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Lecture Comments (6)

1 answer

Last reply by: Professor Hovasapian
Sat Oct 10, 2015 8:00 PM

Post by Kenny Patel on October 10, 2015

At 17:13 did you mean to say myoglobin or am I not understanding correctly?

1 answer

Last reply by: Professor Hovasapian
Sun Jun 29, 2014 6:01 PM

Post by Sitora Muhamedova on June 29, 2014

I cannot describe how much grateful I am for your wonderful lectures!

1 answer

Last reply by: Professor Hovasapian
Sun Sep 15, 2013 6:29 AM

Post by Vinit Shanbhag on September 14, 2013

Hey Raffi,
What is the difference between alpha and (alpha/1-alpha). Alpha is the fractional occupancy as you said in the last lecture, but what does the y axis in the hill plot actually mean??

Protein Function II: Hemoglobin

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Protein Function II: Hemoglobin 0:14
    • Hemoglobin Overview
    • Hemoglobin & Its 4 Subunits
    • α and β Interactions
    • Two Major Conformations of Hb: T State (Tense) & R State (Relaxed)
    • Transition From The T State to R State
    • Binding of Hemoglobins & O₂
    • Binding Curve
    • Hemoglobin in the Lung
    • Signoid Curve
    • Cooperative Binding
    • Hemoglobin is an Allosteric Protein
    • Homotropic Allostery
    • Describing Cooperative Binding Quantitatively
    • Deriving The Hill Equation
    • Graphing the Hill Equation
    • The Slope and Degree of Cooperation
    • The Hill Coefficient
    • Hill Coefficient = 1
    • Hill Coefficient < 1
    • Where the Graph Hits the x-axis
    • Graph for Hemoglobin

Transcription: Protein Function II: Hemoglobin

Hello and welcome back to, and welcome back to Biochemistry.0000

In our last lesson, we talked about myoglobin.0004

In this lesson, we are going to start talking about hemoglobin, so let's just jump right on in.0008

OK, now, whereas myoglobin is involved in the diffusion of the oxygen/O2 in muscle tissue, hemoglobin, that is the protein which actually transports the oxygen/O2 in the blood.0015

Excuse me.0070

Now, hemoglobin has 4 subunits; hemoglobin, which is abbreviated Hb, it has 4 subunits.0082

It is not like myoglobin; myoglobin was just a single subunit protein that has a binding site for 1 molecule of oxygen.0092

It has 4 subunits, and each of these subunits has a heme group, with a heme group.0101

We have - excuse me - 4 sites to bind O2 per protein molecule.0115

Now, the interactions of the subunits, they cause conformational changes that affect hemoglobins' affinity for O2.0131

OK, the conformational changes among these subunits, as 102 binds, there is going to be a conformational change, and that conformational change is going to affect the extent to which another O2 will bind or release depending on the physiological conditions at hand.0177

You remember, we need to actually bind the O2 tightly.0196

We need to take it to another part of the body, to the tissues, and we need to be able to release it.0201

We need that flexibility, that great degree of flexibility, in fact, because the oxygen that is coming...the lungs is going to be a particular concentration of oxygen, but that is not the same concentration of oxygen that is in the tissues.0207

We do not want some molecule that just binds O2 and hangs on to it.0223

It needs to be able to deliver it, and we do not need it to...we need...well, or if we have a molecule that does not bind it very well, well, if it does not bind it very well, then, it is not going to take up the oxygen that is in the lungs.0227

We need that high degree of flexibility based on different concentrations of O2.0242

And again, since it is a gas, we are going to be talking about different partial pressures of O2.0247

OK, now, hemoglobin has alpha and beta subunits.0252

It has 2 alpha subunits, and it has 2 beta subunits, alpha and beta subunits, 2 of each.0263

We have an alpha-1, beta-1 and an alpha-2, beta-2.0275

The alpha-1, beta-1 are...they interact rather tightly, and the alpha-2, beta-2, they interact tightly.0280

Now, there is interaction between the alpha-1, beta-2 and the alpha-2, beta-1, but it is a weaker interaction.0288

The alpha, it has 141 amino acid residues, and the beta has 146 amino acid residues.0298

OK, let's go ahead and take a look at an image of hemoglobin here.0313

We have our alpha-1, beta-1.0319

We have our alpha-2 and our beta-2.0325

Here, you can see, you have your alpha-1, beta-1, that interaction.0331

That forms as 1 subunit, 2; this is the third.0337

This is the fourth; now, you can also make out in each subunit, it has its own heme group.0341

