1. Lecture 2: Genetic mapping and linkage analysis in farm animals
Terminologies: (Quantitative or population genetics: (Population study), Molecular markers Quantitative traits, Variation in quantitative traits, Phenotypic values of the traits, Gene and genotypic frequencies, Additive gene action, Genetic equilibrium and genetic disequilibrium (H.W. Law).
Bovine genetic linkage map on website & internet resources for genetic linkage map of cattle.
Important features on bovine genetic linkage map.
Molecular genetic markers in quantitative Genetics: Microsatellite or short tandem repeat (STR) markers.
GENETIC LINKAGE: Identification of markers linked to quantitative trait loci (QTL).
Linkage and Linkage dis-equilibrium (Hardy-Weinberg equilibrium)
Gene mapping function and genetic distance
Identification and locations of polymorphic markers in analysed population
Identification of linkage between marker and trait of interest by analysing the Linkage between genetic markers.
Detection of linkage
Experimental design to estimate of LOD score.
Detection and localisation of QTL i.e., Mapping of the identified marker in genetic map

2. Terminology commonly used in population genetics
1.Gene and chromosome: Gene is the physical entity transmitted from parent to offspring during the reproductive process that influences hereditary traits.
While, Chromosome is the entity in which genes are arranged in within a cell.
2. Molecular markers: Comprised of alleles that are used to keep track of a chromosome or a gene. For ex: in Past: Blood group, milk protein, enzymes, At present: RFLP, VNTR, STR, EST, and more recently SNP.
3. Quantitative (Polygenic) traits: Phenotypic characters which are control by many genes (Polygenes).
3. Quantitative traits loci: Polygenes that are organised on a fixed locus in the specific chromosmes. OR A gene affecting a quantitative traits.
4. Alleles: Alternative form of a gene. For ex: k Casein A & B allele.
5. Segregation: The separation of two alleles during meiosis.
6. Linked marker allele: Polymorphic allele which segregated according to observed phenotypic traits.
7. H.W. Law: In a random mating population with no selection, mutation, migration, the gene and genotypic frequencies are constant from generation to generation. This properties of a population are derived from a principle, Known as HW law, after Hardy and Wienberg (1908).
8. Gene and genotypic frequency: (AA,AB,BB & A,B respectively).

3. Bovine genetic linkage map on website & Internet resources
Human genetic linkage map on website & Internet resources

4. Bovine genome assembly website & Internet resources
Bovine genome assembly website & Internet resources
Human genome assembly website & Internet resources

5. Important features on bovine genetic linkage map
1. map position (cM)
2. Heterozygosity of marker
3. Primer sequence
4. PCR product size
5. No. of polymarphic alleles
6. Distances with adjacent markers

6. Use of molecular genetic markers in quantitative Genetics: Microsatellite or short tandem repeat (STR) markers
Genetic Markers: In past years, the most commonly used markers were: blood groups, milk protein and biochemical enzymes
Now these above mentioned markers have been replaced by highly polymorphic molecular markers like: RFLP (Botstein, 1980), VNTR (Jaffery et al. 1985), STR (Litt and Luty, 1989), ETS, SNP markers.
Microsatellite markers: Are short tandem repeat of base pairs found through out the mammalian genome. They are classified as:
Dinucleotide type, Trinucleotide type, Tetranucleotide type of STR.
Properties of STR markers: Highly polymorphic in nature. Found abundant through out the mammalian genome. Inherited in a Mendelian fashion. Segregated in a co-dominance fashion.

7. Identification of markers linked to quantitative trait loci (QTL).
GENETIC LINKAGE :
Identification of markers linked to quantitative trait loci (QTL).
In Genetic Linkage study:
For example: At Two loci, A and B, for which the proportion of parental gametes
(1-q ) produced exceeds the proportion of recombinant gametes (q) (violation of Mendel's second law) are said to be genetically linked (1 ).
Counting parental versus recombinant gametes requires knowledge of the
linkage phase, i.e. grouping the alleles of the individual producing the gametes by parental origin.

