## 2008-05-28

## 2008-05-24

### Quantitative genetics in maize

Crow wrote a paper "90 Years Ago: The Beginning of Hybrid Maize" in 1998. It is a nice overview of development of hybrid lines of maize - the theory as well as practice (using genetics and agricultural techniques).

On related topic, Edwards and Lamkey analysed effect of inbreeding from quantitative genetics perspective i.e. they estimated all components of variance due to dominant effect of alleles in noninbred and inbred individuals.

## 2008-05-15

### Towards a new paradigm in quantitative genetics

Peresentation by Daniel Gianola "Alice in Wonderland: towards a new paradigm in quantitative genetics" is available at

http://www.dcam.upv.es/ACTEON/docs/Gianola%20VALENCIA%202008.ppt

http://www.dcam.upv.es/ACTEON/docs/Gianola%20VALENCIA%202008.ppt

### Partitioning of additive genetic variance between relatives

I am deriving covariance between haplodiploids to correct some misunderstandings in the literature (e.g. Liu & Smith, 2000) and to extend their work and work of others (eg. Grossman & Eisen, 1989; ...) for general estimation of genetic parameters and breeding values in haplodiploids.

This is not clear to me, since individual can have only one breeding (additive genetic) value, which is a sum of average effect of alleles. Are then $$ \inline \alpha_j $$ and $$ \inline \alpha_k $$ average effects of alleles? Note that they assumed that $$ \inline E\left(\alpha^2_j\right) = \frac{1}{2}\sigma^2_a $$. I think that this indicates that $$ \inline \alpha_j $$ and $$ \inline \alpha_k $$ are half the breeding values from parents of individual 1, which in turn is the average effect i.e. breeding value is a sum of average effects.

Followup: yes, the $$ \inline \alpha_j $$ and $$ \inline \alpha_k $$ are average effects of alleles! Lange (as well as others) have shown the derivation of genetic covariance in his book (Mathematical and statistical methods for genetic analysis, 2nd edition 2003, page 101).

additive relationship coeffcient = twice the coefficient of coancestry also called kinship or parentage (the probability that a random allele at a particular locus in individual 1 is identical by descent to a random allele at the same locus in individual 2)

dominance relationship coeffcient = the coefficient of fraternity (the probability that both alleles at a particular locus in individual 1 are identical by descent to alleles at the same locus in individual 2)

Liu, F.-H., Smith, S. M. (2000). Estimating quantitative genetic parameters in haplodiploid organisms. Heredity, 85(4):373-382. http://dx.doi.org/10.1046/j.1365-2540.2000.00764.x

Grossman, M., Eisen, E. J. (1989). Inbreeding, coancestry, and covariance between relatives for X-chromosomal loci. J. Hered., 89(2):137-142. http://jhered.oxfordjournals.org/cgi/content/abstract/80/2/137

Grossman & Eisen (1989, page 140) derive genetic covariance for sex-linked genes between two females i.e. between two diploids. This result should be the same as genetic covariance for autosomal genes between two diploid individuals, since females have two sex chromosomes i.e. the result should be $$ \inline Cov\left(g_{jk},g_{j'k'}\right)=a_{12}\sigma^2_a + d_{12}\sigma^2_d $$, where $$ \inline Cov\left(g_{jk},g_{j'k'}\right) $$ is genetic covariance between genetic values of individual 1 with genotype jk and individual 2 with genotype j'k', $$ \inline a_{12} $$ is the additive relationship coeffcient (see bellow), $$ \inline \sigma^2_a $$ is additive genetic variance, $$ \inline d_{12} $$ is the dominance relationship coeffcient (see bellow), and $$ \inline \sigma^2_d $$ is dominance variance. They show the following equation:

