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Michaelis-Menten Kinetics · Acedra
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Michaelis-Menten Kinetics
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In this video we're going to talk about Michaelis-Menten kinetics which is a really important way to think about the rates of chemical reactions in a cellular context. So let's tackle that. So imagine for a second that you have a cell and there you're interested in a particular reaction that's happening in that cell and you want to know how fast is that reaction going given what I know about what's happening in the cell. This is a good point to pause and think about
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it for a second but what sorts of information would you need if you wanted to actually do that if you wanted to guess or to estimate how fast a reaction is going in the cell. So go ahead pause that for a second what do you need to know to guess how fast a reaction is going in the cell. Okay turns out that you can get at this with actually if you know four pieces of information which is how much of the substrate is around, how well does the enzyme recognize the
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substrate, how fast can the enzyme catalyze the reaction when it's bound to substrate, and then how much enzyme is around. It turns out if you know these four pieces of information you can plug this into a really helpful and powerful piece of mathematical machinery called the Michaelis-Menten equation. So it's shown here at the top we'll unpack it a little bit I just want to mention really quickly it's named after Maude Benton and Leonor Michaelis. Maude Benton is
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actually a really fascinating person it's totally worth reading more about her but she did a medical degree in Canada and then they didn't allow her to do research because she was a woman so then she went to Berlin and did research there and then she went to Chicago she had this but she was a very early pioneer in physical biochemistry and she did it the whole time being a pioneer as a woman scientist too so it's really a cool story so totally check that out. But
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with that aside over let's talk about the Michaelis-Menten equation itself. So what are the parts? What is it used for? So I said that you needed four pieces of information. What are those four pieces of information? How are they coded into this? So how much substrate is around is given by this term S naught substrate concentration. We'll talk about details in that in a second. How well does the enzyme recognize substrate? That's given by this thing Km. How fast can the enzyme
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catalyze the reaction is given by Kcat. How much enzyme around is given by this thing E total enzyme concentration and then this is how fast. So I should have put that one on there. And so when you look at this hopefully you can see oh yeah I can see how I could plug numbers in and get an answer. Maybe you don't quite understand how the parts are put together yet so we're gonna take that apart. Alright so let's make a graph of this here what this thing's gonna look
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like. So what I'm gonna do is I'm gonna plot V naught on this so this is the velocity of the reaction and we'll worry about the naught in a second. So for the moment this is just how fast the reaction is going and then down here this is S naught this is the substrate concentration. So as we add more substrate what does the velocity curve look like? This is actually another good point to pause and think about this. So what is the velocity curve gonna look
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like? Is it gonna go up? Is it gonna go down? Is it gonna twirl around? And when I'm teaching a class and personally have students like wave their hands it's really fun. So wave your hand in front of your screen what is this curve gonna look like if I do a graph of S naught versus V naught or V naught versus S naught? Turns out the curve looks like this. Now we talked about this kind of curve in my video about binding if you've seen it I said something about when something does
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a saturation behavior that tells us something is happening what does it tell us? It tells us something is running out. Wow that is terrible writing even for me something is running out. We're running out of something in this situation. What are we running out of here? We are running out of enzyme. That is the key key key thing that this models. It models the fact that there are a finite number of enzymes there's not an infinite number of enzymes in a cell if
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you add enough substrate eventually you're gonna run out of enzymes. We've also seen a curve that has this kind of shape before a saturating curve and that curve looks a lot like this but it has this thing theta and recall that theta would be mx over m plus mx this is the concentration of a macro molecule that's bound to some small molecule x has this kind of shape. Turns out that is given by this x over x plus kd. What do you know that is exactly
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what's here in the middle. This Michaelis-Menten equation has built into it a binding equation. So I said that our velocity as your velocity as your substrate concentration goes up your velocity sort of saturates it runs out because we're running out of enzyme and we can see that that's being modeled in this by a binding equation we're basically seeing that as enzyme binds we run out of enzyme and we can no longer go any faster. Okay so this thing here s
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and km km is measuring substrate affinity it's behaving exactly like a kd it's saying what is the affinity of the substrate for the or the enzyme for the substrate what is the affinity of the substrate for the enzyme we talked previously about steroid receptors we could say what is the affinity of the estrogen receptor for estradiol it's on whatever the number was 0.1 nanomolar here it's the exact same sort of thing we can say what is the affinity of the
