In the last lessons we learned
that the phospholipid bilayer of the cell membrane
electrically acts as a capacitor,
and that transmembrane ion channel proteins
act as conductances for specific ions.
In this lesson, we'll see how the biophysical properties
of electrochemical diffusion
determine the membrane potential.
Let us begin by considering a piece of membrane
with a potassium-selective ion channel in it.
There are two main forces that act upon the potassium ions.
As we've already discussed,
there's an electrical potential across the cell membrane,
the membrane potential.
That membrane potential causes an attraction or a repulsion
for ions of a given charge.
If we have a negative potential on the inside
then potassium ions will be attracted inside,
down the electric-field gradient.
In general, there's a force applied to any charged ion
that depends upon the spatial gradient of the electric field.
So one major determinant of ionic movement across ion channels
is the electric field.
The other major push that moves ions
from one side of a membrane to another
is concentration.
If there's a concentration gradient so that, for example,
potassium might be high inside a cell
compared to the outside of a cell,
then naturally there's a gradient for potassium
to flow down the concentration gradient.
The electrical and the concentration gradients
are the two main forces
that drive electrochemical diffusion
across the plasma membrane.
The two forces: the concentration gradient
and the electrical gradient,
can oppose each other,
and can precisely bounce each other out
so that at one specific membrane potential
it might prevent the overall net flux of ionic flow
induced by concentration gradients.
For example, for the case of potassium,
which is very high inside a cell,
there's a natural tendency
for potassium to want to leave a cell
down its concentration gradient of about 150 mM inside
to about 5 mM potassium outside a cell.
If we make the inside of a cell more negative,
then that will then try to attract positive ions
inside the cell.
If we make the membrane potential very negative,
for example -19 mV,
it's so negative that that precisely counters the flow of ions
induced by the concentration gradient.
The reversal potential is the potential
at which there's no net flux of ions,
and that can be calculated exactly
from biophysical considerations.
The Nernst equation, here described for potassium ions,
precisely describes the relationship of the potential at which an ion,
given the intracellular and extracellular concentrations
are in precise match, so that there's an equilibrium,
and no net flux of ions
across the potassium-selective conductance.
The Nernst equation describes that the reversal potential
is given by the gas constant, Avogadro's gas constant,
the temperature in kelvins, divided by the Faraday constant,
and Z is the charge on the ion.
For potassium, that's +1,
for chloride, it's -1,
and for calcium, it's +2,
so that's the charge associated with the ionic species.
RT over zF multiplies the logarithm
of the ratio of the outside to the inside ionic concentration.
At 37 degrees Celsius,
so that's at physiological temperatures,
RT over zF for a monovalent, positively-charged ion,
and converting this to a base 10 logarithm,
gives us a 61.5 times base 10 logarithm
of the ratio of the outside
to the inside potassium concentrations,
here we've set as 5 mM potassium;
outside, 150 mM of potassium.
Inside, if we do the math,
we find out that that comes out
with a reversal potential of -90 mV.
That's the membrane potential
that's needed to counter
the concentration gradient
that tries to drive potassium out.
We counter that by having a negative potential on the inside
that tries to attract the potassium channels in,
and at this potential, with these concentrations,
it's precisely balanced, it's at equilibrium,
at the Nernst equilibrium potential.
Although potassium is the most important ion
in determining membrane potential,
it is certainly not the only one.
And so we need to find out
what the relevant biological concentrations are
of ions on the intracellular and extracellular side.
And in general, these are approximate values,
but are entirely indicative.
The potassium ions are high intracellularly at about 150 mM,
and extracellularly low at about 4 mM.
Sodium ions are low inside cells, around 12 mM,
and extracellularly they're high,
at somewhere around 140 mM.
Chloride concentration is also low inside cells,
around 4 or 5 mM,
and extracellularly is high at around 120 mM.
Calcium ions are low inside cells, at about 100 nM,
and much higher outside, around 1-2 mM outside cells.
These concentrations, inside and outside,
can then be plugged into the Nernst equilibrium potential
to define the reversal potentials
of these different ionic species.
As before, we see that the reversal potential
for potassium ions is very negative.