$$

Cov\left(g_{jk},g_{j'k'}\right)=\\

Cov(\alpha_j+\alpha_k+\delta_{jk},\\

\alpha_{j'}+\alpha_{k'}+\delta_{j'k'})\\

= Cov\left(\alpha_j, \alpha_{j'}\right) + \\

Cov\left(\alpha_j, \alpha_{k'}\right) + \\

Cov\left(\alpha_k, \alpha_{j'}\right) + \\

Cov\left(\alpha_k, \alpha_{k'}\right) + \\

Cov\left(\delta_{jk}, \delta_{j'k'}\right)\\

= \Pr\left(j \equiv j' \right)E\left(\alpha^2_j\right) + \\

\Pr\left(j \equiv k' \right)E\left(\alpha^2_j\right) + \\

\Pr\left(k \equiv j' \right)E\left(\alpha^2_j\right) + \\

\Pr\left(k \equiv k' \right)E\left(\alpha^2_j\right) + \\

\Pr\left(jk \equiv j'k'\right)E\left(\delta^2_{jk}\right)\\

= 4 r_{12}\left(\frac{1}{2}\sigma^2_a\right) + d_{12}\sigma^2_d\\

= 2 r_{12}\left(\sigma^2_a\right) + d_{12}\sigma^2_d\\

= a_{12}\left(\sigma^2_a\right) + d_{12}\sigma^2_d,

$$

where $$ \inline \equiv$$ means alleles identical by descent, $$ \inline \alpha_j $$ and $$ \inline \alpha_k $$ are additive genetic values of individual with genotype jk and $$ \inline \delta_{jk} $$ is dominance deviation.

$$

Cov\left(g_{jk},g_{j'k'}\right)=\\

Cov(\alpha_j+\alpha_k+\delta_{jk},\\

\alpha_{j'}+\alpha_{k'}+\delta_{j'k'})\\

= Cov\left(\alpha_j, \alpha_{j'}\right) + \\

Cov\left(\alpha_j, \alpha_{k'}\right) + \\

Cov\left(\alpha_k, \alpha_{j'}\right) + \\

Cov\left(\alpha_k, \alpha_{k'}\right) + \\

Cov\left(\delta_{jk}, \delta_{j'k'}\right)\\

= \Pr\left(j \equiv j' \right)E\left(\alpha^2_j\right) + \\

\Pr\left(j \equiv k' \right)E\left(\alpha^2_j\right) + \\

\Pr\left(k \equiv j' \right)E\left(\alpha^2_j\right) + \\

\Pr\left(k \equiv k' \right)E\left(\alpha^2_j\right) + \\

\Pr\left(jk \equiv j'k'\right)E\left(\delta^2_{jk}\right)\\

= 4 r_{12}\left(\frac{1}{2}\sigma^2_a\right) + d_{12}\sigma^2_d\\

= 2 r_{12}\left(\sigma^2_a\right) + d_{12}\sigma^2_d\\

= a_{12}\left(\sigma^2_a\right) + d_{12}\sigma^2_d,

$$

where $$ \inline \equiv$$ means alleles identical by descent, $$ \inline \alpha_j $$ and $$ \inline \alpha_k $$ are additive genetic values of individual with genotype jk and $$ \inline \delta_{jk} $$ is dominance deviation.

This is not clear to me, since individual can have only one breeding (additive genetic) value, which is a sum of average effect of alleles. Are then $$ \inline \alpha_j $$ and $$ \inline \alpha_k $$ average effects of alleles? Note that they assumed that $$ \inline E\left(\alpha^2_j\right) = \frac{1}{2}\sigma^2_a $$. I think that this indicates that $$ \inline \alpha_j $$ and $$ \inline \alpha_k $$ are half the breeding values from parents of individual 1, which in turn is the average effect i.e. breeding value is a sum of average effects.

Followup: yes, the $$ \inline \alpha_j $$ and $$ \inline \alpha_k $$ are average effects of alleles! Lange (as well as others) have shown the derivation of genetic covariance in his book (Mathematical and statistical methods for genetic analysis, 2nd edition 2003, page 101).

additive relationship coeffcient = twice the coefficient of coancestry also called kinship or parentage (the probability that a random allele at a particular locus in individual 1 is identical by descent to a random allele at the same locus in individual 2)

dominance relationship coeffcient = the coefficient of fraternity (the probability that both alleles at a particular locus in individual 1 are identical by descent to alleles at the same locus in individual 2)

Liu, F.-H., Smith, S. M. (2000). Estimating quantitative genetic parameters in haplodiploid organisms. Heredity, 85(4):373-382. http://dx.doi.org/10.1046/j.1365-2540.2000.00764.x

Grossman, M., Eisen, E. J. (1989). Inbreeding, coancestry, and covariance between relatives for X-chromosomal loci. J. Hered., 89(2):137-142. http://jhered.oxfordjournals.org/cgi/content/abstract/80/2/137

