How can I increase the extraction of gelatin and minerals from bones into my stock?

In terms of commercial food processing, there are more efficient ways to extract gelatin than slow-simmering the bones, generally by treating the organic matter with a strong acid prior to boiling, then using commercial evaporation and filtering equipment which is far more efficient than anything a home or even restaurant cook has access to.

According to Gelatin Food Science (see the "Gelatin Manufacture" section) it can also be first treated with a strong base solution before acidifying it, which lowers the isoionic point. It's kind of difficult to explain exactly what that is if you don't have a background in organic chemistry; technically speaking it's a relationship between pH and electrical charge - but applied to extraction it refers to the pH at which the solubility of a protein (such as gelatin) is the lowest. Lowering this is a good thing in extraction, because it means that the gelatin will be easier to filter out in an acidic solution.

Thus I have to point out the question is actually a bit contradictory; if your goal is the extraction of gelatin then you want to decrease the solubility.

But I think this is all going to be beside the point anyway, because none of this applies to stock making; the goal in food processing is to extract the pure gelatin, not to get a flavourful stock. When making a stock you definitely don't want to use an acid solution, it's going to ruin the flavour.

Realistically, when it comes to stock-making, especially at home, the only way you're going to be able to extract more gelatin is to simmer it longer. That's it. When the bones break without any resistance, that means you've denatured all the collagen and you've got all the gelatin you're going to get. I wouldn't worry about solubility because typically in stock-making you're already using more than enough water to dissolve all the gelatin that you could possibly hope to extract.

If you have a pressure cooker, try using that. You only need to cook it under pressure for about an hour, plus 20-30 minutes for it to reach pressure in the first place.

Starting your stock with cold water and 1/4 cup of vinegar should help you get the most goodness out of your bones.

If you want a more gelatinous stock, try adding in extra wings/feet* for poultry, or veal knuckle bones for beef. For a pork stock, pork neck bones work quite well. It's also helpful to break or crack the bones to expose the marrow.

*Chicken feet are awesome. They make a very gelatinous stock, but don't impart any chickeny flavor. They also look creepy. If you get feet that still have skin and nails on, you can just peel the skin off like a glove. The nails have layers that will pop right off. That's the best way to clean them.

Long, slow simmering of the meat and bones - I usually do it overnight - extracts (into the stock) all of the gelatin available, but if you put the aromatic veg in at the beginning of this process, you will lose a lot of the flavor in the long cooking. According to the CIA cookbook, you should put the aromatics in for the last hour.

Purely anecdotal, but I find that boiling turkey bones for hours gives me as much gelatin as I can handle. My stock turns into jello when I refrigerate it.

It's probably worth pointing out that "simmering for hours" for me means at least (or about) 11 or 12 for turkey, and I don't even start getting flavor out of chicken bones until 4 hours in, and that too I keep going for 12.

Some recipes seem to suggest that you can simmer for just 4-8, and that just doesn't work the same. So, yeah, "simmer longer." And if you're simmering for that long already, then I don't know.

I use a pressure cooker with a soup setting. Low pressure, max time ( in my case 240 minutes), repeating up to 4 times. Chicken feet plus necks or backs or legs.