Understand the relationship
The reasoning behind the result
Fit the measured quantity you intend to model
For fixed Km, fᵢ=[S]ᵢ/(Km+[S]ᵢ); Vmax=Σ(vᵢfᵢ)/Σfᵢ²
Direct unweighted least squares treats substrate values as known and gives each observed rate the same weight. At a trial Km, the curve is linear in Vmax, so its best positive amplitude can be found directly. A log-spaced scan and local refinement then search the entered Km interval.
The method is bounded, unweighted and single-curve. It does not supply parameter confidence intervals, replicate-group weights, an errors-in-both-axes fit or a global inhibitor mechanism fit. A result on either search bound means the allowed interval constrains the solution; widening a bound does not itself fix weak experimental information.
The three straight-line rearrangements
LB: 1/v=(Km/Vmax)(1/[S])+1/Vmax; EH: v=Vmax−Km(v/[S]); HW: [S]/v=[S]/Vmax+Km/Vmax
Lineweaver–Burk uses x=1/[S] and y=1/v. Its fitted intercept gives 1/Vmax and slope gives Km/Vmax. Eadie–Hofstee uses x=v/[S] and y=v; the negative slope is Km and intercept is Vmax. Hanes–Woolf uses x=[S] and y=[S]/v; slope is 1/Vmax and intercept is Km/Vmax.
These are algebraic rearrangements of the same exact curve, but ordinary least squares after transformation minimizes a different error sum. Reciprocal rates magnify errors at small rates. Eadie–Hofstee puts the observed rate on both axes, correlating their errors. Consequently noisy data can produce different parameter estimates. Positive Km and Vmax are required; incompatible transformed estimates are rejected, not relabeled as valid kinetics.
Residuals stay attached to the original measurements
Residualᵢ=vobserved,ᵢ−vmodel,ᵢ
Every row retains the observed rate, model rate and signed residual in the original rate unit. A positive residual places a measurement above the fitted curve. All observations, including repeated concentrations, remain in the calculation.
At least three distinct concentrations are required here, but this is only a numerical entry rule. If the measured concentrations lie entirely far below Km, the data mainly constrain Vmax/Km; if they lie far above Km, they may say little about Km. The range relative to the fitted Km is reported without declaring the design adequate.
Follow the numbers
Recovering an exact saturation curve
- At substrate 0, 1, 2, 4 and 10 mM, enter rates 0, 40, 60, 80 and 100 µmol/min.
- Km=2 mM and Vmax=120 µmol/min reproduce every value: at 4 mM the rate is 120×4/(2+4)=80 µmol/min.
- The direct residual sum of squares is zero apart from floating-point roundoff. Excluding zero, the double-reciprocal line is 1/v=(1/60)(1/[S])+1/120, giving the same exact parameters.
Agreement on exact data does not make the transformed error models equivalent on noisy measurements.
Quick guide
How to use this calculator
- Enter substrate and independently established initial rates with consistent unit labels.
- Choose a direct fit unless you specifically need to inspect a named linear transformation.
- For the direct fit, enter a justified positive search interval for Km; inspect any bound-constrained result.
- Review the full observed/fitted/residual ledger. A fitted curve is not evidence that its assumptions hold.
Calculation method
Calculation and interpretation
Estimate kinetic parameters from a real substrate–rate dataset and make the chosen error model and fit limits visible.
Direct fit minimizes Σ(vᵢ−Vmax[S]ᵢ/(Km+[S]ᵢ))²; transformed fits minimize residuals in their stated transformed coordinates.
Worked example
Recovering an exact saturation curve
Agreement on exact data does not make the transformed error models equivalent on noisy measurements.
Direct fit minimizes Σ(vᵢ−Vmax[S]ᵢ/(Km+[S]ᵢ))²; transformed fits minimize residuals in their stated transformed coordinates.
Supported inputs
Precision and limits
An experimental model, not a biological conclusion
These calculations do not establish enzyme identity, purity, active fraction, assay validity or inhibition mechanism. Temperature, pH, substrate, cofactors and preparation must match the experiment behind the entered values.
Keep measurements and mechanisms separate
A time-window slope is not automatically an initial velocity; Km is not generally an equilibrium dissociation constant. No clinical interpretation, dose or treatment decision follows from these outputs.
No automatic experimental acceptance
No outlier removal, fit-quality threshold, parameter uncertainty or assertion of saturation is inferred. Search bounds constrain the direct nonlinear estimator and must be inspected.
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