biometrical geneticsibg.colorado.edu/cdrom2012/eaves...biometrical genetics how do genes contribute...
TRANSCRIPT
![Page 1: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/1.jpg)
Biometrical Genetics
Lindon Eaves,VIPBG Richmond
Boulder CO, 2012
![Page 2: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/2.jpg)
Biometrical Genetics
How do genes contribute to statistics (e.g. means, variances,skewness,
kurtosis)?
![Page 3: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/3.jpg)
Some Literature:
Jinks JL, Fulker DW (1970): Comparison of the biometrical genetical, MAVA, and classical approaches to the analysis of human behavior. Psychol Bull 73(5):311‐349.
Kearsy MJ, Pooni HS (1996) The Genetic Analysis of Quantitative Traits. London UK: Chapman Hall.
Falconer DS, Mackay TFC (1996). Introduction to Quantitative Genetics, 4th Ed. Harlow, UK: Addison Wesley Longman.
Neale MC, Cardon LR (1992). Methodology for Genetic Studies of Twins and Families. Ch 3. Dordrecht: Kluwer Academic Publisher. (See revised ed. Neale and Maes, pdf on VIPBG website)
![Page 4: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/4.jpg)
Ronald Fisher (1890‐1962)
1918: The Correlation Between Relatives on the Supposition of Mendelian Inheritance1921: Introduced concept of “likelihood”
1930: The Genetical Theory of Natural Selection1935: The Design of Experiments
![Page 5: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/5.jpg)
Fisher (1918): Basic Ideas
• Continuous variation caused by lots of genes (“polygenic inheritance”)
• Each gene followed Mendel’s laws• Environment smoothed out genetic differences• Genes may show different degrees of “dominance”• Genes may have many forms (“mutliple alleles”)• Mating may not be random (“assortative mating”)• Showed that correlations obtained by e.g. Pearson and Lee were explained well by polygenic inheritance
![Page 6: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/6.jpg)
0 1 2 3 4 5Y1
0.0
0.1
0.2
0.3
0.4
Distribution of scores produced by two genes(N=1000 subjects)
-2.5 -1.5 -0.5 0.5 1.5 2.5 3.5 4.5 5.5 6.5S1
0.0
0.1
0.2
0.3
0.4
The "smoothing" effect of the environment(N=1000 subjects, 2 gene model)
75 79 83 87 91 95 99 103 107 111 115 119 123Y1
0.00
0.02
0.04
0.06
Continuous distribution of polygenic trait (100 genes with small cumulative effects)
b.
c.
a.
![Page 7: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/7.jpg)
“Mendelian” Crosseswith Quantitative Traits
![Page 8: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/8.jpg)
Mendelian Basis of Continuous Variation?Experimental Breeding Experiments
![Page 9: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/9.jpg)
![Page 10: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/10.jpg)
“Biometrical Genetics”
• Parsimonious specification of genetic influences in terms of effects and frequencies of individual genes (“model‐building”)
• Systematic approach to choosing between different interpretations of the same data (“model‐fitting”)
![Page 11: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/11.jpg)
A “Good” Model
• Fits the data• Explains a lot of different data in terms of relatively few theoretical constructs
• Embraces new data without substantial modification or post‐hoc explanation (“fudging”)
See e.g. Lakatos (1972) “Criticism and the Growth of Knowledge”
![Page 12: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/12.jpg)
Theory Model Data
Model‐building Study designData collection
Model‐Fitting
Fits?Revise Publishestimates
YESNO
“The Logic of Scientific Discovery”
![Page 13: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/13.jpg)
![Page 14: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/14.jpg)
Assumptions (Initially)
• Autosomal inheritance• No epistasis• No sex‐dependent gene expression• Random mating• Genes of relatives (e.g. mothers) do not affect phenotype directly
• No GxE (see Mather and Jinks for GxE)• No G‐E correlation• Simple model for environment• Effects of selection/mutation too small to affect result.
![Page 15: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/15.jpg)
Basic Model for Effects of a Single Gene on a Quantitative Trait
Mid‐homozygote
Homozygous effect
Dominancedeviation
Increasing Decreasing
![Page 16: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/16.jpg)
Derivation of Genotype Frequencies“Hardy‐Weinberg Equilibrium”
![Page 17: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/17.jpg)
Genotype Frequenciesin Randomly Mating Population
“Hardy‐Weinberg Equilibrium”frequencies
![Page 18: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/18.jpg)
What is the mean expected to be?
Note: Effects measured from mid‐homozygote (“m”)
![Page 19: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/19.jpg)
With equal allele frequencies (easier!) put u=v= ½
And the mean is expected to be….
![Page 20: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/20.jpg)
How does A/a affect the variance?
![Page 21: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/21.jpg)
Equal allele frequencies u=v= ½
Additive component
Dominance component
![Page 22: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/22.jpg)
Q: What happens with lots of genes?
A: The effects of the individual genes add up.
