The 5-PL or 5 Parameter Logistic is a nonlinear regression model used for prediction of the probability of occurrence of an event by fitting data to a logistic curve. It differs from the 4-PL or 4 Parameter Logistic model in that it is an asymmetric function which is a better fit for immunoassay or bioassay data.  As the name suggests, there are 5 parameters in the 5-PL model equation:

F(x) = A + (D/(1+(X/C)^B)^E)

• A is the MFI/RLU value for the minimum asymptote
• B is the Hill slope
• C is the concentration at the inflection point
• D is the MFI/RLU value for the maximum asymptote
• E is the asymmetry factor

The 5-PL model equation has the extra E parameter which the 4-PL model lacks and when E = 1 the 5-PL equation is identical to the 4-PL equation.

Anatomy of the 5 Parameter Logistic Model (5-PL)

Parameters A (minimum asymptote) and D (maximum asymptote) are the limits of where you can interpolate or extrapolate your data.  Any MFI/RLU values > D and MFI/RLU values < A simply cannot be calculated because they are out of the function range.

Limits of the 5-PL Standard Curve

What are the differences between extrapolation and interpolation?

Extrapolation occurs when you are inferring or estimating concentrations for points that are within calculable limits (A < x < D) but are outside of the range of our standard curve.  This occurs when the calculated Concentration < Minimum Standard Concentration or when the calculated Concentration > Maximum Standard Concentration.  For typical bioassay standard curves, extrapolating can be very dangerous and quite often be misleading.  The reason being that minute changes in MFI values on the flat parts of the standard curve can lead to huge changes in concentration or dose.

The Dangers of Extrapolation

Interpolation occurs when your MFI/RLU values are within standard range or Minimum Standard Concentration < x < Maximum Standard Concentration.  Ideally, this is the range where you would want all your unknown points to lie.

Extrapolation vs. Interpolation

The 5 Parameter Logistic model equation, by itself, is not equipped to give accurate and precise curve-fitting of bioassay data due to a phenomenon known as heteroscedasticity; the nonconstant variability that arises in almost all fields where chemical and bioassays are no exceptions.  In bioassays, measurement errors increase as concentrations get higher and therefore the variability of a measurement is not constant.

How does this affect curve-fitting? During curve-fitting, all standard samples are given equal freedom to influence the curve.  The only problem is that those points with higher errors (variance) are given the same freedom as those that are more accurate (points at the lower end of the curve).

How can we deal with this issue? One way to counterbalance nonconstant variability is to make them constant again.  To accomplish this, weights are assigned to each standard sample data point.  These weights are designed to approximate the way measurement errors are distributed.  By applying weighting, points on the lower part of the curve are given more influence on the curve again.

MasterPlex quantitative analysis software uses 4 different algorithms of assigning weights:

1. 1/Y^2 – Minimizes residuals (errors) based on relative MFI/RLU values.
2. 1/Y – This algorithm is useful if you know the errors follow a Poisson distribution.
3. 1/X - This choice is rarely used because it minimizes residuals based on their concentration values.  Gives more weight to the right part of the graph.
4. 1/X^2 – Similar to the above.

How can EC50/IC50 be calculated from the 5PL model equation? The formula for calculating the EC50/IC50 from the 5PL is as follows:

x = C((2^(1/E) – 1)^(1/B))

MiraiBio offers 2 powerful curve-fitting quantitative analysis solutions that utilizes our time-tested 5-PL nonlinear regression model as well as many others:

1. ReaderFit.com – This is the free online lite-version of MasterPlex ReaderFit with the same core curve-fitting algorithms.
2. MasterPlex ReaderFit for Elisa or Dose Response Analysis – MasterPlex ReaderFit is our Curve-Fitting Data Analysis Software Package Optimized for ELISA or Dose Response Analyses (e.g. EC50/IC50 or even IC90 calculations).  Download our 14-day free trial today!
3. MasterPlex QT for the Luminex xMAP 100/200 or Bio-Plex Platforms – The World’s First Multiplex Curve-Fitting Software for the Luminex 100 and Luminex 200 xMAP Platforms.  Download our 14-day free trial today!

You may also be interested in reading our blog post on:

• Dummies Guide to Curve-Fitting for Quantitative Analysis
• 4PL nonlinear regression model
• Tips for ELISA analysis
• The importance of weighting with the 4PL and 5PL
• Dose response analysis and EC₅₀/IC₅₀ determination