Biochemistry Enzymes III: Kinetics

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

Today, we are going to start talking about enzyme kinetics.0004

Kinetics is just a fancy word for rate of a reaction, how fast does this reaction go.0009

Kinetics is, sort of, the beginning of our understanding of mechanism.0016

With biochemistry, with all of chemistry, our ultimate goal is to understand the mechanism, how something goes from one thing to another, in the case of an enzyme, how an enzyme takes a particular substrate and how it converts it to product.0022

Each individual little step, every single electron, where it goes, every single atom, every single hydrogen ion that is transferred from here to there- that is our ultimate goal because if we understand the mechanism, then we know how to control it.0038

We know how to fiddle with it, how to do whatever we want to it because we understand how it works.0050

Kinetics is, sort of, the first step of understanding that mechanism.0057

It is the oldest approach to understanding mechanism, and it is still very, very, very important.0062

Let's jump in and see what we can do; now, we are going to deal with the quantitative aspects of kinetics, and I am going to go through a derivation of the Michaelis-Menten equation.0069

Hopefully, it will not be too bad; ultimately, we are not going to be concerned with the derivation.0082

Your teacher will let you know about the extent to which you need to know the derivation or not the derivation, but we will go through it.0086

It is the final form of the equation and how the enzyme behaves; that is what is important.0093

It is more of a global understanding; let's see what we can do.0098

OK, now, as we said, the fundamental approach - let's see here - to studying a mechanism is to determine the rate of an enzyme-catalyzed reaction/rxn and more importantly and how this rate actually changes with changes in substrate concentration.0102

In other words, if I start with a certain amount of enzyme that will hold constant and if I add a certain amount of substrate to this enzyme solution and if I measure how fast these substrate molecules are being turned over into product, that gives me a rate.0175

It will often be expressed in terms of, let's say, millimoles or micromoles per minute or perhaps in molarity, millimolar or micromolar per minute or per second.0195

A rate is some change in some value per unit time.0205

That is what a rate is, how fast something is happening; well, if I start with a different initial concentration, does that change the rate?0209

As it turns out, the answer is yes.0216

Ultimately, substrate concentration controls how fast or how slow an enzyme-catalyzed reaction is going to go.0220

Now, it is never always this simple.0227

With normal chemical reaction, it is not even that simple; you can just imagine how complex it can become in the case of an enzyme-catalyzed reaction where you have, maybe, more than 1 substrate, where you have multiple steps in a mechanism, not just 1 or 2 but perhaps 7 or 8 before the enzyme releases the product.0231

Again, we are going to be making some simplifying assumptions to make it tractable, so that we can actually deal with it, but ultimately, it is about how the substrate concentration affects how fast an enzyme-catalyzed reaction can go.0250

OK, let's see what we have.0266

Now, since the substrate concentration will denote it like that, S with a bracket.0270

Concentration - you remember from chemistry - always has brackets, and those brackets are normally in moles per liter.0277

In the case of biochemistry, it will usually be millimolarity, occasionally micromolarity because you are talking about pretty small amount, at least as far as the enzyme is concerned, and often the substrate is well.0283

Since S changes, as the reaction proceeds, we simplify matters by measuring initial rates.0296

And you remember or you should remember, at least, or I hope you remember initial rates from general chemistry when you were doing the kinetics portion, although I do understand that the kinetics portion of general chemistry is generally that portion of chemistry that one would like to forget.0326

That is the one that seems to leave the mind the quickest because it was so very odd compared to the other concepts because it was heavily mathematical.0339

In any case, as you know, you put a substrate in with an enzyme, the substrate concentration actually changes.0347

So, as an experimenter, how can one...and since the substrate concentration is changing, how can I actually try different substrate concentrations to get some, sort of, a relationship between substrate concentration and speed of the reaction?0355

Well, if we measure how fast the reaction is going in the beginning, right when we put the substrate and the enzyme together, let's say, within the first 30 seconds or, at most, the first minute, we never want to let it go past that.0370

