If You Get a Chance, Read That Article. Let Me Finish up on the Michaelis-Menten Equation

LECTURE #4

If you get a chance, read that article. Let me finish up on the Michaelis-Menten equation, then you'll see why I gave you the article. If there are any free copies, please send them back up.

As I said before, the crux of the Michaelis-Menten equation is v,0 is proportional to ES, and ES is this intermediate, this hypothetical intermediate, that the binding of an enzyme and a substrate in some kind of situation where the configuration is such that it's in a transition state. This is the enzyme substrate complex... and it's a transition state. Linus Pauling is the one who predicted the existence of transition states. Haldane is the one who predicted that at the binding site, there would be weak bonds, and small.

How many people saw Venter on TV with the Celera genome and the other people, the human genome people, on CSPAN? Did you hear what Venter said in the end, with the 26,000 of the 30,000? Did you understand the punchline of everything they said. The punchline is: We have no idea what's going on. It's too small to make the proteins we know exist and we don't know the relationship between these codings and the proteins that exist. So, we don't know what's going on. He mentioned someone's name. Do you remember who it was? I mentioned it to you before. Sidney Brenner. Sidney Brenner is the guy who discovered messenger RNA. He is your uncle. Publish with him. He did the nematode sequence. He did t-RNA. He is working on the fish, the Fugu, which is a blowfish. If you eat sushi, it's very poisonous. The curious thing about this organism is that it has no, very little nonsense DNA; about 95% of DNA is not coding; it's called "junk DNA". It's not clear if that's a good phrase. "Junk" is meant to differentiate it from "garbage". Crick and Brenner created these terms.

Garbage is something you throw away. Junk is something you keep in order to give it a new use later on. So, junk DNA doesn't mean junk, it just means it's sitting there waiting to evolve into something else. And, Brenner found this organism, which also has about 26,000 genes, but very little, maybe twice that, coding that's nonspecific junk DNA. The idea would be to sequence that and compare it. So, everybody's waiting now to try to find some smaller genomes (in DNA size, not in coding capacity) in order to figure out what's going on. It's a whole new ball game. Everything that we know, everything we assumed was true, DNA to RNA to protein and the correspondence of one gene, one enzyme just isn't true anymore. I can't give you the real answer anymore. All I can say is I don't know any better than anybody else, but my ignorance is more profound than theirs, that's all. So, we can leave now, okay?

I'm going to teach you how to think in this area, since we know nothing now. Maybe you can figure out what's going on. It's your generation that will have to do it. Ideally (to what we were talking about before) you want to find a pattern, an equation or a rule for the pattern and then a mechanism to explain that rule. Then, that's science.

In this situation, in the Michaelis-Menten, this is the pattern. It's a graph of the kinetics of an enzyme reaction. I gave you a pattern before that was... what was that for?... What's the gene?... It's the mendelian gene. Watch the pattern. There may be some other patterns of DNA structure that we'll go into that people decipher. Who was the student who figured out that the mendelian gene's on the chromosome?... Sutton. Very important. These are conceptual breakthroughs, even though it seems obvious after the fact.

So, we ended up with this equation, I think, last time it was v,0= Vmax. Do you want me to go through the equation quickly again?... Okay. Let me do that. The assumptions in the equation are that the velocity, v,0 is proportional to ES. To make that an equation, what do we do? To make any proportionality an equation, what do we do?... A constant of proportionality. So, it's some constant. Then, we set up the equation. It's E+S = ES. So, what we're going to do is solve for ES (enzyme + product). Now you know we're going to solve for ES and get rid of ES. That's it. This is the situation that describes this. The only thing we can measure is enzyme concentration, substrate concentration and also product. So, we can...marriage...children is product...back rate is divorce. But, we can't measure love. So, we just assume it exists. We suffer a while and then we throw it out. This is K1, K-1, K2 and K-2.

The other assumption is that the formation of ES is at equilibrium with the breakdown of ES. Then, we need one more equation, that the E total is equal to E1, ????? + enzyme substrate. The total enzyme is equal to the amount of enzyme bound with the substrate and what else?... Enzyme that is floating around free. So, the formation of ES depends on how much free enzyme we have. What we had is K1 x E free x S. That should equal the amount of breakdown. The breakdown is what goes away from this ES, what goes in these two directions, i.e. the breakdown to make product and the breakdown, the divorce rate, to go back to E + S. That's K-1 in this direction and K2 in this direction. So, it's K2, ES, plus K-1 x ES. That's the crux of the setup of the equation of the equilibrium. It's an algebraic equation. It really means that the formation is equal to the breakdown. You have to know the logic of this.

Then, you have to substitute for E free. We don't have a measure of E free, but we can have a measure of E total. So, E free is E total - ES = E free. We'll bring it over here. So when we substitute in that, E total - ES x S = K2 ES + K-1 VS.

