Library / Solution chemistry, equilibria and kinetics / SCH-44
Solution chemistry, equilibria and kinetics
Michaelis–Menten initial rate
Initial rate of a single-substrate enzyme reaction from the substrate concentration S, the maximum rate Vmax and the Michaelis constant Km.
- Ref
- SCH-44
- Rev
- v1
- Field
- Solution chemistry, equilibria and kinetics
- Result
- v0 [mol/(L·s)]
With a steady-state enzyme–substrate complex, the initial rate of a single-substrate reaction is v₀ = Vmax S/(Km + S). At low S the rate is proportional to S, and at high S it saturates at Vmax. At S = Km the rate is half of Vmax, which defines Km. It applies for initial rates, before the substrate or product changes appreciably, and for a single substrate.
Initial rate of a single-substrate enzyme reaction from the substrate concentration S, the maximum rate Vmax and the Michaelis constant Km. It is half of Vmax when S = Km. Steady-state intermediate and negligible depletion.
Calculator
- Fraction of Vmax, S/(Km + S)0.75
- Substrate for 90 % of Vmax, 9Km0.009 mol/L
- Rate at S = Km, Vmax/20.001 mol/(L·s)
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- At 0.003 mol/L of substrate the enzyme runs at 75 % of its maximum rate, 0.0015 mol/(L·s).
- Because the returns fall as S rises, going from 10 Km to 100 Km only raises the rate from 91 % to 99 % of Vmax.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mol/(L·s)) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Vmax Maximum rate | 1e-8 to 10 mol/(L·s) | 7.5e-9 | 7.5 | 7.5 | +10% | +10 % |
| S Substrate concentration | 1e-6 to 1 mol/L | 1.998e-6 | 0.001998 | 0.001996 | +10% | +2.33 % |
| Km Michaelis constant | 1e-6 to 1 mol/L | 5.982e-6 | 0.001999 | 0.001993 | +10% | −2.44 % |
Sweep of Substrate concentration
| Substrate concentration (mol/L) | Initial rate (mol/(L·s)) |
|---|---|
| 1e-6 | 1.998e-6 |
| 5.623e-6 | 1.118e-5 |
| 3.162e-5 | 6.131e-5 |
| 1.778e-4 | 3.02e-4 |
| 0.001 | 0.001 |
| 0.005623 | 0.001698 |
| 0.03162 | 0.001939 |
| 0.1778 | 0.001989 |
| 1 | 0.001998 |
Sweep one variable
Variables and how each one contributes
Initial rate
What the equation gives you. Everything else feeds it.
Maximum rate
The maximum rate at saturating substrate, in mol/(L·s). Raises the rate in direct proportion. Raising it by 10 % (to 0.0022) moves the result by +10 %, from 0.0015 to 0.00165 mol/(L·s).
Substrate concentration
The initial free substrate concentration, in mol/L. Raises the rate, with falling returns as saturation sets in. Raising it by 10 % (to 0.0033) moves the result by +2.33 %, from 0.0015 to 0.001535 mol/(L·s).
Michaelis constant
The Michaelis constant, in mol/L. A larger Km means weaker binding, so the rate falls. Raising it by 10 % (to 0.0011) moves the result by −2.44 %, from 0.0015 to 0.001463 mol/(L·s).
Worked example with the values above
- Denominator: Km + S = 0.001 + 0.003 = 0.004 mol/L.
- Saturation fraction: S/(Km + S) = 0.75.
- v₀ = 0.002 × 0.75 = 0.0015 mol/(L·s).
Assumptions and limits
- Irreversible first-order steps for the consecutive scheme, and a single substrate with a steady-state intermediate for the enzyme.
- Constant temperature, pH and volume, with negligible product and substrate depletion for initial rates.
Common mistakes
- Using the formulas after substantial substrate depletion, where the rate no longer is the initial rate.
- Mixing up k₁ and k₂ in the consecutive formulas.
- Using the total inhibitor concentration in place of the free one.
Statistics and uncertainty
Fitted enzyme constants, Km and Vmax, carry uncertainties of 10 % to 30 % from the data scatter, and a wide range of substrate concentrations is needed to define both.
Where it is used
- Following intermediates in sequential reactions and decay chains.
- Describing enzyme saturation and inhibition.
Maths you need
- Integrated rate law: concentration as a function of time
- Steady state: the intermediate’s concentration stays constant
- Michaelis constant: Km, the substrate concentration at half of Vmax