Calculating Probability of Ligand-Protein Binding at Equilibrium

[Quickdry135]

Homework Statement

A monovalent ligand binds to a protein with six in independent, identical binding sites. What is the probability that a given protein molecule is bound by at least five ligand molecules at equilibrium if K$$^{\mu}_{D}$$ = 1nM and L$$_{0}$$=2nM (constant)?

Homework Equations

I don't really know what equations to use to get started on this.

The Attempt at a Solution

I suppose this would be more of a probability or statistics based problem, but I have to take into consideration the protein binding affinity and ligand concentration. I know K$$_{D}$$ is K$$_{off}$$/K$$_{on}$$, so that would be a measure of the probability of binding to anyone spot. The initial ligand concentration also determines the probability of binding since it allows for more ligand to be bound to the protein binding sites. But I don't know how to use these definitions to create a mathematical way to find the actual numerical probability.

I would appreciate any help.

 

[chemisttree]

Homework Statement

A monovalent ligand binds to a protein with six in independent, identical binding sites. What is the probability that a given protein molecule is bound by at least five ligand molecules at equilibrium if K$$^{\mu}_{D}$$ = 1nM and L$$_{0}$$=2nM (constant)?

Homework Equations

I don't really know what equations to use to get started on this.

The Attempt at a Solution

I suppose this would be more of a probability or statistics based problem, but I have to take into consideration the protein binding affinity and ligand concentration. I know K$$_{D}$$ is K$$_{off}$$/K$$_{on}$$, so that would be a measure of the probability of binding to anyone spot. The initial ligand concentration also determines the probability of binding since it allows for more ligand to be bound to the protein binding sites. But I don't know how to use these definitions to create a mathematical way to find the actual numerical probability.

I would appreciate any help.

Have you studied Scatchard plots or the Scatchard equation or the Eadie-Scatchard equation?

 

[Quickdry135]

The scatchard equation is (r/c) = Ka*n - Ka*r, where r is the ratio of the concentration of bound ligand to total available binding sites, c is the concentration of free ligand, Ka is the association constant, and n is the number of binding sites per protein, right? So through this I could find the ratio of bound ligand to total available binding sites under the given ligand concentration and Kd (the inverse would be Ka). Which would be 4 or 24/6. But how would I use this ratio to determine the probability of a protein binding to 5 or more ligand molecules?

Thanks for replying, by the way.

 

[chemisttree]

Sorry, I think I misunderstood your question. I've never calculated probabilities in that way before. Perhaps someone over at mathematics can help.

 

[Quickdry135]

Ok thanks anyway, I know there's a probability equation pertaining to this in physical chemistry, but for the life of me I can't remember it or find it in my book.