Equilibrium Binding Explorer

Explore when the familiar hyperbolic binding curve is a good shortcut, and when ligand depletion makes the exact quadratic model matter.

A+B Kd AB
A = macromolecule/protein   B = ligand   AB = bound complex

Configure the binding system

Set the true affinity, protein concentration, and ligand range. Concentrations are shown in actual µM.

Binding parameters

Kd, protein, and ligand sliders move on a log10 scale. Concentrations are entered and plotted in µM.

Display mode
Quadratic equation
Y = ([A]o + Kd + [B]o) - ([A]o + Kd + [B]o)2 - 4[A]o[B]o 2[A]o
Hyperbolic equation
Y = [B]o [B]o + Kd

When protein concentration is tiny compared with Kd, free ligand is nearly the same as total ligand, so the shortcut behaves well.

Sampling

Ligand point spacing controls where simulated concentrations are sampled between 0 and ligand [B]o. Simulated data are generated only from the exact quadratic binding equation.

Presets

Use presets to jump between ligand-excess, moderate-depletion, and tight-binding scenarios.

Binding curve

Compare the exact quadratic model with the simple hyperbolic approximation.

Plot summaries

Individual points show every replicate. Mean mode summarizes replicates at each ligand concentration and can show SD or SEM.

Fraction bound vs ligand concentration

exact quadratic hyperbolic shortcut quadratic true hyperbolic true simulated data quadratic fit hyperbolic fit

Fit results

Both fits use the same simulated data. The comparison shows how model choice can shift the apparent Kd.

Plot summaries

These controls change how simulated data are displayed on the fit plot. The nonlinear fit still uses every replicate.

Data with nonlinear least-squares fits

Residuals from fitted models

Residual = observed fraction bound - fitted fraction bound
quadratic residuals hyperbolic residuals
Fitted parameters
ModelKd ± SEYmax ± SESSER2Score
True value0.1001.000--100
Quadratic fit-----
Hyperbolic fit-----

Switch to simulated data mode to generate noisy measurements and fit them. SSE means sum of squared errors; smaller SSE means the fitted curve stays closer to the simulated data points overall.