Question11. If $K_{\mathrm{M}}=2 \mathrm{mM}$ , and $v_{o}=100 \mu \mathrm{~mol} /(\mathrm{mL} \cdot \mathrm{s})$ when [ S $]=2 \mathrm{mM}$ , what is the velocity, $v_{\mathrm{o}}$ , for the reaction when $[\mathrm{S}]=18 \mathrm{mM}$ ? 5

Studdy Solution

STEP 1

1. We are dealing with an enzyme-catalyzed reaction that follows Michaelis-Menten kinetics.
2. The Michaelis constant, $K_{\mathrm{M}}$ , is given as $2 \, \mathrm{mM}$ .
3. The initial reaction velocity, $v_{o}$ , is $100 \, \mu \mathrm{mol} /(\mathrm{mL} \cdot \mathrm{s})$ when the substrate concentration, $[\mathrm{S}]$ , is $2 \, \mathrm{mM}$ .
4. We need to find the initial reaction velocity, $v_{o}$ , when $[\mathrm{S}]$ is $18 \, \mathrm{mM}$ .

STEP 2

1. Understand the Michaelis-Menten equation.
2. Identify known values and the unknown.
3. Use the Michaelis-Menten equation to find $V_{\max}$ .
4. Calculate the new velocity, $v_{o}$ , for $[\mathrm{S}]=18 \, \mathrm{mM}$ .

STEP 3

Understand the Michaelis-Menten equation, which is given by:
$v_{o} = \frac{V_{\max} \cdot [\mathrm{S}]}{K_{\mathrm{M}} + [\mathrm{S}]}$

STEP 4

Identify known values and the unknown:
- $K_{\mathrm{M}} = 2 \, \mathrm{mM}$ - $v_{o} = 100 \, \mu \mathrm{mol} /(\mathrm{mL} \cdot \mathrm{s})$ when $[\mathrm{S}] = 2 \, \mathrm{mM}$ - We need to find $v_{o}$ when $[\mathrm{S}] = 18 \, \mathrm{mM}$

STEP 5

Use the Michaelis-Menten equation to find $V_{\max}$ .
Given: $100 = \frac{V_{\max} \cdot 2}{2 + 2}$
Simplify: $100 = \frac{V_{\max} \cdot 2}{4}$
Multiply both sides by 4: $400 = V_{\max} \cdot 2$
Divide by 2: $V_{\max} = 200 \, \mu \mathrm{mol} /(\mathrm{mL} \cdot \mathrm{s})$

STEP 6

Calculate the new velocity, $v_{o}$ , for $[\mathrm{S}]=18 \, \mathrm{mM}$ .
Use the Michaelis-Menten equation: $v_{o} = \frac{200 \cdot 18}{2 + 18}$
Simplify: $v_{o} = \frac{3600}{20}$
Calculate: $v_{o} = 180 \, \mu \mathrm{mol} /(\mathrm{mL} \cdot \mathrm{s})$
The velocity, $v_{o}$ , for the reaction when $[\mathrm{S}]=18 \, \mathrm{mM}$ is:
$\boxed{180 \, \mu \mathrm{mol} /(\mathrm{mL} \cdot \mathrm{s})}$

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