The X-Chromosome Waltz*

Recent news about a possible ’gene for homosexuality’ on the X chromosome made me think a bit about the special factors influencing selection on the X and Y chromosomes. I may post more about that in future, but, as usual, I got sidetracked by a subsidiary issue.

There are more X than Y chromosomes in the population, and this has been claimed as relevant to their evolution. For example, Matt Ridley’s Genome says: ‘because females have two X chromosomes while males have an X and a Y, three-quarters of all sex chromosomes are Xs; one-quarter are Ys. Or, to put it another way, an X chromosome spends two-thirds of its time in females, and only one-third in males’. Ridley goes on to argue that this affects the competition between X and Y chromosomes. A weightier authority, W. D. Hamilton, also finds it important that an X chromosome spends only one-third of its time in males (Narrow Roads of Gene Land, vol. 1, p.146).

In some sense it is obviously true that on average an X chromosome spends two-thirds of its time in females (assuming equal numbers of males and females, and ignoring rarities like XYY males), but it wasn’t clear to me how the dynamics of chromosome transfer would work. For example, if in a given generation a particular allele on an X chromosome is by chance only found in females, will the frequency in subsequent generations tend towards two-thirds, and if so how?

Anyway, I couldn’t find anything on this point on a cursory look through my books, so I tried working it out myself….

First, suppose there is a newly arisen mutation on an X chromosome. If it arises first in a male, we can be certain that in the next generation, if it survives at all, it will be in a female, since any sperm containing an X chromosome causes the egg it fertilises (if any) to develop as a female.

If on the other hand it arises in a female (or has just entered one), it has a ½ chance of going into a male in the next generation, and a ½ chance of staying in a female. If it stays in a female, there is a ½ chance that it will go into a male in the third generation, and so on. Cumulatively there is a ½ chance that it will spend just one generation in a female, a ¼ chance that it will have just two consecutive generations in a female, a 1/8 chance that it will have three, and so on. The expected average number of unbroken consecutive generations in a female body is therefore 1/2 + 2/4 + 3/8 …. The value of this sum tends towards 2 as the number of generations considered increases [see Note]. It is therefore correct to say that on average a gene on an X chromosome will spend two generations in a female for every one in a male. This is what we would expect given the proportions in the population, but it is nice to be able to confirm it by a more direct line of argument. It also brings out the fact that the 2:1 ratio is only an average, and that from time to time an X gene may spend many consecutive generations in females, whereas it can never stay in a male for more than one. Whether this makes any difference in practice I don’t know, but in theory it might, e.g. because an X gene in a male has no kin-selective interest in that male’s sons.

So far this deals only with a single gene. But what about the frequency of all genes of a given type on X chromosomes among males and females in the population?

First, suppose that there are two gene variants (alleles) at a particular locus on the X chromosome. Call the original form of the gene A, and a recent variant B. Suppose that B is still very rare in the population, so that matings between B-males and B-females can be neglected. The genotypes in the population will therefore be AA , AB, AY and BY (where AY and BY are the genotypes of males with A and B alleles respectively on their X chromosome). For simplicity I make the usual assumption of random mating and separate generations. I use the term ‘B-bearer’ for any individual with a B allele.

Suppose now that in Generation 1 all the B-bearers are male. (Maybe the females all died.) The males have the genotype BY and mate with AA females, producing offspring of genotypes AY (male) and AB (female) in equal proportions. Therefore in Generation 2 the B-bearers are all AB females. These mate with AY males, and produce offspring of genotypes AA, AB, AY and BY in equal proportions. In Generation 3 there are therefore equal numbers of male and female B-bearers. These mate respectively with AA females and AY males (still neglecting the rare possibility of matings between B-bearers). The AA-BY matings produce ½ AB females and ½ AY males. The AB-AY matings produce offspring of genotypes AA, AB, AY and BY in equal proportions. In Generation 4 there are thus 3 AB females to every 1 BY male. Pursuing the same method of analysis, in Generation 5 there are 5 AB females to every 3 BY males, and so on. I have taken this to the 7th generation, with the following results:

