Rich, et al: This is an area of very active research. Do a Google search on Genghis Khan. Recent results claim that 5% of the world's population derives from him or his brothers. (The media claim that this is due to his large sexual appetite; while this probably helped, the truth is that random theory would show that _someone_ would end up with a large number of descendants. A line either dies out within a few generations, or ends up with a huge population; it's actually quite difficult to have only a few descendants for a large number of generations.) Also see David MacKay's (info/coding theorist @ Cambridge) chapter called "Why have Sex? Information Acquisition and Evolution" in his book on Information Theory. MacKay claims that asexual reproduction yields approx 1 bit per generation due to survival of the fittest, whereas sexual reproduction (with exchange of genes) can yield up to sqrt(G) bits, where G is the total number of bits in the genome. (Human genome is 4-6x10^9 ~ 2^32; and you always wondered what was magic about a 32-bit word...) Note that mitochondrial dna is essentially asexual reproduction, which explains why this dna is reasonably stable from generation to generation. The chapter is on the web below, and in fact, his whole book is available for downloading. http://www.inference.phy.cam.ac.uk/mackay/itprnn/ps/265.280.pdf http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=0521642981 Math-funner Tom Knight is also hacking genetics these days -- perhaps he has some comments. At 10:03 AM 6/29/2006, Schroeppel, Richard wrote:

I have a puzzle from c. 1970. Back then, it was science fiction, but now it looks less farfetched. Suppose we have a big computer with everyone's genome data. [Looking less farfetched every day, in fact.] What can we deduce about the genomes of our ancestors? Could we reconstruct an estimated genome of Fermat, Buddha, or Caesar? There's some information loss from generation to generation: Parents with N children will (on average) omit 2^-N of their genes from (the set of) their children. Some of that lost information could be deduced from knowing genomes of cousins (-> aunts & uncles -> grandparents), so it's hard to see exactly how much is erased. If we throw in present-day location information, we can probably figure our who moved where, when, and perhaps deduce population movements from long ago.

Rich

-----Original Message----- From: math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com on behalf of James Propp Sent: Thu 6/29/2006 10:29 AM To: math-fun@mailman.xmission.com Subject: [math-fun] Re: genetics-fun question

...

So figuring out who's mother and who's daughter is as easy as labelling their genomes a,b and c,d and noticing which of them has the property that, eg, ccccccccccccc is the same as aaaabbbaabbbb.

Neat! (And not even all that subtle; if I'd thought a bit harder about what I supposedly learned in high school, I could've figured it out.)

This fact about the asymmetry of the parent-child relationship suggests a more general question: How much can you deduce about the precise way in which two blood-relatives are related simply from looking at their genes?

This can be a real-world issue ("And is it your sworn testimony before this court, as an expert in the field of genetics, that this man cannot possibly be the nephew of Howard Hughes?"), but I could imagine it also giving rise to some mathematically amusing albeit unrealistic puzzles ("Assume an infinite genome..."). Michael?

Jim