The **4 Parameter Logistic** or **4PL** nonlinear regression model is commonly used for curve-fitting analysis in bioassays or immunoassays such as ELISAs or dose-response curves.

The following is the 4PL model equation where **x** is the concentration (in the case of ELISA analysis) or the independent value and **F(x)** would be the response value (e.g. absorbance, OD, response value) or dependent value.

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

Updated 12/10/2013: For those of you who are looking for the back-calculations to solve for **x**, here is the formula.

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

Special thanks to *Pawel S.* for providing this formula!

Not surprisingly, the 4PL model equation comprises of 4 parameters:

**A = minimum asymptote**In an ELISA assay where you have a standard curve, this can be thought of as the response value at 0 standard concentration.**B = Hill slope**The Hill Slope or slope factor refers to the steepness of the curve. It could either be positive or negative. As the absolute value of the Hill slope increases, so does the steepness of the curve.**C = inflection point**The inflection point is defined as the point on the curve where the curvature changes direction or signs. This can be better explained if you can imagine the concavity of a sigmoidal curve. The inflection point is where the curve changes from being concave upwards to concave downwards (see picture below).

**D = maximum asymptote**In an ELISA assay where you have a standard curve, this can be thought of as the response value for infinite standard concentration.

The following are some key characteristics of the 4PL curve-fit model:

**Symmetry**- There is perfect symmetry for the sigmoidal curve around the inflection point for 4PL curve fits.**Monotonic**- A monotonic function is either always increasing or decreasing for all values of x.**Assumptions made by the 4PL model equation**- It assumes that the standard deviation of the scatter is the
**same for all values of x**(homoscedastic data). In the example of a standard curve, this is saying that the standard deviation for all the replicates of a low standard is**equal**to the standard deviation of the replicates for your high standard (see example curve below).Of course, this is rarely the case when dealing with bioassays or immunoassays (ELISAs) where the data is heteroscedastic. We normally see something like this where the standard deviation increases as x increases:

Applying weighting algorithms for 4PL and 5PL curve fitting is something that can be done to offset the assumption that data is homoscedastic.

- The 4PL model equation also assumes that the scatters a normal (or Gaussian) distribution.

- It assumes that the standard deviation of the scatter is the

If you are looking for curve-fitting software with the 4PL model equation that also does weighting we have a couple options:

**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)

**Download Free Trial**

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

I need this software for my OPN elisa reading graph. I am a Ph.D student. Please send me free version of the software so that I can use it.

Hi. We have a free fully functional 14-day trial of MasterPlex ReaderFit (for ELISA analysis). Please let me know if you have any other questions or you can email me directly at aliu at miraibio dot com.

Dear

I did an Elisa test and plot standard values concentration( X axis) vs optical Density( Y axis) in Excel and gave an exponential curve .In order to kit , i need 4pl model for calculation of my results. can i detect results by exponential curve.

Best

Ali

Hi Ali,

That’s a great question. Exponential curve is not a good gauge of biological environment. As you can imagine, in nature, everything has a limit. Thus, you’ll see that all ELISA analysis recommend the use of 4 parameter logistics or 5 parameter logistics equation. These equations will generate a sigmoidal (s-shaped) curve which is a better representation of the biological environment.

I would suggest to try our desktop software MasterPlex ReaderFit or our web app at http://www.ReaderFit.com

Best regards,

Charles

Miraibio Support.

Hy,

i used excel with the 4PL Formular and the ABCD values from the Demo software / your screens here and estimate that i can calc the same values for the curve via excel, as the software, but they are diffrent, why?.

I also tried to plot a curve on excel by changing the numbers from 0.01 till 3 ABS but it don’t looks sigmuid.

I am not sure if i understand 4PL right.

Regards

PhD Student JH

Hi JH,

Even though the 4PL model equation can be identical in 2 different software, the actual curve-fitting algorithms can vary quite a bit leading to different parameters (i.e. A, B, C and D). Here is a post that goes through curve fitting at a very high level. As you can imagine, there can be many ways an algorithm can come up with these parameters. This does not even include weighting (which you should be using) which will have an effect on the parameters as well. Here is a good read on the 4PL and there is a part that talks about why weighting is necessary.

Please let me know if you have any questions.

