I have a chemical phenomenon to model for my research. I'll try to be clear while explaining my problem.
I created little molecules (let's call them "emitters"), which emit a signal. When the emitters are not too close from each other (i.e, when the concentration is low), the signal is proportional to the concentration. However, when they are close, the signal is weakened.
Then, I created a receptor. It's basically a big molecule which can host plenty of emitters. When the emitters are stuck on the receptor, they are close, the "local concentration" is high, so the signal is weakened.
On the picture, the emitters are the blue spheres. And here is a curve of the signal. In x, you have the concentration of the emitters, and in y, you have the intensity of the signal.
During phase A, the signal increases because the local concentration of the emitters is not high enough. Then, it reaches phase B, where adding more emitters on the receptor weakens the signal. As you can see, there is a plateau where adding more emitters change nothing. And then you have phase C, where the emitters are not on the receptor, because the receptor is full. Then the signal is linear.
My goal: Model (which means get a function y=f(x)) the phenomenon. I would need a function describing the process. I'll then fit my experimental points to this function, and hopefully get back some informations:
- the concentration where f(x) is max, before entering phase B
- the concentration where f(x) starts to be linear (indicating the saturation of the receptor)
I know I could get these infos by reading the graph, but it's not really accurate. Plus, if possible, I'll extend this function to include some parameters like the charge of the molecules, or their size.
So, is it possible to do something ? I'm not very good in maths.
Let $X$: the number of emitter molecules bound to a single receptor and $Y$: the strength of the signal from that single receptor.
One way to summarize your experimental result is to see how $Y$ depends on $X$. (Can you count $X$ precisely or approximately?)
Then it is possible to model the receptor as a square lattice, for example, where each small square can be occupied by an emitter molecule. When two emitter molecules are positioned in neighboring squares (or within the distance of three squares etc.), their signal becomes weakened. You can simulate the model numerically and compare it to your above result to fit the parameters. Different experimental conditions can be incorporated into the model by changing the size of a small square or other parameters.