Fundamentals of Biochemistry: Life at the Molecular Level 5th Edition

When $[S] = 100 mM$, the velocity is close to $V_{max}$. Therefore, assume that $V_{max} \approx 50 μM \cdot s^{−1}$ Using the Michaelis–Menten equation: $$v_{\mathrm{o}}=\frac{V_{\max }[\mathrm{S}]}{K_{M}+[\mathrm{S}]}$$ $$K_{M}+[\mathrm{S}]=\frac{V_{\max }[\mathrm{S}]}{v_{\mathrm{o}}}$$ $$K_{M}=\frac{V_{\max }[\mathrm{S}]}{v_{\mathrm{o}}}-[\mathrm{S}]$$ $$K_{M}=\frac{\left(50 \mu \mathrm{M} \cdot \mathrm{s}^{-1}\right)(1 \mu \mathrm{M})}{\left(5 \mu \mathrm{M} \cdot \mathrm{s}^{-1}\right)}-(1 \mu \mathrm{M})=9 \mu \mathrm{M}$$ The true $V_{\max }$ must be greater than the estimated value, so the value of $K_{M}$ is an underestimate of the true $K_{M}$ .
When $[S] = 100 mM$, the velocity is close to $V_{max}$. Therefore, assume that $V_{max} \approx 50 μM \cdot s^{−1}$ Using the Michaelis–Menten equation: $$v_{\mathrm{o}}=\frac{V_{\max }[\mathrm{S}]}{K_{M}+[\mathrm{S}]}$$ $$K_{M}+[\mathrm{S}]=\frac{V_{\max }[\mathrm{S}]}{v_{\mathrm{o}}}$$ $$K_{M}=\frac{V_{\max }[\mathrm{S}]}{v_{\mathrm{o}}}-[\mathrm{S}]$$ $$K_{M}=\frac{\left(50 \mu \mathrm{M} \cdot \mathrm{s}^{-1}\right)(1 \mu \mathrm{M})}{\left(5 \mu \mathrm{M} \cdot \mathrm{s}^{-1}\right)}-(1 \mu \mathrm{M})=9 \mu \mathrm{M}$$ The true $V_{\max }$ must be greater than the estimated value, so the value of $K_{M}$ is an underestimate of the true $K_{M}$ .