BIOCHEM DISCUSSION 3

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Biochemistry: A Short Course Fourth Edition

CHAPTER 7 Kinetics and Regulation

Tymoczko • Berg • Gatto • Stryer

Much of life is motion, whether at the macroscopic level of our daily life or at the molecular level of a cell. Studying motion is what motivated Eadweard James Muybridge to use stop-motion photography to analyze the gallop of a horse in 1878. In biochemistry, kinetics (derived from the Greek kinesis, meaning “movement”) is used to capture the dynamics of enzyme activity. [Image select/Art Resource, NY]

So we're going to start off with a picture of a {horse} here, moving - shows you kinetics because this is about how things move.

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CHAPTER 7 Kinetics and Regulation

And in this case, we're going to be talking about reaction rates. So this is the way the molecules in your solution are moving around - and these are always in solution pretty much, since we're talking about biology - and how you can measure the speed of their interactions. This Michaelis-Menten model is the most popular and most powerful model of enzyme kinetics. And if you're going to learn only one model, that would be the one to learn. So we're going to study it. And we're going to talk a little bit about allosteric enzymes, which are very important in regulation of processes in cells. And then we're going to talk a little bit about some new techniques that are used to study enzymes one molecule at a time.

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Chapter 7: Outline

7.1 Kinetics Is the Study of Reaction Rates 7.2 The Michaelis–Menten Model Describes the Kinetics

of Many Enzymes 7.3 Allosteric Enzymes Are Catalysts and Information

Sensors 7.4 Enzymes Can Be Studied One Molecule at a Time

But I'd like to preface this with a little talk about concentration. Now, when we're talking about concentration in biochemistry, we usually talk about it in terms of molarity. And you should be familiar with this term already. But just to review, it tells us the number of particles in a given volume. And the particles could be molecules, could be ions like sodium ions could be complexes like ribosomes. So it doesn't specifically tell you what kind of particle it is. I suppose it could also be maybe some abstract concept. And it's usually indicated in terms of the moles of the solute per liter of solution. So the solute is the particle and the solution is the volume, basically, usually water that we're talking about. One mole is 6.022 times ten to the 23rd particles - that is Avogadro's number. And when we talk about molarity, usually put it in brackets. So talking about the molarity of substance A, we would put it in the brackets and that would be considered "the molarity of A". You're going to be seeing these notations a lot in this chapter, so get familiar with them. Now, when we are mixing stuff in a lab, we're usually doing molar concentrations, or millimolar - but in the cell, things are a lot less concentrated usually. So usually seeing things in the micromolar or nanomolar range. One of the main reasons I'm saying all this is so you'll be familiar with this notation. uM is micromolar, nM is nanomolar, et cetera.

Now we're going to be talking about the concentrations of enzymes - we will put it in brackets, [E] - and substrates - put in brackets [S]. And in order for a reaction to occur between any two particles, they have to collide. And one of the basic tenets, enzyme kinetics is that if you double the concentration of either of these, then you will double the rate of their collisions. So you will double the amount of reactions that are occurring in {a given} time. And we measure the velocity of a reaction in terms of the concentration per second. So the increase or decrease of the molar concentration per second.

So here we talk about a simple reaction. A goes to P. The A is your starting material and the P is the product. And if we want to talk about the velocity of that reaction, it's a function of time, so it's change. This is a notation that means the change in the molar concentration, the molarity of A, per time, per unit of time, or the change of molar concentration of P per unit of time. And those are opposite of each other, because as A decreases, P has to go up. Now we can measure, perhaps, the disappearance of A, maybe using spectrophotometry. If the A is yellow, then we can measure how much of the yellow light gets absorbed. And that way we can know how much of substance A we have. And if you have more of substance A in your mixture, then you're going to have more disappearing. So we talk about the velocity is related to the concentration of A. And this k there is the number that we use to make it into something general. You can't just say velocity is equal to the molar concentration of A - this gives you "per second". So every k is going to be specific for every thing that you're talking about in the reaction. So this k is specific for this A, this is not some general k. And this would be called a "first-order reaction" because there's only one thing that we're measuring.

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Section 7.1 Kinetics Is the Study of Reaction Rates (1/2)

Learning objective 3: Explain what reaction velocity is. Consider a simple reaction:

• The velocity or rate of the reaction is determined by measuring how much A disappears as a function of time or how much P appears as a function of time.

• Suppose that we can readily measure the disappearance of A. The velocity of the reaction is given by the formula below, where k is a proportionality constant.

