Calculating Km - Online Tutor, Practice Problems & Exam Prep

The Michaelis constant (K_m) can be calculated using two primary methods. The first involves algebraically rearranging the Michaelis-Menten equation, isolating K_m to show it as the substrate concentration at which the initial reaction velocity (v₀) is half of V_max. The second method expresses K_m as the ratio of the sum of dissociation rate constants (k_-1 + k₂) to the association rate constant (k₁). A smaller K_m indicates a stronger enzyme-substrate complex formation.

Calculating Km

In this video, we're going to begin our discussion on the tools that we have to calculate the Michaelis constant or the Km of an enzyme. And so, just like the Vmax, the Km can also be calculated in multiple ways. We're actually going to talk about two predominant ways to calculate the Km of the enzyme. In this video, we're going to focus on the very first way, and then in our next lesson video, we'll talk about the second way to calculate the Km. The Km, just like the Vmax, can also be calculated by algebraic rearrangement of both the Michaelis-Menten and the Lineweaver-Burk equations. Notice down below in our example, it's asking us to algebraically rearrange the Michaelis-Menten equation in order to solve for the Michaelis constant or the Km. Notice we have the Michaelis-Menten equation on the left, and the Lineweaver-Burk equation on the right. From our previous lesson videos, we know these two equations are reciprocals of one another, so they can be converted. This means that we want to isolate the Km variable all by itself.

We can start by getting this Km out of the denominator, so essentially moving it out of the bottom of this fraction. We can do this by taking the entire denominator and moving it up. The way we can do that is by multiplying both sides of the equation by the entire denominator. Doing this, we eliminate the denominator on the right side so that we're only left with the top, which is just Vmax times the substrate concentration. Then, on the left side, what we've done is moved this expression up. So, we will have our v₀ times the denominator, which is Km plus the substrate concentration. Again, we're trying to solve for the Km, so we want to isolate this variable. We can easily get rid of this v₀ by dividing both sides of the equation, moving it to the other side. So, we'll still have our Vmax times the substrate concentration here, but now we have v₀ dividing this entire expression.

On the left, we're left with this expression right here. So, we still have Km plus the substrate concentration. Now, all we need to do is get rid of this substrate, and we can do that by subtracting the substrate concentration from both sides of the equation. What we end up with is the same expression on the left, so we still have our Vmax times the substrate concentration over the initial reaction velocity, but now subtracting off the substrate concentration. What we're left with on the left-hand side of the equation is just the Km. Now, our Km is essentially isolated. However, we can further simplify this expression. We can factor out both of these substrate concentrations to the front. When we do that, what we get is the substrate concentration out in front, and then, of course, we just have Vmax over the initial reaction velocity.

Now, this turns into a 1 when factored out. So, this is the reduced form for rearranging the Michaelis-Menten equation to solve for the Km, just like we wanted to do. Recall from our previous lesson videos, we had defined the Km in several ways. And one of the ways that we defined the Km was the exact substrate concentration where the initial reaction velocity is half of the Vmax. This means that in the bottom, what we have is half Vmax for the initial reaction velocity. If you take Vmax divided by half Vmax, what you end up getting is 2. So, essentially what you end up with is 2 minus 1 here, times the substrate concentration, and this is still equal to the Km. So, 2 minus 1 is just going to be 1 and then, one times the substrate concentration is just the substrate concentration. You can see that the Km will be equal to the substrate concentration, and so you can see that the Km is literally just the substrate concentration that allows for the initial reaction velocity to be equal to half of the Vmax. That is how we define the Km in one of our ways.

That concludes our first method for calculating the Michaelis constant or the Km of an enzyme. In our next lesson video, we'll talk about the second predominant way to calculate the Km. So, I'll see you guys there.

