# new and improved coefficients of relationship

(here we go. me and math again. ruh roh.)

so, just about two months ago i wrote a long and rambling post about calculating the relatedness between various family members. with the generous help of the reluctant apostate, i think i may have finally gotten the math in order. maybe.

before we set off on our magical mathematical journey, let me explain my motivation.

in case you missed it, a couple of researchers took a look at the differential x-chromosome inheritance rates between boys and girls — and then they looked at how this affected how maternal and paternal grandmothers treated their male or female grandkids. they found that the more total dna (autosomal dna + x-chromosomal dna) a grandmother shared with a particular grandkid, the better she treated them. (see my post all grandmas are not created equal. edit: correction – they actually looked at total number of genes shared, not total dna which is what i do here.)

which got me to thinking that this must be true for other familial relationships, too: father -> son or daughter, mother -> son or daughter, sister -> brother, brother -> sister … and, my personal favorite, different cousins -> different cousins.

the coefficients of relationship that are usually used look like this:

0.5 (½) – parent-offspring – exact
0.25 (¼) – grandparent-grandchild – average
0.125 (⅛) – great grandparent-great grandchild – average
1 – identical twins; clones – exact
0.5 (½) – full siblings – average
0.25 (¼) – half siblings – average
0.125 (⅛) – first cousins – average
0.03125 (1/32) – second cousins – average
0.75 (¾) – full hymenopteran sisters (i.e. ants) – average

these are based on the fact that (ignoring the ants) we inherit half of our dna from our fathers and half from our mothers.

only we don’t.

we inherit half of our chromosomes from each parent, but chromosomes come in different sizes, and — for instance — the y-chromosome is a lot smaller than the x-chromosome.

so, i looked up the sizes of the chromosomes on the vega genome brower and worked out the sizes of a man and a woman’s genomes:

Female genome
6068 Mbp
Autosomal DNA: 5758 Mbp (≈94.89%)
X: 155 Mbp (≈2.55%), XX: 310Mb (≈5.11%)

Male genome
5972 Mbp
Autosomal DNA: 5758 Mbp (≈96.42%)
X: 155 Mbp (≈2.60%), Y: 59 Mbp (≈0.99%), XY: 214 Mbp (≈3.58%)

moving on from there, we can work out some new and improved coefficients of relationship based on the facts that:

– a son inherits half of his father’s autosomal dna + his full y-chromosome (virtually) unrecombined
– a daughter inherits half of her father’s autosomal dna + his full x-chromsome (virtually) unrecombined
– both sons and daughters inherit half their mother’s autosomal dna + one x-chromosome (recombined)
– etc., etc. (for more details on the other familial genetic relationships, see the previous post, esp. all the comments — if you can stand it.)

so, here we go. i don’t have all the calculations made yet, but here’s a start (figures rounded to four decimals; key at the bottom):

0.4920 F — s 0.4920
0.5081 F — d 0.5000
0.5000 M — s 0.4951 0.5081
0.5000 M — d 0.5000

and now for other family members (these numbers are probabilities, not exact percentages as between parents and children):

0.5050 B — B 0.5050
0.4951 B — Z 0.4872
0.5127 Z — Z 0.5127

(oops. fogot to calculate the grandfathers. that’ll have to wait ’til tomorrow. or sunday.)

0.2509 PGF — s 0.2509
0.2410 PGF — d 0.2372
0.2540 MGF — s 0.2540
0.2540 MGF — d 0.2500
0.2372 PGM — s 0.2411
0.2628 PGM — d 0.2628
0.2500 MGM — s 0.2476 0.2540
0.2500 MGM — d 0.2500

0.2510 FB — s 0.2510
0.2541 FB — d 0.2500
0.2372 FZ — s 0.2411
0.2500 FZ — d 0.2500
0.2541 0.2476 MB — s 0.2541 0.2476
0.2476 MB — d 0.2436
0.2756 0.2564 MZ — s 0.2606
0.2564 MZ — d 0.2564

