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 or ~0.5 cm for a 96-well plate), 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 is appropriate when R² from linear fitting drops below 0.99 but the data is still monotonically increasing.

Inverse calculation uses the quadratic formula:

The calculator selects the positive real root within the working range. If both roots are positive, the smaller value (closer to the working range) is preferred. If no positive real root exists, the result is reported as "N/A".

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 potentially unreliable and may indicate pipetting errors or well-to-well variability.

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.

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.