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on dilution ("family jewels", continued) [Dec. 31st, 2010|08:11 pm]
matt
A couple of blog posts ago, I had mentioned that I'm fascinated by the concept of dilution.

Part of my duties as a chemist was preparing serially diluted standards for analysis, and the typical approach was to make up a series of 100 ml flasks, create a master flask into which I would dissolve a known weight of a given standard, then dissolve the standard in water, filling the flask carefully to the 100ml line. Then, I would pipet a certain amount- typically 10ml, and transfer that into the next flask, fill to the line... lather, rinse, repeat. The nice thing about such a method is that if I did my job right, you could draw a line between the analytical results and you would get a near perfect correlation coefficient, typically in excess of three or four nines.

The thing about dilution is that if you do it enough times, that last pipet's worth of solution you transfer into the next flask might not contain any of your original substance. You may (or may not) remember Avogadro's number, which states that a "mole" of any molecular substance is it's atomic weight, in grams, and that a mole contains about 6.02 X 10^23 molecules. So if you have a mole or so of a given substance, say, sodium chloride, which weighs 57 grams- if you dissolve a gram of table salt into a 100ml beaker, there are about 10^22 molecules floating around. If you did the procedure above, the next solution would have 10^21 molecules floating around. Twenty one more times, and at last you only have only one molecule left in the flask, and chances are the next time you dilute, and in the next flask, poof! there's no more salt in your salt water.

Now, if you're inclined to believe in homeopathic remedies, you'd recognize that as an "X" dilution. An "M" dilution is one milliliter in 1000 (roman numerals, ftw), and goes even faster- 8 "M", and poof! all gone. Homeopaths believe that the water has a memory of the salt, particularly if the procedure is done by a specialist and shaken "just so". But, if you're to believe science, any time a particulate characteristic is "diluted", at some point the act of dilution fails to transfer the original characteristic, and it becomes "lost". This is a fairly difficult concept to grasp, and I've struggled a bit to find a good way to reconcile all of that in my head.


great-great-whatever-grandparents, vs time, and base pairs of DNA So, I spent some time thinking about the issue of "great-great-whatever-grandparents", and how DNA and genes are passed down through the generations. In my last blog entry, I made what might be a difficult statement to believe, that at some point the act of procreation results in one or more of the grandparents passing nothing to the ultimate child. I wanted to see how fast that could happen. Given the fact that there are about three billion base pairs in the DNA of the human genome, if each parent passed about 1.5 billion base pairs on average, then each grandparent in turn contributed 750 million on average, and so on. It doesn't quite work that way exactly- there are two special cases. The Y chromosome and the mitochondrial DNA are passed (mostly) without recombination or change from the father and the mother and the fathers' father and mother's mother, respectively, (lather, rinse, repeat). There's also random mutations thrown in from time to time, and deletions and additions: but for the purposes of estimation, let's take it to be so. If we assume about 30 years from mother to child, how long would it take to completely dilute things away? (data in spreadsheet, here.)

It was a lot quicker than I thought. The picture here shows the result- in about 30 or so generations, or about 1000 years. Of course, in this picture, some of these "place settings" in your family tree will be occupied by the same person. I have superimposed the approximate world population in each year going back, and this has to happen a lot quicker- back past the year 1150 or in about 850 years, there are more place settings than people on earth on my tree, alone. Of course, if on this tree someone marries their cousin, as must happen to us all, the dilution happens a bit slower, but at some point, we are all ultimately cut from the same cloth. In the end, there is "nothing left behind" of each of us, no matter how many generations we sire- poof!

I've been thinking a lot about generations and children, as I think having genealogy as a hobby is likely to cause you to do. I've often discussed with my spouse what I think about us not having children of our own. I think on reading that last paragraph, some people might feel a little saddened, and assuredly, it's not my intention to bring people down on new year's eve. I think what we leave in our genes is a very small part of the picture, and dwindles rapidly with each generation- but, too, we leave behind our thoughts, and our philosophies, and the impact that we've had on everyone else. If we've been really good, we leave the world a better place- safer, healthier, more easily understood. Leaving behind art, or science- these are things that can last for much more than 1000 years, if you care enough and work hard enough to make them count.

