# Optimizing dynastic succession genetically

The traditional rule for succession in a monarchy is to pass from father to son. Much of King Henry VIII’s spousal folly was over his anxiety for producing an heir. However, if the basis of being a successful ruler has a genetic component then this would be the least optimal way to run an empire. For diploid sexually reproducing organisms, such as humans, the offspring inherits equal numbers of chromosomes from both parents and classically the genetic relationship or kinship coefficient between parent and child is assigned the value of 1/2.  However, there is a crucially important asymmetry in that males are heterozygous in the sex chromosomes, i.e. they inherit an X chromosome from their mothers and a Y from their fathers, while females are homozygous, inheriting an X from both. Now the X is about 100 million base pairs longer than the Y, which accounts for about 2 percent of the (father’s) genome (counting chromosomes separately). Additionally, given that everyone has at least one X while only males have a Y, the Y cannot contain genes that are crucial for survival and in fact there are much fewer genes on the Y than the X (~800 vs ~50). The Y has been shrinking in mammals over time and there is a debate about its importance and eventual fate (e.g. see here).

We can compute the sex chromosome adjusted genetic correlation coefficients between parents and children.  Let the father’s genetic content be $F=F_S + F_D$, where $F_S$ is the genetic content passed to sons (half of the autosomes plus the Y chromosome) and $F_D$ is that passed to daughters (half of the autosomes plus the X) and similarly $M=M_S+M_D$. The son genetic content is then $S=F_S+M_S$ and daughter is $D=F_D+M_D$. We can treat F and M as a string of random variables with variance 1/(length of mother’s genome) and assuming that the genetic correlation between fathers and mothers is zero (i.e. no inbreeding and no assortative mating) then the correlation coefficient between father and son is

$\langle FS\rangle = \frac{ \langle F_S^2\rangle}{\sqrt{\langle F_S^2\rangle+\langle F_D^2\rangle}\sqrt{\langle F_S^2\rangle+\langle M_S^2\rangle}}=\frac{ 1}{\sqrt{1+\langle F_D^2\rangle/\langle F_S^2\rangle}\sqrt{1+\langle M_S^2\rangle/\langle F_S^2\rangle}}$

and similarly:

$\langle FD\rangle =\frac{ 1}{\sqrt{1+\langle F_S^2\rangle/\langle F_D^2\rangle}\sqrt{1+\langle M_D^2\rangle/\langle F_D^2\rangle}}$

$\langle MS\rangle =\frac{ 1}{\sqrt{1+\langle M_D^2\rangle/\langle M_S^2\rangle}\sqrt{1+\langle F_S^2\rangle/\langle M_S^2\rangle}}$

$\langle MD\rangle =\frac{ 1}{\sqrt{1+\langle M_S^2\rangle/\langle M_D^2\rangle}\sqrt{1+\langle F_D^2\rangle/\langle M_D^2\rangle}}$

Now, if you assume that genetic content is homogeneous among all chromosomes then that would mean that the genetic material that fathers pass on to sons is 0.48 of the total and thus $\langle F_S^2\rangle = 0.48$ while $\langle F_D^2\rangle = 0.5$, $\langle M_S^2\rangle = 0.5$, and $\langle M_D^2\rangle = 0.5$ implying that $\langle FS\rangle = 0.49$$\langle FD\rangle = 0.51$$\langle MS\rangle = 0.51$$\langle MD\rangle = 0.5$. Hence, parents are more correlated with their children of the opposite sex and fathers are least correlated with their sons. These numbers also probably underestimate the asymmetry. If genetic relationship is the most important factor for royal succession then a dynasty based on opposite sex succession will be more logical than the father to son model.

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## 4 thoughts on “Optimizing dynastic succession genetically”

1. Interesting thought excercize or experiment, and fairly dense. (I can’t even remember things like whether its the males or the females who are XY vs XX. This is is also interesting (or funny) ‘biblically’ since it seems one could make the case that you could make an XX from an XY as the bible claims—he made eve from adam’s rib, or the reverse—males being made in the lab by a genetic mutation of females. NOI says caucasians were made in the lab by an african scientist —they are genetic mutants, lacking melanin—some say they ar albinos actually).
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It is interesting to consider whether males are neccesary. Some species of lizards,apparently due to climate change, are becoming exclusively females. In the potomac watershed most male smallmouth bass now have female characteristics—some think this is due to pollution. (Everything from caffeine to drugs people ingest).

