Estimation of recombination rates in a
dairy cattle population
A. Hampel, F. Teuscher, D. Wittenburg 01.09.2016
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Background: Half-sib family
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Background: Population parameters
Population structure has influence on population parameters Population LD Population recombination rate Maternal LD Paternal recombination rate
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Background: Linkage disequilibrium
B b
A a 𝐴𝐴; 𝑎𝑎 … two alleles at a locus (allele frequency 𝑓𝑓𝐴𝐴 ,𝑓𝑓𝑎𝑎)
𝐵𝐵; 𝑏𝑏 … two alleles at a locus (allele frequency 𝑓𝑓𝐵𝐵 ,𝑓𝑓𝑏𝑏) Frequencies of combinations in a population:
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Loci are in linkage equilibrium: 𝑓𝑓𝐴𝐴𝐵𝐵𝑓𝑓𝑎𝑎𝑏𝑏 = 𝑓𝑓𝐴𝐴𝑏𝑏𝑓𝑓𝑎𝑎𝐵𝐵
𝐷𝐷 = 𝑓𝑓𝐴𝐴𝐵𝐵𝑓𝑓𝑎𝑎𝑏𝑏 − 𝑓𝑓𝐴𝐴𝑏𝑏𝑓𝑓𝑎𝑎𝐵𝐵
Loci are in linkage disequilibrium: 𝐷𝐷 … disequilibrium coefficient
𝑓𝑓𝐴𝐴𝐵𝐵𝑓𝑓𝑎𝑎𝑏𝑏 ≠ 𝑓𝑓𝐴𝐴𝑏𝑏𝑓𝑓𝑎𝑎𝐵𝐵
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Paternal diplotype
1
2 (1 − 𝜃𝜃)
Daughter generation Probability
A B
a b
A B
Background: Recombination rate
1 2 𝜃𝜃 𝜃𝜃 … Recombination rate a B A b a b 1 2 (1 − 𝜃𝜃) 1 2 𝜃𝜃
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Objective: Estimation of LD and recombination rates
Both methods were applied to an empirical dataset
Verification of the accuracy was performed in simulated half-sib families
EM Method* New Method
Minimization approach with less computation time
Maximization approach with high computation time
*Gomez-Raya (2012): Maximum likelihood estimation of linkage disequilibrium in half-sib families. Genetics
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Parameters
Maternal haplotype frequencies Genotype counts from offspring Paternal recombination rate
Solved by applying the EM algorithm
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Parameters
Empirical covariance between genotype codes for additive/dominant effects at two SNPs
e.g. AA 1, Aa 0, aa -1 / e.g. AA -1, Aa 1, aa -1
Allele frequency in the maternal population LD of dam; LD of sire 𝜃𝜃 = 1−4𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
2
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Empirical data set
Comprised 1295 half-sibs of a dairy cattle population (Fugato-plus “BovIBI” data)
40317 SNP-genotypes (29 autosomes) Estimation on BTA1
Maternal linkage disequilibrium Paternal recombination rate
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Results: LD on chromosome 1
EM Method New Method 𝐷𝐷�𝑑𝑑𝑎𝑎𝑑𝑑 𝐷𝐷�𝑑𝑑𝑎𝑎𝑑𝑑 Fr eque nc y Fr eque nc y13
Results: 𝜃𝜃 on chromosome 1
𝜃𝜃� 𝜃𝜃� Fr eque nc y Fr eque nc y EM Method New Method14
Simulation: Parameters
𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 0.15 𝐷𝐷𝑑𝑑𝑎𝑎𝑑𝑑 = 0.05 𝑁𝑁 ∈#{30,100,1000} 𝜃𝜃 ∈ {0.01, 0.05, 0.10, 0.20, 0.40, 0.50}LD Population size Recombination rate
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Bias of paternal recombination rate Mean squared error (𝐷𝐷𝑑𝑑𝑎𝑎𝑑𝑑)
Computation time Selection of results
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Results: Bias of 𝜃𝜃
