Calculate the Km of an enzyme using Michaelis-Menten kinetics if the forward rate constant for ES formation is 4.3 x 10 6 M -1 s -1 , the reverse rate constant for ES dissociation into E + S is 2.4 x 10 2 s -1 , and the forward rate constant for ES dissociation into E + P is 1.2 x 10 3 s -1 .

Alright. So this practice problem wants us to calculate the Michaelis constant or the kilometers of an enzyme using Michaelis Menten kinetics if the forward rate constant for enzyme substrate Complex formation is this value here, the reverse rate constant for Enzyme Substrate Complex Dissociation into Free Enzyme and Free Substrate is this value right here, and the forward rate constant for Enzyme Substrate Complex Dissociation into Free Enzyme and Free Product is this value right here. And so recall from our previous lesson videos, we talked about 2 predominant ways of calculating the Michaelis constant or the Kilometers of an enzyme. And so the method where we rearrange the Michaelis Menten equation to solve for the Michaelis constant or the k equation to solve for the Michaelis constant or the Kilometers is not going to be useful for this problem since we're not given the values of the initial reaction velocity, the v max, or the substrate concentrations. And so there's just too many variables missing to be able to concentrations. And so there's just too many variables missing to be able to use this equation here to solve for the k m. So that means that we can't use this equation to solve for the k m. And, of course, that means that we're gonna need to use our alternative method, which is essentially related to the rate constants of our typical enzyme catalyzed reaction. And so, just to refresh our memory, recall that with a typical enzyme catalyzed reaction, we have 3 relevant rate constants. And, essentially what we need to do is correspond each of these 3 relevant rate constants k1, k-one, and k2 with the, rate constants that are being described up above in our problem. And so, essentially this one, forward rate constant for enzyme substrate complex formation, that is going to be essentially the enzyme associating with the substrate to form the enzyme substrate complex. So that is gonna be referring to k one. So forward rate constant here is going to be k one. And then, essentially the enzyme substrate complex, the reverse rate constant for enzyme substrate complex dissociation into free enzyme and free substrate is going to be starting with the enzyme substrate complex but dissociating in the reverse direction towards, again, the free substrate and the free enzyme. This is going to be, of course, k minus 1. And then, of course, the forward rate constant for enzyme substrate complex dissociation into free enzyme and free product is just gonna be starting with the enzyme substrate complex and dissociating forward into the free enzyme and free product. So this blue is referring to k 2. Now recall that with our equation over here, we think of the kilometers as being expressed as the dissoci the sum of the dissociation rate constants over the, association rate constant, for the enzyme substrate complex. And so, really, we wanna think of this forward rate constant here, as another dissociation rate constant. So we have 2 dissociation rate constants and we have 1, association or one formation rate constant. Constant. And so the formation rate constant is gonna, of course, be the k one, so we can put that in down below. And then the 2 dissociation rate constants are gonna be k minus 1 and of course k 2, so we can put those in here, k minus 1 and k 2. And, of course, we know that, the dissociation of the enzyme substrate complex is associated with the free enzyme and the free substrate concentrations, and the association of the enzyme substrate complex is gonna be associated with the concentration of the enzyme substrate complex. Now we're not actually given the concentrations for the free enzyme or the free substrate or the enzyme substrate complex, which means that we're not gonna be able to use this portion of our equation here and we're just gonna be focusing specifically on this, rate constant portion. And so, what we can do is just rewrite our equation here. So we have the kilometers is going to be equal to the k minus one rate constant, which we established was this rate constant here, 2.4 times 10 squared, and the units are inverse seconds. So this is our k minus 1, and this is going to be added to our k 2, which is going to be this rate constant here, 1.2 times 10 to the 3rd inverse seconds. And then this is all gonna be over our k one rate constant, which is gonna be 4.3 times 10 to the 6. And, the units are inverse molarity and inverse seconds. And so again, the, just to be clear here, this is our k minus 1, this is our k, our k 2, and then, this here is going to be our k one rate constant. And so all we need to do is type this into our calculators here. So in, your calculators if you do, 2.4 times 10 squared plus, 1.2 times 10 cubed, and you take the answer of this and divide it by 4.3 times 10 to the 6, then you'll get the answer for our k m, which is going to be equal to, 3.35 times, 10 to the negative 4th. And so, because this is the value for our kilometers here, what that means is that we can eliminate option c, which has a different value, and option b, which also has a different value. And so now we're between option a and option d, which both have the same exact value, and the only difference is the unit. So what would the unit be for our kilometers? M? Well, essentially, if we take a look at these units here, at the top here when we combine these two numbers, the units are gonna be inverse seconds. And so the inverse second unit is actually gonna cancel with the inverse second unit from the bottom. So this unit will cancel out both of these units here when we combine this number. And so the only unit that we have left is inverse molarity. And so, maybe, that might make you believe that the unit should be inverse molarity here. However, what we need to consider is that inverse molarity, unit is in the denominator. And so, if we were to just rewrite this over here, let's just say we had a number 1, on the top and we had inverse molarity in the bottom. Well, if we have inverse molarity in the bottom like we do over here, what we need to consider is that inverse molarity is, in the bottom is gonna be the same thing as having 1 over molarity in the bottom. So inverse molarity is just equal to 1 over molarity. And so, this is, when you have a fraction divided by a fraction, it's the same thing as taking, the number on top and multiplying it by the reciprocal of this fraction here. So this is gonna be the same as multiplying it by molarity over 1. So these are just rules of fractions, the the way that it just happens to work. And so one times, molarity over 1 is of course just going to be molarity. And so essentially what we're saying is that, having the inverse molarity in the denominator is the same thing as having just molarity in, the final number here. And so what that means is that the final number unit is going to be molarity and, the correct answer to our problem is going to be option d. And so, d here will be the correct answer and, that concludes this practice problem, and I'll see you guys in our next video.

