weemba@brahms.BERKELEY.EDU (Matthew P. Wiener) (04/20/86)
In article <760@petsd.UUCP> cjh@petsd.UUCP (Chris Henrich) writes: > Now, why should this be so? Why is the space of >organisms ultrametric? I conjecture that the answer will be >generally illuminating for biology. I'd like some mathematical illumination first. I am familiar with the p-adics and ultrametrics in general etc. but I failed to see what your analogy was. ucbvax!brahms!weemba Matthew P Wiener/UCB Math Dept/Berkeley CA 94720
gsmith@brahms.BERKELEY.EDU (Gene Ward Smith) (04/20/86)
In article <13299@ucbvax.BERKELEY.EDU> weemba@brahms.UUCP (Matthew P. Wiener) writes: >In article <760@petsd.UUCP> cjh@petsd.UUCP (Chris Henrich) writes: >> Now, why should this be so? Why is the space of >>organisms ultrametric? I conjecture that the answer will be >>generally illuminating for biology. >I'd like some mathematical illumination first. I am familiar with >the p-adics and ultrametrics in general etc. but I failed to see >what your analogy was. I think it is worth mentioning that a tree each of whose nodes branchs n times has a natural ultrametric topology on it; this is one way of getting this for the p-adics. (The p-adic integers are an inverse limit and the topology and metric is easily derived from a tree branching p times at each node). Does evolution result in trees? Trees with regular branching properties? If the answer is yes, ultrametric norms are no surprise. ucbvax!brahms!gsmith Gene Ward Smith/UCB Math Dept/Berkeley CA 94720 ucbvax!weyl!gsmith "DUMB problem!! DUMB!!!" -- Robert L. Forward
felsenst@entropy.UUCP (04/24/86)
In article <760@petsd.UUCP> cjh@petsd.UUCP (Chris Henrich) writes: >[] > > I am particularly interested in Chapter 12, "A >biochemical echo of typology," because I think I detect a hint >of where such new ideas might be sought. This chapter >deals with a quantitative measure of the difference between >two species: comparisons of DNA sequences, or of important >proteins such as Cytochrome C. > ... > But a surprising and unaccountable regularity emerges, at >several different taxonomical levels. All the bacteria, for >instance, are about equidistant from all the eukaryotes. >It doesn't matter which bacterium you choose to measure, nor >does it matter which yeast or plant or animal you compare it >with. The difference is about the same. ... > Denton regards these clusters as a problem for >evolutionary biology. If we assume that the distinctions >between two flavors of Cytochrome C are unimportant to the >fitness of the organism, then according to him we would expect >the eukaryotes to have drifted different, random, amounts from >their prokaryotic origin. Instead, they have all managed to >move the same amount. If this is a coincidence, it is a big >one. And it keeps getting repeated at other levels. ... (material on p-adic norms and ultrametrics) > Now, why should this be so? Why is the space of >organisms ultrametric? I conjecture that the answer will be >generally illuminating for biology. and Gene Ward Smith of UCB Math Dept. Berkeley adds: > I think it is worth mentioning that a tree each of whose nodes >branchs n times has a natural ultrametric topology on it; this is >one way of getting this for the p-adics. ... > ... Does evolution result in >trees? Trees with regular branching properties? If the answer is >yes, ultrametric norms are no surprise. Gene Ward Smith is correct. Denton's surprise at finding that evolution results in DNA or protein similarity patterns that are ultrametric shows that he has not thought carefully about the matter (or read the relevant scientific literature) before declaring one of his crises in evolutionary biology (his is only the latest of many such non-crises). Consider a sequence of nucleotides (A, C, G, T) which is very long, say many millions of bases (we have 3 billion). Suppose Darwin was right and that there really is a true phylogeny (evolutionary tree) out there. Let's travel down it from humans to the common ancestor of (say) humans and birds, then back up the line to birds. We will enounter many changes in the DNA sequence, some of which will cancel each other out. If we are considering two bird species B1 and B2, we continue up the tree to their common ancestor B0. From there to B1 there are further changes, and from B0 to B2 others (some the same by chance). All that is necessary to find that B1's DNA is as different from ours as B2's is that there has been about the same number of changes on the B0-B1 segment of the tree as on the B0-B2 segment, and that changes on the two lines are equally likely to overlay or reverse changes between humans and B0. If many DNA changes are random "neutral" changes, we don't expect big differences in B0-B1 and B0-B2. We expect the two birds to be equidistant from humans in their similarity of DNA sequences. We expect something like an ultrametric. That is what is observed -- something like (but not an exact ultrametric). For some reason Denton is surprised by it. Charles Sibley's and Jon Ahlquist's article in Scientific American a month or two ago is an accessible source on this. Also there is a very fine article by Steven Jay Gould in Discover in early 1986 or late 1985 on molecular evidence that pandas are bears, and it discusses some of these points. Joe Felsenstein Dept. of Genetics, Univ. of Washington, Seattle, Washington 98195, USA uucp: ... uw-beaver!entropy!uw-evolution!joe ARPANET: entropy!uw-evolution!joe@uw-beaver.UUCP Bitnet: something like that too.