Now, we bring over this, we multiply it out, 1 x E total - K1 ES. We have to multiply the S in there too, minus K1 ES S... That's equal to this. Now, we have to bring the ES over to this side so we can factor it out. We'll just move this over here, plus K1 ES S. Now we factor out ES, so we have (K2 + K-1 + K1S) x ES. Now we can bring this whole thing underneath here. Well, let's bring it up here now. Now we have ES and we want to bring ES on the outside so we can get rid of it. So, if ES is now equal to K1 E total... ES is equal to K1 E total x S over THIS, which is K2 + K - 1 +K1 x S. Now, we're going to divide both denominator and numerator by K1. What we end up with is E total x S/K2 +K-1/K1 +S. This is a constant made up of three constants. This constant is then substituted by a large constant called the Michaelis-Menten constant (KM).

Now, remember what I said. If you copy things three times, usually you've memorized them. You don't have to do it again ever. I'll teach you how to do that with base pair memory. So, if you go over this three times, you'll know this. So, ES is now equal to E total S over KM plus S. Now, there's one other boundary condition, which I didn't give you, that the V max was proportional to E total and therefore, V max is equal to K2 times E total. ES, we know, is... V,0 equals E2, so the constant or proportionality is V,0 is equal to K2, ES. So, ES is equal to V,0 over K2. We're going to take this, this and substitute it into here, so V,0 over K2 is equal to E total times S over KM plus the substrate concentration and that's equal to V,0 times K2 times E total times the substrate concentration over KM plus the substrate concentration. Now, we take this, the Vmax and substitute it for K2 E total and that's equal to Vmax times the substrate concentration over KM plus the substrate concentration. That's the Michaelis-Menten equation: V,0 is equal to Vmax times the substrate concentration... This equation explains this graph. This is Vmax and this is V,O and this is substrate concentration. We can set up a boundary condition and we can ask what happens when V,0 is equal to one half of Vmax? We can go in here and say one half of Vmax is equal to Vmax times S over KM plus S. The Vmaxes drop out. We have KM plus S is equal to 2S. So, I would drop out one of the S's so the KM is equal to the substrate concentration when V,0 is equal to one half of Vmax. So one half of Vmax, the substrate concentration is also KM. It tells you that KM is a concentration.

Now, I'm going to ask you to stretch your minds a little bit into another area. Sometimes when you look at a stock during the day from nine in the morning to four o'clock in the afternoon, 9:30, when the stock market opens... How many of you buy stocks? How many of you made money buying stocks? How many have lost money? Well, a lot of stocks follow this graph if it's a low-volume stock. What I want you to do is think about this equation in terms of mimicking... This is the price of the stock in dollars, let's say, and this is the time of day. If this is the time of day in, let's say, minutes. Let's say this is 9:30. This is 4:00... A lot of stocks follow this characteristic where it hits the price and stays there very early after the morning opening. The question is, what's the logic of why this graph... What's the equation for this, if this is P and this is T. This is V,0 and this is S. Let's call this "maximum P" So, what would be the equation you would write, knowing this is the exact same graph. This is P, this is T. What would they look like?... Right, Max P times T over some constant for the stock plus time. Now, we're DP???????

So, here you have an equation that describes a stock profile that's the same as the Michaelis-Menten equation. Anything that has this asymptotic characteristic, you can tell. But, what does it tell you about the mechanism underneath this. This is the hard part. This is the British part. So, we just substituted there. What does it tell you about what's happening? What does it tell you that the price is proportional to? We know that this equation is derived because velocity, the initial velocity is proportional to an enzyme substrate concentration and we got rid of that intermediate to get this graph and this equation. So, this tells me that the price is proportional (something to do with time because that's this concentration) and some other function which we don't know about because we got rid of it. It tells me there is only one variable and that's called the "specialist". And he's like the enzyme. He's the guy that sits with all the stock. He accepts the stocks and he releases the stocks. He has the bulk of the stocks in his hands and it's proportional. It's saying that the price is proportional to the amount of stocks that this specialist has at any given time. You can do this if you see this kind of pattern in any kind of set of data. You can start figuring out that there's only one variable involved because you can be convoluted back to this equation. And, you know that there's an intermediate involved. This is a little subtle. I like it though.

What is says is that you can get a KM for every stock in the stock market that looks like this as you can get a KM for every enzyme. Especially those stocks that are mainly dependent, not on the buyers and sellers, but on the specialists, depending on one person deciding the price of that stock. This is a characteristic that can prove that.

You can take this equation (it's a hard equation to find it) and you can invert it in what's called a double reciprocal. The double reciprocal is a way of mapping it in linear space, because an asymptotic equation, a hyperbolic equation, inverted, is linear. That's what you kind of want, you want a linear correlation. That's the equation for a line. The question is, how do we convert this into something like this? It's simple. What you do is just invert this and make one axis 1/V,0 and you make the other axis 1/substrate concentration. To do that, you just flip this over, flip them both over. This is 1/V,0 and this is 1/Vmax S/KM plus S, which is equal to KM plus S over 1/Vmax. Let's do it this way, 1/Vmax over S. What happens here is that KM over Vmax times 1/S plus S over Vmax over S. These drop out and that becomes 1/V,0 is equal to KM over Vmax. 1/S plus 1/Vmax. Given that this is the y, this is the slope and this is MX + P, that makes the intercept here 1/Vmax, makes the slope KM over Vmax. It's a very easy way of graphing and identifying your Vmax, where here, the Vmax, on a convergence onto an end point. This is called a Lineweaver- Burk representation of the Michaelis-Menten.

