Adairs equation

Hi,

 

I'm currently struggling with deriving the Adair equation for representing substrate binding cooperativity.

 

So far, I have:

 

p -> PS (reversible reaction, rate of k1) and PS -> P2.S (reversible reaction, rate of k2).

 

y = number of unbound sites / total number of binding sites

 

y = PS + 2P2.S / 2(P + PS + 2 P2.S)

 

I have rate equations for the 2 reactions as

 

k1 = PS / (P * S)

k2 = P2.S / (PS * S)

 

From this, I got

 

PS = k1 * P * S

P2.S = k2 * PS * S

 

From this, I inserted PS into the rate equation for P2.S so I got:

 

P2.S = k2 * (k1 * P * S) * S

 

I multiplied out the brackets to get k2.k1 + k2.P + k2.S^2.

 

I know that from my original equation Y = PS + 2(P2.S) / 2(P + PS + 2*P2.S) that I can insert the rate equations from k1 and k2, replacing PS and P2.S to get:

 

Y = k1.S + 2(k1.k2.S^2) / 2(1+k1.S + k1.k2.S^2)

 

this is Adair's equation, however I can't get from the 1st equation to the last equation when I rearrange the k1 and k2 rate equations and try and cancel out the P's. There's too many P's to cancel, and from what I can see, the whole equation cancels out. I have wrestled with this for several weeks and can't rearrange for formulae to arrive at the last one. Can anybody show me how to derive this equation from 1st principles?

 

Thank you for any assistance.

Edited December 27, 201114 yr by micronaut