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Hello, again.

We're now moving on from our series of lectures on the

cell cycle to another topic, which is mathematical models of action potentials.

And, we're going to start by talking about mathematical models of action potentials

that consist of a system of ODEs, and then we're going to use this

as a way of transitioning from ordinary differential equations models to what are

called partial differential equation models, with

the difference being that partial differential equations.

Things can vary as a function of time, but

they cans also vary as a function of location.

Let's begin today talking discussing how some of the

topics, and some of the themes that we're going to

encounter with our action potential model relate to some of

the themes that, that came up in our previous lectures.

If you remember in the cell cycle model of Novak and Tyson, we showed this plot here,

where when when a cell's undergoing these these

rapid cell divisions, cycling will go up and down.

It will oscillate as function of time.

Pre MPF and MPF will also go up as and down as function of time.

And when you have several variables that are inter, that are influencing one

another, seven variables in the case of the Novak and Tyson cell cycle model.

It can be very difficult to really, to develop the model, to

choose parameters for the model when

they're they're all influencing one another.

And one of the, the key simulations and, and

experiments that we, that we discussed with how, with

respect to how the Novak and Tyson cell cycle

model got developed was this one shown down here.

Where they used nondegradable cyclin.

And they varied the amount of nondegradable cyclin systematically.

And, in the model, they predicted how MPF

activity would, would change the function of nondegradable cyclin.

That was in the system.

And they generated this prediction here of bistability

of MPF activity as a function of non-degradable cyclin.

And the conclusion that we've reached, in discussing

that particular simulation and the experiments that followed it.

Is that the simulation of a simplified experiment, where

it was non-degradable cyclin rad, rather than regular cyclin.

This simulation was critical both for developing the model and

for getting an, an understanding of how the system worked.

By using the non degradable cyclin, this was the only

way they were able to generate this prediction of, of bi-stability.

We're going to encounter something really similar

when we talk about the action potential model.

The output that we're going to get from our action

potential model, this is the Hodgkin and Huxley neuronal

action potential, consists of four variables and they change

as a function of time, as we see here.

And we'll define what these variable mean in subsequent lectures.

But, the reason I showed this here is just to

illustrate that these four variables all have very complex time courses.

And because they influence one another, it's hard to get to

really get the control from just looking at these time courses.

What we're going to, the simulation results

and the experimental results that were going to

discuss consists of experiments that are like this where voltage you can see,

rather than having a complicated time course is going to be, is going

to jump to a particular level, it's going to be held at that level.

And then it's going to jump down.

This is done using a technique that

Hodgkin and Huxley developed called the voltage clamp.

And by clamping the voltage, by controlling it, they

were able to, they were able to isolate particular currents.

In this case, the, the, sodium current.

And, the argument that we're going to make.

The lesson to, that is going to be, that you should take home

from this, is similar to what we saw with the Novak and Tyson models.

That when you do the simulation of a simplified experiment, air voltage

is being controlled rather than, you know, having a complicated time course.

Simulating this simplified experiment is also, in the case of

the Hodgkin-Huxley model, critical for the model development and in understanding.

So, that's going to be a theme that we're going to, that we're going to touch on

that's very similar to what we already

encountered when we discussed the cell cycle.

Before we get specifically to the development of the Hodgkin and

Huxley model, and before we get to the experiments that were done

using the voltage clamp technique, in part one, in our first

lecture on action potential models, we

have to discuss some biological background.

And so what we're going to discuss in this first lecture is

some interesting nonlinear behavior that is observed in, in excitable cells.

Neurons being one example of, of an excitable cell.

But skeletal muscle is another excitable

cell, smooth muscle, cardiac muscle, beta cells

of the pancreas, there are all, all fall under the category of excitable cells.

We have to define some term in order

to understand the biology and then the other topic

we have to discuss in this first lecture

is concepts of electro, electrochemical potential and driving force.

And by this we mean, specifically, driving force for ions.

To cross from one side, one side of

the membrane to the other side of the membrane.

I've used the term action potential already, I haven't

discussed what that really means, I haven't defined it.

And I, alluded to some, interesting nonlinear behavior in the,

the neuron, so let's begin by talking about what this un,

un, what I mean by this unusual nonlinear behavior and that

will give us a chance to define the term action potential.

