I'm working through Mathematical Modelling in Systems Biology: An Introduction by Brian Ingalls, and in Chapter 3 there's an example of a simple allosterically inhibited enzyme with reaction scheme shown here:
$$\ce{S + E<=>[k_1][k_{-1}] ES->[k_2] E + P}\\[2em] \ce{I + E<=>[k_3][k_{-3}] EI}\\[2em] \ce{I + ES<=>[k_3][k_{-3}] ESI}\\[2em] \ce{S + EI<=>[k_1][k_{-1}] ESI} $$
where E is the free enzyme, I is the allosteric inhibitor, S is the substrate, P is the product, and ES, ESI, EI etc. are the various complexes between them. The book goes on to state that the rate of formation of P is
$$\frac{V_{max}}{1+I/K_I}\frac{S}{K_M +S}$$
where
$V_\text{max}$ is the maximal rate of reaction ($k_2$ * Total enzyme concentration), $K_I$ is the dissociation constant of the inhibitor $k_{-3}/k_3$ and $K_M$ is $(k_{-1}+k_2)/k_1$
No proof for this is shown, although the book says the solution is assuming a quasi-steady state of the complexes ES, EI and ESI. I've attempted to work it out, and arrived at almost, the same equation, except I have I/Ki * (k-1+k3)/k1 instead of I/Ki * KM. I have no idea how to get to the equation present by the book. Can someone explain how their equation was derived?