Ratio…………………AB females : BY males
Gen 1…………………………..0 : 1
Gen 2…………………………..1 : 0
Gen 3…………………………..1 : 1
Gen 4…………………………..3 : 1
Gen 5…………………………..5 : 3
Gen 6…………………………11 : 5
Gen 7…………………………21 : 11

It is evident that the ratio of females to males among B-bearers is oscillating around the expected 2:1 ratio, but converging on it quite rapidly. The result is the same whether we start with a surplus of males or females. I do not have a formal proof of the convergence, but the general explanation is fairly clear. Any surplus of males (relative to the 2:1 expected ratio) in a given generation is immediately converted into a surplus of females in the next generation. The surplus of females is then converted into a surplus of males, but as AB females produce equal numbers of AB female and BY male offspring, only half of the female surplus is transferred to males at each stage. This acts as a ’damping’ influence on the oscillation. If the surplus of one sex in the first generation is Q (where Q is a proportion of the whole population) then the surplus in succeeding generations will be -Q/2, Q/4, -Q/8, Q/16, -Q/32, and so on. At the limit, when the ratio reaches 2:1, the proportions are in equilibrium, because the number of BY males created by AB females equals the number of AB females created by BY males. If the absolute numbers are small, there will of course be some random fluctuation around this ratio.

If the B allele is under positive selection, its frequency in the population will rise, and it will no longer be possible to neglect matings between B-bearing males and females. There will be 5 different genotypes in the population (AA, AB, BB, AY, and BY) and 6 different mating combinations (AA-AY, AA-BY, AB-AY, AB-BY, BB-AY, and BB-BY). This makes analysis more laborious, but I have checked that the system is in equilibrium when the distribution of B genes between females and males is in the ratio 2:1, and the genotypes of the females are in the Hardy-Weinberg proportions p^2:2pq:q^2, where p is the frequency of A, and q = 1-p is the frequency of B. Of course, since BB females have two copies of B, a ratio of 2:1 among genes no longer implies a ratio of 2:1 among individuals. At the extreme, when B has gone to fixation under selection, there are equal numbers of BB females and BY males. Matings between BB females and BY males preserve the equilibrium ratio of 2:1 among genes.

I dare say that to someone like G. H. Hardy this would all be obvious at a glance, but I’m not G. H. Hardy, and I found it quite intriguing to work out!

*a Waltz goes one-two-three, one-two-three….

We wish to prove that the limit of the sum 1/2 + 2/4 + 3/8 + 4/16 + 5/32.…. is 2. I found the following way of proving this. First, arrange the terms of the series in a column as on the left-hand side of
the following series of equations:

1/2 = 1/2
2/4 = 1/4 + 1/4
3/8 = 1/8 + 1/8 + 1/8
4/16 = 1/16 + 1/16 + 1/16 + 1/16
5/32 = 1/32 + 1/32 + 1/32 + 1/32 + 1/32
6/64 = 1/64 + 1/64 + 1/64 + 1/64 + 1/64 + 1/64
[Etc.] …………………………………………………………..
Limits: 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 ………[limit of sum = 2]

