How does the Endoplasmic Reticulum scale with Cell Volume in Epithelial Cells?
Please consider this &quot;answer&quot; as an enlarged comment. I can't answer your question, but it's possible that my speculations or considerations could be helpful in your particular situation, in the early phase of building a model.
First of all, I doubt that general formulas that link cell-volume to ER-volume of different cell-types can exist. Dependent on the specific cell-type, (dependent of the protein production need in the rough ER), the ER/total ratio can strongly vary (i.e. compare neurons (strongly protruding ER), red-blood cells (no ER), pancreatic cells (large ER), endothelial-cells..). Thus, any correlation may just be chance, without the general meaning that you truly seek.
Therefore, you should focus on such volume-correlations within the same cell-type at different cell-cycle stages.
In this context, I can think of mathematical considerations, that could help your model: When cells divide, they effectively half their membranes (membrane-lipid count). Mathematically, this means that the overall volume of the cell should reduce to around 35%. In contrast, the volume of the ER as a highly condensed structure could scale linear with its surface area, if we see it as a long tube that gets separated. From this, you could build an easy mathematical model, in which ER volumes are relatively bigger, when cells are small.
However, according to cell-size measurements this is not true; cell-volumes get halved [1] when cells divide, which should indicate relative stretching of the membrane. Unfortunately, I could not find any data on how the stretching of the cell-membrane relieves over time. Further, the volume (quite) linearly increases over time until the next division can take place [1].
This means the cell-membrane keeps a constant weight/V before and after cytokinesis, while the surface-area increases by root 2. Given, that the ER-membrane-mass is also halved when the cell divides, and there is no apparent mechanism that should cause swelling, it is still quite difficult to predict how the ER volume would change. Again, the ER is no simple sphere, but has a highly collapsed shape.
However, I found this source, that deals with changes in shape and volume of the ER during the cell-cycle. Apparently, there is a membrane exchange between nucleus and ER and the ratios in volume between ER and nucleus is constant throughout the cell-cycle. You can study this yourself in more detail, but I would take this observation as an indication, that the ER volume behaves as would be inferred from a mathematical standpoint; to the third power of the diameter of the cell.
To sum it up, I initially expected smaller cells seem to have relatively larger ER volumes, and I suspect, that you had a chance to fall for the same trap. However, if the volume-ratios are not constant, they should rather behave the opposite way: ER takes less cell-volume after division, when the cells are small.