Michaelis-Menten Equation Calculator
Enter any three of the four Michaelis-Menten parameters and this calculator instantly solves for the fourth. Choose what you want to find, fill in the three known values, and see the reaction velocity, saturation curve, and full worked solution. Concentration and velocity units are selectable so you can match your lab data without manual conversion.
Formula
Worked example
An enzyme has Vmax = 100 uM/s and Km = 20 uM. At [S] = 50 uM: v = (100 x 50) / (20 + 50) = 5000 / 70 = 71.4 uM/s. The enzyme is operating at 71.4% of Vmax. Since [S] = 50 uM is 2.5x the Km, the enzyme is well above its half-saturation point.
What is the Michaelis-Menten equation?
The Michaelis-Menten equation, published in 1913 by Leonor Michaelis and Maud Leonora Menten, describes the relationship between the concentration of a substrate ([S]) and the initial rate (v) of a single-substrate enzyme-catalyzed reaction. The equation is v = (Vmax x [S]) / (Km + [S]), where Vmax is the maximum reaction velocity reached when all enzyme active sites are occupied, and Km (the Michaelis constant) is the substrate concentration at which the velocity is exactly half of Vmax. The equation produces a rectangular hyperbola: at very low [S], velocity increases nearly linearly with substrate; as [S] climbs well above Km, velocity asymptotically approaches Vmax and becomes insensitive to further increases in substrate.
Km, Vmax, and what they reveal about an enzyme
Km is often treated as an inverse measure of enzyme-substrate affinity. A low Km means the enzyme reaches half-maximal velocity at a low substrate concentration, implying tight binding. A high Km means the enzyme needs a high substrate concentration to be half-saturated, implying weaker binding or a less favorable reaction pathway. Vmax reflects the catalytic capacity of the enzyme preparation: it scales with the amount of enzyme present (Vmax = kcat x [E]total) and represents the upper limit of the reaction rate under saturating conditions. The catalytic efficiency, often called kcat/Km, combines both parameters and is used to compare enzymes or to identify the physiologically relevant substrate from a group of candidates. The theoretical upper limit for kcat/Km is set by the rate of enzyme-substrate encounter (approximately 10^8 to 10^9 M-1 s-1 in aqueous solution).
How to solve for any of the four parameters
This calculator rearranges the fundamental equation to isolate whichever parameter is unknown. To find v: v = (Vmax x [S]) / (Km + [S]). To find Vmax from a known velocity: Vmax = v x (Km + [S]) / [S]. To find Km when v and Vmax are measured at a known [S]: Km = [S] x (Vmax - v) / v. To find the [S] that would produce a target velocity: [S] = Km x v / (Vmax - v). In each case the relationship is straightforward algebra derived from the core equation. Note that for the Km and [S] solutions, v must be strictly less than Vmax: a velocity equal to or greater than the maximum velocity is physically impossible under Michaelis-Menten kinetics.
Assumptions and limitations of the model
The Michaelis-Menten model rests on several key assumptions. First, it applies to initial-rate conditions, meaning product accumulation is negligible, so the reverse reaction from product to substrate can be ignored. Second, it uses the steady-state approximation, where the concentration of the enzyme-substrate complex (ES) is constant because it forms and breaks down at equal rates. Third, it describes a single-substrate, single-product mechanism with no cooperativity between active sites. Many real enzymes deviate from this model: allosteric enzymes (such as hemoglobin or phosphofructokinase) display sigmoidal kinetics described by the Hill equation; multi-substrate enzymes (including most transferases and oxidoreductases) require ordered or random binding schemes; inhibitors shift the apparent Km and/or Vmax in ways specific to their binding mode (competitive, non-competitive, uncompetitive, or mixed). Despite these limitations, the Michaelis-Menten equation remains the foundational tool for enzyme characterization and drug design.