8. Linkage and Linkage dis-equilibrium (Hardy-Weinberg equilibrium)
HW law is the beck bone of population genetic study. As it is defined as „In a large random mating population with no selection, mutation and migration, the gene and genotypic frequencies are constant from generation to generation”.
Thus, A population with constant gene and genotypic frequencies is said to be in Hardy-Weinberg equilibrium.
      Forces creating linkage disequilibrium:
Random drift
Mutation
Migration
Selection (LD involving the loci underlying the genetic variance for the selected trait = "Bulmer effect")

9. ”Linkage equilibrium”
Linkage and linkage dis-equilibrium
Genes are linked if they are located on the same chromosome.
Linked genes do not show independent assortment.
”Linkage equilibrium”
Allele at locus A
A1 (PA)
A2 (QA)
A1B1 (PAPB)
A2B1 (QAPB)
B1 (PB)
Allele at locus B
A1B2 (PAQB)
A2B2 (QAQB)
B2 (QB)

10. ”Linkage dis-equilibrium”
Allele at locus A
A1 (PA)
A2 (QA)
A1B1 (r)
A2B1 (t)
B1 (PB)
r = PAPB + D
t = QAQB - D
Allele at locus B
A1B2 (s)
A2B2 (u)
B2 (QB)
s = PAQB - D
u = QAQB + D
LD(D) = ru - st
D : selection (Bulmer effect), migration, randam mating
D : recombination

11. At  =0.5, linkage equilibrium
Dt = Linkage disequilibrium in generation t
D0 = Linkage disequilibrium in generation t
 = Linkage disequilibrium in generation t D0
At  =0.5, linkage equilibrium
(Dt)

12. Recombination: The process that generates, new chromosomal
Combinations hat differ from the parental chromosomes,
during meiosis.
For example:
In a genotype A1A2B1B2; we assume that A & B alleles are linked.
A1B1/A2B2 : coupling phase (cis)
A1B2/A2B1 : repulsion phase (trans)
Gametes
A1B (1- )/2
A2B (1- )/2
A1B (1- )/2
A2B (1- )/2
A1B1/A2B2
coupling phase
repulsion phase
= recombination fraction
= 0.5 means genes are linked

14. The location of genes: I II
d = distance between genes in centimorgan (cM).
1cM = the distance between two genes is 1 cM if there is on average
one cross over per 100 meiosis.
Mappng function: Function relating the map distance (d) to
the recombination fraction. (). I II
A B C
AB = 0.20
BC = 0.20
d = , i.e., = AC = = 0.40

15. Experimental example: typing is for genes A and C
A C
AC = ???
Typing for A and C
If typing would have been for A,B and C
Recombinants Ac aC
Abc, Abc, aBC, abC
Non-recombinants AC
ABC, AbC
ABC, AbC are recognised as non-recombinants!!!

16. Calculation for Non-recombinants:
Non-recombinants = non-recombinants + double rcombinants
(1-AC) = (1- AB)(1- BC) + AB BC
(1-AC) = (1- AB)(1- BC) + AB BC
(1-AC) = 1- AB - BC + AB BC + AB BC
AC = AB + BC + 2 AB BC
By putting the values:
AC = – 0.08 = 0.32
when map distance (d) = recombnation rate (),
means linkage or in other words, it does not take into account,
The occurence of double recombination.

17. Gene mapping function and genetic distance
     Genetic distance (d):
The recombination rate (q ) between two loci is a function of the genetic distance between the two loci (d) expressed in Morgan (M).
d corresponds to the average number of crossovers occurring between loci A and B per gamete and per generation.
The distinction between q and d result from the fact that gametes having undergone an even number of crossovers will be mistaken as non recombinant gametes.
Several functions relating q and x are being used, including:

18. Haldane's mapping function:
Haldane’s mapping function assumes that the frequencies of meioses with 0, 1, 2, …,n crossovers between loci A and B are distributed as a Poisson process with mean m:
as all but meioses with 0 crossovers produce 50% recombinant and 50% parental gametes (1 ):
As m = 2x (2d),
(1)
and
(2)

