Try AFM. AFM techs remember.

Also, how about molecular methods, like qPCR? You'd make a standard that you can purify, quantify molecules by spectrophotometry. Then, spike it into your sample at known concentrations, to quantify the plasmids already present there.

John Schloendorn makes a good suggestion, do you really need to visualize and count or can you use an indirect method.

You could also consider tagging the plasmids, for example include a specific protein binding site on the plasmid and use a fluorescently tagged protein to bind that site, so that only the plasmid will specifically fluoresce.

Thanks for the advice. I knew nothing about AFM or qPCR, so I had to look them up. To a country bumpkin like me (at least relative to technology less than 20 years old) it's all Greek. I can't form a clear picture in my mind of exactly how you'd count chromosomes using these tools. But I'll be thinking about it more before I write it off. Thanks again.

Tagging the plasmids is also a good-sounding idea, but seems excessively complicated to me. A device into which one can put a solution of small chromosomes, which device would then immediately report on the particle concentration, would be useful at this juncture. Anyway I thank you both for your interest.

If your solution of plasmids (or small chromosomes) is homogenous and pure, then there are such devices and it is easy to measure. However if your sample has all sorts of other DNA around (such as genomic DNA) then the problem becomes significantly more challenging.

The solution is pure, starting with commercially-available pBR322 DNA. The object of the experiment is to separate the strands of this duplex circular plasmid. Everyone will tell you that that's impossible, but it's not. Tai Te Wu did it in 1996 ( Wu & Wu, Bull Math Biol, 58:1171, DOI:10.1007/BF02458388 ), but no one reads the article. Wu's experiment was very difficult, time-consuming and expensive. No one will ever try to repeat it. After decades of theoretical research into the underlying question, however, I have devised a ridiculously-simple and inexpensive way to repeat Wu's work. But it requires a lab, which I don't have, and a way to identify the products of the strand separation. The question is totally simple: After performing the strand-separation procedure, did the number of particles double, or not? There's no room for ambiguity here, they either double, or stay the same.

Will they be separated in solution? If so, you can detect by fluorescence. There are intercalating dyes, like Sybr Green, that cause bright fluorescence with double but not single stranded DNA. That's routinely used for thermal separation of the strands -- for example in qPCR. (you don't even have to do qPCR to do this, and don't need the machine -- a simple fluorescence spec would do, with thermal melting as control.)

If you must spread them on a surface, and can visualize them by EM, then AFM would be the way to go. It shows you where every atom is on the slide. Just like a high resolution and easy to use version of EM. You can hire a contractor to do it. You can count the particles, but can also distinguish ss from ds DNA by appearance.

Thank you for the additional information. I am aware of the possibility of using fluorescence microscopy for this job, but, in spite of many attempts to enlist support from both scientists and corporate promoters of such equipment, I cannot find anyone who purports to know how to do it. Every once in a while an individual or commercial company expresses a transient interest, but, to date, they nearly all lose interest almost immediately. The AFM idea sounds great, but I wonder what they'd charge. I need to be careful. Here's why: About 5 years ago I spent $5000 on equipment and chemicals, then performed the strand separation protocol in my kitchen (much to the chagrin of my late wife). I sent the products, on dry ice, to a new prospective partner in California, who was going to examine them by fluorescence microscopy. The story from then on is one of complete disaster. First he let the dry ice sublimate, so the DNA was left at room temperature for an unknown period. Then it sat in the refrigerator for weeks while he attempted, in vain, to get his microscopy working, which he never did. This was a fluorescence microscope with an oil immersion lens. To illustrate the pathetic nature of this failed experiment, at one point he reported that the samples I sent had "no DNA". We quickly came to realize that the immersion oil he was using absorbed strongly in the fluorescent light wavelength range! Could my luck really be that bad? With the oil itself strongly fluorescing, all he was seeing was a blinding light and a totally-whited-out field. After weeks of floundering around, he finally gave my samples to a colleague who had some sort of particle-counting device, a 'Zeta Counter' I think it was called. The device destroyed all the samples, and the data it generated was gibberish. The reported DNA concentrations were orders of magnitude off, and the results were neither positive nor negative - they were just strings of alphanumeric nonsense. Apparently I didn't learn my lesson. Several months ago I agreed to pay an Indian scientist-for-hire, from a company called 'Kolabtree', to do my experiment. He asked for $1600. I send half - $800. He claimed to have a fluorescence microscope, and I promised the rest of the money after he sent me pictures of control DNA. Weeks went by, but nothing happened. Eventually I had to badger him for a report. He alleged that Covid 19 had caused his entire university to close down. More weeks went by. Then he reported that his microscope lacked sufficient magnifying power to do the experiment. I'm thankful I didn't sent the whole $1600. So I'm down about $6000, with nothing to show for it. BTW, I just posted a Covid 19 article on ResearchGate. It might interest you. The world is a very strange place - not as reported on television.

Hah, I like your tenacity! It's a good rule in DIY biotech -- No matter what happens, you will end up blowing your entire budget on dead ends in a few years. That's regardless of whether you've got a few $k saved up or run a hundred million dollar fund.

OK but I'm trying to bring you good news. You don't need a microscope. You don't need to resolve single molecules. You can do this in bulk solution. Any old fluorescence spectrophotometer off eBay will do the trick. If you're really determined, you can see it with your own eyes using just a maglite and a Lee filter book.

Get a dye like SYBR green that prefers double over single stranded DNA (I think they make improved versions now with an even greater difference). Look up how to do a "melting curve": You slowly increase the temperature of the solution, until the strands separate. At that point, you'll observe a relatively sharp drop in fluorescence, as your DNA transitions from double to single stranded.

Once you have that working, substitute the temperature change for whatever intervention it is that you're going to use to separate the strands. See if you see the same drop.

Does that sound like a plan?