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I have 9 sites to which X can bind. 6 out of the 9 sites are active and 3 out of the 9 sites are inactive. I need 3 of the active sites to be bound to get the response I am looking for - which we will call EMAX.

So when I add a single X - the chance of it binding to an active site is 6/9 the chance of it binding to an inactive site is 3/9.

My probability knowledge is shaky - bear with me.

Assume that the binding is irreversible. So how many X do I need to add to be sure I have activated 3 active sites. Or more precisely, how many X do I need to add to get a >95% chance that 3 active sites are bound.

Then I want to go more complicated. Say I add 3 Y - which inactivates the sites. The chances are that 2 active sites will be inactivated and 1 inactive site will still be inactive with Y bound.

Now under these new conditions - how much X do I need to add to be sure 3 remaining active sites are occupied?

So I know it will be a probability - so I guess lets say that how much X do I need to add to have a greater than 95% chance that 3 active sites are now bound with X to get EMAX