Standard Curve Fitting Method

The Beer-Lambert law describes the linear relationship between absorbance (A) and the concentration (c) of an absorbing species in solution:

Where ε is the molar absorptivity (L/mol·cm), l is the path length (cm, typically 1 cm for a standard cuvette; in microplates, the effective path length is shorter and depends on plate geometry and sample volume), and c is the molar concentration. This law holds when the solution is dilute, monochromatic light is used, and there are no significant interactions between absorbing molecules.

In protein assays like BCA and Bradford, the "absorbing species" is a protein-dye or protein-metal complex, and the relationship between protein concentration and absorbance is linear within the assay's working range. Beyond this range, the relationship becomes nonlinear due to reagent depletion or dye saturation.

For data within the linear range, the standard curve is modeled as:

Where y is absorbance, x is concentration, m is the slope, and b is the y-intercept. The slope and intercept are determined by ordinary least squares (OLS), minimizing the sum of squared residuals:

This is a closed-form solution requiring no iterative optimization. It is computed entirely in the browser with no server calls.

Inverse calculation (unknown concentration from absorbance):

For extended concentration ranges where slight curvature is observed:

The three coefficients are solved via the normal equations (3×3 system, Cramer's rule), also a closed-form solution. Quadratic fitting may be useful when the assay response shows reproducible curvature over an extended calibrated range. A lower linear R² alone is not sufficient justification without reviewing range selection and experimental quality.

Inverse calculation uses the quadratic formula:

For quadratic inversion, the calculator returns the physically plausible real root that falls within the calibrated concentration range. If no such root exists, the result is reported as extrapolated or N/A.

Monotonicity requirement: Quadratic inversion is only stable when the calibrated region remains monotonic; otherwise a single absorbance value may map to multiple concentrations. If the fitted curve has a turning point within your standard range, consider narrowing the range or using linear fitting instead.

R² measures how well the fitted model explains the observed variation in absorbance:

Where SS_res = Σ(yᵢ − ŷᵢ)² is the residual sum of squares, and SS_tot = Σ(yᵢ − ȳ)² is the total sum of squares. R² ranges from 0 to 1, with 1 indicating a perfect fit.

For n replicate measurements at each concentration, the calculator computes:

The mean is used for curve fitting. The SD is displayed as error bars on the chart. The CV% indicates measurement precision — values above 20% are flagged as a practical screening threshold. This is a heuristic guideline, not a universal acceptance criterion; acceptable CV varies by assay and laboratory protocol.

Note: SD uses the sample standard deviation formula (n−1 denominator, Bessel's correction), which provides an unbiased estimate for small sample sizes typical in assay replicates (n=2–3).

The blank standard (0 concentration) measures the background absorbance of the reagent without any protein. In Auto mode:

Where A_blank is the mean absorbance of the 0 concentration replicates. This correction is applied to all standard and unknown data points before fitting. If no 0 concentration row is present, blank subtraction is skipped automatically.

The BCA assay relies on two reactions: (1) the biuret reaction where Cu²⁺ is reduced to Cu⁺ by peptide bonds in alkaline conditions, and (2) chelation of Cu⁺ by two molecules of bicinchoninic acid to form a purple complex with peak absorbance at 562 nm.

The response is nearly linear within the working range. Common interferents include reducing agents (DTT, β-mercaptoethanol), EDTA (chelates Cu²⁺), and lipids.

The Bradford assay uses Coomassie Brilliant Blue G-250 dye, which binds to protein under acidic conditions. The dye shifts from reddish-brown (465 nm, unbound) to blue (595 nm, protein-bound). The amount of blue color is proportional to protein concentration.

Typical Bradford working ranges depend on the assay format, reference protein, and kit manufacturer. Common approximate ranges are shown below:

The Bradford assay is fast (~5 min), but is affected by detergents (SDS, Triton X-100), and shows protein-to-protein variability (different proteins bind dye differently). BSA typically yields a higher response per μg than immunoglobulins. Consult your kit documentation for the validated range for your specific protocol.