consider these biochemical reactions (simpler than usual michaelis-menten setup):
$E + S \rightarrow^{k_{f}} ES$
$ES \rightarrow^{k_{r}} E + S$
$E$ and $S$ reversibly make $ES$. the forward/reverse rate constants are equal: $k_{f} = k_{r} = k$. how can you solve for steady state concentration of $[ES]$ given initial amounts $[E]_{0}, [S]_{0}$ and assuming $[ES]_0 = 0$?
$d[ES]/dt = k[E][S] - k[ES] = 0$
rate constant cancels:
$[ES] = [E][S]$
then $[E] = E_{t} - [ES]$ where $E_{t}$ is total enzyme concentration. by substitution:
$[ES] = [S]([E]_{t} - [ES])$
$[ES] = [S][E]_{t} - [ES][S]$
$[ES] + [ES][S] = [S][E]_{t}$
$[ES] = \frac{[S][E]_{t}}{1 + [S]}$
but if $E_{0} = 10, S_{0} = 5$ then $[ES] = (5 * 10) / (5 + 1) \approx 8.3$ but the right answer is closer to 4. what's wrong with this?