turpin@cs.utexas.edu (Russell Turpin) (01/25/91)
----- Cross-posted to sci.bio. ----- In article <1991Jan24.220906.16423@usenet.ins.cwru.edu> rpetsche@mrg.CWRU.EDU (rolfe g petschek) writes: > Hmmm. As a statistical physicist I can not help but notice that the > period of time required required for a single lineage to overtake all > other lineages is a random variable with calculable distribution, > dependent on the model. ... Absolutely. You were not the first to notice this. I was not the first to notice this. Ah, well, sometimes we get beat to the punch. > ... Well, I tried lots of models (admittedly crudely) and I did > not get sense from this statement except when (a) teh size of the > breeding population is *very* small (need N^2 generations assuming > neither advantage or growth of size N) or (b) their is significant > advantage passed *only* through the female line ... For biological reasons, forget (b). But for the most obvious model reflecting biological constraints, the number of expected generations is (surprise!) *linear* in N. This discussion came up some months ago in another newsgroup. Not being a statistical physicists, I wrote a model in C rather than attempting an analytic solution. When I posted the results, someone else informed me that the results were well-known, and are known as the Wright-Fisher model. (He did not send me a reference.) Shown below are the number of females in a fixed-size population, the range in the number of generations my simulation required for one female line to push out all others, and the expected number that is analytically calculated. #generations (N) Wright-Fisher F least most average ---- ----- ---- ------- 8 5 37 16 128 172 625 256 1024 2392 4163 2048 (For F=1024, I only ran my model twice, because it took so long on my MacPlus. The other values of F were run several times.) The assumptions in the model are (1) a fixed-size female population, and (2) every woman has a fixed probability of not having daughters. (The number of men and number of sons produced just plain doesn't matter.) In my simulation, I assumed that each "excess" daughter had equal chance of coming from any of the women who did have daughters in the previous generation. A slightly different assumption is made in the Wright-Fisher model. > ... Did you have some model in mind which would answer these > confusions? 10,000 generations is not long (its square root is > 100, and recently, at least, there have been more than 100 > persons in the breeding population.) 2000 generations of early man is only 30,000 years, and that is how long the Wright-Fischer model predicts it took for a single female line to emerge in a population of 1024 women. Playing around with my simulation, I was left with the intuition (which is what a good model provides) that allowing the population to increase and decrease, so that it was only sporadically as low as 1024 women, would actually *increase* the chance of a single female line emerging, especially if one takes into account that small groups of related people, are likely to survive or die together. Russell
joe@evolution.u.washington.edu (Joe Felsenstein) (01/25/91)
The Wright-Fisher model (named after Sewall Wright and Ronald Fisher, the two most important population geneticists ever), is a model in which each gene in the offspring generation is copied from a random gene among the 2N available in the parent generation, and this occurs randomly and with replacement. This is motivated by the idea that each parent produces a very large and equal number of gametes, and the N surviving offspring are the result of random collisions of these gametes, followed by density-dependence trimming them randomly down to N survivors. The haploid version, in which N things are randomly drawn from N, is the one relevant to the Eve question, with N the number of females. The mathematics of the two is identical with 2N replaced by N. You will find it discussed in almost any population genetics text, but a reasonable starting point would be Dan Hartl and Andrew Clark's "Principles of Population Genetics," second edition, published by Sinauer Associates, 1988. The W-F model dates from 1930-1931. If you want something more mathematical that could be made to bear on the Eve issue (or non-issue) try my 1971 paper, "The rate of loss of multiple alleles in finite haploid populations" in Theoretical Population Biology, volume 2, pages 391-403. To (very) good approximation, if we have n females who we are following in a W-F model where there are a total of N females in each generation, the time in generations back until they have n-1 female parents is approximately geometrically distributed with mean N/(n(n-1)). Applying this repeatedly starting with N=n and with n getting smaller and smaller until it reaches 1 you get the required distribution, and can easily calculate its mean and variance. This is only an approximation but is a very close one for large N and darn good even for moderate N. (Asides: I have not posted this to soc.men or soc.women because it seemed more appropriate here. Also: Fisher was English, not German, and thus had no "c" in his last name.) ----- Joe Felsenstein, Dept. of Genetics, Univ. of Washington, Seattle, WA 98195 Internet: joe@genetics.washington.edu (IP No. 128.95.12.41) Bitnet/EARN: felsenst@uwavm UUCP: ... uw-beaver!evolution.genetics!joe