I remember papers by M Kirkpatrick (was in Utah ) on ‘runaway sexual selection’—modeling Darwin’s theory for why peackocks have all that plumage, and other (most) animals have all sorts of ‘secondary sexual characteristics’ which seem to have no ‘adaptive’ value. I was interested in that in those models, they typically used ‘haploid’ (asexual) models—they were just looking at a particular trait (gene)–you could make a convincing argument leaving out sexual details (diploid transmission).
Basically that theory as many know is sexual features are ‘advertizing’ (like driving a BMW, having a degree or cv) and this leads to all sorts of assymetric mating patterns. (I was studying the ‘lek paradox’—which is that in certain species of grouse, which form ‘leks’ (herds) in mating season, typically very few males get to mate with all of the females.) People were trying to decide if this was due to ‘good genes’—ie the ‘ads’ signify health of the males, which were chosen by the females, or just random (spontaneous symmetry breaking or pattern selection). I took the latter view (tho someone scooped me, and i had actually discussed this idea with this person months before i saw it was published in a conference proceedings, which may or not have been correlated with that history). A more complex version of this idea was later published by a physicst, in PNAS, though applied to the more general issue of ‘group selection’ (eg EO and DS Wilson versus Dawkins)—i was using the lek paradox as a ‘stepping stone’ to the larger problem of group dynamics (and there has been a flood of this in the past 10 or 20 years, alot of it on arxiv, and alot fairly redundant—all the models are either closely related or the same, so if you have one for biology you can then just do history next, then philosophy, psychology, etc.
I knew field biologists who despised and ridiculed math biologists—since they had to spend years in the field looking at fish and such, while the latter could just stay at home a write a paper in an hour summariing all that work as well as the work of hundreds of other field biologists.)
This issue of distinguishing random versus value based trajectories (eg why water poured on a floor flows in one path versus another) also occurs in economics–‘path dependence’ or QWERTY (b arthuir, etc,); apple versus mac. coke or pepsi.

The assymetries between MS, MD, FD, FS are fairly small. This does seem to be based on the fact that X, Y chromosomes are a small part of what is passed on (2%). Those little differences I guess make a difference. (Which is why i wonder if males are actually expendible. Sexual reproduction supposedly evolved to create variation—-i forget the theories—‘mueller’s ratchet’, ‘arms race’ (Hamilton, J Maynard Smith—resistance to parasites). The assymtetries might be larger i think if one used a different normalization (ie do as Kirkpatrick did, who used a haploid model to focus on what seemed essential) and did not consider the rest of the genome.
(A recent re-analyses of the ‘hot hand’ phenomena in sports—people seem to operate in ‘winning streaks’. Most said this was just a statistical fallacy (illusion of (non) randomness). The recent view is its a real affect, since the wrong normalization (‘baysian prior’) was used. (The same is said for some analyses of ‘competetive exclusion principle’ in ecology (see oikos blog ‘zombie ideas in ecology’ which says the math (based on lotka-volterra equations) used wrong normalizations, so now students are stuck with zombies (the living dead ideas ) around them. I wonder if .there exist other living dead ideas around in fields like religion, poltics, academia, economic growth, lumps of labor, etc. )

In history one does have queen of england, katherine the great, and more . I wonder if there are examples of dynasties which have alternating succession by sex (perhaps affirmative action? everyone gets a turn , even if they don’t deserve to win the lottery since that should be determined by merit and not chance, as is done with DC lotto). (Details of lotteries are discussed in the youube video called ‘the lottery’ (shirley jackson, part of the house of shirley which includes chisholm and temple).
In leks, some say actually less assymetry than appears—older males get the females, and younger ones eventually get their turn so lifetime income (milton friedman) is much more egalitarian).

(A case of randomness that confuses me is the ‘game of googol’ (Martin Gardner—pick 2 numbers randomly out of a bag and only look at one. Then you decide if its bigger than the other one, and if so keep it. Otherwise you take the other one you havent looked at. M Gardner says there’s no rational choice or winning algorithm–i posted this on azimuth blog where it was discussed. . Some stanford prof has a video claiming the opposite—which seems to rely on some sort of infinite limit as well as possibly a few more unstated or not well discussed assumptions involving ‘prior’ distributions. . )

There may be more factors than genes however. A recent paper looked at whether the height increases in northern europe indicated selection as opposed to environmental factors—they concluded it was environment (though some evidence was presented that taller people have. more children). In Wade’s (Nicholas, not Michael—who worked on group selection) book a ‘troublesome inheritance’ he relied on Gregory Clark’s book suggesting that wealthier people in England had higher IQs, reproduced more, and this explains why continents differ in average IQ. M Feldman gives it a somewhat critical review in PLOS1 Dec 2014 (or perhaps he pans it). Many cultural factors can override genetic transmission pathways.

Hereditary dynasties, even if genetically based, might just be one such pathway for genetic inheritance. It may be like species of dogs (or breeds) or plants—there could be all sorts of different dynasties. (Maybe ones based on ‘logan’s run, brave new world, 1984, the hunger games or ‘lord of the flies’). (Another question is why there are only sexes used in most cases of heredity—slime molds i think do it differently—a classic case of group selection).

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2. Exfernal says:

There is also mitochondrial DNA, which is usually inherited solely in matrilineal fashion.

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3. Jacob says:

Utter irrelevance to kingship. Thank goodness geneticists doesn’t run the show or we’d be cooing over drosophila melanogaster babies on YouTube.

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