Best Regards,

Allen Liu

when we draw a standard curve manually using four known standard concentrations with their response values, we are getting almost accurately the unknown sample concentration if we interpolate the response value in the graph. Where as when I am using 4-or 5-pl nonlinear regression model by using your software the concentrations for unknown samples are not accurate. We are taking 4 standard concentrations and 6-8 unknown samples. can you explain why?

Hi Dr. Prasad,

I’d be more than happy to help you out with your question.

Is it possible for you to give me the MasterPlex ReaderFit project file and the standard curve & results that you did manually so that I can compare. By “manually,” are you using a model equation or are you simply connecting the standard points?

Best Regards,

Allen Liu

Dear sir

I am a research scholar doing my Ph D from . I am working on antihypertensives and want to determine IC50 values for my newly synthesised compounds .I further want to know how following soft ware of yours can help me for this .pl suggest what data I need to provide .

Dose Response Curves with EC50 & IC50 Determination using MasterPlex ReaderFit

The MiraiBio Group Blog.

Regards

Hi Shefali,

In order to use MasterPlex ReaderFit for IC50 determination, you will need the following data:

– Measured response values for your samples from your immunoassay – This value is typically something like OD, absorbance, RLU, fluorescence intensity, etc.

– Known concentrations or dilutions of the synthesized compound used in each corresponding sample. In ReaderFit, this would be your “Independent Value”

Here is a post that goes into more detail on the procedure for determining IC50:

http://www.miraibio.com/blog/2010/03/dose-response-curves-ec50-ic50-determination-masterplex-readerfit/

We also have some tutorial videos on how you can actually use it:

http://www.miraibio.com/masterplex-readerfit/online-demo.html

If you would like, you can also send me your data and I would be more than happy to look at your data for you: aliu@miraibio.com

Thanks for your interest!

Allen

I’m looking for a program that can compute the predction interval for nonlinear regression (the 4PL, for now).

Hi Joe,

I apologize for the delay in my response.

We provide several solutions for nonlinear regression curve fitting with the 4PL. If you have a Windows computer and you are looking for desktop software, then you should take a look at MasterPlex ReaderFit. There is a free 14-day demo that you can download and begin using right away.

If you are looking for a web application, we do have an online solution at ReaderFit.com. This is a light-weight solution compared to our desktop version but all that will change in the near future when we launch with some huge updates!

Please let me know if you have any questions.

Best Regards,

Allen Liu

many thanks for you assistance sorry I got disconnected before I could read the response when I clicked on the readerfit.com link.

Kind Regards,

Pia

Pia,

Glad I could help! I hope readerfit.com was able to give the curve fit you needed.

Regards,

Robert

Hello,

First, I would like to thank you for the helpful and clear explanations about curve fitting.

However, I have a couple of questions:

1. Is it proper to use a different curve fitt model for different ELISA datasets? for instance, If today’s data looks different from yesterday’s, is it ok to use a five parameter instead of a quadratic or four parameter?

2. How do you determine the dinamic range of a non linear curve?

Here is an example of my data after bkg subtraction:

pg/ml

10000 5.626165619 5.400375595

1000 5.824703445 5.767805328

100 5.002458801 5.271297684

10 2.35272049 2.843900909

1 1.277452698 1.559709778

0.1 0.926138153 1.263921967

0.01 1.181835403 0.925813904

Thanks a lot for your help,

Luz

Hi Luz,

1. The decision on which curve fit model to use is usually determined by the assay kit you use or the standard protocol of your lab. If you’re not limited by those restrictions, then you can use different models as it fit better with the sample data. MasterPlex ReaderFit has a “Best Fit” feature which will iteratively determine the best model equation and weighting to use for different data sets.

2. To determine the dynamic range (highest/lowest possible response values), you will need to first fit your data into a model, for example 4PL. Then with the calculation done, a minimum and maximum value will be shown. As described in this blog post above, parameter A is the minimum, and parameter D is the maximum.

I have plugged your data in MasterPlex ReaderFit using a 4PL model. And your minimum is 1.12759 and your maximum is 5.68048.

Regards,

Charles

Hello!

I am a PhD student and going to work with ALPHAScreen assay for Kd determination. Could you be so kind to tell me something about the origins of why do we use 4PL nonlinear regression model for fitting dose-response curves. I mean some physics or mathematics equations, articles with them.