Well, during that timeframe, because the substrate concentration in general is usually going to be so much more than the enzyme concentration - right - enzyme, well, is not used up in a reaction, so the enzyme concentration is very small.0383

One enzyme molecule can handle millions of substrate molecules.0398

Because that is the case, in the first part of a reaction, the first 30 seconds, the first minute, maybe the first minute and a half, the substrate concentration does not really change all that much, especially since the substrate concentration is so much bigger than the enzyme concentration.0401

So, for all practical purposes, the substrate concentration that I start the experiment with, it is going to be held constant.0417

If I start with 10 millimoles or 20 millimoles or 30 millimoles, even though substrate is being used up, compared to the 20 millimoles, 30 millimoles, 40 millimoles, that I start off with, it is going to be completely negligible.0425

Now, we can actually do that; even though substrate concentration changes with an enzyme-catalyzed reaction because of the difference, because there is so much substrate, I can pretend that it does not really change.0440

Now, I can actually run my experiment; I can actually collect some data.0450

Since S changes to the reaction when we simplify matters by measuring initial rates of reaction, how fast the reaction is going at the beginning before other complications start to slow the reaction down.0456

Now, as we said, since the concentration of S is so much greater than the concentration of enzyme during the first minute or 2, the substrate concentration is essentially constant.0470

OK, now, what this does is this allows us to choose various initial concentrations - like I said, 10, 20, 40, 80, 160, however - then, plot the rate of reaction that we measure as a function of these different initial concentrations.0510

In other words, what we are saying is that the rate - I will call it initial velocity, so we will use the symbolism that is pretty common, well, actually, you know what, let me go ahead and just write it as rate first, and I will go ahead and use the symbolism - is going to be some function of S.0568

In other words, you are going to have S on the X axis.0590

OK, that is going to be your independent variable, and the rate is going to be your dependent variable.0594

That is going to be your Y axis.0598

Rate or speed of the reaction, we will often symbolize with v₀, initial velocity.0602

Sort of, this is 0 making reference to that fact that we are dealing with an initial rate.0608

v₀ is equal to some function of the substrate concentration.0613

Different substrate concentrations, different values, and you end up getting this curve.0620

Well, when we do this, when we actually run the experiment, we get the following.0625

We take some enzyme; we start with different substrate concentrations.0631

Let's say we run 10 different initial concentrations, we get 10 data points.0636

We measure the speed at the beginning of the reaction.0640

We get the following graph.0645

We get something that looks like this, so let's take a look at this- very, very, very important graphical representation.0650

This is a plot of substrate concentration on the X axis and reaction rate on the Y axis.0657

Do not worry about what the V_max, half V_max and K_m mean right now.0664

We will get to that in a minute when we talk about this Michaelis-Menten equation that we want to deal with.0668

OK, if I start with an initial concentration of a substrate concentration of 0, well, nothing is going to happen.0673

As I increase my initial substrate concentration, I am going to measure a rate.0682

I try another initial concentration, I am going to measure an initial rate.0688

Another experiment, another concentration, I measure another rate; I do this several times.0693

In this case, it looks like 1, 2, 3, 4, 5, 6, 7, 8, 9, 9 experiments.0698

You can do as many as you like; obviously, the more that you do, the better data you get.0703

When I connect all of those dots, I get this thing.0707

This is a very, very, very characteristic enzyme behavior- reaction rate versus substrate concentration.0712

Let's just talk a little bit about what is happening here just qualitatively; let's get a sense of what is happening.0722

You notice in the beginning almost this...well, that is OK.0728

At low concentrations of substrate, what happens is you have basically almost a straight line here - right - of 2 about right here?0735

For very, very low concentrations, what you have is, sort of, a linear behavior, in other words, just something that is like the rate equals some constant times the concentration of S.0745

That is just some line - that is it - a linear relationship, but as the substrate concentration increases, notice that the speed actually starts to slow down.0760

You might ask yourself, well why is that, why does it slow down?0772

Well, think about it; it actually makes sense.0775

You only have so much enzyme; you keep adding substrate.0778

You keep adding substrate; you keep adding substrate.0781

At some point, you have added so much substrate; around this point, you have added so much substrate that, now, the enzyme is completely tied up with substrate.0784