## 2008-05-10

### Genotypic values under genomic imprinting model

I came across paper by Spencer where he develops the quantitative genetic theory for a single loci with two alleles (A1 and A2) with genomic imprinting. Genomic imprinting is also known as parent-of-origin effect. If there is no imprinting, there are three possible genotypes A1/A1, A1/A2, and A2/A2 and therefore three different genotypic values. However, when imprinting is in place, the mean of genotypes depends on alleles and origin of alleles e.g. maternal imprinting means that alleles inherited from a father are more expressed. Spencer, as well as many others (see bellow), assumed that genomic imprinting changes the mean of heterozygotes. Additionally, heterozygotes need to be distinguished i.e. A1/A2 and A2/A1 are treated separately. He showed the following genotypes: A1/A1, A1/A2, A2/A1, and A2/A2. I was a bit surprised, since I excepted that there should also be some change in homozygotes.

If we assume that there is imprinting, then alleles have different effect when inherited from a particular parent. Naive approach would be to mark alleles as A1, A1+, A2, and A2+, where + means additional effect. However, this is actually a set of four alleles i.e. A1, A2, A3, and A4, from which we can construct 16 ordered genotypes. This is not OK. Let us mark alleles as A1f, A1m, A2f, and A2m, where f means father and m means mother. We can construct the following ordered genotypes: A1m/A1f, A1m/A2f, A2m/A1f, and A2m/A2f. Therefore, there are four possible ordered genotypes and it is clear that only four different genotypic values need to be defined by the model.

Here is a list of some papers on the genomic imprinting - I am sure I missed a bunch of important ones:

Reviews

- 1997 GENOMIC IMPRINTING IN MAMMALS
- 1999 Basics of gametic imprinting
- 2003 What good is genomic imprinting: the function of parent-specific gene expression

- 2001 Assessment of Parent-of-Origin Effects in Linkage Analysis of Quantitative Traits
- 2002 The Correlation Between Relatives on the Supposition of Genomic Imprinting
- 2002 Testing for Genetic Linkage in Families by a Variance-Components Approach in the Presence of Genomic Imprinting
- 2003 Genomic Imprinting and Linkage Test for Quantitative-Trait Loci in Extended Pedigrees
- 2006 Influence of Mom and Dad: Quantitative Genetic Models for Maternal Effects and Genomic Imprinting
- 2007 A statistical model for dissecting genomic imprinting through genetic mapping
- 2007 A simple method for detection of imprinting effects based on case–parents trios
- 2007 A random model for mapping imprinted quantitative trait loci in a structured pedigree: An implication for mapping canine hip dysplasia
- 2008 Maternal Effects as the Cause of Parent-of-Origin Effects That Mimic Genomic Imprinting

- 2003 Linkage analysis of adult height with parent-of-origin effects in the Framingham Heart Study
- 2004 Quantitative trait loci with parent-of-origin effects in chicken
- 2008

"Our [Their] results show that the effectsEvolution of imprinting^{ }of genomic imprinting are relatively small, with reciprocal^{ }heterozygotes differing by 0.25 standard deviation units and^{ }the effects at each locus accounting for 1% to 4% of the phenotypic^{ }variance. We detected a variety of imprinting patterns, with^{ }paternal expression being the most common. These results indicate^{ }that genomic imprinting has small, but detectable, effects on^{ }the normal variation of complex traits in adults and is likely^{ }to be more common than usually thought."

- 1996 The evolution of genomic imprinting
- 1998 Genetic Conflicts, Multiple Paternity and the Evolution of Genomic Imprinting
- 2000 Population Genetics and Evolution of Genomic Imprinting
- 2001 The Evolution of X-Linked Genomic Imprinting
- 2004 The Effect of Genetic Conflict on Genomic Imprinting and Modification of Expression at a Sex-Linked Locus
- 2006 A Maternal–Offspring Coadaptation Theory for the Evolution of Genomic Imprinting
- 2006 A Chip off the Old Block: A Model for the Evolution of Genomic Imprinting via Selection for Parental Similarity
- 2006 Population Models of Genomic Imprinting. II. Maternal and Fertility Selection
- 2007 Sex-Specific Viability, Sex Linkage and Dominance in Genomic Imprinting

### LaTeX in blog posts

Sometimes I would like to include equation into the blog post. I claim that LaTeX is perfect for this job! However, one can not just type LaTeX code directly on Blogger. Wordpress has a plugin for this job. What can Blogger users use? I found the following:

I wil try the jsTeXrender since they have a nice installation howto!