We're going to just talk a little bit about inhibitors, competitive and noncompetitive inhibitors. Is this clear now, the Michaelis-Menten? Are you clear on what the assumption is; V,0 is proportional to the enzyme substrate concentration; that means the velocity, the thing that drives the reaction is proportional to an intermediate, a transition state that may or may not exist, which we're looking for evidence of. That equation is represented by love... Buddy love.

You can have, for an enzyme, different inhibitors. The analogy is: If we have an enzyme plus substrate and it goes to enzyme plus product, we have our intermediate and then enzyme plus product. We can also put an inhibitor. An inhibitor will form an enzyme inhibitor complex. It will pull the enzyme out, but the substrate, if it's in high enough concentration, will compete out the inhibitor. This is called a "competitive inhibition".

If you think about it in this way: Let's say this is love and the product is children, marriage. This is a real situation. There is an old professor at the medical school who is chairman of the department. He has a wife who gave him four children, wrote his grants for him and ran his laboratory. She has a lot of postdocs. She slaved all her life for him. Then, he decides he's getting old and a little crotchety in the head. He decides to go out with a young postdoc. He says to her, "Why don't you give her a bench in the lab?" She is not happy about it, because she is the prime substrate for him (the old enzyme) and they produced all these children. If it is a separate lab bench, then that's a competitive inhibitor.

What happens is, the graph looks like this. You don't change in the end the max, the Vmax. That stays the same. What happens is, it just takes longer to get there because, in the end, the wife, the postdoc, wins out because she's always there. They're going for the same competition point. The substrate is here and the inhibitor has to go into the same point in the old professor. You know the guy?... STUDENT QUESTION... It's a substrate concentration, no. Good point. It's just that the amount of substrate that you need to add to get to Vmax has to increase. I'm thinking about the stock, but it's the same. You're just adding more substrate. If you add more wife, you'll get rid of the inhibitor; if the wife keeps showing up, the postdoc's not going to be around. The professor, being very clever, or thinking he's very clever.... By the way, who do you think they asked to leave the department, the professor or his wife?... STUDENT COMMENT...Why? See, she has the right moral view, but why do YOU think it's the wife? Blame the victim always? She's the victim; throw her out. Anyway, that's Rutgers. He's still the chairman of the department and she had to move to another department. She gave him the kids, the friends, etc.

Then, he says to her, okay, why don't we set it up this way... Oh that was the same bench; she wanted him to ???? but the competitive inhibitor, they're working on the SAME bench, sorry. So, the wife and the postdoc were working on the same bench.

But the noncompetitor inhibition, here you have the enzyme plus the professor plus his wife going through their normal mating, having their children. Then, in comes the postdoc as an inhibitor and he says why don't we just give her another bench and we can both all work together very happily here in the lab together? She doesn't have to have your bench.

So, that forms an enzyme inhibitor complex. Then, you have an enzyme substrate inhibitor complex. Then, this can go back to this, releasing ??????? What happens here is you have another slot and you're asking the substrate, the wife, who is at one place and the inhibitor to stay at another place. And, that's great because they don't interfere with each other and they each have a lab bench. But, the professor gets worn out. Now he has to handle two. So, the amount of product he has, Vmax, will get lower. This stays the same, so what happens is here, KM stays the same, but the Vmax goes down because he has a bit on his hands. He has a lot to deal with; two mother-in-laws, for example. Did you find that offensive? Is that why you're leaving or...? I apologize.

So, that is enzymes and substrates and the Michaelis-Menten equation. You should remember them, the relationship of inhibitors and competitive inhibitors and noncompetitive inhibitors. That means the Vmax will go down if you're wearing up and using up enzymes. STUDENT QUESTION... That's right; KM increases and here the KM stays the same. Here, it stays the same; the Vmax gets lowered. The KM increases because the substrate concentration to get to the higher... increases. One half of the Vmax... STUDENT COMMENT. Well, that couldn't be. The Vmax is the same, so one half of the Vmax has to be the same, KM. Okay, so I'm wrong, OK?... Okay, one half of Vmax, so here if the Vmax stays the same, then the KM has to be the same. Here, if the KM, if the Vmax is lowering, the KM has to go lower because the KM is one half of Vmax. That's why I only gave you one. So what changes here?... In competitive inhibition, the Vmax stays the same and the slope increases the KM over Vmax, (as more inhibitor is put in) so the KM has to increase. In noncompetitive inhibition, the KM stays the same but the Vmax lowers. You want to see on the Lineweaver-Burk ?????????.