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enzyme for the substrate it's some number of bicromole or millimolar usually. Okay that's measuring that but this is giving us a velocity right a rate this has no units actually right this is concentration over concentration plus concentration so this has no units this is a fractional saturation so kcat is giving us a rate and what it's measuring is a catalytic rate once bound so once the enzyme is engaged with the substrate how quickly can it go so this
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is the catalytic rate so km is measuring affinity kcat is measuring catalytic rate. Okay so that's just kind of phenomenologically that's what we're seeing we're seeing all right we can measure velocity we got this guy's his binding curve shape it's got the kcat whatever what are we actually modeling here so the equation we're modeling looks like this enzyme plus substrate is in equilibrium with an enzyme substrate complex which then goes on to enzyme
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plus product this has a relationship kcat so this is the equation that we're modeling I'm going to talk about the assumptions and then what this actually looks like you know what we're actually assuming here so we're making two assumptions in this Michaelis-Menten analysis sometimes people talk about three but really we're making two in the third follow so the first assumption we're making is an enzyme substrate to product I guess enzyme plus
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product make this balanced so the final reaction there is irreversible so we never see product going back to substrate we just see substrate going to product the second assumption we're making is that enzyme plus substrate this first reaction going this is fast and reversible so the last reaction is an irreversible process and this is a fast reversible process and I should say fast here by fast I mean faster than kcat so kcat is relatively slow this is
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relatively fast and this is actually a mathematically exact analogy for this turns out if you go to this YouTube channel you can get a link to actually buy one of these machines so maybe you should go do that maybe I get this guy to sponsor me but the basic idea of that we're seeing here is this is a meth like I said mathematically exact representation of Michaelis-Menten kinetics so what we have are a bunch of balls that are bouncing around they're
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being randomly bounced around they're engaging here with this position which I'm going to call the enzyme substrate complex each of the balls are substrates and so we have this rapid k on k off k on k off right the balls are bouncing in and out of this very very rapidly but then every once in a while we have a slow process that's kcat what is the rate that the reaction happens that the ball comes out of the thing and then once it comes out it is your
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reversible so if you're thinking about an enzyme active site and you're thinking about a substrate binding substrates binding coming out binding and coming out binding and coming out binding and coming out really really fast as it's doing that really fast process occasionally you'll have kcat happen and it will turn into product okay so that's what we're trying to model are those reasonable assumptions well turns out they are with how we're gonna get to
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these values so how do we get to km and kcat remember those are measures of affinity and rate how do we get those values right so we could actually use those to say study if something happening inside of a cell but actually we need to go back before we get there and ask how do we get to v naught actually how do we get to this this thing here that we might experimentally measure it turns out we get to v naught in an experiment so here is the
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experiment we might run what we're gonna do is as a function of time we're going to measure the accumulation of product from a particular starting concentration so I'm gonna say s not equals 1 I just made that number up but what we will see is s naught equals 1 if you imagine you can follow product over time with say a color or something you get a curve that looks something like this now this is a point of confusion and I've decided this is because biochemists hate everybody
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but we have no it's actually part of the physics of the universe um but basically we have two things happening here that have the same kind of shape this curve is not the same same as the curve that we saw previously which was substrate on the x-axis concentration on the y or velocity on the y-axis this is time on the x-axis product concentration on the y-axis these have different shapes or they have the same shape but different underlying things that are happening
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right so what's happening here is we start with no product here we reach equilibrium we're out of we've we've accumulated all the product we're gonna get um okay so cool what do we do with this what we can do is look at the very first step of this the first couple of points if we're say measuring this at that point we know two things we know that s naught equals one and we know that product formation is irreversible right because we don't have any product
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around to begin with the only thing that we're seeing is reactant going to substrate going to product so it turns out this is v naught this is the initial velocity of the reaction when we have some substrate concentration and no product around we know the substrate concentration and we know so we can measure this velocity okay so that was v naught with s naught equals one now let's do s naught equals two so we've added some more substrate and so the
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reaction is gonna go faster actually we're gonna get more product we're gonna get twice as much product right and that's gonna give us another v naught and so that's gonna give us a second value of v naught that's the blue value and then let's do another one so let's do s naught equals three now it's gonna be three times higher and we're gonna have a v naught associated with that right so you can imagine that what we can do experimentally is measure these
