Combinatorics and InBreeding. Goal To provide a rough model which gives a lower bound of organisms...
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Transcript of Combinatorics and InBreeding. Goal To provide a rough model which gives a lower bound of organisms...
![Page 1: Combinatorics and InBreeding. Goal To provide a rough model which gives a lower bound of organisms needed to prevent inbreeding Show that an exponential.](https://reader036.fdocuments.net/reader036/viewer/2022072006/56649d1a5503460f949ef17c/html5/thumbnails/1.jpg)
Combinatorics and InBreeding
![Page 2: Combinatorics and InBreeding. Goal To provide a rough model which gives a lower bound of organisms needed to prevent inbreeding Show that an exponential.](https://reader036.fdocuments.net/reader036/viewer/2022072006/56649d1a5503460f949ef17c/html5/thumbnails/2.jpg)
Goal
• To provide a rough model which gives a lower bound of organisms needed to prevent inbreeding
• Show that an exponential decay of a population can be balanced out with a linear factorial increase in population
![Page 3: Combinatorics and InBreeding. Goal To provide a rough model which gives a lower bound of organisms needed to prevent inbreeding Show that an exponential.](https://reader036.fdocuments.net/reader036/viewer/2022072006/56649d1a5503460f949ef17c/html5/thumbnails/3.jpg)
Problem• Given n genetically distinct starting families, how many generations can they last
before inbreeding
• Assumptions:
• Population is isolated, relatively small so no exponential growth• Every individual replaces him/herself such that each generation maintains the same #
of individuals• At every generation offspring are created when two families merge i.e. :
• Generation 08 Families• Generation 14 Families• Generation 22 Families
1—2 3—4 5—6 7—8
![Page 4: Combinatorics and InBreeding. Goal To provide a rough model which gives a lower bound of organisms needed to prevent inbreeding Show that an exponential.](https://reader036.fdocuments.net/reader036/viewer/2022072006/56649d1a5503460f949ef17c/html5/thumbnails/4.jpg)
Biology Background Info• People, dogs, cheetahs are diploid organisms• DNA inherited maternally and paternally• Each Parent only transfers one set of DNA to offspring
• Mother(2 sets of DNA) Father(2 sets of DNA)
1 set 1 set
– Offspring (2 sets of DNA)
![Page 5: Combinatorics and InBreeding. Goal To provide a rough model which gives a lower bound of organisms needed to prevent inbreeding Show that an exponential.](https://reader036.fdocuments.net/reader036/viewer/2022072006/56649d1a5503460f949ef17c/html5/thumbnails/5.jpg)
Relatedness• Two individuals are related based on probability that they share the same genetic
information called Coefficient of Relatedness (COR)
• COR of 2 identical twins =1• COR of 2 strangers =0• COR of Parent---Offspring =.5 (2 parents half and half)• COR of Grandparent—Offspring=.25 (4 grandparents ¼+1/4 + 1/4 + 1/4)
• The COR of two individuals is directly proportional to # of common ancestors and inversely proportional to how far ancestors are removed
• In general direct ancestors i generations removed will have COR ofi2
1
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Relatedness
• Full siblings COR=.5=1/2
• Cousins with 2 grandparents in common COR=.125=1/8
• In the case of half siblings with 1 parent in common COR=.25 (1/2x1/2)
2
1
2
1
2
1
2
1
(Probability they share from father) + (Probability they share from mother)
4
1
4
1
4
1
4
1
(Probability from sharing with 1 G.Parent)+(Probability from sharing with 2nd G.Parent)
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Relatedness
i
k22
-In general COR= where k=#
of common ancestors i=generations removed
-According to dog breeders inbreeding occurs when two individuals of COR=.0625=1/16 or higher mate to produce offspring
-As such we assume individuals with COR<.0625 does not constitute inbreeding and may reproduce for more generations depending on how far individuals are removed.
![Page 8: Combinatorics and InBreeding. Goal To provide a rough model which gives a lower bound of organisms needed to prevent inbreeding Show that an exponential.](https://reader036.fdocuments.net/reader036/viewer/2022072006/56649d1a5503460f949ef17c/html5/thumbnails/8.jpg)
Solving the Problem
By pairing n distinct starting families, after each ith generation, the total number of distinct families goes down by..
i
n
2
-# of people can only be increased linearly while non-relatives decrease exponentially
![Page 9: Combinatorics and InBreeding. Goal To provide a rough model which gives a lower bound of organisms needed to prevent inbreeding Show that an exponential.](https://reader036.fdocuments.net/reader036/viewer/2022072006/56649d1a5503460f949ef17c/html5/thumbnails/9.jpg)
Solving the Problem--Re-writing n in binary tells us when and where families are in dangerof not passing on their genetic information
e.g. for n=612 226
--In 1st generation, descendents of 2 of the original 6 families cannot pair up to pass on their genes since 6/2=3 3/2= 1 +1 remainder
--Essentially each 2^i term of writing n in binary signifies that at the ith generation, 2^I pieces of the original DNA will be lost
1---2 3---4 5---6--1st generation..Family (5---6) has noone topair with
--2nd generation nobody can pair up
![Page 10: Combinatorics and InBreeding. Goal To provide a rough model which gives a lower bound of organisms needed to prevent inbreeding Show that an exponential.](https://reader036.fdocuments.net/reader036/viewer/2022072006/56649d1a5503460f949ef17c/html5/thumbnails/10.jpg)
Combinatorics
• It is beneficial at those critical generations to not just pair up but rather start creating combinations of families
i
n
2--At the ith generation we have distinct families
--Those families can combine ways
--To create families for the next (i+1)th generation
where some are
related but at least are distinct
2
2in
12in
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Combinatorics
2
2in
2
2#
2
22
chosenfamiliesofdistinctni
---For the (i+1)th generation we have this # of families :
---We know that by excluding any chosen two families out of the total we have the # of families which are completely unrelated to those two:
2
# familiesofdistinct=
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Combinatorics---The (i+1)th generation can provide this many families for the (i+2)th :
!2
2
22
2
2
ii
nn
# of families in (i+1)th generation # of families not related
whatsoever to a chosen family
Divide by 2! since order of choosing family doesn’t matter
= total number of families (i+1)th generation can produce for (i+2)th
![Page 13: Combinatorics and InBreeding. Goal To provide a rough model which gives a lower bound of organisms needed to prevent inbreeding Show that an exponential.](https://reader036.fdocuments.net/reader036/viewer/2022072006/56649d1a5503460f949ef17c/html5/thumbnails/13.jpg)
Combinatorics
37950!2
2
23
2
25
!2
2
22100
2
2100
22
152
2
3
3
5
2
2
2220
2
220
22
22
2012
--The sooner we start combining instead of pairing, the greater the genetic diversity
--n=20 i=2 case, # of families in 3rd generation
--For n=100 i=2 case, # of families in 3rd generation
VS.
122
10012
VS.
![Page 14: Combinatorics and InBreeding. Goal To provide a rough model which gives a lower bound of organisms needed to prevent inbreeding Show that an exponential.](https://reader036.fdocuments.net/reader036/viewer/2022072006/56649d1a5503460f949ef17c/html5/thumbnails/14.jpg)
Conclusion• While this does not prevent inevitable sharing of DNA, it does show
combining families can dilute DNA to enough levels such that if needed, two weakly related individuals can reproduce
• This occurs since combinations of multiple partners leads to many half-siblings
• Given enough time these half siblings can produce offspring which become further and further removed as factorial increase overcomes the exponential decrease
• This model can not only serve to show how combining isolated populations can revitalize a species but..
• It also shows that a drastic drop in population over a short time can do the opposite like cheetahs