Enzyme Kinetics Calculator — Michaelis–Menten Plots & Curve Fitting

About Michaelis–Menten Kinetics

The Michaelis–Menten model describes how the initial reaction velocity \(v\) changes with substrate concentration \([S]\) for many single-substrate enzyme reactions. In its classical form, \( v = \dfrac{V_{max}[S]}{K_m + [S]} \). Here \(V_{max}\) is the maximum rate reached at saturating substrate, and \(K_m\) is the substrate concentration at which the reaction proceeds at half the maximum rate (\(v = V_{max}/2\)). When you vary \([S]\) across a sensible range and measure \(v\), the resulting curve rises quickly at low \([S]\) and then saturates, reflecting the catalytic machinery becoming fully occupied.

Although the nonlinear equation is the primary model, several linear transformations are traditionally used for visualization and quick diagnostics: the Lineweaver–Burk plot (\(1/v\) vs \(1/[S]\)), the Eadie–Hofstee plot (\(v\) vs \(v/[S]\)), and the Hanes–Woolf plot (\([S]/v\) vs \([S]\)). Linear plots can make outliers easier to spot and help you see whether a subset of points deviates systematically. Keep in mind that reciprocal transforms can overweight low-concentration noise; for quantitative parameter estimation, modern practice favors direct nonlinear regression of \(V_{max}\) and \(K_m\) against the original equation.

Assumptions and good practice

• Initial-rate regime: Measure velocities before significant substrate depletion or product buildup.
• Steady-state approximation: The enzyme–substrate complex concentration is approximately constant during measurement.
• Single substrate, simple mechanism: No cooperativity, allosteric modulation, or multiple binding sites.
• Constant enzyme concentration: \([E]_0\) is small relative to \([S]\), and the enzyme is not inactivated over the assay window.
• No significant product inhibition or reverse reaction: Conditions minimize back-reaction.

Designing a useful dataset

To get stable estimates, span substrate concentrations from well below to several times above \(K_m\) (a common rule of thumb is \(0.1\,K_m\) up to \(5\text{–}10\,K_m\)). If \(K_m\) is unknown, pilot runs help bracket the curve. Aim for at least 8–12 concentrations with replicates, keep temperature, pH, ionic strength, and cofactors controlled, and report consistent units (e.g., \([S]\) in mM, \(v\) in µM·min\(^{-1}\) or absorbance units·min\(^{-1}\) calibrated to concentration). Randomize run order to reduce drift, and record blank/background rates to subtract non-enzymatic signal.

Interpreting \(K_m\) and \(V_{max}\)

\(K_m\) is often interpreted as an inverse measure of apparent substrate affinity (lower \(K_m\) → higher apparent affinity), but strictly it also reflects catalytic steps in the mechanism; it is not simply a dissociation constant unless specific conditions are met. \(V_{max}\) depends on total active enzyme and the catalytic constant \(k_{\text{cat}}\) via \(V_{max} = k_{\text{cat}}[E]_0\). When comparing systems, normalize by enzyme concentration to report \(k_{\text{cat}}\) and the catalytic efficiency \(k_{\text{cat}}/K_m\), which captures both turnover and substrate capture in the low-\([S]\) limit.

Extensions you may encounter

Real enzymes can deviate from the simple model. Competitive, noncompetitive, and uncompetitive inhibition modify the apparent \(K_m\) and/or \(V_{max}\); cooperativity (Hill kinetics) produces sigmoidal curves; and bi-substrate reactions require more elaborate rate laws (e.g., ordered/ping-pong mechanisms). Use the basic Michaelis–Menten analysis as a starting point, then consider these extensions if systematic patterns remain unexplained.

This tool plots the Michaelis–Menten curve from your \(V_{max}\) and \(K_m\), lets you paste experimental data, and performs in-browser nonlinear fitting for educational insight. Results are illustrative — for publication-grade analysis, include replication, error models, weighting, and confidence intervals.