I think all this stuff about DNA and genealogy teaches me a lesson, too. Each and every one of us has a massive family. We are all, genetically speaking, cousins, even if the paper trail is really difficult to ferret out. It's pointless to be xenophobic or to hoard your relationships and your wealth when the inexorable act of dilution is constantly reshuffling the deck.

Be good to your family, and have a safe and prosperous new year.
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Comments:
[User Picture]From: nightspore
2011-01-01 02:30 pm (UTC)
This is really great and starts answering a question I've been asking around about, viz. How many generations down are we only averagely related to our descendants -- that is, no more related to them than to non-kin? Do chromosomes play a larger part than you're suggesting? Wouldn't I expect a descendent of mine six generations down to have none of my forty-six? Or does genetic cross-over play a big enough part that it's irrelevant to think in terms of chromosomes? Or are they statistically the same? (I think that's probably the case, but maybe you know.)

Of course you are making a point nicely analogous to homeopathy: culture or the disposition of the gene pool at any moment will contain memories of us, even when we're completely diluted away.
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[User Picture]From: nightspore
2011-01-01 02:42 pm (UTC)
Ah, whoops I see you mentioned crossover in your previous. Another couple of minutes of thought make me think that crossover doesn't matter to the statistical argument: you win sone, you lose some, it foist matter whether the split occurs in a base-pair, a gene, or a chromosome. I guess mutation hastens the process but probably not much? So, yeah, even without considerable relatedness due to interbreeding of third or fourth cousins, our descendants are probably only averagely related to us after 5 generations or so.

I'm curious about this because of Fisher's attractive sons argument. That suggests that our genes strategiize to produce fit grandchildren, rather than children. Where do the opposite curves of reproduction and dilution sum up to our optimal goal? Better, genetically, to aim for fit children or grandchildren or great-grandchildren? Any ideas much appreciated.
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[User Picture]From: hbergeronx
2011-01-01 06:32 pm (UTC)


By and large, "I don't know". But you have to remember that for each randomizing event, if the majority of the chromosome is the same, crossing over isn't going to change a lot. I've attached a visualization showing my brother's genome compared with mine at a somewhat random sampling of base pairs known to carry mutated differences, called SNPs (single nucleotide polymorphisms). For example, there might be a stretch GATCGGTC vs GATCGATC on each of the pair of chromosomes, in which case they would not be shown as having identical regions when compared. If my brother had two GATCGATC and I had the above, we would be shown as "half identical" (HIR) in that range because the SNP at the sixth position (which is the only thing that was actually tested) is typically the only thing that is "different". The tests don't show all of the SNPs, and don't show repeats or deletions. (of course, our Y is identical, and there's only one of them generally for males, so it's not even measured in this visualization)

I'm writing these posts for crossposting to a more public area as well, where I don't post some of my more controversial beliefs or feelings (read- family friendly content) but I don't buy into a "fitness" mechanism. Again- as far as I can tell it's random, and any evidence of an "invisible hand" or some such in the process is not there for me. Note: I realize the oxymoron in looking for an invisible hand. I'm pretty sure much of the so-called "fitness" happens long after meiosis, when you're drowning the inconvenient truth in the river because you can't take it.