19. Haldane's mapping function:
Where x(d) = -1/2 In (1- 2)
-1/2In (1-2*0.20) = = dAB = dBC
= = dAC
½ (1- e–(2*0.511)) = 0.32 = AC

20. At map distance 1.0,
The  value is 0.43
0.5
 = 0.5 (1- e–2d)
0.4
Observed 
0.3
0.2
0.1
Map distance (d)
The relation between the observed  and map distance (d in Morgan)
Haldane mapping function takes into account the occurence of multiple Recombinants based on the passion distribution
f(i) = e-µ . µi i!
So, 1.0 Morgan (100cM) corresponds with  = 0.43

21. 100 cM = on an average there are 100 cross overs per 100 meiosis.
By putting the values;
f(i) = e-µ . µi i!
f (0) = e-1 . µ0 = that is 37% no recombination. 0!
f (0) = e-1 . µ1 = that is 37% 1 recombination. 1!
f (0) = e-1 . µ2 = that is 18% 2 recombination. 2!
f (0) = e-1 . µ3 = that is 6% 3 recombination. 3!
f (0) = e-1 . µ4 = that is 2% 4 recombination. 4!
By summing up all: 37(0) + 37(1) + 18(2) + 6(3) + 2(4) = 99
That is approximately 100 recombination.

22. When we correspond d (1M or 100cM) value to  value = 0.43.
Observed as recombinants:
1 recombinant (i=1)
+ 3 recombinant (i=3)
= = 43
Another mapping function (Kosambi’s) widely used in estimation of
genetic distance in mammals, especially with reference to
”Interference” during meiotic recombination.

23. Kosambi's mapping function:
Kosambi’s mapping function allows for interference (I), whereby one crossover tends to prevent other crossovers in the same region:
The amount of interference allowed in the Kosambi map function decreases as the loci get further apart, and is zero for unlinked loci:
Kosambi's mapping function is the preferred mapping function in mammals.
Interference (I) defined as ”the effect in which the crossing ovr in a certain region reduces the probability of a crossing over in a adjacent area.”

24. For example: for 10% interference, i.e., I = 0.1,
Interference (I) defined as ”the effect in which the crossing ovr in a certain
region reduces the probability of a crossing over in a adjacent area.”
For example: for 10% interference, i.e., I = 0.1,
The Frequency of expected double recombination will be:
I = 0.1 = 1 – [Observed no. Of double recombination / 0.043]
So, Oberverved no. of double recombination = 0.9 x 0.04 =

25.  The genetic distance, d, between two loci is a function of:
The physical distance between the two loci
The recombinogenic potential of the intervening sequences: the genome is characterized by recombination hot- and cold spots (1).
The sex of the parent: generally speaking, the recombination rate between two loci is expected to be higher in the homogametic sex than in the heterogametic sex. In some chromosome regions, however, such as subtelomeric and imprinted regions, this ratio may be inverted (1).
Individual variation in recombination rates.

28. The total genetic map length (m = Morgan), and therefore the average kb / cM ratio, differs between species:
Species
Sex-averaged map
Female map
Male Map
Human
 43 M
28.5 M 
Mouse
 16 M
Cattle
 29.9 M
Pig

29. Identification of linkage between marker and trait of interest by analysing the Linkage between genetic markers. (i.e., Linkage between microsatellite marker linked to the QTL loci)
Estimation of recombination fraction ().
a)Estimates of linakge phase (information on Cis or trans phase of parents)
b) information of the gemetes transmiteed by the parent.
c) idntification of gemetes in the progeny.
d) maximum likelihood estimation of  .
2. Detection of linkage.
a) estimates of LOD score
3. Experimental design for estmating . (half-sib and fullsib design).
a) quanitative measures of LOD score values by maximum likelihood estimates.

30. Estimation of recombination fraction ().
a)Estimates of linakge phase (information on Cis or trans phase of parents)
b) information of the gemetes transmiteed by the parent.
c) identification of gemetes in the progeny.
d) maximum likelihood estimation of  .