In other words, it is bound to the substrate; it is converting it to product.0795

The minute it releases that product, yes, it is a free enzyme again, but immediately, another substrate binds to it.0798

So, for all practical purposes, at some point, you have tied up all of the enzymes that can turn molecules over, that can turn substrates over.0804

It does not matter how much more substrate you add; you can keep adding substrate.0813

You are not going to make the reaction go any faster because there is only so much enzyme there that can react, that can actually do the job.0816

That is why you see this tapering off behavior, and you know intuitively that at some point, you are just going to keep adding so much substrate- that is it.0824

There is only so fast that the enzyme is going to turn these substrates over.0835

That is what this is here; this is an upper limit on the speed of the enzyme.0840

Enzymes achieve this, what we call, maximum velocity, maximum speed.0844

There is some maximum speed faster than which the enzyme...it is not just going to go.0850

No matter how much more substrate or no matter what else you do to it, it is just not going to go any faster than it can go, than it can turn over the molecules.0855

Let's write all this out.0864

At low substrate concentrations, the rate increases.0870

So, there is an increase; as you increase concentration, the rate, it is faster.0878

The rate increase linearly, or I will just say "almost linearly", which seems to make sense.0882

I mean it is reasonably intuitive; now, as the substrate concentration increases, the rate slows down with each increasing concentration, and notice, you are having a big increase in the initial concentration; but you are getting a very, very little increase in rate.0890

It is definitely slowing down here; the rate increase slows until a point is reached where increasing the substrate concentration has no effect on the rate.0912

OK, this V_max is exactly what you think it is, this maximum velocity, V_max, the fastest this enzyme can go.0950

OK, it is here that the enzyme is saturated.0976

In other words, it is completely tied up with substrate; it is completely saturated, and no matter how much more S is added, it will only go as fast as it can turn substrate over or convert substrate to product.0989

Again, what we would like here is to have some, sort of, quality, be able to explain this behavior.1031

We explain this behavior by saying that at some point, your initial substrate concentration, the minute you put it into this enzyme solution, it just binds up all of the enzyme.1037

Now, there is no free enzyme floating around; here, there was free enzyme.1048

The more substrate you added, the more free enzyme actually started to operate.1052

So, you have got a faster turnover overall, but at some point, your substrate is going to completely saturate the enzyme.1059

Now, all of the free enzyme is bound up, and since there is no more enzyme that is free, you can add as much substrate as you want.1066

It is not going to make it go any faster than it is already going; that is its maximum velocity.1074

Now, when you do these experiments, when you do reaction rate and substrate concentration experiments, you are probably wondering where does this V_max come from.1079

OK, if your data is reaction rate versus substrate concentration, this maximum velocity is something that you are going to have to estimate.1087

You are going to have to take a look at your data, and you are going to have to extrapolate some maximum velocity.1095

Usually, it is not a problem; you just go a couple of units above something like that, and you draw a line.1100

Now, in the next lesson, we will talk about a way to actually get an analytical method for coming up with the V_max, the maximum velocity, which is a really, really fantastic method.1106

Do not worry about that; we will be talking about that, but when you have just reaction rate and concentration data, this is something that is just extrapolated.1116

It is estimated; you look at your graph, and you just draw a line based on experience or whatever other information you have at your disposal.1122

OK, now, what we have here is empirical behavior.1132

You took some enzymes; you took some initial concentrations, measure the rates; this is how an enzyme behaves.1138

What Michaelis and Menten did is they postulated a few things based on what is happening, and they decided to derive an equation to see if they can come up with some equation that explains this behavior- OK, that is it.1143

They wanted to have some mathematical formula that matches this.1156

That is usually how it works in science; you come up with some data.1162

You graph it, and then, you try your best to go back and theoretically see if you can find an equation which fits the data.1166

That is usually how it works; we, sort of, learn it in reverse especially when we are learning things like math and physics.1174

We learn the formulas, and then, simultaneously we will talk about the data, but this is how it really happens.1178

