does Cl- fit into all of this?

2) Hey we’re not at equilibrium anymore! What gives?


IN OUT

K+

150mM K+

170mM Ac-

20mM Na+

5mM K+

145mM Ac-

140mM Na+

Ac-

Ac-

Ac-

Ac-

K+

Ac-

Ac-

Ac-

K+

Na+

Na+ Na+

Na+Na+

Na+

Na+

Na+

Na+

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

Ac-

K+

K+K+

K+

K+

Ac-

Ac- Ac-

K+

K+

K+

K+

Na+

Na+

Na+

----------

++++++++++

-81mV 1) Most cells are permeable to Cl- too, so where does Cl- fit into all of this?

As much as I would like to explain this one with another animation, these slides are already getting too “busy” with just two ions.

And as you may already know, there’s really not much difference.

Instead (click), let’s take advantage of the fact that we’ve gotten pretty good at using the Nernst and GHK equations.

Let’s use the following values for our calculations (somewhat similar to what you might find in a real cell).

Permeabilities: PK = 1, PNa = 0.01, PCl = 0.2

Inside: [K+] = 150mM, [Na+] = 20mM, [Cl-] = 6mMOutside: [K+] = 5mM, [Na+] = 140mM, [Cl-] = 120mM

(next)



First we’ll calculate the three equilibrium potentials.

=(approx.) -87mV

=(approx.) +50mV

=(approx.) -76mV

(next)

RT [K+]OUT

zF [K+]IN

lnEK = (8.314)(298) 5

(1)(96500) 150ln =

RT [Na+]OUT

zF [Na+]IN

lnENa = (8.314)(298) 140

(1)(96500) 20ln =

RT [Cl-]OUT

zF [Cl-]IN

lnECl = (8.314)(298) 120

(-1)(96500) 6ln =


And now we’ll use the GHK equation to calculate the new membrane potential.

=(approx.) -80mV

*Notice that instead of dividing by the valence of chloride (-1) like we did for the Nernst equation, we’ve moved [Cl-]IN

to the numerator and [Cl-]OUT to the denominator.

There you have it, a membrane potential using concentrations and relative permeabilities that are similar to those of a real cell!

(next)

RT PK[K+]OUT + PNa[Na+]OUT + PCl[Cl-]IN

F PK[K+]IN + PNa[Na+]IN + PCl[Cl-]OUT

lnVm =

(8.314)(298) (1)(5) + (0.01)(140) + (0.2)(6)

(96500) (1)(150) + (0.01)(20) + (0.2)(120)ln =

Another Note – In some cells the chloride equilibrium potential is exactly equal to the resting membrane potential (i.e. ECl = Vm)

(next)

2) Hey we’re not at equilibrium anymore! What gives?

Very perceptive!!!

Recall that the equilibrium potential for an ion (EI) is the potential at which the electrical and chemical forces for that ion are equal in magnitude and opposite in direction. Thus, EI is the potential at which the net flow of that ion across the membrane will be zero.

Notice however that in the situation that we just outlined the membrane potential (-80mV) not equal to the equilibrium potentials of any of the ions ( EK = -87mV; ENa = +50mV; ECl = -76mV)

Thus, none of these ions are in equilibrium (This was of our previous experiments involving multiple ions as well)

That means that although we’ve reached a steady membrane potential, there will still be a net flow of potassium, sodium, and chloride.

(next)


That is, at this potential, potassium will continue to flow from IN to OUT, while sodium and chloride will continue to flow from OUT to IN (click).

What effect do you think this will have on the ionic gradients and membrane potential if we leave things like this for a while?

Well, eventually the concentration gradients will degrade and the membrane potential will degenerate.

Now while this is really not such a big deal for our beaker. For a real cell, this would be devastating.

In fact, It would kill the cell!

Well…this situation exists for living cells. That is, in most circumstances, none of their ions are in equilibrium either. So how do real cells keep their concentration gradients and membrane potentials from dissipating?

(next)


Yet Another Note – Unlike the situation we’re discussing now, in those cells in which ECl = Vm, chloride will be at equilibrium when the cell is at rest, and therefore there will be no net flow of chloride at rest.

(click)

If you guessed MEMBRANE PUMPS, you’re absolutely right!

The major function of ATP-dependent pumps like the Na/K-ATPase is to maintain the concentration gradients of ions across the membrane and thereby keep the cell alive.

We’ll address some of the other nuances of real cells in

“Part III. The in vivo Membrane Potential”

But first let’s tie things together for Part II.

(next)


Brian

dependent


• Some take home points from Part II…

– An ion that can influence the membrane potential (Vm) “tries” to make the membrane potential equal to its equilibrium potential (EI).

– Typically, when multiple ions influence Vm, the ion(s) to which the membrane is most permeable will have the greatest influence.

– The GHK equation predicts Vm based upon concentration gradients and relative membrane permeabilities.

– Typically, when multiple ions are involved in establishing Vm, none of the ions in question are in equilibrium.

– Real cells have membrane pumps that prevent the concentration gradients across their membranes from dissipating, thus maintaining their membrane potentials.

membrane & action potentials module part ii: a multi-ion membrane potential in vitro

Documents

membrane permeability

membrane permeable

membrane potentials

multiion membrane potential

red membrane

singleion membrane potential

na ac slide

kac solution b