Here is the heme group for this one, and this one is a little hidden.0346

You can see the heme group right here; let me go ahead is right there, and you have the heme group right there for that one, and it is right there for this one, and here it is for that.0350

Now, I do not know if you can actually see the oxygen because it does not look like oxygen is bound, or maybe it is.0366

It is hard to tell, but that is it; this is what it looks like, so alpha-1, beta-1, alpha-2, beta-2- that is it, nice and straightforward multi-subunit protein.0372

You have this gap in between, which is going to become very, very important in just a little bit.0383

OK, now, our alpha-1, beta-1 and alpha-2, beta-2 - OK - their respective interactions are quite strong.0390

You have about 30 amino acids between alpha-1, beta-1, alpha-2 and beta-2 that are interacting with each other, so 30 amino acids interacting.0412

That is pretty strong interaction; now, as far as the alpha-1, beta-2 and the alpha-2, beta-1 down here, alpha-1, beta-2, alpha-2, beta-1, the interactions in that region there, they are less strong.0423

I will put "the interactions are less strong", somewhere in the neighborhood of about 18 or 19 amino acid residues that actually end up interacting.0447

When we use chemical agents to actually separate these subunits, as it turns out, they break up not into 4 separate subunits.0465

We can do that if we want to, but from mild treatment, what ends up happening is the alpha-1 and beta-1 actually stay together, and the alpha-2 and beta-2 stay together.0473

The interactions are quite strong between those subunits.0480

Alright, hemoglobin - let's see - it has 2 major conformations.0485

One is called the T-state, and T stands for tense.0505

This is where no oxygen has bound to it, so this is deoxyhemoglobin.0513

This is deoxyhemoglobin, and then, there is something called an R-state.0519

In this case, R stands for relaxed, and this is deoxyhemoglobin.0523

When oxygen is actually bound to it, the more oxygen that is bound to it, the hemoglobin is in more of an R-state.0532

So, when we say it has 2 major conformations, we are not saying it goes from T to R, boom.0540

There is a transition; you have 4 oxygen molecules that are binding to the hemoglobin.0547

As one of them binds, there is some conformational change, and it starts to move away from the tense state; and it starts to make the transition over to the R-state.0553

Now, it makes the second molecule of oxygen a little easier to bind.0561

That second oxygen causes more conformational changes, shifts it more towards the relaxed state, which makes it more easy to bind the third oxygen.0566

By the time the fourth oxygen molecule binds, it is completely in the relaxed state.0575

The relaxed state, the R-state, has a high affinity for oxygen.0579

The T-state has a low affinity for oxygen.0584

Now, these terms high and low, these are relative terms.0588

We are not saying that this T-state hemoglobin has no oxygen attached to it.0592

That is not it at all; in fact, it is about 60% saturated with oxygen.0596

The difference is the R-state is about 96% saturated with oxygen.0600

It is not 0, 100; it is not all or nothing.0606

It is just relative degrees of saturation.0610

The R-state- very, very saturated with oxygen; that is the arterial blood.0614

That is the red blood that has all that wonderful oxygen in it.0618

The T-state, about 60% saturation for the hemoglobin, that tends to be more of a bluish-purple.0622

That is the venous blood; that is the blood that is actually coming back from the tissues because it has delivered its oxygen.0628

Now, it is depleted oxygen, so the color actually changes.0635

OK, now, the binding of O2...OK, as we just said, the binding of O2 to a subunit of T-state hemoglobin causes a conformational change toward the R-state.0640

And again, it is a transition - it is not just, boom, it goes to the R-state - which has a greater affinity for O2.0678

It begins to bind more O2, and the more O2 it binds, the more it wants to bind.0704

That is this, sort of, domino effect, if you will, of this thing called cooperative binding, which we will talk a little bit about a little bit later.0714

OK, now, the transition from the T-state to the R-state, the alpha-1, b-1 and the alpha-2, b-2, what happens is they slide past each other.0724

I mean, the biggest change that we can see on this protein, they slide past each other, and narrow the gap between the beta-1 and beta-2.0745

OK, let's see if we can take a look and see what that, kind of, looks like here.0778

Here, again, we have our alpha-1 and our beta-1, and then, we have our alpha-2 and our beta-2.0784

What is going to actually happen is this alpha-1, b-1 and this alpha-2, b-2, are actually going to slide past each other, and they are going to narrow this gap right here.0796

