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My textbook in chemistry ("Biochemistry", Stryer et. al 9th ed p. 246) states that:

$$[ES]=\frac{[E][S]}{K_m}$$

where [ES] is the concentration of the enzyme-substrate-complex, [E] is the concentration of enzyme and [S] is the concentration of substrate. For the purposes of this discussion, we'll just regard $K_m$ as a constant I think.

The book then claims that:

$$[E]=[E]_T-[ES]$$

Substituting this into the first expression yields:

$$[ES]=\frac{([E]_T-[ES])[S]}{K_m}$$

And that I can live with. In the next step however, "Solving [the last equation] for [ES]" is supposed to give:

$$[ES]=\frac{\frac{[E_T][S]}{K_m}}{\frac{1+[S]}{K_m}}$$

How exactly do we arrive at that step?

Magnus
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1 Answers1

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Your last expression is not entirely correct. Nevertheless, this is an elementary school problem: $$ \left[ {ES} \right] = \frac{{(\left[ E \right]_T - \left[ {ES} \right])\left[ S \right]}}{{K_m }} = \frac{{\left[ E \right]_T \left[ S \right]}}{{K_m }} - \frac{{\left[ {ES} \right]\left[ S \right]}}{{K_m }} \\\Rightarrow \left[ {ES} \right] + \frac{{\left[ {ES} \right]\left[ S \right]}}{{K_m }} = \frac{{\left[ E \right]_T \left[ S \right]}}{{K_m }} \Rightarrow \left[ {ES} \right]\left( {1 + \frac{{\left[ S \right]}}{{K_m }}} \right) = \frac{{\left[ E \right]_T \left[ S \right]}}{{K_m }} \Rightarrow \left[ {ES} \right] = \frac{{\frac{{\left[ E \right]_T \left[ S \right]}}{{K_m }}}}{{1 + \frac{{\left[ S \right]}}{{K_m }}}}, $$ or $$ \left[ {ES} \right] = \frac{{\left[ E \right]_T \left[ S \right]}}{{K_m + \left[ S \right]}}. $$

Gary
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