# Thread: Fibonnacci series appears in X chromosome analysis

1. ## Fibonnacci series appears in X chromosome analysis

A Genealogical Look at Shared Ancestry on the X Chromosome
An old article (2016) my apologies if it has been previously posted.

Amusingly
In general, a present-day female’s X genealogical ancestors are growing as a Fibonacci series (Laughlin 1920; Basin 1963), such that k generations back she has Fk+2 X genealogical ancestors, where Fk is the kth Fibonacci number [where k is 0 indexed and the series begins F0=0,F1=1,…; We can demonstrate that one’s number of X genealogical ancestors (⁠n⁠k) grows as a Fibonacci series by encoding the X inheritance rules for the number of males and females (⁠mk and fk, respectively) in the kth generation as a set of recurrence relations:

fk=nk−1 Every individual receives an X chromosome from his/her mother
mk=fk−1 Every female receives an X chromosome from her father
nk=fk+mk
Rearranging these recurrence equations gives us nk=nk−1+nk−2, which is the Fibonacci recurrence. Starting with a female in the k=0 generation, we have initial values n0=1 and n1=2, which gives us the Fibonacci numbers offset by two, Fk+2. For a present-day male, his number of X ancestors is Fk+1,i.e., offset by one to count the number of X ancestors his mother has.

Vince Buffalo, Stephen M Mount, Graham Coop
Genetics, Volume 204, Issue 1, 1 September 2016, Pages 57–75, https://doi.org/10.1534/genetics.116.190041
Published: 01 September 2016
Abstract
Close relatives can share large segments of their genome identical by descent (IBD) that can be identified in genome-wide polymorphism data sets. There are a range of methods to use these IBD segments to identify relatives and estimate their relationship. These methods have focused on sharing on the autosomes, as they provide a rich source of information about genealogical relationships. We hope to learn additional information about recent ancestry through shared IBD segments on the X chromosome, but currently lack the theoretical framework to use this information fully. Here, we fill this gap by developing probability distributions for the number and length of X chromosome segments shared IBD between an individual and an ancestor k generations back, as well as between half- and full-cousin relationships. Due to the inheritance pattern of the X and the fact that X homologous recombination occurs only in females (outside of the pseudoautosomal regions), the number of females along a genealogical lineage is a key quantity for understanding the number and length of the IBD segments shared among relatives. When inferring relationships among individuals, the number of female ancestors along a genealogical lineage will often be unknown. Therefore, our IBD segment length and number distributions marginalize over this unknown number of recombinational meioses through a distribution of recombinational meioses we derive. By using Bayes’ theorem to invert these distributions, we can estimate the number of female ancestors between two relatives, giving us details about the genealogical relations between individuals not possible with autosomal data alone

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3. An important thing to remember here is that this only determines the potential number of X-chromosome contributors in any generation of ancestors. What the "real" number may be changes every time a potential X-chromosome contributor can be ruled out.

For example, I have one brother and one sister who inherited an X chromosome from our mother that appears to contain DNA only from our mother's father. This is based on many X-chromosome matches that collectively span the entire chromosome.

In addition, I match both of these siblings on the entire X, except for an approximately 20 cM region surrounding the centromere. I share DNA with a number of relatives of my maternal grandmother in this region. Otherwise, all of my matches on the X chromosome are from relatives of my maternal grandfather.

What this means is that in the case of my brother we can see that the maximum number of 4th great grandparents who contributed to his X chromosome is the fourth number in the series 1, 2, 3, 5. For me this number is the sixth number in the series 1, 2, 3, 5, 8, 13.

I made use of this fact in determining the most recent common ancestor who may have been the source of the X-chromosome segments shared by a 3rd or more distant cousin and myself, by following our respective (potential) pathways to find a point of intersection.

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5. Originally Posted by geebee
An important thing to remember here is that this only determines the potential number of X-chromosome contributors in any generation of ancestors. What the "real" number may be changes every time a potential X-chromosome contributor can be ruled out.

For example, I have one brother and one sister who inherited an X chromosome from our mother that appears to contain DNA only from our mother's father. This is based on many X-chromosome matches that collectively span the entire chromosome.

In addition, I match both of these siblings on the entire X, except for an approximately 20 cM region surrounding the centromere. I share DNA with a number of relatives of my maternal grandmother in this region. Otherwise, all of my matches on the X chromosome are from relatives of my maternal grandfather.

What this means is that in the case of my brother we can see that the maximum number of 4th great grandparents who contributed to his X chromosome is the fourth number in the series 1, 2, 3, 5. For me this number is the sixth number in the series 1, 2, 3, 5, 8, 13.

I made use of this fact in determining the most recent common ancestor who may have been the source of the X-chromosome segments shared by a 3rd or more distant cousin and myself, by following our respective (potential) pathways to find a point of intersection.
Right on geebee,
In a very recent intense discussion on X DNA we were getting all fascinated by the prospect of using these statistics when two participants pointed out that they had this sort of thing - one actually had two instances among close kin.
This ~14% (roughly 1 in 7) phenomenon in female to female transfers really throws a hand grenade into calculations on any wide scale.
So try the stats, but just understand that every now and then you get an all-or-nothing transfer of one chromosome.

X DNA can work really well for close kin estimations.
Having a match can also help exclude those ancestors that can't possibly contribute.
Beyond that it can be tricky.
(All the big successes I have seen back a few generations had a tree attached! So look for one of those.)

And hey, after all of that negativity in discussion, I went home, reviewed my data and found one significant connection I could not have made any other way.
But I also realized that one or two of my other matches could easily be back 500 years plus seeing from our trees their ancestors and mine must have separated at least that long ago. The Fibonacci sequence really didn't help with those.

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