IF… the genes are independent (“linkage equilibrium”)
Requires random mating, complete admixture
![Page 23: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/23.jpg)
So:
Additive Genetic Variance Dominance Genetic Variance
![Page 24: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/24.jpg)
Additive and Dominance Components:Unequal allele frequencies.
Can show (see e.g. Mather, 1949)
VA VD
Q: What happens when u=v?
![Page 25: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/25.jpg)
Bottom line:
With unequal allele frequencies can still separate VA and VD but their
definitions change
![Page 26: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/26.jpg)
d<‐1 # Homozygous effect ("additive")h<‐1 # Heterozygous deviation ("dominance")u<‐seq(0.01,0.99,by=.01) # Vector of frequencies of increasing allelev<‐1‐u # Frequencies of decreasing alleleVA<‐2*u*v*(d+(v‐u)*h)^2 # Additive genetic varianceVD<‐4*u*u*v*v*h*h # Dominance genetic varianceVP<‐VA+VD # Total (genetic) variance# Plot resultsplot(u,VP,type="l",main="VA (red) and VD (green) as function of increasing allele frequency",xlab="Frequency of increasing allele",ylab="Variance component")# Add line for VAlines(u,VA,col="red")# Add line for VDlines(u,VD,col="green")
Plotting Effect of Allele frequency on Genetic Variance Components (“R”)
![Page 27: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/27.jpg)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
VA (red) and VD (green) as function of increasing allele frequency
Frequency of increasing allele
Var
ianc
e co
mpo
nent
VA+VD
VA
VD
![Page 28: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/28.jpg)
What about the environment???
![Page 29: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/29.jpg)
Two main sources of environment
• Individual experiences – not shared with siblings:
VE• “Family” environment – shared with siblings:
VC
![Page 30: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/30.jpg)
So: the TOTAL variance(Genes + Environment) is:VP = VA+VD+VE+VC
![Page 31: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/31.jpg)
“Heritability”
“Broad” heritability: h2b=(VA+VD)/VP
Proportion of total variance explained by genes
“Narrow” heritability: h2n=VA/VP
Proportion of total variance explained by additive (homozygous) genetic
effects (predicts response to selection – Fisher, 1930)
![Page 32: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/32.jpg)
So far: have looked at effects on total variance…
How do VA and VD affect the correlations between relatives?
![Page 33: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/33.jpg)
Contribution of genes to correlation between relatives (r):
r = C/VPWhere C=Covariance between
relative pairs
“C” depends of kind of relationship (sibling, parent‐offspring, MZ twin
etc)
But can also be expressed in terms of VA and VD
![Page 34: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/34.jpg)
Approach1. For a given relationship, work out expected frequencies of
each type of pair (AA, aa etc.)2. Write phenotypes of each type of relative3. Compute cross‐products of phenotypes of members of
type of pair4. Each cross‐product by the corresponding frequency5. Add the result of “4” across all pair types
The answer is the covariance you want (if you have donethe algebra right!)
![Page 35: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/35.jpg)
For equal allele frequencies….
![Page 36: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/36.jpg)
![Page 37: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/37.jpg)
Contribution of one gene to covariance:
![Page 38: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/38.jpg)
Notice that terms in d2 and h2 are separated – but their coefficients
change as a function of relationship
![Page 39: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/39.jpg)
Can add over all genes to get total contribution to covariance
Cov(MZ) = VA + VDCov(DZ) = ½VA + ¼VD
Cov(U)= 0
![Page 40: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/40.jpg)
Can use the same approach for other relationships
![Page 41: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/41.jpg)
RelationshipContribution to Covariance
VA VD
Total variance 1 1Sibling (DZ twin) ½ ¼ MZ twin 1 1Half‐sibling ¼ 0First cousin 1/8 0Parent‐offspring ½ 0Avuncular ¼ 0Grand‐parent 1/8 0Unrelated 0 0
Contributions of VA and VD to covariancesbetween relatives (ignoring environment)
![Page 42: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/42.jpg)
Adding effects of Environment
VP = VA + VD + VE + VCCov(MZ) = VA + VD + VCCov(DZ) = ½VA + ¼VD + VCCov(UT) = VC
Etc.
![Page 43: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/43.jpg)
To get the expected correlations
Just divided expectations by expected total variance
Results are proportional contributions of VA, VD etc. to total variance
![Page 44: Biometrical Geneticsibg.colorado.edu/cdrom2012/eaves...Biometrical Genetics How do genes contribute to statistics (e.g. means, variances,skewness, kurtosis)? Some Literature: Jinks](https://reader036.vdocuments.us/reader036/viewer/2022062416/6108452956cd63099a1b242f/html5/thumbnails/44.jpg)
Practice (paper and pencil)
• Pick a “d” and “h” (e.g. d=1,h=1; d=1,h=0)• Pick a frequency for the increasing (A) allele (e.g. u=0.2, u=0.7)
• Work out VA and VD• Tabulate on board