What is the expected distance to the nearest molecule? 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:  Added simulation
This is a simulation of the distance of the nearest neighbor to the center of a micron cube. The center was named "RNA". This represents N=501 molecules.



 A: Consider $d$ dimensions.  The distribution to the nearest neighbor of any point can be approximated by supposing $N$ neighbors are independently, uniformly, and randomly situated within a radius of one unit from that point (where the distance unit and $N$ are chosen to reproduce the molecular density; preferably $N$ is large).
The chance that one given neighbor is further than a distance $r$ (for $0\le r \le 1$) is the relative volume of the spherical shell between the balls of radii $r$ and $1,$ equal to
$$S_{d}(r) = 1 - r^d.$$
Since the neighbor positions are independent, the chance that all are further than distance $r$ is
$$S_{N;d}(r) = (1-r^d)^N.$$
This (the survival function) determines the distribution of the distance $R$ to the nearest neighbor.  Its expectation is the integral of the survival function,

$$E[R] = \int_0^1 S_{N;d}(r)\,\mathrm{d}r = \int_0^1 (1-r^d)^N\,\mathrm{d}r = B(N+1,1/d)/d$$

where $B$ is the Beta function
$$B(a,b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}.$$
For instance, here is a histogram of 50,000 simulated values of $R$ with $N=500-1$ neighbors.

On it I have superimposed the graph of the density function $-d/dr\, S_{499;3}(r)$ in red to show the agreement and I have plotted a vertical line to show $E[R].$
