Last updated: July 3, 2026
Michaelis–Menten Equation Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
The Michaelis–Menten equation v = Vmax[S]/(Km + [S]) models initial enzyme velocity as a function of substrate concentration. Km is the substrate concentration where v = Vmax/2, and v/Vmax shows fractional saturation. This calculator solves for velocity, substrate concentration, Vmax, or Km and flags interpretations such as half-saturation and near-saturation.
The Michaelis-Menten equation says reaction velocity equals V max times substrate concentration divided by K m plus substrate concentration. When substrate concentration equals K m, velocity is one half of V max.
Key Takeaways
- Michaelis–Menten kinetics relates initial velocity to substrate concentration: v = Vmax[S]/(Km + [S]).
- Km is the substrate concentration where v equals one-half of Vmax.
- The fraction of maximum velocity is v/Vmax = [S]/(Km + [S]).
- The equation can be rearranged to solve for [S], Vmax, or Km when the other quantities are known.
- As [S] becomes very large, v approaches Vmax asymptotically but does not exceed it.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
v = (Vmax × [S]) / (Km + [S]); [S] = v×Km/(Vmax − v); Vmax = v(Km+[S])/[S]; Km = [S](Vmax−v)/v
Where:
- v=Initial reaction velocity(rate units)
- Vmax=Maximum reaction velocity at saturating substrate(rate units)
- Km=Michaelis constant; substrate concentration at half Vmax(concentration units)
- [S]=Substrate concentration(concentration units)
- v/Vmax=Fractional velocity or saturation(dimensionless)
Worked Examples
Half-saturation: [S] equals Km
When substrate concentration equals Km, the Michaelis–Menten equation gives exactly half of Vmax.
- 1Start with v = (Vmax × [S]) / (Km + [S]).
- 2Substitute Vmax = 100, Km = 5, and [S] = 5.
- 3v = (100 × 5) / (5 + 5) = 500 / 10 = 50.
- 4The fraction of Vmax is 50/100 = 0.5, so [S] = Km gives v = ½Vmax.
Moderate substrate concentration
Doubling substrate above Km increases velocity but not linearly because the enzyme is approaching saturation.
- 1Use v = (Vmax × [S]) / (Km + [S]).
- 2Substitute Vmax = 100, Km = 5, and [S] = 10.
- 3v = (100 × 10) / (5 + 10) = 1000 / 15.
- 4v ≈ 66.67, which is 0.6667 of Vmax.
Near-saturating substrate
At ten times Km, the velocity is close to Vmax but still below it.
- 1Apply v = (Vmax × [S]) / (Km + [S]).
- 2v = (100 × 50) / (5 + 50) = 5000 / 55.
- 3v ≈ 90.91, so the enzyme is operating at about 90.9% of Vmax.
- 4As [S] becomes very large compared with Km, v approaches Vmax asymptotically.
Solve Km from a half-maximal velocity
Rearrange the equation to find Km from v, Vmax, and [S].
- 1Use Km = [S] × (Vmax − v) / v.
- 2Substitute [S] = 5, Vmax = 100, and v = 50.
- 3Km = 5 × (100 − 50) / 50 = 5 × 50 / 50 = 5.
- 4Because v is half of Vmax, the solved Km equals the substrate concentration.
Solve substrate concentration for 80% of Vmax
Find the substrate concentration required to reach v = 80 when Vmax = 100 and Km = 5.
- 1Use [S] = v × Km / (Vmax − v).
- 2Substitute v = 80, Vmax = 100, and Km = 5.
- 3[S] = 80 × 5 / (100 − 80) = 400 / 20 = 20.
- 4The fraction of Vmax is 80/100 = 0.8, so this is 80% saturation.
Introduction
The Michaelis–Menten equation describes how the initial velocity of an enzyme-catalysed reaction depends on substrate concentration: v = Vmax[S]/(Km + [S]). It is the core model for steady-state enzyme kinetics, separating the maximum catalytic rate (Vmax) from the substrate concentration that gives half-maximal velocity (Km). This calculator solves the equation forward for reaction velocity and rearranges it to solve for S], Vmax, or Km. It complements the [enzyme activity calculator, which focuses on activity units from assay data, and the calibration curve calculator, which helps convert absorbance signals to concentrations. Authoritative definitions follow the IUPAC Gold Book and classic enzyme-kinetics literature such as Michaelis and Menten's original paper.
Michaelis–Menten equation and variables
The standard form is v = (Vmax × [S]) / (Km + [S]). Here v is the initial reaction velocity measured before significant substrate depletion or product inhibition occurs. Vmax is the limiting rate when the enzyme is saturated with substrate, Km is the substrate concentration at v = Vmax/2, and [S] is the free substrate concentration. The ratio v/Vmax = [S]/(Km + [S]) is often called the fractional velocity or saturation.
If [S] = Km, then v = Vmax/2 exactly.
If [S] ≪ Km, velocity is approximately first order in [S].
If [S] ≫ Km, velocity approaches Vmax asymptotically.
Km has concentration units; Vmax and v share rate units.
