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I'm working my way through the chapter on enzymes in "Biochemistry" with Stryer et. al, 9th edition. In the current paragraph it's claimed that the fraction of active sites bound to substrate in an enzyme can be approximated by: $$f_{ES}=\frac{V}{V_\mathrm{max}}$$

And fair enough, that sounds reasonable. It also claims, however that:

$$f_{ES}=\frac{[S]}{[S]+K_M}$$

This i can't quite grasp. Why would this be the case?

I tried to arrive at this expression by rewriting $V_0=V_\mathrm{max}\frac{[S]}{[S]+K_M}$ in terms of $V_\mathrm{max}$ and then dividing the expression for $V_0$ with the expression for $V_\mathrm{max}$. So far, no such luck.

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    – Poutnik
    Oct 5, 2021 at 10:38

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If you already know that

$$V_0=V_\mathrm{max}\frac{[S]}{[S]+K_M}$$

you divide by $V_\mathrm{max}$ to get

$$\frac{V_0}{V_\mathrm{max}} = \frac{[S]}{[S]+K_M}$$

Both sides of the equation tell you what fraction of active sites is occupied by substrate.

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