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I am working on a mathematical model of a biological tissue (drosophila pupal notum; an epithelial tissue) where the tissue is built up from cells all described by the same cellular-model. The tissue is generated using a somewhat random process, and so each cell in the tissue has a different size (volume, surface area, etc.)

The model focuses on the movement of particles both between cells as well as between the cytosol of each cell and internal stores (like the Endoplasmic Reticulum (ER)). In order to use the same parameters across the entire tissue, I need to know the ratios of various volumes and surface areas, such as the ratio of the volume of the cell to the volume of the ER, or the ratio of the volume of the cell to the surface area of the ER.

So my question is, if we have a cell of some volume $V_1$, and we look at another cell with volume $V_2$, what can we say about how the volume of the ER of cell $1$, $V_{\text{ER},1}$, or the surface area of the ER of cell $1$, $A_{\text{ER},1}$, compares to the volume and surface area of the ER of cell $2$, $V_{\text{ER},2}$, and $A_{\text{ER},2}$? For example, would we expect $V/V_\text{ER}$ to remain constant across all cells in the tissue? Or maybe we would expect $V/A_\text{ER}$ to be a constant across cells in the tissue? Or maybe both of these ratios follow some other sort of scaling law?

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  • $\begingroup$ Do we speak about volume-ratios within the same cell-types at different cell-states or do we compare different cell-types? I think this question is too specialized and you won't receive an answer. It could help if you provided allowed generalizations; i.e. you tell us what cell-types (or what one specific cell-type) you are interested in, and I am quite sure there will be people who can tell how those ratios behave in similar cell-types/tissues, that are not exactly drosophila pupal notum. $\endgroup$ – KaPy3141 Mar 22 at 9:09
  • $\begingroup$ @KaPy3141 I'm not thinking about various cell-states. As for cell type, do you mean like epithelial cells? $\endgroup$ – BioPhysicist Mar 22 at 10:29
  • $\begingroup$ But what would case different cell-volumes? Different points in the cell-cycles? Different cell-types that naturally have different sizes? $\endgroup$ – KaPy3141 Mar 22 at 12:12
  • $\begingroup$ @KaPy3141 Or just cell-cell variability. The tissue I'm considering has the same cell type throughout. $\endgroup$ – BioPhysicist Mar 22 at 12:14
  • $\begingroup$ Ok, I see. At least that is now defined. Maybe someone has data. $\endgroup$ – KaPy3141 Mar 22 at 12:37
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Please consider this "answer" 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.

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