Solving for v, [S], Vmax, or Km
The same equation can be rearranged depending on which quantity is unknown. To solve for substrate concentration, use [S] = vKm/(Vmax − v); this only makes physical sense when 0 < v < Vmax. To solve for Vmax, use Vmax = v(Km + [S])/[S]. To solve for Km, use Km = [S](Vmax − v)/v. These rearrangements are algebraic, so units must be internally consistent.
If v is equal to or greater than Vmax, the solved [S] would be infinite or negative; that usually indicates experimental error or an underestimated Vmax.
What Km means in enzyme kinetics
Km is not simply an affinity constant, but it is often interpreted as an apparent substrate affinity under Michaelis–Menten assumptions: a lower Km means half-maximal velocity is reached at a lower substrate concentration. For the mechanism E + S ⇌ ES → E + P, Km = (k−1 + kcat)/k1. Only when product formation is much slower than complex dissociation does Km approximate a dissociation constant. For biomolecular charge-state context, the pKa calculator and isoelectric point calculator can help interpret pH effects on enzymes.
Compare Km values only under the same pH, temperature, ionic strength, and assay format.
Vmax, saturation, and the hyperbolic curve
The Michaelis–Menten curve is a rectangular hyperbola: velocity rises steeply at low substrate and gradually levels off near Vmax. The asymptote matters experimentally because adding more substrate produces diminishing returns once [S] is several times larger than Km. For example, [S] = 10Km gives v/Vmax = 10/11 ≈ 0.909, while [S] = 100Km gives 0.990.
[S] = 0.1Km gives about 9.1% of Vmax.
[S] = Km gives 50% of Vmax.
[S] = 4Km gives 80% of Vmax.
[S] = 9Km gives 90% of Vmax.
Assumptions and limitations
The classical equation assumes initial-rate measurements, a single substrate, steady-state ES concentration, no allostery, no cooperativity, no substrate inhibition, and no significant reverse reaction. It is not appropriate for sigmoidal enzymes, multi-substrate mechanisms without simplification, or assays where substrate is depleted during measurement. Use initial slopes from well-mixed reactions, and confirm rates with replicate concentrations spanning below and above Km.
Lineweaver–Burk double-reciprocal plots are historically important but overweight low-substrate error; nonlinear regression of v versus [S] is preferred.
Practical workflow for using the calculator
Measure initial rates at several substrate concentrations, convert instrument signals to concentration or product rate, then fit Vmax and Km. The molarity calculator helps prepare substrate stocks, while the ph calculator helps document buffer conditions. For deeper derivations and nomenclature, see LibreTexts Biochemistry, NCBI Bookshelf biochemistry chapters, and IUPAC recommendations.
Blank-correct every rate before fitting.
Keep enzyme concentration low enough that substrate depletion is negligible.
Use at least 6–8 substrate levels around the expected Km.
Report temperature, pH, buffer, ionic strength, and enzyme concentration with Km and Vmax.
Quick Reference Card
Michaelis–Menten — Quick Reference
Quick reference • Michaelis–Menten Equation Calculator
v = Vmax[S]/(Km + [S]); [S] = vKm/(Vmax − v); Km = [S](Vmax − v)/vValid range: Use for initial-rate, single-substrate, non-cooperative enzyme kinetics with 0 ≤ v < Vmax and nonnegative [S] and Km.
Common Values
⚠ Watch Out
- •Do not solve [S] with v ≥ Vmax; the denominator Vmax − v becomes zero or negative.
- •Do not mix units: Km and [S] must be in the same concentration units.
- •Use initial velocities only; later time points may suffer substrate depletion or product inhibition.
- •Classical Michaelis–Menten kinetics does not describe allosteric or cooperative enzymes.
Pro Tips
- →Choose substrate concentrations below and above the expected Km for reliable fitting.
- →Nonlinear regression of v versus [S] is usually better than Lineweaver–Burk plots.
- →Report assay pH and temperature because both Km and Vmax can shift strongly.
- →Use v/Vmax to compare saturation across enzymes measured in different rate units.
FAQs
What does the Michaelis–Menten equation calculate?
It calculates the initial reaction velocity v from Vmax, Km, and substrate concentration [S]: v = Vmax[S]/(Km + [S]). This calculator can also rearrange the equation to solve for [S], Vmax, or Km.
Why is Km the substrate concentration at half Vmax?
Set [S] = Km in v = Vmax[S]/(Km + [S]). The numerator becomes VmaxKm and the denominator becomes 2Km, so v = Vmax/2. This is why Km has concentration units.
Can velocity ever exceed Vmax?
Not in the classical Michaelis–Menten model. Vmax is the asymptotic maximum at saturating substrate, so a measured v greater than Vmax suggests an underestimated Vmax, inconsistent units, or an assay artifact.
Is Km the same as binding affinity?
Not exactly. Km depends on binding and catalytic rate constants: Km = (k−1 + kcat)/k1. It approximates a dissociation constant only in special cases where catalysis is slow compared with ES dissociation.
What units should I use?
Use any consistent units. Vmax and v must share rate units, such as µmol/min or U/mg. Km and [S] must share concentration units, such as mM or µM. The fraction v/Vmax is dimensionless.
When should I avoid this model?
Avoid simple Michaelis–Menten analysis for allosteric enzymes, cooperative binding, substrate inhibition, multi-substrate mechanisms without simplification, or data collected outside the initial-rate linear range.