The sodium ions have positive reversal potentials,
and the reason is that the relative concentrations
of sodium and potassium are opposite to each other.
Potassium concentrations are high inside the cell,
and so potassium wants to leave the cell.
In order to counter that concentration gradient
of potassium wanting to leave the cell,
we can apply a negative potential
to try and attract those positive ions back
to reach a equilibrium situation
where potassium has no net flux
through potassium-selective ion channels.
For sodium, the situation's different.
We have a high extracellular concentration of sodium.
Sodium wants to flow inside a cell,
and in order to prevent that flux
we would have to have a positive potential inside a cell
that would then repel the positively-charged sodium ions,
and so we have a positive reversal potential for sodium.
For chloride we have a similar situation as with sodium,
in that it's high extracellularly, and low intracellularly,
so chloride, through its diffusion down the concentration gradient,
wants to enter a cell,
and bring negative charges inside a cell,
and in order to prevent that,
we would then have to have a negative potential
that would then repel the negatively-charged chloride ions
from entering the cell.
And, finally, calcium has an enormous concentration gradient,
10,000 times more calcium outside a cell than inside,
and that then naturally means
that whenever a calcium conductance opens,
calcium flows inside the cell,
and you need to go to very positive potentials
in order to reverse the effect of a calcium conductance.
In general, a cell has permeability to many different types of ions.
And so there are permeabilities to potassium, sodium and chloride
that are all mixed on the plasma membrane of a single cell.
And indeed, an individual ion channel,
although primarily selective for, say, potassium,
would also have some degree of permeability for other ions,
and so even a single ion channel opening
doesn't have a reversal potential
that's exactly that of an individual ionic species,
but rather it would be a mixed permeability
of several different ions.
In order to calculate the equilibrium potentials
when there are permeabilities to several different ions,
one can use the Goldman-Hodgkin-Katz equation,
which is a modified version of the Nernst potential.
And it states that the equilibrium potential
is equal to RT over F again
times a natural logarithm of the permeability to potassium
times the extracellular potassium concentration,
plus the permeability to sodium
times the extracellular sodium concentration,
plus the permeability to chloride
times the intracellular chloride concentration
divided by the permeability to potassium
times the intracellular concentration of potassium
plus the permeability to sodium
times the intracellular sodium concentration
plus the permeability to choride
times the extracellular chloride concentration.
Note that whereas here we have the extracellular concentration
for the positively charged cations of potassium and sodium,
here we have the intracellular concentration of chloride,
because this is negatively charged.
As an example we can consider a cell
that has a relative permeability ratio of potassium, to sodium to chloride
of 1 to .04 to .45.
So this would be a cell that has a high potassium conductance,
a much lower sodium conductance,
and an intermediate chloride conductance.
If we plug the numbers in,
and the concentrations from the previous slide,
we reach a situation where the equilibrium potential
for this particular cell, with a high potassium permeability
and a high chloride permeability
has a membrane potential of around -76 mV.
And so in this case we have a negative membrane potential,
and that's because our permeability
is dominated by potassium and chloride.
Potassium has a high concentration inside a cell, it wants to leave,
and so that creates negative charges.
Chloride has a high concentration outside cells,
and therefore wants to enter, again bringing in negative charges.
And so you can see that at resting conditions, at equilibrium,
we would expect hyperpolarized, negative membrane potentials,
sitting somewhere close to the potassium and chloride reversal potentials,
in this case -76 mV.
We can now also return to our electrical model of a cell.
As before, we have membrane capacitance
formed by the lipid bilayers.
We have ionic conductances
where we now add the reversal potentials for the individual species,
so we have a potassium conductance
with a reversal potential of potassium,
and that's here, this potassium ion channel.
We have sodium conductances with the reversal potential of sodium
going through the sodium ion channels,
and we have chloride conductances,
chloride ion channels,
again with their own reversal potential.
We can use our two simple electrical equations
to derive some interesting facts
about how the membrane potential behaves.
We can use Ohm's law,
thinking about the ionic conductances
where we have the voltage is equal to the current flow times the resistance,
and we know that the resistance is 1 over the conductance,
so we can rewrite that, and say that the current flow
is equal to the voltage times the conductance.