Thank you,

Best regards!

Dear Tata,

I’m not quite sure what the origins of 4PL nonlinear regression model came from. But the reason why the majority of immunoassays and bioassays use 4PL nonlinear regression is that, in nature, nothing behaves in a straight (linear) line. Most living organisms have a minimum and maximum reaction towards an agent. Thus a sigmoidal / non-linear model represents nature much better.

Unfortunately, I don’t have any articles to reference, but if you do come across anything good articles in your research, please feel free to add it to our blog. I trust it will be greatly beneficial to all others who are looking for the same answer.

Thank you,

Charles Ma

-Miraibio Group

Hi,

Was going through this equation and there seems to be something wrong. If I am correct isn’t the equation:

y= min + [(max-min)/(1+(x/EC50)*b))].

But here it is mentioned as

y = max + [(min-max)/(1+(x/EC50)*b))], where D= max and a = min.

Am I missing something here? Please advice.

Thanks

Hi Sharath,

The correct formula for Four-Parameter Logistic (4PL) should be F(x) = ((A-D)/(1+((x/C)^B))) + D. May I ask where did you obtain your equation? It would be nice if I can research that source. As a secondary reference for our model equation, please check out the article by Bio-Plex at http://www.dartmouth.edu/~dartlab/uploads/Bio-RadTechNote2861_principles_of_curve_fitting.pdf. There you’ll find the reference equations for both 4PL and 5PL.

Thank you,

Charles Ma

-Miraibio Group

Hi Charles,

Thanks for the reply and the link. Here is the link where I came across the equation:

http://www.sigmaplot.com/products/sigmaplot/productuses/prod-uses43.php (scroll down to 4 parameter logistic equation)

Based on the link you send me, if I am correct, D = max and A= min. Could you please explain and verify for me and correct me if I am making a silly and stupid mistake!!

Thanks!!!

Hi Sharath,

I see your equation and “raise” you a negative sign! (A little poker reference) =)

This is actually a little known fact, but there are several different derivatives of the 4PL equation. The one that we use is more popular, and used by many other immuno/bio related software; with that said, both will end up giving you similar results.

The formula you described, “y= min + [(max-min)/(1+(x/EC50)*b))]” is actually miss-quoted. If you look closely to your referenced page, there is a negative sign on the “raised” Hillslope. Which should change the whole formula.

I hope this can help shine a light to your question.

Best regards,

Charles Ma

-Miraibio Group

Hi Charles,

A good one. Thanks a lot for pointing it out to me. One last help with the same. Here are my readings for my set of standards which were transformed into log scale:

Log of Standards OD @ 450nm OD @ 450nm

1.30103 (20ng/ml) 0.037 0.04

1 (10ng/ml) 0.042 0.037

0.69897 (5ng/ml) 0.053 0.054

0.39794 (2.5ng/ml) 0.087 0.094

0.09691001 (1.25ng/ml) 0.157 0.167

-0.20412 (0.625ng/ml) 0.245 0.245

-0.5044557 (0.313ng/ml) 0.343 0.301

-0.8068754 (0.156ng/ml) 0.412 0.372

-1.107905 (0.078ng/ml) 0.427 0.417

(0ng/ml) 0.509 0.473

I used these on GraphPad for analysis and came up with these values:

Bottom 0.02738 – so this is D value based on the link you send me (estimated response at infinite concentration)

Top 0.4507 – this is A value based on the link (estimated response at zero concentration)

LogEC50 -0.1905

HillSlope -1.242

EC50 0.645

AM I CORRECT HERE WITH THESE? Or am I burning out my brains.

Your help here will be greatly appreciated!!

Best,

Sharath

I’m not too familiar with who SigmaPlot calculate it’s standard curves. But we usually use the raw values from the plate readers instead of the log values. But if your value/description of each of the parameter is correct, then, yes this should work out fine.

I would highly recommend you to try MasterPlex ReaderFit for fitting your standard curve. This will at least give you a second opinion on your answer.