That gap is going to close in on itself, kind of, like this.0808

So, if I have alpha-1 and beta-1 and alpha-2 and beta-2, what is going to end up happening, they slide past each other.0813

The betas actually start to come together and narrow this gap.0825

This is beta; the thumb is beta.0828

They will slide past each other, and beta gap will actually close; and that is going to be the transition from the T-state to the R-state.0830

The more that gap closes, the more of an affinity is has for the oxygen until by the time you get to the third bound oxygen, it is almost completely in the R-state.0839

OK, now, hemoglobin/Hb must bind O2 tightly in the lungs.0851

Alright, that is where the concentration of O2 is high, and the partial pressure is about 13 kPa in the lungs.0869

Under high concentration, it needs to bind the oxygen very, very tightly in the lungs because from there, it is going to take the blood to the tissues, then, bind it loosely, in other words, release the oxygen that it has bound, release it.0880

When we say "loose binding", we are talking about releasing the O2, not just that it does not bind the O2.0909

We are talking about actually releasing the O2, release it at the tissues.0916

Now, at the tissues, the concentration is about 4 kPa.0925

Again, partial pressure, concentration, it is the same thing; the oxygen concentration at the lungs is very high.0930

Oxygen concentration at the tissues is very low.0935

In the lungs, we need it to bind oxygen; when it gets to the tissues, we need it to no longer want the oxygen.0939

We need it to give it away.0944

OK, now, a protein such a myoglobin, which we discussed previously, with a singular affinity, it cannot accomplish this task.0948

Myoglobin is a single polypeptide protein.0983

Yes, it has a heme group, and yes, it binds oxygen; but it has - you remember the binding curve - 1 affinity.0988

It has a certain Kd, but that is it.0995

It just binds the O2; it is a great place to store oxygen, but it does not really experience the kind of changes in affinity, which are required under certain physiological conditions in the lungs.1001

We need this hemoglobin to have a high affinity for oxygen so that it can actually bind them, and then, when it gets to the tissues where the situation is completely different, where pH is different, where the concentration of oxygen is different, we need it to actually let go of its oxygen so that the tissues can use it.1015

Hemoglobin will not do that; hemoglobin has 1 affinity.1033

So, it makes sense that we have this multi-subunit protein, which is very finely-tuned and can adjust its affinity for oxygen depending on the circumstance, depending on how much O2 is there to bind, which is really the key thing that drives this.1038

As an O2 binds, the conformational changes cause it to transition from the deoxy state to deoxy state, from the T-state/tense state to the relaxed state that binds more oxygen- that is it.1055

We need it to be in the R-state in the lungs so it can bind it, and when it gets to the tissues, we need it to make a transition from the R-state to the T-state; and as it gives up each oxygen, it makes the transition back from the R to the T-state, and at that point, it is just giving up its oxygen to the tissues.1068

And again, we are talking about 96% saturation versus 60% saturation.1085

We are not talking about 96 versus 3% saturation.1090

It does not give up all of its oxygen, only what is required.1095

OK, now, let's go ahead and go to - yes - the next page here.1101

As we said, if the affinity of the protein for ligand is high...well, we do not have to say ligand, we can just say O2 because we are talking about oxygen here.1112

If the affinity of the protein for O2 is high, it simply would never release the oxygen when necessary.1137

It would not release the O2.1155

Well, if the affinity were low, then, the protein would never/not bind the O2 when necessary.1160

Both of these situations are depicted diagrammatically as follows.1188

Both these situations - I should say "are shown below", excuse me - are shown below in the following binding curves.1198

And again, we have seen this binding curve; we saw it in the previous lesson, that hyperbolic binding curve.1227

It is going to look something like this; you have got here, and you have got that.1233

This is going to be ligand concentration, and as we said, this is going to be the fraction that is actually the fraction of the binding sites that are actually occupied.1239

I believe we called it alpha; different Greek letters are used.1249

I think some books will call it theta; in fact, I think theta is the standard.1253

I just happen to use alpha; it does not really matter what you use.1257

We said because this is a fraction, 1 is going to be our highest.1261

Now, low Kd is high affinity.1266

High Kd is low affinity; now, if you have a protein that has a high affinity for oxygen, its binding curve is going to look something like this.1273

Let me go ahead and mark at least the maximum here so that I have asymptote to which I can get close.1285

It is going to be something like this; it is going to show some really, really heavy hyperbolic behavior, something like that.1292