Typical Km values for common enzyme-substrate pairs
| Enzyme | Substrate | Km (approximate) | Notes |
|---|---|---|---|
| Hexokinase (brain) | Glucose | 0.05-0.1 mM | High affinity - important at low blood glucose |
| Chymotrypsin | Glycyltyrosinamide | ~108 mM | Low affinity substrate |
| Carbonic anhydrase | CO2 | ~8 mM | Turnover rate ~10^6/s |
| Acetylcholinesterase | Acetylcholine | ~0.09 mM | Near-diffusion-limited enzyme |
| Lactate dehydrogenase | Pyruvate | ~0.17 mM | Varies by tissue isoform |
| Urease | Urea | ~10-25 mM | pH-dependent |
| Catalase | H2O2 | ~25-93 mM | One of fastest known enzymes |
| Alcohol dehydrogenase (yeast) | Ethanol | ~13 mM | Secondary alcohol substrate |
Km values vary widely between enzymes and reflect their affinity for their substrate. Lower Km means higher affinity.
Frequently asked questions
What does Km represent physically?
Km, the Michaelis constant, is the substrate concentration at which the reaction velocity equals half of Vmax. Mechanistically, it equals (k-1 + k2) / k1, where k1 is the rate of ES complex formation, k-1 is its dissociation back to E and S, and k2 is the catalytic rate (kcat). When k-1 is much larger than k2 (a slow catalytic step), Km approximates the dissociation constant Kd and therefore reflects binding affinity. In general, a lower Km indicates the enzyme operates efficiently at lower substrate concentrations.
Why must v be less than Vmax when solving for Km or [S]?
The Michaelis-Menten equation predicts that reaction velocity can never exceed Vmax, because Vmax is approached only asymptotically as [S] goes to infinity. If you input v equal to or greater than Vmax, the algebraic rearrangements for Km and [S] produce zero or negative denominators, which have no physical meaning. This calculator validates this condition and returns no result rather than a nonsensical number.
What is a Lineweaver-Burk plot and how does it relate?
A Lineweaver-Burk plot, also called a double-reciprocal plot, is obtained by taking the reciprocal of both sides of the Michaelis-Menten equation: 1/v = (Km/Vmax)(1/[S]) + 1/Vmax. Plotting 1/v on the y-axis against 1/[S] on the x-axis gives a straight line whose y-intercept is 1/Vmax, whose slope is Km/Vmax, and whose x-intercept is -1/Km. This linearization was historically used to estimate Km and Vmax graphically before nonlinear regression software became available. It is still used to identify inhibition mechanisms: competitive inhibitors change only the slope (raising the apparent Km), uncompetitive inhibitors change only the y-intercept (lowering apparent Vmax), and non-competitive inhibitors change both proportionally.
What units should I use for my inputs?
All concentration inputs ([S] and Km) must use the same unit, and the velocity inputs (v and Vmax) must match in both concentration and time units. This calculator handles the internal conversion automatically: choose your concentration unit (nM, uM, mM, or M) and time unit (per second or per minute) using the selectors at the top, then enter all values in those chosen units. The result is returned in the same units.
How is the saturation fraction calculated?
The saturation fraction is v divided by Vmax, expressed as a percentage. It tells you how close the enzyme is to operating at its maximum rate. A saturation of 50% corresponds to [S] = Km by definition. Below 50% the enzyme is in the approximately linear region of the hyperbola, where small increases in [S] yield nearly proportional gains in velocity. Above 90% the curve is nearly flat, so large increases in [S] give only marginal improvements.
Can this calculator handle enzyme inhibition?
This calculator solves the uninhibited Michaelis-Menten equation. Inhibitors alter the apparent Km and/or apparent Vmax in specific ways depending on their mechanism. Competitive inhibitors increase apparent Km without changing Vmax; uncompetitive inhibitors decrease both apparent Km and Vmax; non-competitive inhibitors decrease apparent Vmax without changing Km. To analyze inhibition, measure kinetics at multiple inhibitor concentrations and compare the apparent parameters to the uninhibited values.