Good questions. I'll try to answer, but I'm still learning about this as I go.
Wikipedia page for kappa casein:
https://en.wikipedia.org/wiki/K-Casein Wikipedia page for casein:
https://en.wikipedia.org/wiki/Casein Questions about the structure of micelles is hard to answer because it's still a topic of active research. In other words, nobody really knows exactly how it is set up. In fact if you try to find out exactly why cheese melts or stretches, you will be on a nice ride of scholarly papers on the subject
Casein is largely hydrophobic (it repels water). I'm not at all knowledgeable about soaps (especially how they work at a molecular level). However, I don't think it is generally related to the way soap works. I'll be honest and say that I don't really understand how anything is hydrophobic. Just to give a bit of background, though, water is a "polar" molecule. It has one oxygen atom and 2 hydrogen atoms and is shaped a bit like a boomerang with the oxygen in the middle. The hydrogen is positively charged and the oxygen is negatively charged. The way water "dissolves" salt is that the negative ions are attracted to the hydrogen and the positive ions are attracted to the oxygen. This is enough to wrench the salt apart and for the ions to wander away.
Kappa casein (k-casein) is not polar as far as I know. It is positively charged. "Polar" means that there is a different charge on one end than the other. The micelle itself is round (and very, very big compared to other molecules -- enormous, really) and the whole thing will be positively charged when the K-casein is attached. The whole thing is negatively charged when the k-casein is removed (cleaved). I suspect its size and charge have something to do with the reason it is hydrophobic, but I don't really know.
k-casein is not particularly interesting with respect to cheese making because we cleave it off and it gets drained away in the whey. I wouldn't get too worked up in the name, either. Proteins are just large chains of molecules with nitrogen in them. Proteins are made up of building blocks called "amino acids". The way these amino acids are put together in the protein determines what kind of protein it is. K-casein is just a casein molecule with a slightly different "shape" (combination of amino acids). The rest of the micelle contains some other shapes as well (and since they got to kappa in the Greek alphabet I guess there are alpha, beta, gamma, delta, epsilon, zeta, eta, theta and iota caseins at the very least
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Because it is a marketing term, some people know about A1 versus A2 beta casein. I have absolutely no idea what role beta casein plays (if anything) in cheese, but some people are allergic to A1 beta casein. A2 is just the same as A1, but has a different amino acid somewhere in the chain (Wikipedia has more info in the casein article). I suspect this is what your farmer friend is talking about. Maybe not??? Anyway, Jersey cows have more A2 beta casein while most other cows have mostly A1 beta casein.
I'm not entirely sure how the isoelectric point works. Here's another Wikipedia page:
https://en.wikipedia.org/wiki/Isoelectric_point My very loose understand is this: "acid" has a very specific definition. I don't want to write a book on ionic chemistry, but essentially water will "dissociate". In other words instead of H2O, it will split into an H+ ion and an OH- ion. In water that has a pH of 7 (neutral), these ions are in balance. When they get unbalanced the water becomes either acidic or basic. More H+ ions than OH- ions and it is acid. More OH- ions than H+ ions and it is basic. This is literally what the pH scale measures. It's important to understand that the H+ and OH- ions will react with other substances in the water and then "precipitate out" (i.e. come out of solution). This strips either H+ or OH- ions from the the water. Normally if we add an acid to water, that acid reacts with the OH- ions in the water, creating a salt. This leaves more H+ ions than OH- ions, making the water more acidic.
Some judicious cutting and pasting from Wikipedia: "The pI value can affect the solubility of a molecule at a given pH.... Amino acids that make up proteins may be positive, negative, neutral, or polar in nature, and together give a protein its overall charge. At a pH below their pI, proteins carry a net positive charge; above their pI they carry a net negative charge."
In all of these ionic systems there is a free exchange of ions. Things bind a positive ion and then dissociate (and the same with the negative ions). Everything is in flux and it can be a bit hard to say "What is it now". These big proteins just keep exchanging ions and the net result is either having an excess of positive charges (in which case the proteins are positively charged) or having an excess of negative charges (in which case the proteins are negatively charged). But in any case, when the pH hits the pl for the protein, it no longer has a excess of either positive or negative charges. It's neutral.
Just a quick sentence to say that I've often wondered how this affects whether or not the protein is hydrophobic. Not sure... because I don't know if that is related to charge at all.
Sorry for the crap explanation. I don't understand it myself...
You asked what about the fat? Fat is *not* held in the micelle. The micelle is just a tightly bound ball of protein and calcium phosphate salt (which may or may not play a role in keeping the micelle together depending on what paper you read). The fat exists as globules floating around in the liquid. When the casein micelles stick together (either mechanically or through boding with calcium) there are spaces in between. That's where the fat gets trapped. This should give you a good idea of how big these micelles are: they are big enough that when you stack them all up, the space between them is enough to trap globs of fat!
Finally about the stickiness: no idea. I don't believe it is hydrogen bonding, but then I don't know why *anything* is sticky
I actually suspect that nobody has studied the reason why yogurt gels as it does, given the fact that nobody really knows how a casein micelle is put together. It might sound weird, but I remember about 30 years ago going to a physics seminar discussing the topic of the formation of the head on a beer. Before that nobody knew how it worked in detail. The same is true about a lot of the physics and chemistry of cheese making. It's difficult stuff to study, I think.
Hope that helps. I even hope more that someone will come by and fix my mistakes :-)