OK, and again, the 0.5 where it meets the graph, and if you go down, this is going to be your Kd.1298

Well, low Kd, high affinity.1309

A high affinity protein for its ligand demonstrates this kind of behavior.1313

OK, let me go ahead and mark some things here; I am going to go ahead and put, let's say, 4, 8, 12 and 15.1319

So,13 kPas is there, and about 4 kPas is there.1339

Remember we said this is the region; tissues are about 4 kPa than the concentration of O2.1344

The lungs are at about 13 kPa for the concentration of O2.1350

This is not just ligand; we are actually talking about O2.1358

I am going to put O2 concentration...well, you know what, that is fine.1365

We are talking about pressure; I will put PO2.1369

This is partial pressure O2 concentration; I do not know.1371

I like the bracket symbol, but that is OK; we are talking about a gas.1374

OK, now, a high affinity protein demonstrates this kind of behavior right here.1377

A low affinity protein demonstrates this kind of behavior, something like that.1385

Now, your Kd is like over here.1394

OK, we have a really high Kd, low affinity.1402

A high affinity protein, the binding curve would look like that.1409

A low affinity curve would look like that; let me go ahead and label these.1414

Let me do this in blue; this is a low affinity protein.1418

It displays this kind of behavior; this up here, this is a high affinity protein.1428

What we need is something like this; we need a protein that in the range of - let me go ahead and do this in...that is fine, I can do it in blue - somewhere in this range, 4 kPa in the tissues and about 13 kPa in the lungs.1439

We need a protein that displays the following behavior.1466

Under conditions of low concentration, we need its Kd to be low.1470

We need it to follow this trajectory; we want it to follow this trajectory, the low affinity curve.1476

We need it that way; OK, because again, in the tissues, it needs to have a low affinity for oxygen so that it can actually push away its oxygen.1483

It does not want it; it wants to release it into the tissues because the tissues need oxygen, and then, the oxygen that is there, it does not want to just grab it up.1492

So, we need it to be a low affinity protein, to behave like a low affinity protein, so we want it to follow a low affinity protein trajectory.1500

However, in the lungs, we need it to be a high affinity protein.1510

We need it to display this kind of behavior right here at 13 kPa.1515

We need for it to actually make a transition and follow that trajectory right there.1521

We need a protein that displays this kind of behavior, not up and over high affinity, not over, over, over, slowly reaching its fraction, but we need it to be at low concentrations.1532

We need it to display low affinity; at high concentrations, we need it to display high affinity.1548

We need a hybrid between a high affinity curve and a low affinity curve.1553

That is what we want; OK, that is what we would like to see.1558

This right here - let me go back to blue - this is what we would like to see, the protein display, this kind of behavior.1564

This makes sense; in the low pressure of the tissues, low affinity.1598

We do not want it to grab the oxygen; we want it to give it away.1604

In the high pressure of the lungs, we want it to display high affinity.1607

We want it to grab up as much oxygen it can, so that when it goes back to the tissues, it will give up as much oxygen as it can.1612

That is what we want; we want hybrid behavior.1620

We want this protein to make a transition from a high state to low state, not just one affinity, boom- that is it.1624

I hope this makes sense; this is something that we want the protein to display.1631

Now, let's go ahead and go over here, do a little bit more discussion.1638

In the lungs - let's see here - the hemoglobin becomes about 96% saturated with O2.1644

In blood returning from tissues, where it has delivered its O2, as we said, it is about 60% saturated.1675

Once again, high affinity curve, that is a protein that demonstrates high affinity.1710

A low affinity curve, it demonstrates low affinity; we need a protein that demonstrates both depending on the physiological conditions.1718

Under conditions of low pressure in the tissues, we need the affinity to be low.1727

We need it to follow this trajectory, but as the pressure becomes high, as it starts to move towards the lungs, the concentration of O2, the pressure of O2, is going to be high about 13.1732

At that point, we need it to make the transition from a low affinity protein to a high affinity protein.1744

When we connect these 2 in hybrid fashion, we end up getting this curve.1751

This is called a sigmoid curve; this S pattern is characteristic of this thing called cooperative binding.1756

This is what hemoglobin demonstrates; hemoglobin satisfies this.1765

When we take a hemoglobin molecule, it actually displays this kind of behavior, which is exactly the kind of behavior we need in order to fulfill the function that it actually fulfills- binding O2 tightly, not binding O2, releasing it, binding O2 tightly, releasing it.1770