You have the data first, the graph first, and then, you try to fit the equation to the graph by making some assumptions and fiddling around with a bunch of equations until something works.1186

Alright, now, this is empirical behavior.1198

OK, Michaelis and Menten postulated the following; let me actually move here.1210

Again, any theoretical discussion begins with a series of postulates, and then, you, sort of, take it from there; and you see where your math leads you, or you see where your chemistry leads you theoretically, and then, you compare it with the data.1227

That is how it works; Michaelis and Menten postulated the following.1240

They postulated that the enzyme and the substrate, there is some k₁ - rate constant - and k_-1, and they form something called an enzyme substrate complex.1247

Enzyme substrate bind, you have this enzyme substrate complex.1262

They assumed that this was the fast step, that this happens very quickly, and then, of course, the enzyme substrate complex, that is the one that actually breaks down into enzyme and product.1267

So, there is another equilibrium here.1279

They call this k₂, and they call this -2 into enzyme and product.1283

Their postulate was that this was the slow step, and if you remember from general chemistry, the slow step in a mechanism, in a series of steps, that is the step that controls the rate of the reaction.1288

As it turns out, now, because step 2 is slower, the overall reaction rate depends on this concentration of enzyme substrate rather than just substrate- that is it.1301

That is what they postulated, that this was the slow step, and this behavior actually depends on the concentration of enzyme substrate, not necessarily just the concentration of substrate.1341

They decided to take it from there.1357

Let's see; let's go ahead and take a look at this one more time.1364

Again, under low concentrations of sub...well, let's write all this out.1374

When the concentration of substrate is low, it is still going to be mostly free enzyme.1382

In other words, the enzyme substrate complex has not formed.1399

It is not saturated; the enzyme is not saturated with substrate.1403

It is still mostly free enzyme.1406

The rate still depends just on substrate concentration.1411

Remember what we had?1417

Let me draw it over here; we said E + S goes to E_S, and then, from here, goes to enzyme + product.1420

At low concentrations of substrate, we are still mostly free substrate and free enzyme.1430

The speed of the reaction is still just contingent on the substrate concentration.1436

The rate is still just a direct function of substrate concentration/S- the substrate concentration.1442

That is what explains the linearity; when it is just depending on one thing, you are generally going to get some, sort of, a linear behavior.1460

OK, now, but as the concentration of substrate/S rises, the enzyme, now, is mostly in its bound state, right?1468

The enzyme becomes saturated; the enzyme becomes bound up with the substrate.1498

Now, it is no longer here; now, it is mostly this, thus, the second step of this Michaelis-Menten postulate takes over, and that is where we experience deviation from linear behavior.1502

We have deviation from a linear behavior, and this deviation comes from the fact that it is no longer contingent just on substrate concentration; but it depends on enzyme substrate concentration.1535

Clearly, from here to here, things get rather complicated because now, it is the enzyme substrate that is controlling how fast the reaction is going.1552

OK, now, a second assumption here or postulate, if you will, I will say in these proceedings, is the steady state postulate, steady state assumption.1562

OK, I will just call it assumption.1590

The steady state assumption...again, we have to, sort of, make things as easy as possible for us that we can deal with the mathematics in reasonably decent way.1602

The steady state says this; the formation of the enzyme substrate complex is quick - right, a fast step, this is the fast step - and once it is formed, the concentration of this enzyme substrate complex remains constant.1611

That does not fluctuate; it achieves a steady state.1644

It does not mean things stop; it just means it has achieved a steady state.1648

This concentration is not going wildly up and down.1654

If that were the case, there is no way we can handle this mathematically, at least not in any simple fashion.1658

A second assumption of proceedings is the steady state assumption; this is a very important assumption.1664

The formation of the enzyme substrate complex is quick, but once it is formed, the concentration of enzyme substrate, it stays reasonably constant.1669

OK, in other words, that is the rate of enzyme substrate breakdown is the same as the rate of enzyme substrate formation- that is it.1691

In other words, you saturated your enzyme; now, you have enzyme substrate complex.1720