A neuron, in this case a, a squid giant axon, if

no, no electrical stimulus is applied to it and you have

an electrode in the, in the neuron, you're measuring the voltage

from the inside of the neuron to the outside of the neuron.

When the neuron is at rest, the voltage will be constant.

And it'll be constant at a level of around

minus 60 millivolts is, are the units in this case.

So V in this case is for voltage, and minus 60.

The units in this case are, are millivolts.

Now, if a single electrical stimulus is applied to this neuron, what will happen?

This is a single stimulus, here, called I, little i stem, big I stem rather.

I in this case is short for current.

If the current stimulus is applied to this neuron, what will happen to

the voltage is the voltage will go up, and it will come down.

It will go down to a level that's, of greater magnitude than minus 60.

It will go down to around minus 70 or so, and then it will come back to minus 60.

And this will all ha, happen within a few milliseconds.

This voltage wave formed here is what, what we define as action potential.

These are seen in, in all of the neurons

in your body in, in response to these electrical stimuli.

And this is very interesting nonlinear behavior, this

this is, happens after the stimulus is applied,

so this is autonomous to the, the neuron

and it's this characteristic a time course error.

In contrast, if a sustained stimulus is applied, this is a stimulus

that's over and done with in, in, say the course of a millisecond.

If a stimulus current is applied for, for many milliseconds.

What you can see, in some neurons, are these repeated action potentials.

One, and then a second one, and then a second one, and then another one here.

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Another interesting and unusual nonlinear behavior that's observed

in the neuron is the sharp response threshold.

This is a paper from Hodgkin.

We're going to discuss Hodgin and Huxley.

So the mathematical model there was published in 1952 but

they were working on this for a very long time.

And in 1938, Hodgkin published this paper that showed this sharp response threshold.

Showing that if a weak stimulus is applied, there

will be a, a small change in the voltage,

a little bit greater, a little bit greater, a

little bit greater causes a little bit bigger of change.

And then, as you apply the voltage a little bit

more, all of a sudden you see this, this threshold behavior.

You see this sharp response.

And if this threshold, behavior looks familiar to you, it should.

Because this is very similar to what we saw in,

in some of the other, biological systems that we encountered.

Now all of this is, all this background is

just to convince you that this is unusual non-linear behavior.

It's not very trivial to describe this mathematically to say,

okay, how can you get a neuron that rests and under

normal conditions, without a stimulus, okay, and it can exhibit either

single stimulus or sustained stimulus, and it can exhibit a threshold.

So, that's what we want to address in these

lectures on action potential models and on the Hodgkin-Huxley

model, is how can we account for this

type of interesting, nonlinear behavior in a quantitative way?

Before we discuss electrical behavior in neurons in,

in more detail, we need to define some terms.

You're going to hear me use terms

such as depolarization, repolarization, and, and hyperpolarization.

And I think it's important to note that we use these in a, in a fairly loose way.

That might not be technically correct.

[NOISE] If you have a membrane that's at zero millivolts.

That doesn't have any voltage.

What I mean by that is, if you measure the

voltage on the inside, minus the voltage on the outside.

This would be an unpolarized membrane.

Because, you would have no voltage

difference between the inside and the outside.

Now, what I showed on the previous slide, is on a resting

neuron you can measure a voltage of around minus 60 millivolts and

so, because there's a voltage difference between the inside and the outside,

this is a membrane that we would, we would call a polarized membrane.

So, where the term depolarization comes from is when the voltage goes up,

when it goes from minus 60 millivolts

towards zero millivolts, you're losing this polarization.

It's moving from being polarized towards being unpolarized,

and so that's why we call it a depolarization.

It's because, it's losing polarization.

And therefore, if you went more negative than

minus 60 millivolts, that would be called hyperpolarization.

But where the terms that we use

in, in physiology are, are technically not correct.

Is that what would happen if you went from minus

60 millivolts to 0 millivolts, and then you kept going?

You went above 0 millivolts.

Then, you would go from depolarized to unpolarized,

and then you would polarize the membrane again.

And then, if you went from zero millivolts up to some value, say plus 30

millivolts, and then you came down towards zero

millivolts, that would be depolarizing the membrane again.

