I need help with these questions!!

Question

Two proteins, Pepperoni (Pe) and Pizza (Pi) bind to each other with a K D value of 3.4uM

1.) Assume that Pepperoni binds to Pizza independently. If [Pizza] = 1uM, what is the concentration of Pizza bound at two sites to Pepperoni if [Pepperoni]= 10uM?

2.) Studies have shown that pepperoni binds to pizza with a cooperativity factor of 6. What

is the affinity for the second binding event if there is already one pepperoni bound to pizza?

Answer 1

Question 1:

If the dissociation constant (K_D) is given as 3.4 µM, this defines the affinity between Pepperoni (Pe) and Pizza (Pi). The formula that relates the concentration of the bound complex, which we can call [PePi], to the free concentrations of Pepperoni and Pizza is derived from the law of mass action, which can write as:

K_D = [Pe][Pi] / [PePi]

We need to calculate the concentration of Pizza bound at two sites to Pepperoni under the given conditions.

Assuming independent binding of the two sites (since there is no cooperativity initially mentioned for this first question), the binding at each site is treated individually but will still be under the influence of the existing free and bound concentrations.

This makes it a sequential binding scenario and we have to calculate the concentration of the bound complex in two parts.

First binding event: Calculate the concentration of the singly-bound Pepperoni-Pizza complex (call it [PePi₁] ) when [Pi] = 1 µM and [Pe] = 10 µM

You can rearrange the above equation from the law of mass action and then plug and chug to find [PePi₁]:

[PePi₁] = [Pe][Pi] / K_D

Second binding event: Now, the free [Pe] will be reduced by the amount bound in the first event, so you'll have to subtract the answer you got for [PePi_1] from the initial [Pe] which was 10 µM.

This makes sense because you had to take some of that initial pepperoni to bind the first spot on the pizza. There will inherently be less pepperoni floating around, and you have to account for that. The amount of pizza hasn't changed though, so that concentration will stay the same.

We can call this new concentration of pepperoni [Pe' ]. Then you can again just plug and chug with the above equation (since there is no cooperatively yet) to find the final concentration of pepperoni bound at two spots to pizza (which we can call [PePi₂] ).

[PePi₂] = [Pe' ][Pi] / K_D

This final answer for [PePi₂] will be your concentration of Pizza bound at two sites to Pepperoni.

Question 2:

This will be pretty straightforward with the right equation. We are given a cooperativity factor of 6. For your understanding, this factor affects the affinity for the second binding event, typically making it easier (increased affinity, or a lower K_D) for the second Pepperoni to bind if one is already bound.

The new binding affinity (call it K'_D) for the second binding, given a cooperativity factor n = 6, can be calculated by reducing the original K_D (which the question stem already told us was 3.4 µM) by this factor:

K'_D = K_D / n

You just have to solve for K'_D using those numbers you've been provided in the prompt.

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Hope this helps! If you have any questions please let me know, I am happy to help!