Based on comparisons of my chromosomes with a third cousin once removed, with whom I only share two tiny HIR regions with, and who is "1.65%" shared as opposed to my brother who is "52.46%" shared, I would say that after five generations there's at least a good probability that there's about 3% of the genome left around in common, and based on what I'm seeing out of 23andme.com, which is where I've been tested, you can find potential cousins out about 6 generations or so, or more, but it's unclear how much of that is "false positives" in terms of actual, blood-line relatedness. The fact that I knew nothing about this cousin prior to 23andme.com is about as "blind" an experiment as I can perform. If you procreate with someone who is very similar to you in genetic makeup, there might be less measurable dilution, but in looking at genomes of someone who has a known history of cousin intermarriage within 5 generations or so, the picture is quite a hodgepodge of dozens of HIR's. It's not like the randomization across the human genome is super dramatic, we're not dropping chickens or kittens out the other end. I'm mostly just idly musing about dilution, and randomization, and not producing "serious science" here. But I do welcome the feedback.

Edited at 2011-01-01 07:03 pm (UTC)
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[User Picture]From: nightspore
2011-01-02 12:37 am (UTC)
Great -- thanks for this. What I'm trying to think about is how many generations down my descendants I become as indifferent to my descendants as to non-kin.

The attractive sons argument explains sexual display as a handicap or costly signal, while also explaining why sexual display can change in isolated sub-populations. It's the biological version of a Keynesian Beauty Contest. There seems to be good evidence that selectors take receiver psychology into account and pick mates who they anticipate other selectors will pick, apparently because the next generation's selectors will pick similar mates.

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[User Picture]From: hbergeronx
2011-01-02 03:31 am (UTC)
I don't think I buy Fisher's attractive sons argument. By that, I'm not suggesting that having an "attractive" son as a sexual selection strategy is invalid, per se, but I think that there are numerous strategies, each more complicated than the last, and they're derivative of the original thesis of sexual selection. At some point, it's like, yeah, the bowling ball can drop to release the canary and boil the teakettle to etc etc but just because it's complicated doesn't mean there's a theory driving it. It could be that there was sufficient food or insufficient competition to allow for unconstrained variation, and that strategy got lucky.

Furthermore, genes don't necessarily have anything to do with it. Why does the host of the parasitic cuckoo take care of the invader? Why are non-paternity events as common as 10%? Mostly, in my opinion, because all animals (including humans) are pretty dumb, and mostly, doop-de-doo, just going about the act of living. Life happens to the dumb.

It also depends how much actual genetic variation actually exists. Fruit flies are the common example- reproduce by the millions, live for a few days, and every possible mutation will eventually be seen. Even through we might have a genome of 3 billion or so base pairs, the amount of actual variation between people is quite small. You might, for example, have trouble detecting that your brother swept in and cuckolded you. Even harder to detect, maybe, a twin. At some point, it doesn't really matter what comes out the other end, unless you're going to let that bother you, and unless you're facing some other survival pressure, like having to choose who to feed and who to drown. Or, living off that fat Maury check.

I think it's a case of failing to distinguish what is statistically true, with what is actually true. Statistically, over many generations, a characteristic which can survive, will. We can't predict what's a successful strategy for survival because, like the bowling ball, teapot, and canary, sometimes improbable mechanisms work. If you're a Chinese aristocrat, you might have your family names planned out for a thousand years, even though by that point, QED, they're almost certainly indistinguishable from the Fong dynasty across the river, especially if you marry a few daughters off to them to keep the peace. Or, like Tevye, you're ready to have a daughter be dead to you in a generation. They're all successful strategies.

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[User Picture]From: hbergeronx
2011-01-03 11:57 pm (UTC)

not sure

if this is citable or usable to you wrt what you wanted to know, but I'll let you read it while I mull it some more:

http://tedlab.mit.edu/~dr/Papers/Rohde-MRCA-two.pdf
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[User Picture]From: nightspore
2011-01-04 12:16 am (UTC)

Re: not sure

Thank you -- I'll try to read it this (busy) week, since the whole thing looks interesting.

I also remember that I meant to say that the question of dilution came up for Darwin and bothered him. He saw that his "gaseous" theory of inheritance meant that new traits should get diluted each generation qua traits. Mendel saved the day by showing inheritance wasn't gaseous. James Schwartz talks about this in his really good book In Pursuit of the Gene.
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