Best regards,

Charles Ma

-Miraibio Group

Can we use the 4 PL to to fit and forecast incidence rate of a disease?

thanks,

Amal

Hi Amal,

Yes, I would think rate of diseases follow a sigmoid curve, and so a 4PL model would work. As you can imagine, most biological systems will have it’s limits, so naturally it will fall into a sigmoid shape. But with that said, I would suggest you to dig deeper into your specific topic of research to make sure that your case follows the norm.

Thank you,

Charles Ma

Mirabio Support

Hello,

I’m a graduate student doing some analysis on bone markers in pigs. The analysis requires the 4 parameter curve software so I downloaded the free trial. I have a few question here that I hope you can help :

1. After entering my plate optical density value and calculation is done, I obtained the calculated values ( x-concentration ). However, as my assay kit is for human research while my research is on pigs, the values are way higher compared to the standards. Eg standard range from 0-29 ng/ml while the range of my samples are above 100 ng/ml. The calculated values from the extrapolation of the graph only give to a max value and then it will be for eg >150. Therefore, all my samples have the same calculated value which is >150ng/ml. Any way that I can actually see the real value for each sample or do I have to calculate them manually using the inverse 4-parameter curve equation ?

2. How much is the software ? license per year or one time payment ? how many computer can install the purchased software ?

Thanks

Dear Felina,

This is a special case. I would try to normalize your standard curve to accommodate the difference in the value. You will need to find out the accepted ratio of optical density one is compared to the other and multiply/divide your standard curve to bring your curve up to a comparable range.

As a disclaimer. I have not tried this before, so I would definitely consult with your professor to find the best method to go about this type of analysis.

Thanks,

Charles Ma

Miraibio Group

what is the software that can be used to estimate the parameters for the 4 PL model

Hi Moen,

We have two software modules that can calculate 4PL model. It will depend on the instrument you are using. If you are working on a Luminex or BioRad instrument, then you will want to try our MasterPlex QT. If you are working with an ELISA plate reader, then it will be MasterPlex ReaderFit.

Charles Ma

Support.

Pingback: MiraiBio - Top 10 Tips for Meso Scale Discovery (MSD) Assay Prep and Analysis

Pingback: MiraiBio - Dose Response Curves with EC50 & IC50 Determination using MasterPlex ReaderFit

I am performing a bioassay using Y=(A-D)/1+(X/C)^B)+D for curve fitting. I am curious if you have any recomendations on estabishing specification ranges on each of the points. For example, the following ranges are being used for A, B, C and D values:

A: 4.0-10.2 (x10^3)

B: 0.697-1.27

C: 2.2-7.94 (x10^5)

D: 4.08-6.21 (x10^7)

Numerous assays fail due to a high C-value. Any input would be great. Thanks!

Hi Riki,

That’s a tough one. I’m not quite sure how to answer this. Normally in our experience the A,B,C,D parameters are determined by the standard curve. It usually is a single value (instead of a range). Have you tried using our software to fit your standard curve? I think this can help you determine the four parameter values.

Please let us know what if you get an answer. I think that will help other visitors to this blog! Thanks!

I would also like to invite anyone else reading this blog to help Riki with this question. I’m sure there are many others who has much more experience than I do.

Best Regards,

Charles Ma

Hi, I’m trying to find a peer reviewed paper to reference the four parameter logistic model that is the basis for the 4PL package/algorithm. Is there such a paper? Thanks.

Hello,

I have a question about the updated equation (as I am using the same equation at the moment).

“Updated 12/10/2013: For those of you who are looking for the back-calculations to solve for x, here is the formula. x = C*(((A-D)/(F(x)-D))-1)^(1/B) Special thanks to Pawel S. for providing this formula!”

When this is all tidied up I get a formula:

x = C*( (A – F(x)) / (F(x) – D) )^(1/B)

Has anyone encountered scenarios where F(x) = A, F(x) larger than A, F(x) = D or F(x) less than D?

If so how do you deal with this? I have seen F(x) = A or D being substituted with 0 (which is mathematically invalid according to maths help sites) or taking absolute values where F(x) larger than A or less than D (I am worried that this is also not valid).

Any help or suggestions appreciated!

Thank you

Heather

Hi Heather,

When it is mathematically impossible to calculate the point, we usually suggest our customers to exclude this point from their research. But there could be other ways of dealing with this that we don’t know about. Let’s open it up to other readers to see if anyone else has a better solution.

Thanks,

Charles Ma