BIOCHEMISTRYVerified Reference Parity

5-Parameter Logistic (5PL)

Asymmetric sigmoidal bioassay model incorporating an asymmetry factor (S) for non-symmetric dose-response curves.

Primary Disciplines:PharmacologyImmunoassay CalibrationBioassays

Mathematical Formulation

$$y = D + \frac{A - D}{\left(1 + \left(\frac{x}{C}\right)^B\right)^S}$$
Plaintext:y = D + (A - D) / ((1 + (x / C)^B)^S)

Parameters & Physical Meanings

Detailed parameter breakdown, standard units, valid mathematical constraints, and initial guess heuristic algorithms.

SymbolNameUnitConstraintsPhysical InterpretationInitial Guess Heuristic
$A$Zero-Dose AsymptoteOD / SignalunconstrainedBaseline response at zero analytemin(y)
$B$Slope FactorDimensionlessB > 0Transition slope steepness1.0
$C$Transition ParameterConcentrationC > 0Apparent midpoint parameter (related to EC50 via S)median(x)
$D$Infinite-Dose AsymptoteOD / SignalD != AUpper plateau saturation responsemax(y)
$S$Asymmetry FactorDimensionlessS > 0Quantifies asymmetry: S = 1 is symmetric 4PL; S != 1 fits asymmetric slope transitions.1.0

When to Choose This Model

  • Immunoassays exhibiting asymmetric transition slopes
  • Receptor-ligand binding with multiple non-equivalent binding sites

Typical Scientific Applications

  • Asymmetric ELISA bioassays
  • Pharmacological dose-response calibration

Expected Fit Profile & Curve Morphology

Shape: sigmoid
Fitted ModelRaw Data

Example Dataset & Expected Fit Output

Synthetic 12-point asymmetric ELISA bioassay curve with asymmetry factor S = 1.48

R²:0.9998RMSE:0.0160

Expected Converged Parameters

  • A0.048
  • B0.950
  • C4.120
  • D2.990
  • S1.480

Sample Experimental Vectors (12 points)

#x (Independent)y (Observed)
10.0100.051
20.0500.064
30.2000.098
40.5000.165
51.0000.312
62.5000.684
75.0001.152
810.0001.745
+ 4 more points available in AltaiPlot preset

Python / SciPy Reference Implementation

Reference library: scipy.optimize (Trust-Region-Reflective).Exact parameterization parity verified.

import numpy as np
from scipy.optimize import curve_fit

def logistic_5p_model(x, A, B, C, D, S):
    x_safe = np.maximum(x, 1e-15)
    return D + (A - D) / ((1.0 + (x_safe / C)**B)**S)

p0 = [np.min(y_data), 1.0, np.median(x_data), np.max(y_data), 1.0]
popt, pcov = curve_fit(logistic_5p_model, x_data, y_data, p0=p0)
Numerical Verification Tolerance: Max Relative Error < 3.6e-6Verified: 2026-08-16

Comparative & Alternative Models

Key decision trade-offs between this model and related functional alternatives:

Alternative Model

4-Parameter Logistic (4PL)

The gold-standard symmetric sigmoidal bioassay model for quantitative ELISA and ligand binding calibration.

$y = D + \frac{A - D}{1 + \left(\frac{x}{C}\right)^B}$

Related Mathematical Models

Models sharing mathematical formulations or physical domains in AltaiPlot:

biochemistry

Hill Equation (Dose-Response)

The definitive non-linear model for sigmoidal cooperative binding and pharmacological dose-response relationships.

$y = y_{\min} + \frac{y_{\max} - y_{\min}}{1 + \left(\frac{K}{x}\right)^n} = y_{\min} + (y_{\max} - y_{\min})\frac{x^n}{K^n + x^n}$
biochemistry

4-Parameter Logistic (4PL)

The gold-standard symmetric sigmoidal bioassay model for quantitative ELISA and ligand binding calibration.

$y = D + \frac{A - D}{1 + \left(\frac{x}{C}\right)^B}$

Frequently Asked Questions (5-Parameter Logistic (5PL))

When should 5PL be preferred over 4PL?

When residual plots of a 4PL fit display systematic curvature around the asymptotes, the bioassay is asymmetric. 5PL resolves this asymmetry by fitting parameter S.

Scientific References & Citations

  • Gottschalk, P. G., & Dunn, J. R. (2005). The five-parameter logistic: a characterization and comparison with the four-parameter logistic. Analytical Biochemistry, 343(1), 54-65.[Source / DOI]