And this would be really confusing, because here

depolarization means that the voltage is, is going up.

And here, depolarization means that the voltage is going down.

So, these terms would be technically correct.

But what we actually use in, in physiology, when we're

discussing electrical behavior and, and neurons and myocytes, is whenever the

voltage is going up, we call that depolarization, and then

when the voltage is coming back down, we call that repolarization.

And again, if it becomes even more negative than the

resting level we started with, we can call this hyperpolarization.

So, these are the terms we're going to use.

Depolarization for going up.

And then repolarization, for coming back down.

Even if these terms are not, precisely correct.

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Now, we want to ask, where do these changes in voltage come from?

What causes the membrane to depolarize, or to repolarize, or to hyperpolarize.

And the answer is that voltage changes in,

voltages changes in excitable cells result from ion movements.

So that, if a cation, a positive

ion, flows in, that will depolarize the membrane.

But if a cation flows out, that will repolarize, or hyperpolarize the membrane.

So, if we look at an action potential in a, in a

neuron such as this, where it starts at minus 60, it goes

up, it comes down, it goes to a value that's even more

negative than minus 60, and then it comes back to minus 60 again.

When the voltage is going up, when it's

depolarizing, that's because the cation is going in

or an anion is, is going out, when

it depolarizes, that's because a cation is going

out or an anion is going in and then, when it comes back to minus 60

again, after the hyperpolarization, that's because a cation

is coming in or an anion is going out.

And this leads us to two questions that we're going to address in these lectures.

One is, how can we describe

the depolarization or repolarization process quantitatively?

And then, the second question we want to address

is, which ions are the most likely candidates?

So, if I said the voltage is going up here

and this is because the cation is, is going in.

One, you can imagine several different

cations, potassium, magnesium, lithium, calcium, sodium.

So, we want to address which ions are the most

likely candidates for this phase, this phase or this phase.

To address the first question, how

do you account for voltage changes quantitatively,

we have to understand that the cell

membrane basically behaves as an electrical capacitor.

Capacitor's are something that many of you

may have encountered in, in say physics class.

And what you learned in freshman physics classes for instance.

That if you connect the battery to a com, a capacitor

is basically two parallel metal plates, that are separated by, by air.

If you collect, connect the batteries to these two parallel

plates, positive charges will come off the battery onto this plate.

Negative charges will go off the negative terminal

of the battery onto this bottom plate, here.

And the way that you describe voltage changes

on this capacitor are through this equation here.

Capacitance times the change in voltage with respect to time,

is equal to the ionic current that flows onto the capacitor.

And we if we think about cell membranes, the structure of

cell membranes, we can understand how this can behave as a capacitor.

A cell membrane is a bunch of phosphor lipids that are arranged in a bilayer

and the, the cell membrane is hydrophobic on the inside where the lipid groups are.

And that's hydrophilic on the outside, where the charged

parts are the, are the phospholipids are, are located.

And so, this basically behaves as two parallel plates, where the

charges can't easily move across from one side to the other.

But the charges kind of collect either on a inside face of a

cell membrane or on the outside face of the, of the cell membrane.

So, this does have this sort of parallel plate arrangement similar to what

you see in an idealized capacitor because the cell membrane is a bilayer.

And this, because the cell membrane behaves as a capacitor, this

is where we get our differential equation for, for membrane voltage.

The equation that we're going to have to voltage in this

case looks very similar to the equation we have up here.

Capacitance times the change in voltage with respect to time is equal,

in this case it's going to be the negative of the ionic current, but

that's just convention based on how we define one direction as negative and

another direction as positive, and we will discuss that as we go on.

The first question we addressed is accounting for

changes in membrane potential or membrane voltage quantitatively.

The second question we said we wanted to

address is, which ions are the most likely candidates.

And to understand which ions are the most likely candidates,

we need to know what the ionic concentrations are in cells.

In a mammalian ventricular myocyte, for instance, these

are typical of the equations you might observe.

On the outside of the cell you might see five millimolar potassium, 140 millimolar

sodium and two millimolar calcium, and on the inside you would see, much

greater concentration of potassium, a much lower

concentration of sodium and then, a much,

much, much lower concentration of calcium on the inside compared to on the outside.