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6. Enzymes and Enzyme Kinetics

Calculating Km

Biochemistry

6. Enzymes and Enzyme Kinetics

Calculating Km

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Multiple Choice

Calculate the Km of an enzyme using Michaelis-Menten kinetics if the forward rate constant for ES formation is 4.3 x 10⁶ M^-1s^-1, the reverse rate constant for ES dissociation into E + S is 2.4 x 10² s^-1, and the forward rate constant for ES dissociation into E + P is 1.2 x 10 ³ s^-1.

3.35 x 10^-4 M^-1.

3.58 x 10³ M.

3.85 x 10³ M^-1.

3.35 x 10^-4 M.

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Verified step by step guidance

Understand that Km, the Michaelis constant, is a measure of the substrate concentration at which the reaction rate is half of its maximum value. It is calculated using the rate constants of the enzyme-catalyzed reaction.

The formula to calculate Km using the rate constants is: \( K_m = \frac{k_{-1} + k_{2}}{k_{1}} \), where \( k_{1} \) is the forward rate constant for ES formation, \( k_{-1} \) is the reverse rate constant for ES dissociation into E + S, and \( k_{2} \) is the forward rate constant for ES dissociation into E + P.

Substitute the given values into the formula: \( k_{1} = 4.3 \times 10^{6} \text{ M}^{-1}\text{s}^{-1} \), \( k_{-1} = 2.4 \times 10^{2} \text{ s}^{-1} \), and \( k_{2} = 1.2 \times 10^{3} \text{ s}^{-1} \).

Calculate the numerator of the Km formula: \( k_{-1} + k_{2} = 2.4 \times 10^{2} + 1.2 \times 10^{3} \).

Divide the result from the previous step by \( k_{1} \) to find Km: \( K_m = \frac{2.4 \times 10^{2} + 1.2 \times 10^{3}}{4.3 \times 10^{6}} \).