Combinatorics and InBreeding

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

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

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)

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

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)

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.

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

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

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

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=

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

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.

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