Calculating Km

So in our last lesson video, we talked about one way to be able to calculate the Michaelis constant or the Kilometers of an enzyme, and that was to algebraically rearrange the Michaelis-Menten equation. Now, in this video, we're going to talk about a second alternative way to calculate the Kilometers of an enzyme. And so recall in our previous lesson videos where we talked about the Km of an enzyme, we said that the Km of an enzyme can be defined in multiple ways and it can be expressed with rate constants. And so recall that the Km is really just the ratio of the sum of the enzyme substrate complexes to dissociation rate constants, which are k−1, and k2, over its association rate constant k1. And so notice down below in our image on the left here, we have our typical enzyme-catalyzed reaction at the very very very beginning of the reaction. So we have the initial v0 here to remind us of that, and that's because we can see that the rate constant k−2 is being ignored here, that's why it's not being shown. And so, recall that, the enzyme substrate complex can dissociate in 2 different ways. It can dissociate backwards, via k−1 to form the free enzyme and free substrate, and it can also dissociate forwards via k2 to form the free enzyme and free product. And so the enzyme substrate complex dissociation rate constants are just going to be k−1, backwards, and k2 forwards. And so, we know that this is going to be over the enzyme substrate complex association rate constant, So that's going to be the free enzyme and the free substrate associating with each other via k1 to form the enzyme substrate complex. So that would be k1 down below in the bottom. And so this is one way to be able to or another way to be able to calculate the Kilometers of the enzyme is to just, take the sum of the dissociation rate constants over the association rate constant as stated up above. And so, also recall from our previous lesson videos, we additionally had said that this also correlates with the ratio of the free enzyme concentration times the free substrate concentration over the concentration of the enzyme substrate complex. And so, that's because the free enzyme and the free substrate is a measure of the enzyme substrate complex dissociation, whereas the concentration of the enzyme substrate complex is a measure of the enzyme substrate complex association. So, over here, we can say that it's a measure of the free enzyme, concentration times the free substrate concentration over the concentration of the enzyme substrate complex. And so, this is really correlating, to, other ways that we are calculating our Kilometers. And so, again, all of this is review from our previous lesson video. There's no new information here; it's just a refresher. And so as also recall from our previous lesson videos that a small value of the Kilometers actually means that there's going to be less dissociation of the enzyme substrate complex, and there's going to be more enzyme-substrate complex formation or association. So essentially, if the Kilometers value is really, really small, as this is indicating, the only way that we can get the Kilometers value small is if, we say that there is a little bit of free enzyme and free substrate, and a lot of the enzyme substrate complex or more of the enzyme substrate complex. So again, this is all refresher from our previous lesson videos, and we'll be able to utilize these different methods of calculating the Kilometers in our practice problem. So, I'll see you guys in our next video.

Calculating Km Example 1

Alright. So here we have an example problem that says the following rate constants were measured for a simple enzyme-catalyzed reaction. Determine the Michaelis constant or the kilometers for the enzyme, and notice that we're given the values for the three relevant rate constants for a simple enzyme-catalyzed reaction. And then we've got 4 potential answer options for the Kilometers of the enzyme. So recall from our previous lesson videos, we talked about 2 predominant ways to calculate the Kilometers of an enzyme.

And so notice the first way, which is the algebraic rearrangement of the Michaelis-Menten equation. Notice that we're not actually given the values for the initial reaction velocity, the v, or the substrate concentrations, which means that we cannot use the first method here of algebraically rearranging the Michaelis-Menten equation to determine the Km. And so, for that reason, we can actually just go ahead and just remove this completely and, essentially what that means is we're going to need to use our second method of calculating the kilometers. And, recall that the second method of calculating the Kilometers is related to the rate constants and expressing Kilometers with rate constants.

And so, we know that a simple enzyme-catalyzed reaction, shown here is going to have 3 relevant rate constants, and we know that the kilometers is going to be the rate constants for the enzyme substrate complex dissociation over the association of the enzyme substrate complex. So, essentially, what we need to recall is that the enzyme substrate complex can dissociate in 2 different ways. It can dissociate backwards via k-1 to form the free substrate and free enzyme, and it can dissociate forwards via k2 to form the free enzyme and the free product. And so it's going to be the dissociation over the association, so these two dissociations here, it's gonna be the sum of them, so it's gonna be k-1 + k2, and then it's gonna be over the rate constant for the association of the enzyme substrate complex. So that would be via k1 to form the enzyme substrate complex.