As quickly as the enzyme turns over the substrate and kicks out a product, another substrate actually binds.1724

You are never going to end up with some enzyme substrate concentration that goes from 10 to 50 to 3 to 5 to 47 to 8.1731

Once it gets there, that is it; it usually just stay there.1740

That is the steady state assumption; it allows us - in a minute, as you will see - to set rates equal to each other so that we can make the mathematics a little bit more tractable.1743

OK, now, let's go ahead and rewrite what we have here.1753

We have our enzyme plus our substrate.1758

It is going to form our enzyme substrate complex, and that is going to break down into enzyme plus product.1762

This is the Michaelis-Menten 2-step mechanism; now, we say mechanism here.1768

Mechanism means individual steps in a reaction- what atoms move, which electrons move.1773

We talk about mechanism a little bit more broadly in terms of steps.1782

I will call it a mechanism; this is not a mechanism formally.1786

This is more just reaction steps because lots of things might happen between here and here.1789

Even though this is one step at the molecular level, at the atomic level, several things might be going on.1795

Again, there is a rate constant in this direction; there is a rate constant in this direction- the breakdown of the enzyme substrate complex.1800

Well, the breakdown of the enzyme substrate complex to product has its own rate constant, K, and the formation of enzyme product.1811

Well, enzyme and product might come back together to form enzyme substrate, right?1822

This is always in equilibrium; things go back and forth.1825

This we will call it k_-2; now, in this case, the k_-2, this step, it can be ignored.1828

It can be ignored because in the beginning of a reaction, which is really all we are concerned with, the concentration of product is negligible.1842

The first minute or 2 of a reaction, the concentration of product does not matter.1858

Because the concentration of product does not matter, the reformation of enzyme substrate complex from product and enzyme, we can pretend that it does not really happen.1862

We can just, sort of, knock this out of consideration, so when we do our mathematics, we are only going to be concerned with this step forward, this step backward and this step forward.1872

Enzyme substrate forming enzyme substrate complex, enzyme substrate complex breaking down into enzyme substrate free or breaking down into enzyme product- that is it.1884

That is all we are going to be concerned with when we deal with mathematics.1894

Rate is determined by how fast this enzyme substrate complex breaks down- that is it.1901

That is how fast it is; it is controlled by that thing.1917

It breaks down to product; we know that the rate, which we are calling v₀, initial velocity.1923

Now, because the Michaelis-Menten postulate was that this second step is the slow step, that is the rate-determining step.1934

That means that the rate law for this, the rate equation that equates concentration of this to how fast a reaction is going, is going to be this: the rate constant of this step times the concentration of the reactant.1942

That is how we do it if you remember from chemistry and kinetics.1959

The rate of a reaction is equal to some rate constant times the concentration of the particular reactant in that reaction.1963

In this particular case, this is the fast step; that does not control the rate.1971

The slow step, this step, from E_S to E + P, that is what controls the rate.1976

Therefore, this is the reaction we are concerned with- this way, right?1980

We said we can ignore this; we are concerned this way; the rate of our ultimate overall reaction is going to be the rate at which this breaks down.1987

That rate is given mathematically by k₂ x S; k₂ is the rate constant.1994

E_S is the concentration of the enzyme substrate complex- basic, first order kinetics.1999

This exponent here is 1; we are not going to get into what the order of the reaction is right now.2005

We are just going to stay nice and simple, so let me go back to black.2010

OK, now, we have this; this is our starting point right here.2015

Now, the thing is, the concentration of enzyme substrate complex, this is not easily measured experimentally.2024

We have to find different expressions for E_S so that we can substitute into this expression.2047

In other words, we need to express E_S in terms of things that are easily measured, and we can do that.2055

Let's find some alternatives for substituting - I hope I spelled that correct - into this E_S thing.2062

OK, excuse me.2086

Now, we are going to get into the mathematical derivation, and again, please do not worry if you do not necessarily follow this derivation.2090

The only reason I am actually doing the derivation is so you have some sense of what is actually happening, that these things do not just fall out of the sky.2097

