You may assume any easy model without collisions/attractions, molecule diameters of 0 and perfect mixing.
If you have 500 copies per micro-m^3 of molecule A , then each molecule A has a box of 125 nm x 125 nm x 125 nm. However, according to my intuition, the expected distance to the closest neighbor should be significantly less than 125 nm, as the shapes of the theoretical volumes will not be perfectly cubic/spherical.
So my question is:
- What is the expected distance between molecules A (to the closest molecule A)?
Background According to this source, we have 1^6 proteins per micro m^3 in huma cells. If we assume 20.000 genes and that my protein is ~10x more expressed than average genes (which might not be expressed at all), then this would be 500 copies per micro-m^3.
EDIT: I asked the wrong question
Actually I need to know the expected distance between 1 RNA to the closest of 500 proteins within the cubic micron (center to center, no collusion). I did a small 3d random-walk simulation, but the result was quite different from @whuber result. I wonder if the reason is the slightly different question or that my simulation is inaccurate.