Dose-response curves are generally strictly increasing : when dose is increased, the response will be increased as well. Logit and Probit transformations could be used to linearise such curves. But sometimes, dose-response curves begin with a decreasing part and could have a U-form. My question is how to model such curves ?
You must make plenty of assumptions to model these. The best way is to start with a model that you find reasonably satisfying. Fort instance, a cell that expresses two receptors that bind the same ligand: R1 activates a function, R2 inhibits said function. Then the dose response becomes the sum of two dose response curves, with shapes that depend upon the relative affinities of R1 and R2 for the ligand, the relative levels of expression of the receptor, and the efficacy of R1 to activate the function and of R2 to inhibit it. Some software packages used to analyze binding contain some modeling functions that you can use. For example:
http://www.graphpad.com/guides/prism/6/curve-fitting/index.htm?reg_bellshaped_dose_response.htm
But the shapes depend on what actually is going on. For example, the drug may induce dimerization of the receptor, but the monomer is a lot more active than the dimer. The dose response will increase at small doses of drug, but when the dimer becomes predominant as you increase the drug concentration, the activity will start decreasing. This will give you a bell-shaped curve with very different shape shape parameters than the model described above!!
In general, curve shapes are very much model dependent. The way we deal with these things in my lab is to first derive the equations for a number of potential, reasonable and realistic models; then enter these equations in a curve fitting program; then see what happens. Some commercial packages allow you to enter user-defined equations for nonlinear fits, and most are good enough. I still use sometimes an old program written in Fortran77 by Mike Johnson (University of Virginia). No, I cannot distribute it; and anyhow 1) you need a Fortran77 compiler; and 2) you cannot get pretty graphs out of it.
Finally, although several commercial packages will allow you to fit data to these complex models, your skills in deriving the correct mathematical equations that describe the model are irreplaceable.
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