Again, it is good to see a derivation; it is good to listen, to think about what is happening just to, sort of, see it even if it is only passively.2107

At least your mind is starting to wrap itself around what is happening.2116

The ultimate form of the Michaelis-Menten equation is what we are going to be concerned with, but I think the derivation is just important to see.2120

We are going to introduce some thing; actually, let me go ahead and do this derivation in blue.2129

Let's introduce some notation here, introduce this thing called E_T.2135

This is the total enzyme concentration; OK, well, the total concentration of enzyme comes from 2 sources.2145

You are going to have free enzyme, and you are going to have the enzyme that is actually bound up as the enzyme substrate complex, right?2157

That is it; those are your sources that comes up to the total enzyme concentration.2164

Our total enzyme concentration is equal - oops, let me make this a little bit better - to free enzyme concentration plus the enzyme that is tied up as enzyme substrate.2172

These things we can measure; this we can measure.2189

That is easy; we know how much enzyme we are actually using.2193

This is very, very easy; notice, we have already started to express E_S in terms of things that we can measure.2198

OK, now, rate formation, now the rate of E_S formation...OK, actually let me do these a little bit separately.2205

Now, we are going to write 2 rate equations.2229

And again, a rate equation is some rate constant times the concentration of the reactants.2233

Now, the rate of enzyme substrate formation is equal to enzyme substrate formation comes from here.2239

It is this step right here, this step.2254

It is equal to k₁ times the concentration of free enzyme times the concentration of S- that is it.2258

Again, the rate of a given reaction is the rate constant in the direction of that reaction times the concentration of the reactants.2267

In this case, you have that reactant and that reactant, so it is this, this.2276

Well, it is equal to k₁.2281

Let us rearrange E; E is equal to the total enzyme concentration minus the enzyme that is tied up as substrate.2288

It is equal to E total - E_S, and that times S- that is it.2296

I just substituted; I have just moved this over here.2309

That is E; I have substituted to here, and I end up with this.2312

That is my rate of formation of enzyme substrate complex.2316

Now, my rate of enzyme substrate breakdown because that is what is happening here.2321

Enzyme substrate is forming; enzyme substrate is breaking down.2331

Breakdown of the enzyme substrate complex can happen in 2 directions.2336

It can happen in this direction to form enzyme and product; it can happen in this direction to form enzyme and substrate.2340

We account for both; in this direction, it is equal to k_-1 because it is now this direction, so we use that rate constant, and the reactant is if our reaction is going in this direction, that is the reactant.2345

It is k₂ times the enzyme substrate complex.2364

Now, the reaction in this direction, again, E_S is the reaction, but k₂ - that is it, they are additive - k₂ times enzyme substrate concentration.2368

I hope this makes sense; again, anytime you have some reaction, AA + BB going to, let's say, CC, and if there is some equilibrium, if the reaction that we are looking at in that direction, well, it is going to be the rate constant - k₁ - times the concentration of this and times the concentration of that because we always deal in terms of the concentration of reactants.2385

If we are looking at the reaction in this direction, well, the rate constant, we call it k_-1 - that is the symbol - but now, in this direction, this is the reactant.2413

Because that is the reactant, it is going to be k_-1 times the concentration of that because now, the reaction is going in this direction.2422

We are just not writing it left to right; the forward reaction formation, that is this one.2430

The breakdown happens in 2 ways; it can break down in that direction.2438

It can break down in that direction; each of those is represented here.2442

OK, now is when we are going to apply the steady state assumption.2447

The steady state assumption says that the rate of breakdown of E_S is the same as the rate of formation of E_S.2452

I am going to set this equation equal to that equation.2458

That is the steady state assumption; here is where the math comes in.2465

We need to simplify the math; if I do not have the steady state assumption, I cannot do anything here.2470

Let's go ahead and do that one.2475

OK, now, I have got k₁ times the total enzyme concentration minus enzyme substrate concentration times substrate concentration is equal to k_-1 enzyme substrate concentration plus k₂ enzyme substrate concentration.2479