FBD – s 0.1205
FBS – s 0.1304
FZD – s 0.1205
FZS – s 0.1205
MBD – s 0.1270
MBS – s 0.1205
MZD – s 0.1303
MZS – s 0.1303

FBD – d 0.1314
FBS – d 0.1186
FZD – d 0.1250
FZS – d 0.1250
MBD – d 0.1250
MBS – d 0.1186
MZD – d 0.1282
MZS – d 0.1282

(another oops! don’t have calculations yet for the cousins→sons OR for cousins→daughters and aunts, uncles or cousins, either. jeez, i’m such a slacker!)

well, there you have (some of) it.

i’m not going to make a lot of comments about this tonight (esp. since i don’t have all the calculations finished), but here are some interesting notes: 1) out of all his cousins, a son is most closely related to his FBS — that’s ’cause they share a full y-chromsome; 2) out of all his female cousins, a son is most closely related to his MZD — that’s kinda interesting ’cause, i think, the most common form of cousin marriage is MBD — and then we have FBD marriage in the formerly-part-of-the-caliphate world and, yet, that’s one of the cousins a guy is least related to — but he’s really related to her brother.

ok. that’s it for now. more on this anon.

– key –
F = father
M = mother
s = son
d = daughter
B = brother
Z = sister
PGM = paternal grandmother
MGM = maternal grandmother
FB = paternal uncle
FZ = paternal aunt
MB = maternal uncle
MZ = maternal aunt
FBD = father’s brother’s daughter
FBS = father’s brother’s son
FZD = father’s sister’s daughter
FZS = father’s sister’s son
MBD = mother’s brother’s daughter
MBS = mother’s brother’s son
MZD = mother’s sister’s daughter
MZS = mother’s sister’s son

edit: oh — if anyone (is a glutton for punishment and) wants to check my math, feel free. please! (pretty please?)

update 05/29: corrections have been made, numbers have been added (grandfathers, daughters, but not yet cousins→sons OR for cousins→daughters — those will have to wait for another day.) many, many thanks to the AWEsome, albeit reluctant, apostate for his help. he’s really cool! (^_^)

(note: comments do not require an email. calculator, yes. email, no.)

1. I think it would be theoretically possible to do a calculation like this and find what level of homogeneity and economic surplus was required for the majority of the population to support a health service funded from taxation.

The more “you” everyone else is the lower the threshold needed before it is in people’s interests to chip in to their health costs.

2. I uploaded a spreadsheet to Google Docs here. I set the sharing permissions so that anyone who has the link can edit it, so feel free to play around with it a bit. It’s essentially the spreadsheet I used last time with updated Mbp numbers and a few more relations (though I didn’t upload it last time).

I also put in a key based off the one in this post for the sake of comparability, though it may be a few differences in practice.

Just a note on the key I used. The leftmost column is the sex of the subject denoted by the ♂ and ♀ symbols. Obviously, since the last post on this matter was focused on cousins, those are the first group “X” = cross and “ll” = parallel. In terms of relations ↑ is a parental relationship ↓ is for offspring and → is for siblings. If the first part of the relationship is parental, I omitted the ↑.

As for the columns, column A “sex” is the sex of the subject, column B “A” is the autosomal DNA shared with the relative, column C “X” is the number of X chromosomes shared, column D “Y” is the number of Y chromosomes shared, column E “pct” is the overall proportion of DNA shared, with the final two columns as the keys for which relationship is which.

I haven’t done a thorough comparison between my results and the results in the post.

3. @r.a. – you’re awesome!! thnx! (^_^)

“I haven’t done a thorough comparison between my results and the results in the post.”

you might wanna skip that for now ’cause i already found some errors a little while ago (had to stop for a food break). but, if it amuses you….

4. @g.w. – “The more ‘you’ everyone else is the lower the threshold needed before it is in people’s interests to chip in to their health costs.”

exactly! (i think.)

5. i’ve made some changes to the post above (see the update in the post).

thnx, r.a., for your help! gotta run now, but i’ll be working on this some more in a day or two. thnx for that spreadsheet! (^_^)