• When the velocity of a reaction is directly proportional to reactant concentration, the reaction is called a first-order reaction and the proportionality constant has the units s−1.

Now, second-order reactions occur when you have two things that you're measuring. And here we have 2A goes to P - but remember that's just A plus A. Or A plus B goes to P. You have to multiply the amounts by each other. So if you have concentration of A, and you have two A's that have to collide to go to P, then you end up squaring that concentration. A little bit easier to understand in this context, maybe - you have A and B and you have to multiply them against each other, because if you double the concentration of A, you double the velocity. Same if you double the concentration of B, you double the velocity. So multiplying them by each other is the way to determine what the velocity is.

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Section 7.1 Kinetics Is the Study of Reaction Rates (2/2)

• Many important biochemical reactions are bimolecular or second-order reactions.

• The rate equations for these reactions are

• The proportionality constant for second-order reactions has the units M−1s−1.

A + A -> P

V = k[A][A]

So now we're going to talk about the Michaelis-Menten model of enzyme kinetics. Now if you're going to be investigating enzyme kinetics, you would normally use a fixed amount of enzyme just to make it simple. And the form of this reaction would look something like this. Now remember you're going from S to P, or substrate to your product. But there is this intermediate step where the enzyme and the substrate are bound together. And there are rates of these reactions for the binding of enzyme and substrate, and also the unbinding of enzyme and substrate, and for the changing of enzyme and substrate to enzyme and product. And also for the reverse reaction, of changing enzyme and product back to enzyme and substrate. So there are four different reactions occurring here simultaneously with four different rate constants. Now to simplify things a little bit, we are going to talk about the reaction just as it's starting. So there is no product, which means there is no rate of reaction going backwards here. And we can just get rid of that k to the negative two. It's important to remember - that is an assumption that is made when you are talking about Michaelis-Menten model. It's not always true. In fact, usually it's not. But when we're studying these things, we try to simplify them as much as possible. Now, if there's no product, the initial velocity - the velocity of the reaction would be called the "initial velocity" because you're just getting started with this reaction.

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Section 7.2 The Michaelis–Menten Model Describes the Kinetics of Many Enzymes (1/3)

Learning objective 4: Explain how reaction velocity is determined and how reaction velocities are used to characterize enzyme activity.

A common means of investigating enzyme kinetics is to measure velocity as a function of substrate concentration with a fixed amount of enzyme.

Consider a simple reaction in which the enzyme E catalyzes the conversion of S to P.

with k1, k2 being the rate constant for the indicated reaction steps. To ignore the reverse reaction of PàS, we measure activity when [P] ≈ 0.

Under these conditions, the velocity is called the initial velocity or Vo. In other words, the initial velocity is determined by measuring product formation as soon as the reaction has started.

Now here's a few words about these terms. The k1 here, rate reaction, is a second- order reaction because you have to talk about the concentration of the enzyme and the concentration of the substrate. But the other two, k-1 and k2, are first-order reactions because you've only got one thing to start with, and it breaks apart. And so the units are different. For k1, the units are "per molarity per second". And for the other ones the units are simply "per second". Now, on the next slide we're going to be talking about something called Km. And that is the rate constant for k-1 plus k2 divided by k1. And the units of Km - if you look at the units of these individual rate constants are (s-1), (s-1), and (M-1)(s-1). Add these together, the units are the same. Cancel out the (s-1), you get one over (M-1), which is just M. So Km is actually a molarity.

And way back, these guys figured out that that term, Km, can be used to calculate the initial velocity of a reaction. Now a couple of things to remember here (or actually I'm introducing this now I guess). There is a maximum velocity of a reaction. So when you have your substrate and your enzyme, you only have so much enzyme. Remember, we're talking about reactions with a fixed amount of enzyme here. And there comes a point, if you keep adding substrate, that all of the enzyme is bound to substrate. And at that point, you can't go any faster because there's no more enzyme to go any faster. So you can add as much substrate as you want after that point, and you won't be able to go any faster. And that point, that maximum velocity, is called Vmax. And we see down here, Vmax equals k2 times the total enzyme concentration, E sub t. Because if you have all of the enzyme is bound, then your ES will equal your total enzyme concentration. And Vmax would then be directly related to the total enzyme concentration. So that's not something we need to worry about too much, because we're going to be talking mostly about situations where the enzyme concentration doesn't change. But this term Vmax is important to think about. Also, we have the Km in here, as we see. And this reaction then tells us what fraction of the maximum velocity we should be able to achieve for the initial velocity. Now, some very interesting features of this equation are, if you take the substrate, and remember Km is a molarity, if the concentration of substrate is a lot higher than the Km, what happens? Well, S and S both go up and they continue to go up. And as they go up, this Km term becomes less and less important. And S over S becomes closer to one. So

