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List of symbols.

 
Symbol Definition†
a Vector of breeding values
Vector of estimated breeding values
A Pedigree or genomic numerator relationship matrix
ak Additive effect at QTN k simulated for a given trait
αk Average allele substitution effect at QTN k computed for a given trait
b Vector of SNP effects
Estimated effect of SNP j
dk Dominance effect at QTN k computed for a given trait
δk Dominance degree at QTN k simulated for a given trait
e Vector of residual effects
ei,r Residual effect for individual i and trait r
gEBVi Genomic estimated breeding value of individual i
H2 Broad-sense heritability
h2 Narrow-sense heritability
i Indicates a given individual, varies from 1 to nIndiv
j Indicates a given SNP, varies from 1 to nSNP
k Indicates a given QTN, varies from 1 to nQTN
LA, LE Lower triangular matrix obtained from the Cholesky decomposition of VA and VE
Entry of LA and LE between traits r and s
λ Penalty factor applied on the loss of genetic diversity in optimal contribution selection
mδ User-specified mean dominance degree
μ Intercept of the regression models (Eq. [10] and [12])
μ0 Mean value of the base generation of the pedigree
nIndiv No. of individuals
nQTN Total no. of unrestricted or frequency-restricted QTN in the genome
nSNP Total no. of SNP in the genome
nTraits No. of simulated traits
pEBVi Pedigree-estimated breeding value of individual i
pk, qk Frequencies of the nonzero and zero alleles, respectively, at QTN k in the base generation of the pedigree
r, s Indicate two distinct traits: r varies from 1 to nTraits, and s varies from 1 to r
RandDev Random deviate sampled from a Gaussian or Gamma distribution
A priori additive genetic variance specified by the user for a given trait
Additive genetic variance computed for a given trait in the base generation
Additive genetic variance computed in the training population using the TBV of the training individuals.
Variance of the SNP effects
Dominance genetic variance computed for a given trait in the base generation
User-specified variance of the dominance degrees
Residual variance computed for a given trait
Genotypic variance computed for a given trait in the base generation
TBVi, TDVi, TGBi True breeding value, true dominance value, and true genotypic value of individual i for a trait characterized by a given set of QTN, unrestricted or frequency-restricted, and a given distribution, Gaussian or Gamma
VA, VE Additive genetic and residual correlation matrix, respectively, dimensions nTraitsnTraits
x Vector providing the contribution of each selection candidate to the next generation
X Incidence matrix linking phenotypes to b
xi,k, xi,j Genotype of individual i at QTN k or SNP j, coded as 0, 1, or 2 according to the number of copies of the nonzero allele
y Vector of phenotype records
Z Incidence matrix linking phenotypes to a
QTN, quantitative trait nucleotide; SNP, single-nucleotide polymorphism.



View Full Table | Close Full ViewTable 2.

Benchmarking of AlphaSim through the simulation of three distinct plant breeding programs.

 
Base Scenario 1 Scenario 2 Scenario 3
User features
 Number of chromosomes 7
 Number of segregating sites 30,356
 Start–stop generation 1–2 3–5 3–9 3–13
 Number of individuals 52 300 504 708
Results
 Genetic variance F4 stage 0.333 0.110 0.002
 Mean gEBV† F4 stage 1.315 2.182 2.584
Computational feature
 Running Time 0 m 6 s 0 m 4 s 0 m 7 s 0 m 10 s
gEBV, genomic estimated breeding value.



View Full Table | Close Full ViewTable 3.

Benchmarking of AlphaSim through the simulation of two distinct scenarios run with or without requesting the full genome sequence to be written out.

 
Scenario 1 Scenario 2
User features
 Write out the full genome sequence No Yes No Yes
 Number of segregating sites 70,140 72,240 162,780 164,400
 Number of SNP† 5000 20,000
 Number of QTN‡ 2500 10,000
 Pedigree size (no. of individuals) 1210 234,500
 Size of the training population 1000 30,000
Computational feature
 Running Time 1 m 34 s 3 m 39 s 4 h 9 m 54 s 19 h 6 m 11 s
SNP, single-nucleotide polymorphism.
QTN, quantitative trait nucleotide.