The nth term of the series on the left-hand side has the form n/2^n, while the equations on the right-hand side expand each term in the manner shown. Now look down each column on the right-hand side and note that it is itself a regular series, with its sum tending to the limit 1/2^(c -1), where c is the column number (numbered from left to right, on the right-hand side of the equations). [Added: the columns come out ragged in this text format, so this is not as clear visually as I hoped.] The limits are shown on the bottom line. Moreover, these limits themselves form a series, the sum of which has the limit 2. Since all the terms on the right-hand side are positive, their sum must increase with increasing n, but it can never exceed the overall limit to the sum of the columns, i.e. 2. The sum 1/2 + 2/4 + 3/8 + 4/16 + 5/32.… therefore has a limit not greater than 2. To prove that the limit actually is 2, note that for any chosen finite value of n the sum of the terms in each column down to that value of n falls short of its limit by 1/2^n. There are n of these columns, so their total ’remainder’ is n/2^n. We must also allow for the columns that would appear further to the right if the series were extended beyond this choice of n. The total value of the terms of these ’invisible’ columns cannot exceed the limit of the sum of the last ‘visible’ column. Since the column number of the last ‘visible’ column is n, this limit is 1/2^(n -1). For any given finite n the sum of all the terms on the right-hand side, down to this value of n, therefore falls short of the upper limit 2 by not more than n/2^n + 1/2^(n -1). But it is evident that as n increases this maximum ’remainder’ tends to zero, therefore the series 1/2 + 2/4 + 3/8 + 4/16 + 5/32… tends to the limit 2. Q.E.D.

Posted by David B at 04:21 AM

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Halloween is no escape from politics

So in this superheated political season it should come as no surprise that political bloodlines are mingling with Halloween themes. Huh, you say.

In a roundabout way, let me explain. Surely you recall Razib’s post that made the point that 8% of Asian men are descendents of Genghis Khan. Perhaps you also saw the Vice-Presidential debate, and by some stroke of luck you may also have read Daniel Ruth’s column in the Tampa Tribune, where he wrote:

It was Edwards’ mission to suggest to the electorate that Cheney is Vlad the Impaler, only without the sense of humor.

This is our first clue that something eerie was afoot. Either Ruth was prescient by catching a foggy vision a month before the rest of us, or Senator Edwards was missing his targets by one pay grade, for it turns out that the intrepid folks over at have determined that both President Bush and Senator Kerry are related to Vlad the Impaler. Here are the family trees.

For those who doubt that both physical and behavioral traits are encoded in DNA they need look no further than to the two candidates. It seems clear that some heretofore undiscovered variant of Creutzfeldt-Jakob Disease must have passed down the Bush family tree, quite evidently from all of the imbibing of human blood that Vlad partook of in his day. And look at the cadevorous pallor, demeanor and humor that infects Kerry, surely the result of Vlad being one of the living dead.

Happy Halloween.

Posted by TangoMan at 11:21 AM

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Ethnic tensions in China?

Ethnic Clashes Erupt in China, Leaving 150 Dead. The story details the strife and rioting that occurred after a Hui taxi driver struck a Han girl and killed her. The Han are who we generally think of when we speak of “Chinese,” and form between 90-95% of the China’s population (some of the same tendencies that have occurred in the United States with Native American ancestry claims being boosted by cultural change and preferential policies have probably inflated the number of minorities in China). The Hui are “Chinese Muslims”, in that unlike Uighers or Kazakhs, who are Turkic speaking Muslim citizens of China, the Hui speak the local Chinese dialect and are physically indistinguishable from the Han.1 In times past, there have been tensions and violence between Han and Hui, and during the late 19th century the Hui set up warlordships in Yunnan, but the Hui have also been used by the Chinese gtovernment as proxies in Turkic Muslim areas. This illustrates the peculiarity of Hui identity, among other Muslims their similarities to the Han are stark, while around the Han their Muslim religion (and pork avoidance) is most salient.

Why does this matter? China matters. No elaboration needed.

1 – The Hui are probably the descendents of Muslims who migrated to China and intermarried with local women. As such, some do display a phenotype that shows an influence from their Central and West Asian ancestors. Many Hui in fact claim that these atypical individuals are the archetypical exemplars of their group in terms of appearance. Additionally, there is evidence that a nontrivial number of Hui have been assimilated into the Han population, therefore the maxim “…the ancestors have eat no pork.”

Posted by razib at 02:21 PM

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No more contemporary poltical posts until November 15th

Posts that deal too closely with the kind of stuff you can find at Daily Kos, Polipundit, Andrew Sullivan or Kausfiles will be deleted. Long commentaries on foreign policy are fine, ruminations on the relationships between political ideologies are fine, etc. etc. etc.