this whole term becomes closer to one. And that means your initial velocity is equal to your V max. And that's approximately equal, of course, because you're always going to have some small amount of Km in there that's going to reduce the initial velocity. But we say that if the concentration of the substrate is much higher than Km, then the initial velocity is going to be very close to your maximum velocity. Another interesting feature of this - if the concentration of your substrate is equal to Km, then you've got one substrate here, one here, and another one there. All of these three terms are equal, it becomes one over two. And that is one half of the Vmax. So if you know that your concentration of substrate is equal to Km, then you can calculate what the initial velocity is. And another important feature here is that lower Km means a higher initial velocity. And you can see because this term, this Km, is in the denominator, the higher it gets, the smaller this amount is going to be. And by the same token, the lower it gets, the higher this amount is going to be. The maximum amount this could be is one. But a lower Km, then, will give you a higher, faster initial velocity.

Figure 7.3 Michaelis–Menten kinetics. A plot of the reaction velocity, V0, as a function of the substrate concentration, [S], for an enzyme that obeys Michaelis– Menten kinetics shows that the maximal velocity, Vmax, is approached asymptotically. The Michaelis constant, KM, is the substrate concentration yielding a velocity of Vmax /2.

So here's a graph that shows the relationship between substrate concentration and Vmax. And as your substrate concentration increases, then your initial velocity of the reaction will also increase. And at the point where your substrate concentration equals Km, Then you'll be at half the maximum velocity. And your velocity of the reaction, the initial velocity, can increase, but it never will get higher than Vmax. It's asymptotic to that line at Vmax.

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Graph of Michaelis–Menten Kinetics

So you're going to have variations in Km that change that physical abilities of your metabolism. And alcohol is an example. There are two enzymes, alcohol dehydrogenase and aldehyde dehydrogenase. And remember, I don't know if you know this, but it's the aldehyde that, that gives you the headache.

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Clinical Insight (1/2)

CLINICAL INSIGHT Variations in KM Can Have Physiological Consequences

• Two enzymes play a key role in the metabolism of alcohol.

So there are two different acetaldehyde dehydrogenases. One that has a low Km, and one with a high Km. Remember low Km means it goes fast, the high Km means that it goes slower. Now, if you have the low Km inactivated, then your other enzyme can't keep up and process all of the aldehyde. So you get some acetaldehyde in your blood and that's when you get a headache.

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Clinical Insight (2/2) CLINICAL INSIGHT Variations in KM Can Have Physiological Consequences

• Some people respond to alcohol consumption with facial flushing and rapid heartbeat, symptoms caused by excessive amounts of acetaldehyde in the blood.

• There are two different acetaldehyde dehydrogenases in most people, one with a low KM and one with a high KM. The low KM enzyme is inactivated in susceptible individuals.

• The enzyme with the high KM cannot process all of the acetaldehyde, so some acetaldehyde appears in the blood.

So Km and Vmax are different for every enzyme. And they are important ways of characterizing what an enzyme does. And it is an interesting point actually, that the Km values for enzymes in biological systems often are right around the substrate concentration that's found in those systems. And that makes a good amount of sense. That means that your enzymes are not being overly taxed, but they're also not sitting around doing nothing. They have evolved, let us say, so that they can process the amount of substrate with maximum efficiency.

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KM and Vmax Values Are Important Enzyme Characteristics (1/2)

• KM values for enzymes vary widely

• Evidence suggests that the KM value is approximately the substrate concentration of the enzyme in vivo.

And here are some values of Km. You can see lysozyme is pretty fast. And these other ones are a little slower. And this is in micromolar. So just remember that. So these would be the initial velocities, as far as in terms of the maximum velocity. So... I say this is fast, six, which means it actually starts off close to the maximum velocity.

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Table of KM Values of Some Enzymes

TABLE 7.1 KM values of some enzymes

Enzyme Substrate KM (µM) Chymotrypsin Acetyl-L-

tryptophanamide 5000

Lysozyme Hexa-N- acetylglucosamine

β-Galactosidase Lactose 4000 Carbonic anhydrase

CO2 8000

Penicillinase Benzylpenicillin 50 Lower KM means higher V0

And remember, if you know what the total concentration of your enzyme is, then you can calculate the Vmax. And there's this term called Kcat, which is the "turnover number" of the enzyme. Remember from that equation, the k2 is the rate at which the enzyme and substrate turns into the product.

