# Is there a way to determine corresponding differential equations from fitted Hill equation parameters

I am currently doing some work with GCaMP, a genetically encoded calcium indicator. According to the literature the Hill equation parameters for GCaMP6m is $$K_d=167\pm3\, \text{nM}$$ and $$n=2.96\pm0.04$$. Usually the "derivation" of the Hill equation assumes that a number of binding sites equal to the the Hill coefficient, but I understand that the Hill Coefficient should be interpreted more as an effective "interaction coefficient" rather than an actual indicator of how many calcium binding sites GCaMP6m has. This is certainly true here, since GCaMP is a fusion of GFP and Calmodulin, which has 4 calcium binding sites.

However, given that $$K_d=167\pm3\, \text{nM}$$ and $$n=2.96\pm0.04$$, is there a way we can formulate a set of differential equations describing the kinetics of calcium binding to GCaMP? Both "intuitive" ways I can think of don't seem to be valid. If we take the naive approach similar to the Hill equation "derivation" we would end up with equations like $$\frac{\text dc}{\text dt}=-k_\text{on}c^n(g_T-g_c)+k_\text{off}g_c$$ $$\frac{\text dg_c}{\text dt}=k_\text{on}c^n(g_T-g_c)+k_\text{off}g_c$$

Now, $$n=4$$ is invalid because this would not produce the correct equilibrium behavior determined experimentally. However, $$n=2.96$$ would also appear to be invalid, as it doesn't correctly give the actual number of calcium ions taken up per GCaMP molecule.

Therefore, my question is how can we set up a correct (or reasonably correct) set of differential equations for a reaction given experimentally determined $$K_d$$ and $$n$$ that are more of "effective" parameters rather than literal ones?

• @com.prehensible I am not asking a genetics question. Did I say something to make you think so? Jun 8, 2020 at 21:03
• I study digital signal processing, I'd have that in code for a graphical illustration. I can't figure out if the 6m version is very different from the GCaMP. Jun 9, 2020 at 1:34