This is typical of, of mammalian cells the

specific numbers come from a ventricular myocyte but, these

this general sort of general pattern here is very

similar to what you see in all mammalian cells.

High potassium on the inside compared to on the outside.

High sodium, and calcium on the outside compared to on the inside.

In the model that we're going to discuss, of the squid giant

axon, that was derived and, and built by Hodgkin and Huxley.

We had, the concentrations are a little,

are different, although the pattern's the same.

Remember that squids live in the ocean, they live in salt water, so their

concentration, their osmolarity is, is much higher

on both the inside and the outside.

But the pattern is the same, where you see a

higher concentration of potassium on the inside than on the outside.

Here it's 400 and 20, rather than 140 and five, but much higher in

the inside than on the outside with

potassium, but the reverse is true with sodium.

Much higher sodium on the outside than on the inside.

What we can conclude from this is that

diffusion is going to want to drive sodium from

outside to inside and diffusion is going to

want to drive potassium from, from inside to outside.

And these movement are going to

either depolarize or hyperpolarize the membrane respectively.

So, if sodium is going in, you have cations

going into the cell, that's going to depolarize the membrane.

And then if potassium comes out, that potassium coming

out is going to depolarize slash hyperpolarize the membrane.

We just discussed ionic concentrations that are present in cells, either

mammalian cells or in squid giant axon in the case of the Hodgkin and Huxley model.

And so, let's consider what happens when we have a membrane that

might be permeable to one ion and not permeable to other ions.

In other words, some ions can cross the membrane readily and other ions can't.

So, this this is a thought experiment here which we call a concentration cell.

Imagine we have 100 millimol of potassium chloride in

this chamber here, which we're calling i for inside.

And we have 20 millimol of potassium chloride in

this chamber here, which we're calling o for outside.

So we can say, by diffusion, potassium is going to

want to move from this side to this side.

Let's also imagine that these two chambers,

the inside and the outside chamber, are separated

by a membrane and the membrane is permeable

to potassium but is not permeable to chloride.

If the membrane were permeable to both, then

both potassium and chloride would, would cross the cell

membrane and you would end up with 60 on either side, 100 plus 20 divided by 2.

But, in this case, the membrane is

permeable to potassium and not permeable to chloride.

So, let's think about what happens if you started with 20 and 20 and then you

instantly raise this left hand side, the inside

chamber, from 20 to 100, what would occur?

Well, because you'd have more potassium ions on the

left, you'd have potassium ions flowing from left to right.

But you would have an excess of positive charge on

the right chamber, than you would on the left chamber.

So, excess positive ions on the, on the right would reduce the voltage

difference because, remember, this membrane is a,

is a capacitor, as we already discussed.

And, because you have a buildup of positive charges on the, on the

right hand side here, you've developed

a voltage difference across the cell membrane.

And the voltage difference, having more positive charges on the

right, would oppose the left to right movement of potassium.

And eventually, you'd have an equilibrium.

You'd have an equilibrium where the voltage difference is

trying to push potassium ions from the right to the

left would balance the diffusion, which is trying to push

potassium ions from the, from the left to the right.

So, we can understand this just by going through a thought experiment like this.

Now the question becomes, how can we

understand this process in more quantitative terms?

In order to, to understand ion movement across cell membranes in

quantitative terms, we need to introduce

this concept called the electrochemical potential.

Electrochemical potential of a species can be calculated like this, mu equals mu 0,

mu naught, plus RT times natural log of concentration plus z, times F times V.

What is the significance of each term?

Mu zero, in this case, is what

we called the standard electochem, electrochemical potential.

But we, we're interested in potential differences from the inside

membrane or the membrane to the outside of the membrane.

And mu zero is going to be the same on both sides of the membrane.

So, from now on, we're going to ignore this.

R times T times natural log of C.

This term describes diffusion.

R and T, in this case, are the

gas constant is R, the absolute temperature is T.

And this is taking the natural log of C being the concentration.

And this term describes diffusion.

In other words, a higher concentration

leads to a higher electrochemical potential.

If you have more concentration, then this term,

natural log of C, is going to get bigger.

Therefore, your electrochemical potential is going to get bigger.

In the third term here, little z times F times V.

Little z refers to the valence of your ion.