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KM and Vmax Values Are Important Enzyme Characteristics (2/2)

• If the enzyme concentration, [E]T, is known, then

and

• k2, also called kcat, is the turnover number of the enzyme, which is the number of substrate molecules converted into product per second.

[E]T = Total enzyme

Now we're going to talk about allosteric enzymes. And they are very important because they are able to sense the amount of a substance in your cell and change the amount of enzymatic activity in relation to that. So they're very important in metabolic pathways, and can make a big difference in enzymatic activity.

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Section 7.3 Allosteric Enzymes Are Catalysts and Information Sensors

Learning objective 5: Identify the key properties of allosteric proteins and describe the structural basis for these properties.

• Allosteric enzymes control the flux of biochemical reactions in metabolic pathways.

• Because of their regulatory properties, allosteric enzymes allow for the generation of complex metabolic pathways.

So this is an example of just a metabolic pathway, some general metabolic pathway. Let's say A gets turned into B, B gets turned into C... And these are the enzymes that mediate these processes. Now, in this one, the conversion of A to B is a committed step. Once you get B, then it will just go to F. That's the only way that that can go. A, on the other hand, might go off into other directions and do other things. But that makes this step very important. And in order to regulate that step, we usually find that these important steps are mediated by allosteric enzymes. And the rest of this is done by Michaelis-Menten, which is just kind of, you know, automatic stuff that happens.

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Allosteric Enzymes Are Regulated by Products of the Pathways Under Their Control (1/3)

• The conversion of A to B is the committed step, because once this occurs B is committed to being converted into F.

• Allosteric enzymes catalyze the committed step of metabolic pathways. Michaelis–Menten enzymes facilitate the remaining steps.

And the way that this might work is that the product, the end product, will feed back on this enzyme and it will reduce its activity. So the more of this product you get, the slower this goes. And that way it prevents you from creating too much of the product.

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Allosteric Enzymes Are Regulated by Products of the Pathways Under Their Control (2/3)

• The amount of F synthesized can be regulated by feedback inhibition.

• The pathway product F inhibits enzyme e1 by binding to a regulatory site on the enzyme that is distinct from the active site.

Now this can become really complex. You can have a product that inhibits your enzyme or stimulates your enzyme.

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Allosteric Enzymes Are Regulated by Products of the Pathways Under Their Control (3/3)

• The regulation of metabolic pathways can be quite complex.

• Allosteric enzymes may be inhibited or stimulated by several regulatory molecules.

Figure 7.8 Two pathways cooperate to form a single product.

So in this example, we once again have F coming back to inhibit e1, but also K comes back and inhibits e1. We also have F goes here to increase the activity of this enzyme e10. And “I” here will go and increase the activity of enzyme e1. So all of these things have their own rates and effects and it becomes very complicated. And it's one of the jobs of computational biologists right now is to try to come up with equations that can take all of this into account.

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Model of Two Pathways Cooperating to Form a Single Product

Allosteric enzymes are extremely important. And if you lose allosteric control, it can cause pathological conditions. One of these examples is phosphoribosylpyrophosphate synthetase. That's a mouthful. And it creates a purine nucleotide synthesis. And if you lose control of that, then it can create an overproduction of those, which leads to gout, large inflammation in the joints.

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Clinical Insight: Loss of Allosteric Control May Result in Pathological Conditions

CLINICAL INSIGHT Loss of Allosteric Control May Result in Pathological Conditions

• Phosphoribosylpyrophosphate synthetase (PRS) is an allosteric enzyme in the purine nucleotide synthesis pathway.

• A mutation leading to the loss of regulatory control without an effect on catalytic activity leads to the overproduction of purine nucleotides.

• The overproduction results in the painful disease gout.

Figure 7.14 A gout-inflamed joint. [Medical-on-Line/Alamy Images.]

So very important to keep those things under control.

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Image of a Gout-inflamed Joint

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Section 7.4 Enzymes Can Be Studied One Molecule at a Time

• Studies of individual enzyme molecules suggest that some enzymes may exist in multiple conformations that are in equilibrium.

• These different conformations may have different catalytic or regulatory properties.

Figure 7.15 Single-molecule studies can reveal molecular heterogeneity.