And so, what we can do is we can put the k1 on the bottom. And so we know that these rate constants here, the dissociation of the enzyme substrate complex is associated with the free enzyme and the free substrate, and the association of the enzyme substrate complex is, of course, associated with the concentration of the enzyme substrate complex, and so this is another way to express this equation. Now, notice that, we're not actually given the values for the concentrations of the enzyme or the substrate or the enzyme substrate complex. So we're not gonna be able to use that portion of the equation and we're gonna be focusing specifically on this portion right here.

And so, notice that we're given k1, as 2×109 inverse molarity, inverse seconds, and then we're given, the value of k1 and k2 as well. So all we need to do is plug in these values into our expression here to calculate for the Kilometers. So, essentially what we see here is that the Kilometers is going to be equal to, k-1, which is given to us as 1×103 inverse seconds, plus the value of the k2, which is 5×103 inverse seconds. And then, of course, this is all going to be divided by k1, the association rate constant, which is 2×108, and units of inverse molarity, inverse seconds. And so, essentially, if you take your calculators and add 1×103 plus 5×103 and divide that answer by 2×108, what you'll get is the answer for our Km, which is equal to 3×10-5 M. And so we can see that 3×10-5M matches with answer option a. So we can go ahead and indicate that a here is the correct answer for this example problem. And that concludes this example problem, and we'll be able to apply these concepts moving forward in our practice problem. So I'll see you guys there.

To determine the Km from a Lineweaver-Burk plot you would:

• A. Multiply the reciprocal of the x-axis intercept by -1.
• B. Multiply the reciprocal of the y-axis intercept by -1.
• C. Take the reciprocal of the x-axis intercept.
• D. Take the reciprocal of the y-axis intercept.

The V_max for an enzyme is 9 mg/min. Calculate the K_m if the [S] = 5 mM when the V₀ = 3 mg/min.

• A. 100 M.
• B. 10 M.
• C. 10 mg.
• D. 10 mM.

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.

• A. 3.35 x 10^-4 M^-1.
• B. 3.58 x 10³ M.
• C. 3.85 x 10³ M^-1.
• D. 3.35 x 10^-4 M.

Here’s what students ask on this topic:

What is the Michaelis constant (Km) and how is it calculated?

The Michaelis constant (Km) is a measure of the substrate concentration at which the reaction velocity is half of the maximum velocity (Vmax). It can be calculated using two primary methods. The first method involves algebraically rearranging the Michaelis-Menten equation to isolate Km. The second method expresses Km as the ratio of the sum of the dissociation rate constants (k_-1 + k₂) to the association rate constant (k₁). A smaller Km indicates a stronger enzyme-substrate complex formation.

How do you algebraically rearrange the Michaelis-Menten equation to solve for Km?

To algebraically rearrange the Michaelis-Menten equation to solve for Km, start with the equation: $\frac{V=\frac{V}{}}{}$_max⋅SK_m+S. Multiply both sides by (Km + S) and then isolate Km by dividing by V and subtracting S. The final expression is: $K$_m=V_max⋅SV-S.

What is the significance of a small Km value?

A small Km value indicates a high affinity between the enzyme and its substrate. This means that the enzyme-substrate complex forms more readily and dissociates less frequently. In other words, a lower Km suggests that the enzyme is highly efficient at low substrate concentrations, which is often desirable in biological systems.

How is Km related to the rate constants of enzyme reactions?

Km can be expressed in terms of the rate constants of enzyme reactions. Specifically, Km is the ratio of the sum of the dissociation rate constants (k_-1 + k₂) to the association rate constant (k₁). The formula is: $K$_m=k_-1+k₂k₁. This relationship helps in understanding the dynamics of enzyme-substrate interactions.

What are the two primary methods to calculate Km?

The two primary methods to calculate Km are: 1) Algebraically rearranging the Michaelis-Menten equation to isolate Km, and 2) Expressing Km as the ratio of the sum of dissociation rate constants (k_-1 + k₂) to the association rate constant (k₁). Both methods provide insights into the enzyme's affinity for its substrate and its catalytic efficiency.