We want this; we have it in hemoglobin.1790

Hemoglobin demonstrates this transition; it is not just a single binding affinity.1792

It has multiple binding affinities; very, very low under low concentration, very, very high under high concentration and everywhere in between.1796

Again, it is a transition, not just boom, boom, and that is what this graph represents.1805

Let's see here; OK, now, let's see.1812

...and there is a hybrid, so yes, as long as it finished off with this part, this section at least, this graph is a hybrid of the high affinity graph and low affinity graph and is exactly what hemoglobin demonstrates/displays.1822

In other words, hemoglobin, when we subject it to this analysis, we get a sigmoid curve.1868

We get what we want.1874

This is called a sigmoid curve.1878

OK, let's see what we have got here...a little bit about that, that, that.1889

OK, now, let's go ahead and take a look at it.1894

This is what it looks like; a sigmoid curve, notice, it is not this way, and it is not this way.1902

It is a little bit of both; it starts to go in the low direction, but then, as it rises, it starts to get a little higher, higher, higher, boom.1908

That is what you get; that is what is happening.1916

When we measure the oxygen partial pressure percent saturation, when we measure the partial pressure, this time at 26, 8, 80, low, 50.1922

That is what is happening here; the sigmoid curve under low concentrations, the affinity, it is going to actually end up being low.1932

It is going to follow this trajectory; the high concentrations, it has a very high affinity.1942

OK, that is what is going on; that combination, that hybrid curve, gives us this curve.1949

OK, now, hemoglobin does this by what is called cooperative binding.1957

Now, as hemoglobin binds 102 molecule, it begins the transition from T-state to R-state.1979

And again, it begins the transition; it does not just jump over there.2008

It begins the transition from T-state to R-state; that is why you have this S-behavior, this little concave thing here instead of just straight up or that way.2012

It brings the transition from T to R, and the R-state, which binds O2 more tightly has a greater affinity for it.2026

As we add more O2, more O2 wants to bind tightly.2043

It makes the transition from a low affinity to a high affinity.2048

And again, things in nature do not just - boom - jump; they happen slowly.2054

You get curves; you do not get sharp edges.2060

Nature does not behave that way mostly.2064

OK, now, hemoglobin is an allosteric protein.2068

We just said that the binding of 1 molecule actually affects the binding of another molecule of O2.2078

This is allosteric regulation; this is allosteric behavior.2086

It is an allosteric protein; now, it is one where the binding of 1 molecule affects the binding of the protein for its natural ligand.2091

Now, again, protein-protein reaction, the general definition of allosteric behavior is if you have some ligand that the protein binds, there might be some other molecule that attaches to another part of the protein, and by attaching, now, the protein wants to bind more of its natural ligand or less of its natural ligand.2137

The 2 things do not have to be the same; in the case of hemoglobin, the natural ligand, which is oxygen and the thing which controls the binding of its natural ligand, also happens to be oxygen.2162

It does not have to be that way.2175

OK, now, for hemoglobin, O2 is both the allosteric modulator.2179

OK, it is the molecule which controls the behavior of the protein and how well it binds its natural ligand, and it is also the natural ligand.2200

This is called homotropic allostery.2219

It is when the natural ligand and the molecule controlling the binding of the natural ligand affecting it happen to be the same molecule.2231

If the molecules are different, it is called heterotropic allostery.2238

OK, and again, the effect can be positive or negative.2244

If I have some protein and let's say the natural ligand binds here and let's say there is an allosteric site someplace else, again, if it happens to be the same as this, it is called homotropic allostery, but if it is a different molecule that binds to the protein and has an effect on how the natural ligand actually binds to the protein, changes the affinity, it is called heterotropic allostery.2250

Again, allosteric just means it binds at a different site than the natural ligand.2275

That is all that means; OK, let's see.2280

Now, the sigmoid curve is typical of cooperative binding.2287

Now, here is what is great.2311

Cooperative binding...where, do I have it on the next page?2315

Nope, not quite; now, cooperative binding can also be described quantitatively, not just regular binding like myoglobin.2322

Cooperative binding can also be described quantitatively.2333

For a protein with N binding sites - in the case of hemoglobin, n is equal to 4 because it has 4 binding sites - we have the following equilibrium.2349