Now, after some algebra, I will just write algebra…, and you can check the algebra for yourself if you want.2515

You are going to get something that looks like this; you are going to get the enzyme substrate.2523

We are going to solve for this E_S thing; it is equal to the total enzyme concentration times the substrate concentration all divided by k₂ + k₁ / k_-1 + S.2527

I will go ahead and put this, something like that.2552

Now, this thing, this is defined - it is a definition right now, we will give a better definition in a minute - as the K_m.2556

We call it a capital K_m; this is the Michaelis-Menten constant.2570

OK, now, I am going to rewrite this.2588

Because we call it K_m, I am going to put K_m in there, so I have, now, E_S is equal to the total enzyme concentration times the concentration of S over K_m + S.2592

Now, I have an expression for this E_S; that is what I wanted.2607

Well, all of these things are actually pretty easily measured.2612

That is hard to measure, but E_T is easy to measure; that is easy to measure, and this one we will see in just a minute.2616

Now, we said - let me go back to blue here - the initial velocity is equal to k₂ x enzyme substrate concentration.2624

Well, now, I am going to take this, which is this, this expression, and I am going to substitute into here.2638

Therefore, I have v₀ is equal to k₂ times total enzyme concentration times substrate concentration over this thing called K_m + S.2644

Notice, this K_m constant, it is just a combination of the 3 constants that are involved in the reaction.2656

You have reaction 1, reaction 2, reaction 3- that is it.2663

That is all you are doing; we are just calling it 1 constant because we do not want a bunch of Ks floating around.2667

Alright, now, let's take a look.2674

Now, V_max, the maximum velocity occurs when the enzyme is saturated, right?2679

When the enzyme is fully saturated, we have that curve, that line up at the top when the enzyme is saturated.2694

That is when the total enzyme concentration happens to equal the enzyme substrate concentration.2710

In other words, when the enzyme is saturated, that means it is all enzyme substrate complex.2720

Well, that means that is the same as all of the enzyme concentration that I started with.2727

So, E_S happens to equal E_T; the total enzyme concentration, all of the enzyme is now enzyme substrate, so I can set those equal to each other.2734

The maximum velocity equals k₂ E_T, right?2751

We said that v₀ is equal to k₂ E_S.2762

Well, E_S is E_T, so V_max, now, because at maximum velocity, the enzyme substrate concentration equals E_T, so v₀, which happens to be the maximum velocity now - not just some initial rate - is equal to k₂ E_T, right?2766

That just comes from this; this, at maximum velocity, E_S is equal to E_T.2785

I put E_T in here, and instead of calling it initial velocity because the velocity is now maximum when these 2 are equal, I just call it V_max.2791

Now, I can write it.2801

Now, v₀ is equal to V_max x S/K_m + S.2807

This is what we wanted; OK, we had k₂ E_T up at the top.2823

We call that the V_max, maximum velocity; this is the Michaelis-Menten equation.2832

This is what we were hoping for.2837

This equation is the equation for that thing, that line that we say, which we are actually going to see in a second again.2843

This equation expresses that; now, each of these - this is what is nice - quantities, in other words, the initial rate that we measure the velocity, the maximum velocity, the substrate concentration and this Michaelis-Menten, this thing, are all easily experimentally determined and/or measurable.2853

We took that expression that involved the enzyme substrate complex.2897

We fiddled with it because we did not want to deal with the concentration of enzyme substrate complex, and we expressed it in terms of something that we can measure.2903

We know the substrate concentration; that is what we are controlling.2912

These other things are all easily measurable- that is it.2916

This is the Michaelis-Menten equation; this equation represents, under the postulates, the steady state postulate and the Michaelis-Menten postulate of the second step being the rate-limiting step, this describes that curve that we saw.2922

This says that the initial rate is contingent on...it depends on a maximum velocity.2937

It depends on substrate concentration, and it also depends on something called the Michaelis-Menten constant, which we will talk about a little bit more in just a second.2945

Now, let's see what happens when the v₀ is equal to 1/2 of V_max.2954

Again, special cases, we like to just, sort of, see what happens mathematically.2970