So, that would be plus one for,

a monovalent cation, such as sodium or potassium.

F is a term called Faraday's constant.

And V, in this case, it the voltage.

This terms describes the electrical effects.

Greater voltage means you have a greater electrochemical,

electrochemical potential when you have a positively charged species.

For instance, z is greater than, zero.

So.

What you can conclude from looking at this equation for electrochemical potential,

is that when the concentration gets

bigger, your, your potential's going to go up.

And then, when the voltage gets bigger, your potential

goes up, assuming that you have a positively charged, ion.

And the reason that we care about this is we

can understand how ions move

by calculating this electrochemical potential.

And it's analogous to what you have with a ball rolling down a hill.

You probably, you may have learned about potential energy in,

in the context of things like, you know, lifting something up.

When you raise, when you, you know, take a ball,

take a ball or a rock and you life it up

to the top of the hill, you are increasing it's

potential energy, because then you can roll it down the hill.

Just like a ball rolling down the hill, an ion will want

to move from higher electrochemical potential

to lower electrochemical potential, just like

the potential energy that you you impart to a rock or to

a ball when you lift it up to the top of a hill.

When we compute electrochemical potential, the units

in this case are in joules per mole.

It's a way of calculating energy, but it's energy normalized to how much is present.

Let's go back now to our example of the concentration cell, where we

had high potassium on the inside

compartment, low potassium on the outside compartment.

We said that potassium would move from left to right,

due to diffusion, we'd get a build up of charge and

the electrical field, the voltage difference this created, would put,

would be acting to push potassium ions from right to left.

So, we can compute that at equilibrium the potential on the

inside compartment has to equal the potential on the outside compartment.

And therefore we can set R times T

times natural log of concentration on the inside,

plus zF times voltage on the inside equals

this term here, with inside replaced with, with outside.

And remember, we've ignored the standard electrochemical potential, mu naught.

Because we decided that that was the same on both the inside and on the outside.

So, in this case, we can rearrange terms.

We can take the two voltage terms, and move them to one side of the equation.

And the two concentration terms, and move them to the other side of the equation.

And then we can compute voltage on the inside

minus voltage on the outside is equal to RT

divided by little z times F times natural log

of concentration on the outside over concentration on the inside.

This is a definition of the equilibrium potential,

or what is commonly called the Nernst potential.

What this means is that this is the voltage

that you can be at where the diffusion term pushing

the potassium from left to right exactly, exactly balances the

electrical term that's pushing ions from, from right to left.

So, when you're at this particular voltage, that's calculated

as follows, then you're at equilibrium, and because this was

this was originally def, derived by someone, a man

named Nernst, this is what we call the Nernst potential.

>> Now, if we go back to the squid giant axon, which

we said had electric, ionic concentrations like this, high potassium on the

inside, high sodium on the outside, each ion, both sodium and potassium,

are associated with their own Nernst potentials which we can compute like this.

E sub X equals RT over zF times the natural log of whatever the

species X is on the outside, over whatever the species X is on the inside.

And if we plug in these numbers that

we have for potassium and, and sodium up here,

we can compute that, E sub K, or

the equilibrium voltage for potassium is minus 72 millivolts.

Th equilibrium voltage for sodium is plus 55 millivolts

and what this means is that if you're exactly at

mi, minus 72 millivolts, potassium doesn't want to move in

or move out and if you're exactly at plus 55

millivolts, sodium doesn't want to move in or move out.

You can take that a step further and say that however far away you are

from minus 72 millivolts is however much potassium wants to move in or move out.

In other words, the distance away from the

reversal potential, if you take the current voltage minus

the reversal potential, E sub X, that is what we call the driving force for ion X.

What I mean by that in, in simple quali, qualitative terms is, if we went from

minus 73, from minus 72 to minus 73, how much would potassium want to move in?

Well, it might want to move in a little bit,

but not that much, because you're almost at equilibrium, right?

What if we went from minus 72 all the way to say, minus 172?

What if we went if we hyperpolarized the voltage by

100 millivolts, well then we would be very far away

from equilibrium, and then potassium would want to move in

and it would want to move in quite a bit.

So, the way that we would specify that quantitatively is

we compute the voltage minus the Nernst potential, and that

tells us what our driving force is, how much does

this, this ion want to move in or move out.