Protein + n ligands is in equilibrium with protein ligand complex and Ln.2370

Once all of the 4 in the case of hemoglobin, it is going to be Hb + n.2379

O2 is going to be in equilibrium with Hb O2 4, not n.2388

This is 4 because I have 4 O2 molecules bound to 1 hemoglobin molecule.2395

Again, 4 things bound to hemoglobin, this is a general case.2400

This is a specific case for hemoglobin.2404

OK, let's go ahead and do the same analysis that we did before, equilibrium constants, things like that.2408

Let's see if we can come up with something; well, the association constant here is going to be P Ln, products over reactants over P, Ln - right - because now, we have a coefficient there.2412

Well, as it turns out, when I do the same analysis, I end up with the following.2432

I end up with an expression for alpha, and alpha was the percentage of the binding sites that are actually occupied.2436

Alpha is going to equal, again, the ligand concentration this time to the nth power, over the ligand concentration to the nth power plus Kd.2445

OK, this reduces to what we had from myoglobin because myoglobin only has 1 site.2456

So, n is equal to 1, L/L + Kd.2461

Here, it is just Ln/Ln+ Kd, so it is actually the same expression.2465

Now, we have an expression for cooperative binding, a multi-subunit protein for which we can actually draw a binding curve, and the binding curve is going to be, again, kind of hyperbolic; but now, because of the n here, it is actually going to be sigmoid.2473

What this describes is the sigmoid curve.2492

It is a hybrid between a low affinity, hyperbolic curve and a high affinity, hyperbolic curve, follows the low trajectory, and then, goes to the high trajectory.2496

That is what this equation describes; let's go ahead and actually play with this a little bit.2508

Let's write it again; we have alpha is equal to Ln/Ln + Kd.2514

When I multiply this through, I am going to get alpha Ln + alpha Kd is equal to Ln.2525

I am going to move this over, so I am going to get alpha.2540

Kd is equal to Ln - alpha Ln.2545

I am going to factor out, so I am going to get alpha Kd; I am going to factor out my Ln, Ln x 1 - alpha, and now, I am going to end up doing a little bit of division here.2552

I am going to divide both sides by 1/alpha and divide both sides by Kd.2567

I get alpha/1-alpha is equal to Ln divided by Kd.2571

That is the equation that I get.2583

OK, now, I am going to end up taking the logarithm of both sides; when I take the log of this side, I am going to leave it as log of this alpha/1-alpha.2587

The log of this side is going to be n times the log of the ligand concentration minus the log of the Kd.2600

This is a Y = mx + b.2610

I am going to end up with a linear plot.2617

This thing, this is called the Hill equation, and the plot that it represents is something called a Hill plot.2621

OK, now, let's take a look at what this actually looks like when I graph this.2634

On the Y axis, again, I have log of alpha/1-alpha on the Y axis.2640

The X axis, I have log of ligand concentration, not ligand concentration.2647

I fiddled with the other one; the other equation from which this is derived is this one right here, where alpha is the percentage of binding sites that are occupied, and L on the X axis is the ligand concentration.2654

Now, the X axis is log of L, and the Y is log of alpha/1-alpha.2667

Let's take a look and see what this actually looks like.2675

OK, you are going to end up with something that ends up looking like this.2684

Let's see what we have got; here, on this axis, we have the log of the ligand, and on this axis, we have the log of alpha/1-alpha.2689

They did it this way; let me just rewrite this.2700

This is alpha/1-alpha, and this is just the ligand concentration.2703

Again, let me rewrite the equation.2710

Let me write it up here; that is fine.2715

I guess I can write it down here; we have log of alpha/1-alpha = n x the log of the ligand concentration - it was minus, right - minus the log of Kd.2717

Notice, this n right here, that is the slope.2733

The slope is going to give you, if you have 4 binding sites, n should be 4 for fully cooperative binding, but as it turns out, well, first of all, let's see.2739

First, the equation predicts that the line that we get will have a slope of n, the number of binding sites.2755

Again, all we have done is taken that binding curve and found a way to represent it in linear form by taking a logarithmic version of it.2774

That is all we have done here; there is nothing strange going on.2782

The equation predicts that the line should have a slope of n.2786

Experimentally, it is always going to be the slope ends up being less than n.2791

When we actually run the experiment, collect the data, it does not follow the theoretical.2796

So, experimentally, the slope is never n, but actually less than n.2804

It is actually not a bad thing; this is a pretty good thing.2820

Do not worry about it; OK, the reason this is the case is the following.2823

What the slope actually reflects is not the number of binding sites.2829

It represents the degree of cooperativity among the binding sites.2834

What the slope reflects...I should not say "reflects".2844

What the slope represents is not the number of binding sites, but the degree of cooperation among the binding sites.2850

