non-linear regression time response model


I need to choose a non linear regression model for, protein phosphorylation over time response, for this kind of data

Is not like enzyme kinetics, the michaelis menten model does not behave properly, becouse, I am not dealing with a susbtrate concentration, instead I got a signal response over time, for receptor stimulation.

So, please help me to get a working model for this kind of non linear regression.
I do not understand your question? Is there a drug response at t = 10 or just unperturbed development in time without a drug?

Maybe look at Earp and Jusko 2008 or Koch and Schropp 2012 for non-linear PD models describing such unperturbed (no drug) data.

In these papers cytokines like IL1, … ,TNF, GM-CSF are modeled by a non-linear model to describe such a typical behavior
The experiment was a time course, for a drug, so I get phosphorylation response behaviour in time

In enzyme kinetics or a binding model, you do not take into account time, instead you plot the substrate concentration Vs Vmax or radio ligand concentration (X) Vs total binding (Y). Here I got time (X) Vs kinase phosphorylation (Y), so I do not find a proper model to do the fit.
Okay. The first question before building a realistic model is what to do you expect for large time.

When looking at the grey data one could assume that the measurements run back into the initial state (baseline condition). This behavior is not really true for the black one. However, maybe the measurement interval is too short.

If possible you should present more details. Does the drug stimulate the protein production? (If so I have the perfect model for you) Do you have measurements from the drug concentration?

Nevertheless, in the cited papers you find among other things mathematical models (based on differential equations) to describe the situation if the measurements saturates one a higher level than the first measurement (what one could expect when just looking on the black data). I would expect that these models could properly describe the data (black one). But in these models the physiological interpretation of the parameters is difficult, if this is one of your aims. And more precisely these models just describe the data with only a limit realistic biological interpretation.

I am not expecting to stimulate the protein production instead Y data, is a rate for protein phosphorilation fold change Vs control (time 0).

The cell culture was stimulated, with a fixed drug concentration (optimal for this receptor), at 2, 5, 10, 15 & 30 min, the the cells was harvested and resolved by western blot, for kinase phosphorylation detection.

The samples for the black line was a control, there is only receptor present, and the grey line represent samples with a over expressed protein, involved in receptor internalization.

For the non linear regression, for example we have the michaellis menten equation
Y = Vmax*X/(Km + X)

X, is for substrate concentration, but I am dealing with protein phosphorylation fold change. So Can I apply this kind of equation?

Looking for a paper I found this one for ERK and Akt kinases

But I dont known how to apply the model

Thanks for your help
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Dear friveroll

Maybe it is annoying but before modeling data you have to define your goals. I am not a biologist but developed a large amount of models in the last years together with biologist, pharmacologist,… . You have to clearly state what you actually want:

If you just want to describe the data by a mathematical model with a slight biological background I could help you. But as I said before if you want to physiologically interpret the model parameters such a model will not really help. So the main question is, for what you will use the model (pharmacological parameter interpretation, simulation,..)

If you are also interested in drug response on the target (I am still not sure if or if not), then you have to apply the concept of pharmacokinetic / pharmacodynamics modeling. This is the standard concept in pharmacological modeling in industry and academics. If I understand you right there is an influence of a drug on your data. So without taking the drug into account in your model every model will be worthless. Note that PKPD tries to mimic the underlying biological mechanism and as a result they describe the data. The models are not developed with just looking at the data. Therefore, you cannot just take a model from literature to apply it to your situation.

I would like to help you but to get a better overview please answer the following questions ;)

Are you familiar with PKPD?
Do you know what differential equations are?
Did you ever develop a mathematical model?
Which mathematical models you already applied?
Which software do you use?
What is your background (PhD….)