And the answer is, it depends on how far away you are from equilibrium.

What this means in, in, in physical terms,

is we're basically converting electrochemical potential from units

of joules per mole into units of volts,

and volts are defined as joules per, per coulomb.

So, we take our voltage difference, our, our driving force, voltage minus

Nernst potential, that's our dif, difference

in electrochemical potential divided by Faraday's constant.

And if you work out the, the units there, you'll see that that's

going to be units of joules per coulomb, rather than in joules per mole.

Finally, why do we go to all this trouble

of defining the Nernst potential, defining the driving force?

Well, the answer is, knowing the driving force is

going to help us to mathematically compute ionic currents.

We said that driving force is defined

as a voltage minus a particular Nersnt potential.

So, you have one driving force for potassium.

Another driving for for sodium.

Here, we're just using sodium as an example.

But the same considerations would apply, to potassium.

If the voltage minus the Nernst potential is greater than 0.

Then the, difference in electrochemical potential is greater than 0.

And that means that sodium is going to move out of the cell.

Most of the time in the in the context in the squid giant

axon, we're going to have the second condition, where the voltage minus the

sodium Nernst potential is less than zero, then delta mu is going to

be less than zero and then sodium is going to move into the cell.

Now, let's think about this in terms of units, we

have a voltage here and we want to compute the current.

The way we can compute a current from a voltage difference,

from voltage minus a Nernst potential, what the term you need in

order to get the units to work out is a conductance,

a conductance in this case is the reciprocal of, of a resistance.

So, this is equation we're going to use for our, calculate our sodium current

in the Hodgkin-Huxley model, where the current

resulting from sodium, is a conductance for sodium,

g sub NA times the voltage minus the Nernst potential for sodium, V minus

E Na and conductance times voltage is going to give us currant, in this case.

And so, this is just to, to review how the units are going to work out.

Usually voltage and Nernst potentials are given in millvolts in, in physiology.

Our currant, in this case, is in fact going

to be a currant density, which is going to be

in units of micro amp here is in per

centimeters squared and this is a way of normalizing it.

To the membrane area.

Conductance is, therefore, also going to have to be normalized.

This is going to be in units of millisiemens per centimeters squared.

The other thing we see here is, we said that if V minus ENa is less than zero.

Sodium is going to move into the cell.

In that case, if V minus ENa is less than

0, than our current is also going to be less than 0.

And this a convention that we use in physiology.

By convention, inward current is negative.

So, this seems like a really simple equation here.

If you know what your voltage you're at,

you know how that relates to the Nernst potential.

You can compute the current just by multiplying it by the conductance.

What we're going to see in the next lectures

on this topic is that this is simple

in principle, but the things makes it complicated

is your conductance, your g sub X, in

other words, your g sub Na or your g sub K can be dependent on both

voltage and time, and it's this voltage dependence

and this time dependence that makes this complicated.

And this voltage dependence and time dependence is also what gives

us the interesting non-linear behaviors that

we already illustrated at the beginning.

Now, to summarize this first lecture on the,

action potential model development by Hodgkin and Huxley.

We see that neurons can exhibit complex non-linear behavior.

And this is complex non-linear behavior

that is very challenging to describe mathematically.

So, that's what we're going to go through in the next couple of lectures

is how can we develop a

mathematical representation of this complex non-linear behavior.

The second thing, thing we've seen is changes in the

membrane potential or voltage, we're going to use these terms interchangeably.

Membrane voltage and membrane potential.

These change can result from ion movements across the cell membrane.

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The final thing we've seen is how can we decide if

an ion is going to move in or it's going to move out?

Well, that is determined by electrochemical potentials,

electrochemical potentials which determine which direction ions move.

And we can compute a Nernst potential for each ion.

And the Nernst potential represents the equilibrium where

the ion doesn't want to move in or move out.

And it's when you get away from that Nernst potential

that the ion is going to move either in or out.

And these concepts that we've talked about in

this first section are going to be critical.

For understanding the Hodgkin Huxley model in, in more quantitative terms,

which is what we're going to do in the next couple of lectures.

Lecture 20 - Mathematical Models of Action Potentials - Part 1

系统生物学的动力学建模方法

与讲师见面