OK, now, the higher the slope, the greater the degree of cooperation of those binding sites- that is it.2889

That is all it means.2915

Experimentally, if we get a slope of 1, there is no cooperation.2922

If we get a slope actually equal to the number of binding sites, and in the case of hemoglobin, let's say the slope is 4, that is full cooperation, full complete cooperation.2926

As it turns out experimentally for hemoglobin, the slope is going to be somewhere around 2.8-3.2937

There is a lot of cooperation but now complete cooperation.2944

This idea of the slope being the number of binding sites, that is a theoretical maximum.2949

It will never be more than that, but it can get up to that; and the more the slope, the more cooperative binding is taking place.2955

If the slope is 1, that means that there is no cooperation among the subunits.2965

Each subunit is just grabbing whatever ligand it can as it needs to.2970

There is no cooperation among them.2977

OK, well, now, let's see.2981

OK, the higher the slope, the greater the degree of cooperativity.2985

Yes, now, therefore, we designate the slope because it is not n as nH, and call it the Hill coefficient.2989

This Hill coefficient, the slope at any given time in the progress is a measure of how cooperative the individual subunits are being among each other to make the protein do what it is supposed to do, which I will repeat.3008

The Hill coefficient is a measure of the degree of cooperativity.3037

Hill plus can be a little strange, and they can be a little strange to interpret.3048

Hopefully, we can offset some of that bizarre nature of it because you are not really accustomed to seeing something like this especially for some multi-subunit protein, but do not worry about it.3054

We will deal with it as needed.3065

When you have a slope, when nH = 1, this means, no cooperativity.3069

If you do a Hill plot of, let's say, a myoglobin, well, myoglobin only has 1 binding site.3082

It only has 1 affinity; there are not multiple binding sites, so there is no cooperative binding.3088

If you looked at the graph, a Hill plot of myoglobin, you are going to see something like this.3095

Because myoglobin, it has a slope of 1, that is it.3105

One binding site, there is absolutely no cooperativity involved.3109

It binds O2 when it can; it releases O2 when it can- that is it.3112

There is no variation; it is just 1 straight line.3116

The binding curve is - boom - like this; the Hill plot is a straight line - that is it - with a slope of 1.3120

OK, now, notice, hemoglobin, that is what is happening in red.3128

This is the Hill plot for hemoglobin.3142

OK, let's take a very, very close look at this.3147

OK, notice, for hemoglobin, there are 2 places where the Hill coefficient does equal 1- here and here.3152

What this means, even for multi subunit proteins - I will say multi-subunit allosteric proteins - multi subunit allosteric proteins, there can be situations where the subunits do not affect each other or there is no cooperative binding.3181

There can be situations where there is no cooperative binding, where binding is just plain straight binding.3215

Now, again, these are experimental conditions.3239

Physiological conditions are not like this; it does not have this broad range of really, really low concentration, really, really high concentration.3243

It does not like that; the physiological conditions are mostly here where there actually is some transition between low state and high state, and I will talk a little bit more about that in just a minute.3250

Now, if nH = n, if the Hill constant, the slope actually equals the number of binding sites, which is the theoretical limit, this means that the protein is engaging in complete cooperativity.3262

It means complete cooperativity.3292

What this implies is that all sites bind simultaneously.3300

All sites, they bind simultaneously, and no partial saturation ever occurs.3308

No partial saturation ever occurs.3325

That means that hemoglobin, which is automatically grabbed for oxygen molecule simultaneously, and it will be 100% saturation- that is it.3329

There is no 1 bound, 2 molecules bound, 3 molecules bound.3337

There is no real cooperativity; it is complete cooperativity.3343

Boom, it is all or - boom - it is nothing- that is it.3347

That is complete cooperativity; we never see that in practice, so we do not have to worry about that, and let's see, the final situation just for the sake of completeness.3350

If nH is less than 1, this is negative cooperativity.3364

It is negative modulation- not altogether that important.3368

OK, now, here is what is important; let's see.3372

Now, where the graph hits the X axis, where it hits the X axis, right there, this is where the log of alpha/1-alpha = 0, right?3375

That is the 0 on the Y axis; well, if we get rid of the log, if we exponentiate this, 10 raised to both power, we end up with the following.3412