31. Which individuals are recombinants? (Coupling Phase) (repulsion phase)
a)Estimates of linakge phase (information on Cis or trans phase of parents)
What is important?
Example:
Sire genotype M1M2N1N2
Which individuals are recombinants?
(Coupling Phase) (repulsion phase)
M1N1/ M2N2 M1N2/ M2N1
M1N * (1-) * 
M1N *  * (1-)
M2N *  * (1-)
M2N * (1-) * 

32. And offspring genotype : M1N1M1N2
b) information of the gemetes transmiteed by the parent.
Example:
Sire genotype M1N1M2N2
And offspring genotype : M1N1M1N2
Both gemetes can be obtained from the sire. Progeny is not informative.
c) identification of gemetes in the progeny.
M1M2 * M1M2 =results in M1M2 offpring.
M1 can be inheritated from the father and from the mother. The same
thing holds for the M2 allele.

33. d) maximum likelihood estimation of  :
no-fam 2 2
no-off
L () =     P(gkl/gs,gd,phsi,phdj, )
i=1
j=1
i=1
k=1
Where:
gkl = genotype of kth ofprng in the ith family
gs = genotype of sire of family l
gd = genotype of dam of family l
phsi = ith phase of sire
phdj = jth phase of dam
P(gkl/gs,gd,phsi,phdj) =prbability of observing an offpring with genotype gkl given the genotypes ofthe parents and the linkage phases of the parents
No-fam = number of unrelated full sib families
No-off = number of offsprings

34. Detection of linkage
LOD score values of more than 3 are considered as linkage between STR marker and QTL.
Formula’s for Lod-score:
Z() = log10 [ L() / L(0.5);
= log10 [(L()] - log10 [(L(0.5)];
 Differs significantly from 0.5 if lod score exceed value 3.

36. To be Continued in
NEXT LECTURE:

37. Detection and localisation of QTLs
Introduction
Establishing linkage between genetic markers
Detection of linkage between QTL and genetic markers
Expected distribution gene and genotypic frequecies of QTL
in a population.
5. Simulation study on a segregating population considering a
Marker allele linked to a QTL locus (for half-sib family).
6. Model
a) equation.
b) Marker gene is a QTL
c) A QTL linked to the marker has an effect.
d)Estimates of gene and genotypic frequencies
e) Significance of marker & QTL effect.
7. Model estimates in a half-sib segregating population.
a) Marker – QTL genotype frequences.
b) mean and variance estimates of these genotypes.
8. Required size of experiment.
9. How to reduced the size of experiment?
a) grand daughter and daughter design.
10. Overview

38. Introduction:
Linkage between genetic markers: M N
Data : genotypes for M and N
Problem: are M and N linked?
Detection of QTL=detection of linkage between QTL and genetic markers. M
QTL
Data : information on marker genotypes M and phenotypic
Observations on thequantitative traits.
Problem: Is there a QTL linked to the marker?

39. How can wedetect QTLs with the aid of genetic markers?
==== Linkage disequilibrium approach.
Linkage disequilibrium generated by:
Selection
Random drift
Mutation
Migration
Linkage disequilibrium reduced by:
Recombination
Dt = D0 (1-)t
Dt = Linkage disequilibrium in generation t
D0 =Linkage disequilibrium in generation 0
 =Recombination fraction between A and B.

40. ”Linkage equilibrium” Allele at locus A
A1 (PA)
A2 (QA)
B1 (PB)
A1B1 (PAPB)
A2B1 (QAPB)
Allele at locus B
A2B2 (QAQB)
A1B2 (PAQB)
B2 (QB)
”Linkage dis-equilibrium”
Allele at locus A
A1 (PA)
A2 (QA)
A1B1 (r)
A2B1 (t)
B1 (PB)
r = PAPB + D
t = QAQB - D
Allele at locus B
A2B2 (u)
A1B2 (s)
B2 (QB)
u = QAQB + D
s = PAQB - D
D = ru - st
D : selection (Bulmer effect), migration, randam mating
D : recombination

41. Notation: Marker genotypes: M1M1, M1M2, M2M2
QTL genotypes Values for quantitative traits
Q1Q1 a
Q1Q2 d
Q2Q a
 : recombination fraction between marker and QTL.