I am developing an immunoassay where I am depositing a fixed quantity of protein on a finite surface, and I want to determine the spacing between protein molecules. Assuming that the majority of the protein molecules in solution bind to the surface, and assuming** that they do so in an evenly spaced monolayer, is there a formula for calculating the 1) maximum density (i.e. maximum packing) on the surface and 2) the distance between molecules at less than maximum density, given the number of molecules that bind, the surface area and the effective radius of the protein?
Question 1) is addressed by a branch of mathematics known as packing theory. It’s a fairly recently-developed branch, so you may not be able to find many results, but I think that the very basics of packing circles on the plane is known.
I think that Question 2) might be addressed by stochastic geometry, but I don’t know that for certain.
Then again, if you get it wrong, you could just say,
"I have HAD IT with these muthaphukkin molecules on this muthaphukkin plane!!1"
sorry, couldn’t resist.
Not a solid-state guy, so take this with a monolayer of salt, but here are my thoughts:
The simplest BOTE thing you could do would be to pretend the molecules arrange themselves in a nice lattice, with nearest-neighbor spacing given by twice the molecular radius. The most obvious choice for lattice is hexagonal (the two-dimensional closest-packed structure), with a packing density of 1 molecule per R[sup]2[/sup]sqrt(12) of area. Square might be a reasonable guess too; it’s got 1 molecule per 4R[sup]2[/sup] of area, of course.
What really happens? At maximum density, the molecules probably will arrange themselves into some sort of lattice. (Though it would be cool if they were Penrose proteins and arranged themselves into an aperiodic structure.) The shape and size of the unit cell will depend on the particular protein and on its interaction with the surface. If all you have is an effective radius, then you might guess it will be a hexagonal lattice. (At any rate, the hexagonal-lattice approximation will give you an estimated upper bound for the packing density, assuming the “radius” is a reasonable measure of the protein size.)
At lower densities, there are different things that might happen. The proteins might be attracted to one another, so you end up with condensation into regions (each arranged in the protein’s preferred lattice) on the surface, separated by regions of empty surface. This is a complicated situation; the sizes of the empty regions will depend on the various interaction strengths and maybe even on how the molecules were deposited. If they don’t strongly attract, the surface might have imperfections or other locations which bind the proteins more strongly, so they might still form some regular structures. Otherwise, I’d expect to see somewhat random patterns. You could compute mean or RMS distances based on an assumption of randomness, or on some uniform arrangement if you want to. There are methods in statistical mechanics for computing this sort of thing, for particles with pairwise interactions (here, the approximate hard-sphere radius).
If you want the probability distribution of items, then you have stumbled onto a problem that was so interesting that I included an appendix in my thesis on it.
For particles distributed in space with average density rho (number per unit volume), you can guess by dimensional analysis alone that the probable spacing ought to be on the order of (1/rho)^1/3. The probability that the closest separation between a random particle and its nearest neighbor is actually given by P®, where
P® = (4pirho*r^2)exp(-((4pi)/3)*r^3))
This function has its maximum at about 1/2 the value you’d guess from dimensional analysis (and about an equal width). People solved this problem over and over, apparently not aware that it had already been tackled. Some got it wrong. The best proof was by Chasndrasekhar, who proved it in about two lines (he was applying it to the distributions of galaxies).
I once derived the equivalent formula for nearest neighbor distances on a plane. It’s something like:
P® = Alambdarexp((-piLambda*r^2))
Here A is a constant and Lambda is the density of items on the plane (in units of Length^-2)
