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 (Mean Fluorescent Intensity)/RLU (Relative Light Unit) 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.

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.

**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.

**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.

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/Y^2**– Minimizes residuals (errors) based on relative MFI/RLU values.**1/Y**– This algorithm is useful if you know the errors follow a Poisson distribution.**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.**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 **3 powerful curve-fitting quantitative analysis solutions **that utilizes our time-tested 5-PL nonlinear regression model as well as many others:

**ReaderFit.com** – Free online curve-fitting application

**Sign Up for Free Account**

**ReaderFit Desktop** – Robust curve-fitting, quality control and reporting desktop software

**Download Free Trial**

**MasterPlex QT** – Robust curve-fitting, quality control and reporting desktop software for multiplex ELISA data (Luminex, Bio-Plex, Meso Scale Discovery and Applied BioCode platforms)

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

If it is dangerous to extrapolate the grapt then what will the values of A and D in the 4 parameter equation .

@anshuman: The concentrations cannot be calculated at A and D because those are the limits of the function. Basically, anything less than or equal to A or greater than or equal to D will be out of limits of the function.

Thanks for the reply, but then how do one find the values of A and D for the 4PL equation,besides thee values values for B and C can be calculated from the graph. Kindly sight one example and explain.

Thanks for the reply, but then how do one find the values of A and D for the 4PL equation,besides thee values for B and C can be calculated from the graph I suppose. Kindly sight one example and explain.

@anshuman: I apologize but I think I initially misread your question. Actually calculating the values for A and D is definitely beyond the scope of this post and is a really complicated process. Essentially, one begins with a “guess” of the parameters and depending on the fit of those parameters the algorithm will try to improve of the initial values and so on and so on. I apologize that this is not really straightforward but that is the reason why we have algorithms and software that do it for us.

Thank you very much for the assistance, as I have to answer to my project incharge about the

calculations associated with the 4PL equation, as I am presently dealing with the ELISA techniques.

This 5PL form has a lot of drawbacks. Please see the paper by Liao, et al. (2009), “Re-parameterization of Five-parameter Logistic Dose Response Curve”, J. of Chemometrics, 23, 248 – 253.

Thanks for the reference! I read through the paper discussing the reparameterized 5PL and it seems to me that the main advantage this new function is the ease of interpretation of the parameters such as the C* (ED50) which is no longer a function of E or the asymmetry factor. Are there references to this relatively new reparameterized 5PL function in other published articles?

the formula for calculating the EC50/IC50 from the 5PL doesn’t look right to me. shouldn’t it look like

x = C*((2^(1/E) – 1))^(1/B))???

Hi J,

You are absolutely right! I should know my PEMDAS much better than that. It has been corrected. Thanks for the heads up.

Allen

It is difficult to find a proper definition of a 5PL-er. I was wondering if anybody in this group could give me the definition?

Hi Fica,

Can you please be more specific on what you mean by “5PL-er”?

Thanks,

Allen

How can I get linear regression of the 5PL function, I am performing leptin quantification but I cannot do it.

Hi Carla,

I am not aware of any logistic 5 parameter functions that are linear. May I ask why you require linear? If so, you can can try a simple linear equation such as: f(x) = ax + b

Have you tried a nonlinear 5PL fit yet? If not, I would recommend you try that first in our free online curve fitting application. You will also have the option of trying the linear fit as well.

I hope this information helps.

Allen

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Hi,

The export for Bioplex exports the LLOQ and ULOQ based on the % recovery >70% and <130% rec. is there any way to export this data in the table format

Hi Wendy,

Yes, by using MasterPlex QT does have an export feature that can export your data in a Table format. We also have a custom report feature that we can help you with developing a customized report in virtually any format you need.

Regards,

Charles Ma

Senior Technical Support

Sorry, could you explain me how can I solve X in the equation?

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

Thank you very much!

Hi Pablo,

Thanks for reaching out to us with your question. As a disclosure, I’m not a mathematician so this back-calculation formula may be incorrect. Please double check with your data to see if this correct.

`x = C*(((D/(F(x)-A))^(1/E))-1)^(1/B)`

Hello again.

Formula is perfect!!

Thank you very much and congratulations for your work.