This implies that alpha/1-alpha, well, 100 is equal to 1.3422

Well, this is the same as alpha = 1-alpha, 2 alpha = 1, alpha = 1/2 or 0.5.3430

Where this Hill plot actually touches, that is going to be equivalent to the log of the Kd.3444

We said that the Kd was the concentration of ligand that allows for half saturation.3453

Well, here, if we set the log of alpha/1-alpha equal to 0, we end up getting the alpha = 0.5.3460

What this represents is the log of Kd.3472

OK, it represents the log of Kd.3476

Now, let me go ahead and draw an actual graph for hemoglobin and just to see some numbers to see what is going on here.3483

Let me go one more page.3493

Alright, in the case of hemoglobin, what is actually happening is this.3496

It is actually starting with 0 cooperativity.3502

If I extend this line, I end up here.3507

I end up with a high Kd; high Kd is low affinity.3511

OK, this represents the T-state of hemoglobin, its low affinity state.3516

At a certain point, at a certain range of concentration like from here to here, it demonstrates cooperative binding.3525

The sigmoid curve, it starts to behave a little differently, or it actually starts to demonstrate movement towards a higher binding.3535

So, now, this slope here has, now, gone from 1, 0 cooperativity, low binding state.3544

Now, the slope has gone up to about 2.8, 3 range high cooperativity.3551

Now, there is a lot of cooperativity going on as it is making the transition from the T-state to the R-state.3556

At a full R-state, again, if I extend this down here, notice, again, this has its 1.3564

Here, the Hill constant is 1, 0 cooperativity.3574

Now, it is fully in the R-state; well, in the R-state, it is almost fully saturated, so there is no more cooperativity to be had.3579

Again, it goes back to no cooperativity behavior, but it made the transition to the R-state.3587

If we extend this graph down here, and we find this Kd, this is going to be a low Kd, which is a high affinity.3594

This right here is the R-state.3603

In the case of myoglobin, when you have just 1 affinity, in the case of hemoglobin under very low concentrations, you have virtually no oxygen binding.3608

There is no cooperativity; at some point, that is going to change cooperativity.3618

It is going to start cooperating to bind more oxygen as the concentration of oxygen actually rises in the lung area, and at that point, it is going to make the transition; and once all of the hemoglobin are actually saturated with O2, at that point, again, it reaches a saturation point.3623

Most of the hemoglobin molecules, all of the binding sites, are occupied.3644

So, it no longer demonstrates cooperative binding.3649

There is no other oxygen to bind; there is no site for it to bind.3653

Now, it goes back to this non-cooperative behavior, but it has made the transition from the low affinity state, which is represented by extrapolating this and hitting a high Kd to the high affinity state, which is the R-state, which is represented by low Kd when we extrapolate this.3657

This is what makes this Hill plot actually very, very confusing, and it is confusing.3676

I am not going to suggest that it actually is not, but what you want to remember in a Hill plot, the important thing is the slope.3681

When a slope is 1, there is no cooperativity.3689

When a slope rises like in this range right here, the slope - boom - just jumped up, that is when you have cooperative binding.3693

That means the hemoglobin molecule, the individual subunits are starting to bind more oxygen.3700

As more oxygen binds, the affinity rises, so it is making its transition to the high affinity state.3706

Once it reaches its full high affinity rate, boom, it stops there.3712

Now, cooperativity can no longer operate its fully high affinity state, low affinity state.3716

The Hill plot represents the transition from low affinity state to high affinity state- T-state to R-state.3724

The slope of the Hill plot while it is making that transition is a measure of the degree of cooperativity.3735

If it were something like maybe 1.2, you would not see a lot of cooperativity- a little bit.3740

If you saw something like 3.8, you are seeing almost complete cooperativity.3745

What you are seeing is, sort of, a middle ground, more towards the high end.3750

Again, 2.8-3.0 which is perfect for what the hemoglobin molecule wants to do.3754

I hope this helps; again, let me repeat one last time.3763

The slope of the Hill plot tells you about the degree of cooperativity.3766

It will always be greater than or equal to 1 but less than the number of binding sites.3772

A slope of 1 means no cooperativity is active, either low state or high state- that is it.3781

They cannot cooperate anymore - the binding sites - because they are either mostly all empty or mostly all full.3787

This, the transition in between, here is where cooperativity is taking place.3795

This right here is what gives rise to the sigmoid behavior of the binding curve itself.3799

And again, this is just a variation of the sigmoid behavior binding curve.3805

I hope that makes sense; thank you so much for joining us here at

We will see you next time, bye-bye.3815