And for the capacitance we know that the charge stored in a capacitor
is equal to the capacitance times the voltage.
We also know that a current
is equal to the rate of change of charge.
And so we can differentiate this equation
to give us the capacitative current flow,
which is equal to the derivative of this.
The capacitance times dV by dt.
If we now think about the electrical circuit of the cell
we can say that the total transmembrane current, Im,
can be composed of four different channels.
There's the capacitative current, Ic.
We have a potassium current, Ik.
We have a sodium current, INa,
and we have a chloride current, ICl.
And that's what we have here,
we simply sum up these different conductances.
Now we've already said that the capacitative conductance
is equal to C, dV, dt.
We can, by inspection, see that the potassium conductance
is equal to the difference between the membrane potential
and the reversal potential of potassium
times the potassium conductance,
and so on for sodium and for chloride.
At the membrane potential that is equal to the reversal potential,
there's no current flux, as expected.
Now, we can put these equations together
and solve for membrane potential,
and we can do it under one interesting condition,
the steady-state equilibrium condition
where the total transmembrane current is zero,
and by definition, the rate of change of membrane potential is also zero.
In order just to simplify the writing of our equations,
we can consider the total conductance of the cell
as being the sum of the potassium, sodium and chloride conductances.
The solution to the equation we presented on the previous slide
then has that the membrane potential
is equal to the ratio of the potassium conductance,
relative to the total conductance
times the reversal potential of potassium
plus the ratio of the sodium conductance to the total conductance
times the reversal potential of sodium
plus the ratio of the chloride conductance to the total conductance
times the reversal potential of chloride.
So in order to change the membrane potential,
what we have to do is to change the relative importance
of each of these ionic conductances.
If we want to have a membrane potential
that's quite close to the reversal potential of potassium,
then the potassium conductance should dominate the total conductance.
Similarly, if we want to have a membrane potential
that begins to approach the reversal potential of sodium,
then the sodium conductance should dominate
the total conductance of the cell.
By inspecting this equation
you can see that it's very easy to change the membrane potential of a cell.
All you need to do is to change the conductance
of different ionic species,
and you'll change the membrane potential,
forcing it towards these different reversal potentials
that are being driven to by the conductances.
So we've now seen that simple biophysical laws
determine electrochemical diffusion.
Concentration gradients and electrical fields
determine the flux of ions across the plasma membrane.
We can define reversal potentials given by the Nernst potential,
and we can see that it's easy
to move the membrane potential from one value to another
by simply changing the relative permeability
of the plasma membrane
to specific ionic species,
each of which have their own reversal potential,
determined by their concentration gradients.
Let's next consider some specific examples,
and get towards some numbers and dynamics
as to how membrane potential changes
when ion channels open.
Let's consider what happens if potassium channels
suddenly increase their open probability.
Let's imagine we start off with a simple situation
where the membrane potential is at 0 mV initially.
We have a potassium-rich intracellular,
and lower concentration of potassium outside the cell.
A potassium conductance suddenly opens,
and of course potassium ions want to leave,
and go down the concentration gradient,
and create a negative potential inside.
We've just learned that there's a potential
to which the membrane potential will tend to.
The membrane potential will go
towards the reversal potential for potassium,
which is around -19 mV.
But let's think about the dynamics of the situation.
If we have time running here,
we begin and think about the potassium conductance.
It starts off low, and at some stage it suddenly jumps up.
Membrane potential we say starts at 0 mV,
and we know that at equilibrium
it's going down to the reversal potential of potassium,
at around -19 mV.
What are the dynamics?
How does the membrane potential
reach the equilibrium potential for potassium?
Let's return to our electrical equivalent of a cell,
where we have a capacitance,
and we have a conductance.
And together they define the membrane potential and its dynamics.
We then have a potassium reversal potential,
and we have a resistor or a conductor,
depending on how we want to think about it.
We know from capacitors equations that Q is equal to CV,
and that the current flowing in the capacitator is a derivative,
the rate of change of charge over time: C, dV by dt.
We also know that the membrane potential,
from considering this arm of the equation,
that the membrane potential is equal to E + IR.
That's simply just the Ohm's law here,
if we consider this as a resistor here,
just for the sake of simplicity.