So if you're looking at enzyme activity of a specific enzyme, you might actually be looking at the activity of several different versions of that enzyme. And when you add them all up, you end up with your enzyme activity of 1.9. But it's important to remember that there is often more going on than meets the eye.

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Diagrams of Possible Enzyme Structures and Activity Levels

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Extra slides

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Allosterically Regulated Enzymes Do Not Conform to Michaelis–Menten Kinetics

• The reaction velocity of allosteric enzymes displays a sigmoidal relationship to substrate concentration.

Figure 7.9 Kinetics for an allosteric enzyme. Allosteric enzymes display a sigmoidal dependence of reaction velocity on substrate concentration in contrast to the hyperbolic curve seen with Michaelis–Menten enzymes.

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Graph Model of Kinetics for an Allosteric Enzyme

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Allosteric Enzymes Depend on Alterations in Quaternary Structure (1/3)

• All allosteric enzymes display quaternary structure with multiple active sites and regulatory sites.

• One model that explains the behavior of allosteric enzymes is the concerted model.

• Features of the concerted model: – The enzyme exists in two different quaternary

structures, designated T (tense) and R (relaxed). – T and R are in equilibrium, with T being the more stable

state. – The R state is enzymatically more active than the T

state. – All active sites must be in the same state.

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Allosteric Enzymes Depend on Alterations in Quaternary Structure (2/3)

• The binding of substrate to one active site traps the other active sites in the R state and removes the substrate- bound enzyme from the T ⇌ R equilibrium.

• This disruption of the T ⇌ R equilibrium by the binding of substrate favors the conversion of more enzymes to the R state.

Figure 7.10 The concerted model for allosteric enzymes. (A) [T] >>> [R], meaning that L0 is large. Consequently, it will be difficult for S to bind to an R form of the enzyme. (B) As the concentration of S increases, it will bind to one of the active sites on R, trapping all of the other active sites in the R state (by the symmetry rule). (C) As more active sites are trapped in the R state, it becomes easier for S to bind to the R state. (D) The binding of S to R becomes easier yet as more of the enzyme is in the R form. In a velocity-versus-[S] curve, V0 will be seen to rise rapidly toward Vmax.

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Diagram of the Concerted Model for Allosteric Enzymes

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Allosteric Enzymes Depend on Alterations in Quaternary Structure (3/3)

• Allosteric enzymes are more sensitive to changes in substrate concentration near their KM values than are Michaelis–Menten enzymes.

• This sensitivity is called the threshold effect.

Figure 7.11 Allosteric enzymes display threshold effects. As the T-to-R transition occurs, the velocity increases over a narrower range of substrate concentration for an allosteric enzyme (red curve) than for a Michaelis–Menten enzyme (blue curve).

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Graph Model of an Allosteric Enzyme Displaying Threshold Effects

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Regulator Molecules Modulate the T ⇌ R Equilibrium

• Allosteric regulators disrupt the R ⇌ T equilibrium when they bind the enzyme.

• Inhibitors stabilize the T state, whereas activators stabilize the R state.

• The disruption of the T ⇌ R equilibrium by substrates is called the homotropic effect.

• The disruption of the T ⇌ R equilibrium by regulators is called the heterotropic effect.

Figure 7.12 The effect of regulators on the allosteric enzyme aspartate transcarbamoylase. ATP is an allosteric activator of aspartate transcarbamoylase because it stabilizes the R state, making it easier for substrate to bind. As a result, the curve is shifted to the left, as shown in blue. Cytidine triphosphate (CTP) stabilizes the T state of aspartate transcarbamoylase, making it more difficult for substrate binding to convert the enzyme into the R state. As a result, the curve is shifted to the right, as shown in red.

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Graph of the Effect of Regulators on the Allosteric Enzyme Aspartate

Transcarbamoylase

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Quick Quiz 2

QUICK QUIZ 2 What would be the effect of a mutation in an allosteric enzyme that resulted in a T/R ratio of 0?

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The Sequential Model Also Can Account for Allosteric Effects

• The sequential model for allosteric enzymes proposes that subunits undergo sequential changes in structure.

Figure 7.13 The sequential model. The binding of a substrate (S) changes the conformation of the subunit to which it binds. This conformational change induces changes in neighboring subunits of the allosteric enzyme that increase their affinity for the substrate. The K1, K2, etc., represent rate constants for the binding of substrate to the different states of the enzyme.

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Diagram of the Sequential Model