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v naught these initial velocities for reactions as a function of substrate concentration and that will let us construct a v naught versus s naught curve and so I was actually doing it there in that form so we're gonna transfer this way we're basically gonna say all right our s naught equals here and this is our v naught that's the initial rate of this reaction right so for zero I'm gonna do an orange our v naught s naught is zero for red maybe
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it's there this is one and then I had two let's put that there and then three let's put that there now I made up these numbers right schematically but you can imagine here that this is what you see right so you measure what's the initial velocity under s equals one under red you measure the initial velocity under s equals two blue you measure the initial velocity and s equals three the black one you then plot those initial velocities versus the concentration and
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you get out one of these Michaelis-Menten curves okay so remember though our goal is to get to Km and Kcat and remember that those things have meaning this is the affinity and this is the rate or the sort of rate constant for the reaction how do we get those from one of these curves well it turns out the Km we already know how to do so remember that this relationship this equation looks like this s naught over s naught plus Km times Kcat times ET this
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block here looks a heck of a lot like what we've seen before right with X over X plus Kd how do we get a Kd from a curve well we go to the 50% line and whatever exactly that is in this context we drop it down and Km is s naught at 50% okay now you might be saying yeah but 50% of what and it's going to be 50% of V max V max is saturated velocity that's when all of your enzyme is bound up with substrate there's nothing limiting it other than the amount of enzyme around
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and the rate of the reaction that is the V max it would be the equivalent of the theta equals 1 it's the point where the enzyme is fully saturated so what is V max on this graph here well V max is when the enzyme is completely saturated what that means is this here should be 0 right the concentration you're you're at a very high concentration of s relative to Km so the V max equals Kcat times the enzyme concentration so you can get to the V max by asking where does it go
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what is the highest curve on there that's your V max what's your Km Km is the s not at 50% of whatever the V max value is and then what is your Kcat your Kcat is V max over your enzyme concentration because if you're doing the experiment presumably you know your enzyme concentration okay so what is that not what does that mean that means that we can describe any enzyme any Michaelis-Menten enzyme in terms of a Km which is the affinity for substrate
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which we get the same way we would get the affinity for any other molecule we look at the concentration of half of its activity and we can get Kcat which is the fastest the enzyme can go if the enzyme is fully loaded if this this fractional saturation is at one so it's fully loaded up how fast does it go how was the Kcat times the amount of stuff we have okay last little a couple of cleanup points that are cool and interesting the first is that one of the
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things that biochemists often describe is Kcat over Km and this is basically a ratio that says how often do the enzyme and the product I'm sorry sorry the enzyme the substrate collide productively so this is in rates so this is in per seconds and this is in concentrations and moles so it's basically the seconds per mole per molar per second how how quickly is this happening and it turns out that you can actually discover when a reaction is
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diffusion limited or not based on this reaction so basically if you know the size of the molecules you know how fast they can diffuse there are some really nice physics equations that say how quickly you know big things versus small things to fuse into each other and if that rate is up there as high enough what that says is you actually have a diffusion limited reaction basically if this Kcat over Km is really really high that says you're now instead of being
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limited by you know the formation of this transition state and all these things you're just limited by the stuff running into itself last bit of cleanup on putting here and then we're actually gonna be done with this is we've talked about Kcat and Km as though they're kind of a feature of the enzyme but really they're a feature of the enzyme with a particular substrate so for example this is the enzyme fumarase which is an important enzyme in the citric acid
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cycle it interconverts between fumarate and malate and hopefully if we can do those are kind of small I'm realizing but you can look at these and say oh what is Kcat it's 8 times 10 to the 2 for fumarate it's 9 times 10 to the 2 for malate and then they have pretty different Kd's the Km sorry the Km is 5 times 10 to the minus 6 for fumarate and 2.5 times 10 to the minus 5 for malate so what this means is the enzyme is recognizing these things slightly
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differently because they have slightly different structures okay that was a lot of stuff but what do we say well first of all we said that enzyme kinetics are described using this thing called the Michaelis-Menten equation in reactions in a cell the number of enzymes is often limiting which means we have curves that look like this as we increase our substrate concentration our velocity ends up leveling out the Km is a measure of the affinity for the substrate so that's something like
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right there that's the Km the Kcat is the rate of the reaction that's catalyzed by the enzyme once the substrate is bound and you can read out these curves these Kcat and Km values by looking at these V0 versus S0 curves okay so that's what I have on Michaelis-Menten kinetics hope that was helpful we're gonna be doing enzyme inhibitors and then we'll be done with our enzyme series so talk to you later
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