Now let us try with double $ tags: $$\Pr\left(A_1\right)=\frac{1}{2}$$

with pre tags:

and with code tags:

I wil try the jsTeXrender since they have a nice installation howto!

Now let us try with double $ tags: $$\Pr\left(A_1\right)=\frac{1}{2}$$

with pre tags:

`\Pr\left(A_1\right)=\frac{1}{2}`

and with code tags:

`\Pr\left(A_1\right)=\frac{1}{2}`

## 2008-05-09

### Another paper on the genetic complexity

By Brem and Kruglyak

The landscape of genetic complexity across 5,700 gene expression traits in yeast

http://www.pnas.org/cgi/content/full/102/5/1572

The landscape of genetic complexity across 5,700 gene expression traits in yeast

http://www.pnas.org/cgi/content/full/102/5/1572

### The Importance of Genealogy in Determining Genetic Associations with Complex Traits

Newman et al. (2001) have assesed the importance of genealogy in determining genetic associations with complex traits in Hutterites. In my opinion the key parts of their note is:

"In general, the significance of association with a given marker was considerably inflated when pedigree structure was not included, although in some cases the reverse was true."

"In addition, many more loci showed evidence of association when the pedigree structure was not included (see fig. 2). In fact, 10%–22% of all markers appeared to have a strong association (P<.01) with the phenotype, when pedigree structure was not included." "In an association study in which it is not possible to take into account all familial relationships, as we have done with the Hutterites, another option is to use genomic controls (Devlin and Roeder^{1999}). Otherwise, naÃ¯ve approaches to genetic-association analysis could result in an enormous amount of time and of money spent in following up artifactual associations."

The same conclusions (more or less) were obtained by Kennedy et al. in year 1992!. The paper is "Estimation of effects of single genes on quantitative traits".

## 2008-05-07

### What If We Knew All the Genes for a Quantitative Trait in Hybrid Crops?

Bernardo has assessed potential of additional information from knowing the genes (in quantitative genetics, one can do a lot with knowing nothing about the genetic architecture beyond simple assumptions about resemblance between relatives) in maize. He concluded that not much can be gained for traits controlled by many loci. More ...

P.S. There is much more literature on this topic!

P.S. There is much more literature on this topic!

## 2008-05-01

### Another classics: THE NUMBER OF DAUGHTERS NECESSARY TO PROVE A SIRE

http://jds.fass.org/cgi/reprint/14/3/209

### SEWALL WRIGHT’S “Systems of Mating”

Have you every tried to read a series of five papers (see bellow) on Systems of mating by Sewall Wright? They are were hard to read. Nice summary about this series of papers and their effect on animal genetics was given by Hill in perspectives section of Genetics journal.

SYSTEMS OF MATING. I. THE BIOMETRIC RELATIONS BETWEEN PARENT AND OFFSPRING

Genetics 1921 6: 309-s [Abstract] [PDF]

SYSTEMS OF MATING. II. THE EFFECTS OF INBREEDING ON THE GENETIC COMPOSITION OF A POPULATION

Genetics 1921 6: 309-t [Abstract] [PDF]

SYSTEMS OF MATING. III. ASSORTATIVE MATING BASED ON SOMATIC RESEMBLANCE

Genetics 1921 6: 309-u [Abstract] [PDF]

SYSTEMS OF MATING. IV. THE EFFECTS OF SELECTION

Genetics 1921 6: 162-166. [PDF]

SYSTEMS OF MATING. V. GENERAL CONSIDERATIONS

Genetics 1921 6: 167-178. [PDF]

SYSTEMS OF MATING. I. THE BIOMETRIC RELATIONS BETWEEN PARENT AND OFFSPRING

Genetics 1921 6: 309-s [Abstract] [PDF]

SYSTEMS OF MATING. II. THE EFFECTS OF INBREEDING ON THE GENETIC COMPOSITION OF A POPULATION

Genetics 1921 6: 309-t [Abstract] [PDF]

SYSTEMS OF MATING. III. ASSORTATIVE MATING BASED ON SOMATIC RESEMBLANCE

Genetics 1921 6: 309-u [Abstract] [PDF]

Genetics 1921 6: 162-166. [PDF]

Genetics 1921 6: 167-178. [PDF]

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