In the Michaelis-Menten equation, KM = Ks only when the equilibrium assumption is valid. Which of the following conditions would have to be satisfied for Ks to be within 10% of KM?
1. k-1 = 0.9 × k2
2. k2 = 0.09 × k1
3. k2 = 0.9 × k-1
4. k-1 = 9.0 × k1
5. k2 = 0.09 × k-1
IM soo confused on how to do this .....can someone plz help me ? :(
To determine the condition in which Ks is within 10% of KM, we need to compare their values and find the appropriate relationship between the rate constants (k1, k-1, and k2) in the Michaelis-Menten equation.
Let's start by understanding the Michaelis-Menten equation:
v = (Vmax * [S]) / (KM + [S])
Where:
- v is the initial reaction velocity.
- Vmax is the maximum reaction velocity.
- [S] is the substrate concentration.
- KM is the Michaelis constant, which represents the substrate concentration at which the reaction velocity is half of Vmax.
Based on this equation, the relationship between KM and Ks is given by:
KM = (k-1 + k2) / k1
Now, let's analyze each option:
1. k-1 = 0.9 × k2
Substituting these values into the equation for KM, we have:
KM = (0.9 × k2 + k2) / k1 = (1.9 × k2) / k1
This relationship does not give us any information about the relationship between KM and Ks; thus, we can eliminate this option.
2. k2 = 0.09 × k1
Similarly, substituting these values into the equation for KM, we get:
KM = (k-1 + 0.09 × k1) / k1
This option also does not provide any information about the relationship between KM and Ks.
3. k2 = 0.9 × k-1
Substituting these values into the equation for KM, we have:
KM = (k-1 + 0.9 × k-1) / k1 = (1.9 × k-1) / k1
This relationship does not give us any information about the relationship between KM and Ks.
4. k-1 = 9.0 × k1
Using these values in the equation for KM, we obtain:
KM = (9.0 × k1 + k2) / k1
This relationship does not provide any information about the relationship between KM and Ks.
5. k2 = 0.09 × k-1
Substituting these values into the equation for KM, we get:
KM = (k-1 + 0.09 × k-1) / k1 = (1.09 × k-1) / k1
Comparing this result with the equation for Ks, we can see that Ks = (1.0 × k-1) / k1
Now, we need to determine the condition in which Ks is within 10% of KM. For this to be true, Ks must be equivalent to (1 ± 0.1) × KM. Therefore, we can conclude that Ks will be within 10% of KM if k2 = 0.09 × k-1.
Therefore, the correct answer is option 5: k2 = 0.09 × k-1.