Which theoretical function to fit for a sigmoidal melting curve?

15 November 2021 5 6K Report

Dear community,

Currently I try to figure out a reliable way to determine the melting temperature of my denaturation curves. They are not perfectly sigmoidal since the upper and lower plateaus, which we regard as the state of completely folded or unfolded molecules, have slopes higher than zero. My intention is to find a good fit function for my data and determine the melting temperature with the derivative of the fit function. I tried several ways and would like to know your opinion or suggestion on how to proceed on this.

The standard method, which many use, is do a linear regression fit to the upper and lower plateaus and determine the median curve. Unfortunately in many of my denaturation curves, the plateaus are not always present in the range I am measuring. In this case a linear fit is not possible (see uploaded figure).

I tried a five-parameter function for asymmetric sigmoidal curves and determined the parameters via gradient descent: f(x) = d + (a-d) / (1 + exp[(x-c)*b] )^g. While doing so I tried to minimize the sum of squared residuals (SSR). Unfortunately this method cannot handle plateaus with a slope. I tried to cut out the plateau, but the resulting melting point of the fit function shifts when I cut too much or too less. So determining the melting temperature is a bit arbitrary here.

I tried a 6-parameter function, which takes slopes of the plateaus into account. This one I found in the internet with references of Santoro and Bolen 1988; Clarke and Fersht 1993 for thermal denaturation curves: theta = (αN +βN*T) + (αD + βD)*exp[-(∆G/(R*T))] / (1+exp[-(∆G/R*T)]), where αN is the native state signal at 0 K, βN is the slope of the native state baseline, αD is the denatured state signal at 0 K, βD is the slope of the denatured state baseline, T is the temperature in Kelvin, R is the ideal gas constant and ∆G is the free energy. Furthermore ∆G = ∆Hm ( 1 − T/ Tm ), where ∆Hm is the enthalpy at Tm and Tm is the melting temperature. I thought this one seems reasonable, but somehow the fit is not satisfying. It overfits the plateau region, but I see clear deviations in the denaturation region (see uploaded figure).

Finally, I tried a ninth order polynomial. This method is super easy to apply, does not need constant adjustment of the initial values and shows a very nice fitting function to the data (see uploaded figure). Also the SSR value is the smallest one here. But I guess a polynomial was never used for this case. So is it scientifically reasonable to use it?

Now the questions which arised is, do I need a fit function with a clear theory as in point 3? Or can I also use an arbitrary function (here a polynomial), which is differentiable and has the smallest deviation from the data.The differences of the melting temperature are roughly 2°C. I know, the deviation is quite small. But when I have a more accurate formula for my data and when this one is also much more time saving, I would like to go with the polynomial.

Sorry for the long text, but I am really interested in the details here.

Bharat Singh Rawat

This plot looks like an IV curve of a Langmuir probe . There is one generalized MatLab function that can be used to approximate the entire curve.

you can check it on

https://in.mathworks.com/matlabcentral/fileexchange/19312-langmuir-probe-data-analysis-code

Zeynep Atris

Bharat Singh Rawat Thank you for your recommendation. But your advice does not solve my problem. I already have a curve which fits my data. My question is, whether I have to go with the theoretical expressions for thermal denaturation or whether I can use a random polynomial function for the best fitting result.

Bharat Singh Rawat

Zeynep Atris you say "But I guess a polynomial was never used for this case." Do you mean no one has ever used a 9th order polynomial to fit such kind of data?. One way to check is to do it for some known curves. Fit the 9th order polynomial for data of which you already know the melting temperature. Take its derivative and compare the error with the actual values. I think as long as the function is smooth are represents your actual data well, the derivative should give you the melting point.

Koen Van de Moortel

Polynomes are never good for understanding the physical reality.

Could you send me some raw data? I would like to take a look and throw them in my software "FittingKVdm" (see: www.lerenisplezant.be/fitting.htm).

Zeynep Atris

Thank you for your answers. I am replying to my own post, because I think I found my own solution to this problem. I derived the thermodynamics of protein folding and unfolding according to Norma Greenfield (DOI: 10.1038/nprot.2006.204). I additionally added a "double baseline correction" for the fully folded and unfolded states. By playing with some parameters in the fitting procedure I could fit the theoretical 6-parameter model very well to my data. Furthermore, this model also contains the physics behind, which scientifically makes more sense than using a random function.

How can I measure MMP-1 protein activity or protein level from liver tissue?

Binary Logistic Generalized Linear Mixed model in SPSS. Is it doable?

Looking for Materials with high thermal expansion coefficient and high temperature resistance?

Guidelines for survey (question type) analysis?

What are the values needed for gabor filter's parameter for cifar10 data?

Can we block unwanted images during the x-ray prediction?

No title

What techniques can I use to determine the miscibility of polymer blends?

Sample dilution for quantitative analysis using ELISA?

The procedure compute?

Looking for Materials with high thermal expansion coefficient and high temperature resistance?

What's the best way to measure growth rates in House sparrow chicks from day 2 to day 10?

12% ledeburitic cold work tool steel, etched for revealing retained austenite, any help for clarifying?

What are anisotropic electrical and optical properties of a semiconductor due to?

1 nm thick MgO thin-film to be deposited on Si wafer using thermal evaporation. What will be the optimum deposition conditions?

How to find the right number of integration points in second order 3d finite elements?

Why do a weight decrease for thermal degradation in nitrogen gas condition?

Can thermal cycler machines be problematic during PCR?

How to increase solubility of Poly acrylic acid in Dioxane solvent?

Standard curve shifting over time. How do I stabilise the ELISA?