Monday, November 23, 2009

Pedigree Collapse and the impossibly spreading family tree

This article allegess that all US presidents (except Martin Van Buren) share common ancestry with John “Lackland” Plantagenet, King of England at the beginning of the 13th century.

Do all U.S. Presidents share a common ancestor?

Of course, this is all based on the assumption that when we go back far enough to have millions of ancestors, there is bound to be a common ancestor.

Only rarely do we see these alleged “relationships” properly proven. For example, genealogists have proven that Sara Belcher was a common ancestor of FDR, Winston Churchill and Doug Macarthur.

As a genealogist, I find this nonsense about a recent common ancestor to be idiotic, but it brings up a good point about the assertion that large populations of people may share ancestry with famous people who procreated frequently.

The numerical puzzle of ancestry

Everyone has four grandparents, eight great grandparents, and so on, ad infinitum.

It’s the same 2**n (powers of two) equation used with binary computing, and it has some important ramifications because it does not take long to generate more ancestors than the world population.

Even 20 generations ago, everyone has over a million distinct ancestors! (2**20= 1,048,576), and at four generations per century, 20 generations only takes us back a mere 500 years.

For example, if we go back forty generations ago (about 1,000 years), we see we have over one trillion distinct ancestors (2**40 = 1,099,511,627,776), a number that far exceeds the number of humans who ever lived.

From this mathematical certainty, can we conclude that sharing of the gene pool (marrying cousins) was necessary to explain the over-lap in family trees?

For populations living in isolation (i.e. Madagascar circa 800 A.D., Colonial America), this is even more pronounced, and everyone, by mathematical certainty, was sharing common ancestors. Before the great immigrations (China, Ireland) during the 1800's, almost all multi-generation Americans were cousins by virtue of sharing a common great-great-great grandparent.

Pedigree Collapse: marry your cousin

So, how to we resolve the numerical certainty that 40 generations ago, we must have descended from a trillion ancestors? Some use the concept pedigree collapse to explain this problem.

The term “pedigree collapse” is a polite word for marrying your cousin, and they show that isolated island populations have a high rate of pedigree collapse because of the lack of un-related partners.

This chart
shows that 10th cousins share only 0.0000238419% of having a common ancestor ten generations ago (about 200 years):

In this case, marrying a 10th cousin would eliminate a large chunk of common ancestors, starting only 10 generation before when the common ancestor appears.

This article has a great discussion of this issue, and this paper from Yale claims to have developed a mathematical model to explain why we must, by certainty, share ancestry with everyone else:

“all individuals who have any descendants among the present-day individuals are actually ancestors of all present-day individuals.”

Conclusions on pedigree dilution

In sum, within the same general population, the chances are almost 100% that your spouse is a distant cousin, and you share a common ancestor from 20 generations ago. Once you go back 20 generations (a million distinct ancestors), you are almost guaranteed to see a common ancestor, and all ancestors of the common ancestor reduce the overall number of unique ancestors.

In animal husbandry, breeding of cousins is an acceptable practice, and it can greatly help increase desirable traits, even though it dramatically narrows the total number f distinct ancestors.

For example, consider the pedigree of one of our champion stallions, Tha Dude.

Tha Dude – Champion Arabian Black stallion

Dude does not have 16 distinct great grandparents, since the famous Khemosabi appears as both a paternal and maternal great grandparent.

The great Khemosabi

Also notice that the equally famous stallion *Bask++ appears several times in his ancestry, appearing as both his grandfather and as Dude’s great grandfather.

In this case, the line breeding was a smashing success, since Dude is almost a carbon copy of Khemosabi, inheriting both his stamina and stunning looks. These are the lines that are commanding millions of dollars, as with this *Bask++ line horse that sold for $2.8m . . .