New Method EM Method New Method EM Method
𝜃𝜃 = 0.20; N = 30 𝜃𝜃� 𝜃𝜃 = 0.01; N = 30 𝜃𝜃�
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Results: MSE of 𝐷𝐷
𝑑𝑑𝑎𝑎𝑑𝑑𝑀𝑀𝑀𝑀 𝑀𝑀 𝐷𝐷� 𝑑𝑑𝑎𝑎 𝑑𝑑 𝜃𝜃
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Results: Computation time
N=30 N=100 N=1000 FBN-Method Gomez-Raya tim e i n se conds 20 40 60 80 100 120 0 1 1 𝜃𝜃� 1
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𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐷𝐷𝑑𝑑𝑎𝑎𝑑𝑑
Results: Likelihood function
Two maxima of likelihood function Example: 𝑓𝑓1𝑑𝑑𝑎𝑎𝑑𝑑 = 𝑓𝑓 2𝑑𝑑𝑎𝑎𝑑𝑑 = 0.5 𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 0.15 𝐷𝐷𝑑𝑑𝑎𝑎𝑑𝑑 = 0.05 𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 0.05 𝐷𝐷𝑑𝑑𝑎𝑎𝑑𝑑 = 0.15
20 Simulation 𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 0.15 𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 0.05 𝐷𝐷𝑑𝑑𝑎𝑎𝑑𝑑 = 0.05 𝐷𝐷𝑑𝑑𝑎𝑎𝑑𝑑 = 0.15 𝜃𝜃 = 0.20 𝜃𝜃 = 0.40 𝑓𝑓𝐴𝐴𝐵𝐵 = 0.40 𝑓𝑓𝑎𝑎𝐵𝐵 = 0.10 𝑓𝑓𝑎𝑎𝑏𝑏 = 0.40 𝑓𝑓𝐴𝐴𝑏𝑏 = 0.10 recalculation Complementary case 𝑓𝑓𝐴𝐴𝐵𝐵 = 0.30 𝑓𝑓𝑎𝑎𝐵𝐵 = 0.20 𝑓𝑓𝑎𝑎𝑏𝑏 = 0.30 𝑓𝑓𝐴𝐴𝑏𝑏 = 0.20
Results: Complementary case
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𝜃𝜃� 𝜃𝜃=0.20; n=100 𝜃𝜃�
Results: Simulation of complementary case
𝜃𝜃=0.40; n=100
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Outlook and summary
Simulation
The New Method had more accurate estimates for higher recombination rates
New Method
Based on a minimization approach
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Empirical data set
The comparison of results showed that both methods had a
similar distribution of the maternal LD but the distribution of the recombination rate differed.
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Maximization function
Two possible solutions two maxima Found in both New and EM Method
Final solution depends on starting values
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Next steps
Criterion for distinction of both maxima (likelihood values) Estimation of good starting values
(combination of minimization and maximization approach)
26 𝑔𝑔2: = 16𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝐷𝐷𝑑𝑑𝑎𝑎𝑑𝑑 + 4𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 1 − 2𝑓𝑓̂1𝑑𝑑𝑎𝑎𝑑𝑑 1 − 2𝑓𝑓̂1𝑑𝑑𝑎𝑎𝑑𝑑 𝑔𝑔1: = 𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 + 𝐷𝐷𝑑𝑑𝑎𝑎𝑑𝑑 𝑐𝑐𝑐𝑐𝑐𝑐� 𝑑𝑑𝑜𝑜𝑑𝑑 = 16𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝐷𝐷𝑑𝑑𝑎𝑎𝑑𝑑 + 4𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 1 − 2𝑓𝑓̂1𝑑𝑑𝑎𝑎𝑑𝑑 1 − 2𝑓𝑓̂1𝑑𝑑𝑎𝑎𝑑𝑑 𝑐𝑐𝑐𝑐𝑐𝑐� 𝑎𝑎𝑑𝑑𝑑𝑑 = 𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 + 𝐷𝐷𝑑𝑑𝑎𝑎𝑑𝑑 𝑄𝑄 𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠, 𝐷𝐷𝑑𝑑𝑎𝑎𝑑𝑑 = 𝑐𝑐𝑐𝑐𝑐𝑐� 𝑎𝑎𝑑𝑑𝑑𝑑 − 𝑔𝑔1 2 + 𝑐𝑐𝑐𝑐𝑐𝑐� 𝑑𝑑𝑜𝑜𝑑𝑑 − 𝑔𝑔2 2 arg min(𝑄𝑄) 𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠,𝐷𝐷𝑑𝑑𝑑𝑑𝑑𝑑 = 𝑄𝑄 𝐷𝐷 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠, 𝐷𝐷𝑑𝑑𝑎𝑎𝑑𝑑 Minimization approach
Appendix: New Method
Leibniz-Institut für Nutztierbiologie FBN Wilhelm-Stahl-Allee 2 18196 Dummerstorf contact Hampel, Alexander Telefon: +49 38208 68 908 E-Mail: hampel@fbn-dummerstorf.de Internet: www.fbn-dummerstorf.de
