Nerves, the heart, and the brain are electrical. How do these things work? This course presents fundamental principles, described quantitatively.

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Bioelectricity: A Quantitative Approach

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Nerves, the heart, and the brain are electrical. How do these things work? This course presents fundamental principles, described quantitatively.

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Energy into Voltage

This week we will examine energy, by which pumps and channels allow membranes to "charge their batteries" and thereby have a non-zero voltage across their membranes at rest. The learning objectives for this week are: (1) Describe the function of the sodium-potassium pump; (2) State from memory an approximate value for RT/F; (3) Be able to find the equilibrium potential from ionic concentrations and relative permeabilities; (4) Explain the mechanism by which membranes use salt water to create negative or positive trans-membrane voltages.

- Dr. Roger BarrAnderson-Rupp Professor of Biomedical Engineering and Associate Professor of Pediatrics

Biomedical Engineering, Pediatrics

So now we come, in segment 5 of week 2, to the most complex topic that we'll discuss in the present week.

How does the membrane make use of these ionic differences

that have been established by the sodium-potassium pumps

in order to create a membrane voltage?

It does so in the following series of steps;

First, it adjusts itself, so that it is

permeable to potassium only; in other words, not sodium.

Now of course, it can't do that perfectly,

But what we mean by that is it adjusts itself

so that the permeability of potassium is much greater than the permeability of sodium

or, in other words, ions of potassium can pass through the membrane relatively

easily as compared to ions of sodium.

It's what it does first. It's not an easy trick to do,

because what we're asking the membrane to do--here is the membrane

, and here's a

potassium channel...right here...

that's the channel, here comes a potassium!

HEre comes a potassium, and we're asking the membrance to let that potassium go right on through.

The potassium is coming fast. Fast, fast, fast!

It's not just drifting over there; think in terms of a car traveling very fast.

It's coming at high speed; the membrane recognizes it, and lets it

though the channel.

On the other hand, what we're asking the membrane to do

is if we have a sodium coming, a sodium ion coming, up to that same channel

well, we wanted to X out the sodium and bounce it back

so that teh sodium ion does not get through.

It's not an easy trick to do,

because these are both small ions

Both a positive charge--just has to recognize it and act quickly.

But it is able to do it, so it's a

great accomplishment.

So the membrane begins by being permeable to potassium

by a much higher degree than it is to sodium.

Then, the memrbane lets potassium ions <i>leak</i> back

out.

through these channels where it is permeable to ]

potassium only.

So, in a, uh, conceptual or pictorial sense, you'd say,

"What's happening here

is that the membrane

The membrane is letting potassium leak, from the inside to the outside.

And when it lets it leak from the inside to the outside,

it carries a positive charge with it.

So the result of the potassium ion moving out

is that there is excess positive outside, excess

negative inside.

The corollary is that an electric field is established,

pointing inward, from higher potential to lower potential.

So I've drawn an arrow here representing the direction of the

electric field.

Every time a potassium ion goes out,

The electric field gets stronger

If one describes this with a mathematical equation,

one has the equation that is, uh, given here.

The movement of potassium outward because of the diffusion gradient

is the term here on the left.

The electric field, on teh other hand,

is pushing potassium back in

That's the term over here on the right.

This is inward.

Here.

This is outward. There.

Both processes are occurring at the same time,

to the degree that they are not in balance--

there is a flux of potassium ions, indicated by the J<u>p symbol</u>

that is moving across the membrane.

So now let's look at this slide and take this process as a whole.

The membrane begins by

being permeable to potassium only.

There's way more potassium inside then there is outside.

So diffusion pushes potassium ions out

because the membrane will allow potassium

to move.

The electric field pushes potassium back in.

(!)

At some point, enough potassium has moved out

that the electric field very strongly is pushing back in.

So an equilibrium results

when the flow from diffusion equals the flow because of the

electric field.

The result is equilibrium,

and in a one-ion system,

a system in which only one ion

can move across the membrane, this equilibrium

is called the Nernst V<u>m</u>

So the Nernst transmembrane voltage for potassium occurs

when the two forces, diffusion and the electric field,

are in balance.

What is the voltage at which that equilibrium

will occur for potassium?

So, if we had more weeks in our course :),

I would derive this result mathematically.

Since our time is limited, I will just show teh result.

There will be an equilibrium when the transmembrance

voltage Vm is equal to RT/F.

You remember that from last week.

times the logarithm of the concentrations:

the extracellular concentration over the intracellular concentration.

In the particular case of squid,

that means that the logarithm 20/397

and you recall that R<u>TO/F from last week is about 26 millivolts.</u>

So, adjusting our slide just slightly,

the membrane is permeable to potassium only,

Diffusion pushes potassium out,

The electric field pushed potassium in,

and there is a Dynamic equilibrium near the Nernst Vm,

the Vm for K+ at equilibrium is about -77mv

for squid

And this value of Vm is normally given a special symbol, E<u>k</u>

If you write it out step by step like that,

then it's just one step following another.

You think of the process as a whole and you say,

This is just amazing!

This membrane takes advantage of the fact

that there is a potassium concentation

difference.

It's <i>letting</i> potassium ions move outside the membrane!

That's creating an electric field.

Diffusion, the force of diffusion,

the process of diffusion is being used to create an electric field!

It all comes to equilibrium, and in the case

of squid axon that's at

-77mV

That means the inside is negative by 77 mV

with respect to the outside.

Now the fact that it has a special symbol is to be expected.

What's extraordinary is that it happens at all!

The thing is just sitting there,

there's no electricity, and it charges itself up.

If we talk about sodium, it's all the same things,

but the sodium concentration starts outside instead of inside

but same thing, same equation--

the result is different because the concentration inside and outside is at a different ratio,

But the result now is that the equilibrium value is <i>positive<i>.</i></i>

(plus 57mv)

Wow.

Potassium equilibrium was negative, and the sodium equilibrium is positive.

Quite a voltage shift!

Of course, the membrane to accomplish this, has to become

permeable to sodium,

no longer permeable to potassium,

so that the system is now a sodium-ion system

instead of a potassium-ion system.

Just as before, the system depends on the

concentration gradient

being established first

so that it can be exploited to create the voltage difference.

We have not discussed chlorine ions,

but there's a similar process.

for chlorine.

One might ask, can you do the mathematics

so that all the different ions appear in the same system?

And in fact you can, and that's this

multi-ionic expression, sometimes called

"Goldman's expression."

And it appears on the slide.

The thing that you

notice is that the ratios <i>ahem</i>

the expression is written in terms of

these permeabilities, and if you work with the equation

somewhat, you'll find that what's important is the permeability <i>ratios</i>

more than the permeabilities.

You can do that, for example, by taking the expression as it's given

and dividing each term by, say, P<u>k.</u>

And if you do divide each term by P<u>k,</u>

you'll get a result that will be in terms of the ratios.

Once again, the mystery factor, RT/F

There it is.

Once again the mystery factor RT/F

shows up as our coefficient or, if you will, our scaling factor.

So it's essentially Vm is 26 mV

times the ratio of these concentrations

each multiplied by their permeability.

Thank you for your attention, and we'll go on

to the next segment.