Well, when you set v₀ to 1/2 V_max, you end up with V_max/2 - you just plug it into that equation - equals V_max x the concentration of S over K_m plus the concentration of S.2974

The V_maxs go away; you can cross multiply.2996

You end up with K_m plus the concentration of S equals 2 times the concentration of S.3000

Go ahead and subtract, and you get K_m equals the concentration of S.3008

This is beautiful; this says that when we measure an initial rate, which is half as fast as the enzyme is ever going to get half the maximum velocity.3014

It is at that, when that is the case, the K_m is actually equal to the substrate concentration at that point.3028

Whatever substrate concentration I happen to pick, that is my Michaelis-Menten constant.3040

That is this thing; this is the definition we want.3044

That is beautiful; this K_m is equal to the substrate concentration, which gives us a velocity equal to 1/2 V_max.3048

Again, let's go ahead and take a look at our...alright.3080

Yes, alright, now, we can talk about V_max, 1/2 V_max, K_m.3095

We go ahead and we run our reaction data with different initial concentrations.3101

We end up getting this thing; we go ahead and we take a guess, an estimate, of this maximum velocity.3105

For this particular enzyme, whatever it happens to be, I go ahead and once I actually estimate a maximum velocity, I take half of that maximum velocity.3111

I come down; I draw a horizontal line.3120

I see where it touches the graph, and then, I go down, then, whatever this is, it is that substrate concentration, which will allow me to be at half my maximum velocity for this enzyme- that is it.3123

That is the definition of K_m; that is the best definition of K_m.3139

It is a nice experimental definition of K_m; again, K_m is the substrate concentration - moles per liter, millimoles per liter, micromoles per liter - at which the enzyme is at half maximum velocity.3142

OK, you run the experiment; you can find the maximum velocity from that, then, from maximum velocity, you can take half of it, and then you can find the K_m.3157

Now, you have got yourself a perfectly valid equation: v₀ = V_max x the concentration of S/K_m + S concentration.3167

Now, you can use whatever you want.3180

Let's say you want to slow it down a little bit for your experiment, whatever you are doing, or let's say you want to speed it up faster than half the velocity max, let's say you want the velocity to be up here, you can control the substrate concentration; and you have this equation.3185

In other words, you found K_m, and you found V_max- that is it.3200

You have an equation for that particular enzyme; that is what we want.3205

We want some sort of a quantitative description; now, let's take a look at this equation.3209

When the substrate concentration is a lot less than K_m on the denominator, we can ignore the S.3215

OK, because it does not make that big of a contribution to the denominator, we can ignore S in the denominator.3230

The equation becomes v₀ = V_max/K_m x the S concentration.3250

As we said before, this is a linear equation, thus, the almost linearity for small concentrations of S.3266

In other words, under small concentrations of S, the Michaelis-Menten equation reduces to a linear equation.3276

It supports the data; now, when the concentration of S is a lot greater than K_m, now, K_m can be ignored.3282

We ignore K_m in the denominator, and the equation becomes v₀ = V_max x S/S.3298

These cancel implying that v₀ = V_max, which is exactly what this graph says.3320

As your concentration rises really, really high, so much higher than what the K_m is, the K_m becomes negligible.3326

In this equation, the velocity becomes V_max- asymptotic behavior.3332

As you go in this direction, you actually approach your maximum velocity.3340

This equation confirms low-end behavior, high-end behavior, and it also matches everything else in between.3346

Now, the question is this: “Do enzymes actually behave this way?".3355

The answer is yes; enzymes actually behave this way even for situations that are not necessarily postulated by Michaelis and Menten, where you have not this simple 2-step process, where the 2nd step - the breakdown of the enzyme substrate complex - is the slow step.3359

It is very, very interesting; this is characteristic enzyme behavior- steady state kinetics.3377

It is called Michaelis-Menten kinetics; the names in and of themselves do not really matter all that much.3385

What matters is the actual behavior.3390

Thank you for joining us here at Educator.com; we will see you next time for a further discussion of enzyme kinetics, bye-bye.3394