Also, many of the blogs run by posters are GNXP have political content, so I suggest you check those out.

By the way Aziz, why does Houston smell like a pig’s anus so often?

Posted by razib at 01:45 AM

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A topic for debate…

For those who have been monitoring all of the various comments sections at this site, I’m sure you have noticed that there has been raging a rather strong debate here and there about what journals are credible sources for material and which are not. That brings me to a great question that I simply cannot allow to not be asked any longer:

What are the prerequisites for determining whether or not a journal should be used as source material?

There are a whole slew of journals out there; tens of thousands in the United States, at the very least. Every little college has a journal it seems, some created for the simple reason that the work of their academics is so pathetic that the larger journals don’t feel it’s good enough for publication. Many colleges have student edited journals such as the one published by my own university, the University of Florida International Review (which has published some of my work). As for this blog, a few journals have been discussed as of late: The Occidental Quarterly, Mankind Quarterly, and The Journal of Social, Political, and Economic Studies.

The Occidental Quarterly is a favorite of Thrasymachus and is published by the Charles Martel Society, a grouping of paleoconservative, white nationalist, and European “New Right” intellectuals. While I consider this to be a fascinating journal (mostly because I love studying political ideologies and it’s one of the best hard-right journals out there) and sometimes even agree with them, I have been fairly critical of the content of the journal, such as:

1. The publication of and favorable reviews of the beliefs of Richard McCulloch, a racial separatist and “Nordishist.”
2. Their favorable opinions of Francis Parker Yockey, a somewhat neo-Nazi intellectual and anti-Semite, and author of Imperium (which he dedicated to Adolf Hitler). For more information on this individual, be sure to check out Kevin Coogan’s book, .
3. Their regular (too regular to post all the links to, except a listing on his web page) publication of Jew-skepticist material by Kevin MacDonald.
4. Their publication of M. Raphael Johnson, editor of The Barnes Review (a pro-Nazi publication) and regular contributor to The Idyllic. For those who don’t know, The Idyllic is published by the Lawrence Dennis Institute, which is named after the American fascist intellectual.
5. Their sympathetic attitude towards William Pierce, former member of Lincoln Rockwell’s American Nazi Party, editor of their National Socialist World magazine, and founder of the National Alliance.

I could continue on and on, but I think I’ve made my point well enough. This is definitely a publication that advocates some fairly far-right viewpoints, even though they do occasionally publish some good material.

This brings me to Mankind Quarterly and The Journal of Social, Political, and Economic Studies. Both of these publications were founded by Roger Pearson, a former extreme-rightist turned pro-American nationalist after getting a Ph.D. in anthropology, who also founded The Journal of Indo-European Studies, a rather prestigious anthropology, archaeology, and linguistics journal. All three of these journals are peer-reviewed and are published by either the Institute for the Study of Man or the Council for Social and Economic Studies.

Mankind Quarterly became rather famous after the publication of , in which both the authors cited essays published in it. This was subsequently used to attack the authors, because the journal publishes essays by eugenicists and people who actually believe in race or, for that matter, racial differences (the horror!!!!!). The special issue (for those of you who have access to EBSCOhost, the October 31, 1994 issue) of The New Republic that was dedicated to launching vitriolic and silly assaults on the The Bell Curve used this as an attack against the book, especially in the hysterical article by Jeffrey Rosen and Charles Lane titled “Neo-Nazis!”

Mankind Quarterly has been compared by numerous readers and bloggers to Stephen Jay Gould, except being right-wing instead of left-wing. Over the past few weeks I’ve been reading essays published in the journal on EBSCOhost and I must say that it is absolutely nothing like The Occidental Quarterly. It is intelligent (being peer-reviewed helps) and from the past 10 years I can detect little or no anti-Americanism or anti-Semitism (which does exist in earlier issues). Sure, maybe some of the studies in it have been tilted in a Gouldian fashion, but there really aren’t all that many journals out there that delve into controversial topics like this one does. Also, the journal regularly publishes essays by Richard Lynn and stuff on H-BD! It’s a lot like American Renaissance, actually. And just like American Renaissance, there are a few extreme elements here and there, but nothing like The Occidental Quarterly.