Now this current that's flowing through here,
in this particular case that we're thinking about,
is actually the capacitative current.
We can therefore rewrite this equation as V equals -RC, dv by dt,
and the minus sign is because they're flying in opposite directions,
in terms of the electrical circuit diagram.
You can solve this equation,
and because it's a differential equation
you expect to find exponentials present in it,
and you find out that the solution can be written in this form.
Here 𝞃 (tau) is a membrane time constant,
which is equal to the resistance times the capacitance.
You can check this equation, of course, by plugging it in
and seeing that it comes out correct.
So what we see, from this equation here,
is that at times zero, the membrane potential is indeed 0.
At time infinite, this disappears,
and we reach the reversal potential for potassium,
and in between we have an exponential time course
that then brings us to the reversal potential of potassium,
and that is characterized by a time scale 𝞃 (tau),
over which it goes to 1 over E of the membrane potential,
so that's about 63 percent.
That time course, 𝞃 (tau),
is given by the resistance times the capacitance,
which is also the capacitance divided by the conductance.
So the rate of change of membrane potential in a cell
is determined by a number of different factors.
If the cell has a high resistance, a low conductance,
then the time course is long,
it takes a long time to change.
If the capacitance is high,
then it also takes a long time to change,
and both of those factors make sense.
If there's very little conductance then it takes time to move
sufficient charge to change the membrane potential.
If the capacitor's very large, we need a lot of ions
to change the membrane potential.
This equation, then, tells us also
that membrane potential can't change instantaneously,
rather it changes with a time scale
that is equal to RC, the membrane time constant.
In general, of course, potassium conductances
aren't the only important conductance in a cell,
and we have to consider both potassium, sodium,
and chloride conductance changes
if we want to understand how the membrane potential moves.
We know that at equilibrium membrane potential
will be driven towards the potassium reversal potential,
if the potassium conductance increases
it'll be driven towards the sodium reversal potential.
If sodium conductance increases
it'll be driven towards the chloride reversal potential,
if a chloride conductance increases.
And, typically, this is sitting at +60 mV minus 90 mV for potassium
and around -70 mV for chloride.
If we then begin with a membrane potential sitting at -55 mV,
and that's our membrane potential,
and we open a sodium conductance
then with an exponential time cross
we head towards the sodium reversal potential.
Again, this membrane potential doesn't occur instantaneously,
but depends upon the membrane time constant RC.
If we open a chloride conductance
then the membrane potential
tends towards the chloride reversal potential,
and so on, also, for the potassium conductance.
So we can move membrane potential
by simply opening the conductance for sodium,
and then it goes towards the sodium reversal potential,
towards the chloride reversal potential,
or towards the potassium reversal potential.
In general, if we make the membrane potential more positive,
this is called a depolarization.
And if we make the membrane potential more negative,
then it's called a hyperpolarization,
and these are terminologies that we'll consider
and talk about a lot in the next lessons.
Depolarization for more positive membrane potentials,
and hyperpolarization for more negative potentials.
And so if you want to hyperpolarize a neuron
we need to open a chloride or a potassium conductance,
and if you want to depolarize a neuron
we want to open a sodium conductance.
In this lesson we've learned
a number of important biophysical principals
that ultimately lead us to find out
how the membrane potential of a cell is determined.
We've seen that electrochemical diffusion
describes ion flow through ion channels,
and so there are two important forces
that act upon ions.
There's a concentration gradient, where ions want to move
from high concentration to low concentration,
and there's an electric field,
which also acts upon the individual charged ions.
Together, these two forces, electricity and concentration gradients,
can mathematically be formulated into the Nurnst equilibrium potential,
which is the electrical potential
that precisely counters the concentration gradient,
and leads to no net iron flux.
By independently regulating the ion channel conductances
with selective permeability
we can control the membrane potential.
If we open a sodium conductance,
the membrane potential will move towards the sodium reversal potential,
and depolarize the cell.
If we open a chloride, or a potassium conductance
we will cause negative charge to accumulate inside the cell
causing a hyperpolarization.
These basic principals of ionic conductances
driving membrane potential changes
are fundamental to understanding how neurons work.