Note that the symbol K is already predefined, so we use k

??K

y = (Subscript[R, P,ss])/(Subscript[R, T])

As derived, we must solve these two equations, for y, eliminating x :

GKrule=Solve[{x+y==1,(u x)/(J+x)==(v y)/(k+y)},y,x]

GK1=y/.GKrule[[1]]

eq .7 in PNAS 1981, ref [30], see http://www.nat.vu.nl/~ivo/SB/dynamic_models/MM/GoldbeterPNAS1981_78_ 6840.pdf
 is arrived at by division through v in numerator and denominator:

GK1/.v->1/.u->(u/v)

To derive the Goldbeter - Koshland function we follow the derivation in 
http : // en.wikipedia.org/wiki/Goldbeter - Koshland_kinetics

any Mathematica expression has an internal representation, which you can see using TreeForm or FullForm

GK1//FullForm

GK1//TreeForm

On the first level GK1 is a multiplication of three terms

GK1[[1]]

GK1[[2]]

GK1[[3]]

GK2=y/.GKrule[[2]]

Likewise, on the first level GK2 is a multiplication of three terms, and we need the third one for further simplification:

GK2[[3]]

Simplify[GK1 GK2[[3]]]/GK2[[3]]

and this is the form we find in the Sniffers paper.

Exercise : prove eq.1.9 from the lecture notes

First study the functions.nb by executing the commands and reading the Help pages. Next define (using SetDelayed) the GK function with four arguments u, v, J and k using the above expression

Now prove eq. 1.9 from the lecture notes using Simplify