I have been particularly fascinated by The Journal of Social, Political, and Economic Studies, mostly because I’m a political scientist and this is more up my alley. When I first heard about the journal and Roger Pearson, I thought when I looked it up that I would find a journal similar to The Occidental Quarterly with strong underlying anti-American, anti-Semitic, and far-right tendencies. This has not been the case. In fact, I agree with much of what they have to say. If anything, the journal is strongly pro-American and pro-science. Some of the things that I’ve found and agreed with in recent issues of the journal are:

1. An essay on suicide bombings in the former-USSR (Fall 2004) and many essays on comparative politics type material.
2. Favorable and insightful reviews of books by evolutionary psychologists in almost every issue.
3. Interesting essays by Dwight D. Murphey and .
4. Numerous studies on the history and development of a ballistic missile defense system (Spring 2004 and Fall 2004).
5. Essays on how to begin large scale space colonization and exploration (Spring 2004).

…and a heck of a lot more. Pearson’s transition from far-right to more towards the middle is unusual, since normally people move to the far-right and stay there becoming gradually more anti-American and anti-Semitic as time passes
. Since the phenomenon is so unusual, I had to understand it. I tried to find the e-mail of Pearson a few weeks ago, but failed. It’s sort of like the conversion of neocons from far-left Trotskyites. Oh well… I guess I’ll figure out what happened eventually.

But back to the topic at hand, these journals, while biased in many instances, have arisen as a result of the fields of anthropology and genetics being taken over by Marxists and Boasians (just look at the statement by the American Anthropological Association on race). When I go into my library to find a journal, I see dozens of journals like Dialectical Anthropology and “mainstream” journals that publish the same claptrap. Why can they have journals dedicated to their biases and we can’t? Why shouldn’t we cite essays out of Mankind Quarterly? How and what source material should we cite when the dominant publications are controlled by people whose politics matter more than the science?

Posted by Arcane at 04:52 PM

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Opposites attract

St. John, Warren and Rachel L. Swarns. “Politics Makes Strange Bedfellows”. The New York Times. 2004 October 31.

In towns big and small across the country, couples and family members on opposite sides of the political fence are struggling to maintain amicable relationships as a highly polarized political season reaches its apex. With the presidential race so close and emotions so raw, their homes are microcosms of the sharply divided electorate, places where a kitchen-table conversation can quickly devolve into the bitter back and forth of an episode of “Crossfire” or worse.
For Laurice Pearson, a Democrat who works at a Manhattan legal services company, and her husband, Mihai Radu, an architect who defected from communist Romania in the early 1980’s and came to view Ronald Reagan as a kind of liberator — and by extension the Republican Party, too — political arguments were initially a courtship ritual.

But as their disagreements became more intense, she said, they agreed not to talk politics over breakfast, for fear they would commence an argument they couldn’t resolve before heading to work. Ms. Pearson said she also encouraged her husband to argue himself out with others, so she wouldn’t have to engage.
Gene and Adam Ortiz, the Republican father and Democratic son, said they were groping for ways to fight the political fight while keeping the peace at home.

They’re called blogs, people.

Posted by jeet at 09:20 AM

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Osama endorses Kerry

We knew he would, a Kerry victory is in the best interest of Al Qaeda.

Update Ok everyone, let me first state that this was a joke. I have seen several leftie-bloggers seriously say that the terrorists prefer Bush and endorse him, so this is my way of thumbing my nose at the left. I do not think that what OBL says should have any effect on this election, and I would be very disappointed if anyone changed their vote because of this.

Posted by scottm at 05:17 PM

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