library(simplePHENOTYPES)
library(dplyr)
library(tibble)12 Mating designs and crossbreeding
Suppose you have two groups of inbred lines, say two heterotic pools, and want to plan the crosses among them, run them all in one call, trace each progeny’s expected ancestry, and ask how much of the hybrid advantage survives a backcross or a rotation. This chapter writes mating plans with mating_design(), executes them with mate(), follows ancestry with breed_composition() and crossbreed(), and checks the heterosis produced by a dominance architecture against its expected value.
12.1 Worked example
12.1.1 Two founder pools from the maize panel
The maize panel has no breed labels, so we form two pools of eight inbred lines each by taking the first sixteen lines of the table. The pool argument of as_population() records the label in the pedigree; it is the only thing that distinguishes the pools, and it is what breed_composition() and heterosis() trace. The pools are therefore an arbitrary split of one panel, not two diverged populations, and the heterosis they show is whatever their allele-frequency differences happen to produce.
data("SNP55K_maize282_maf04", package = "simplePHENOTYPES")
geno <- as_numeric(
SNP55K_maize282_maf04,
to_r = TRUE, to_file = FALSE, verbose = FALSE
)
ids <- colnames(geno)[-(1:5)]
pool_a <- as_population(geno, individuals = ids[1:8], pool = "A")
pool_b <- as_population(geno, individuals = ids[9:16], pool = "B")
pool_a#> <Population>
#> Individuals: 8 Markers: 10650 Chromosomes: 10
#> Genetic map: 1497 cM total span (109-219 cM per chromosome)
#> Origin: founder
#> Pedigree: 8 recorded individuals; generation 0
#> IDs: 4226, 4722, 33-16, 38-11, A188, A239, ... (2 more)12.1.2 Write a mating plan
mating_design() returns a data frame with mother, father, and n columns. The standard designs differ only in which pairs they list.
mating_design(pool_a[1:3], pool_b[1:2], design = "factorial",
progeny_per_cross = 4)| mother | father | n |
|---|---|---|
| 4226 | A6 | 4 |
| 4722 | A6 | 4 |
| 33-16 | A6 | 4 |
| 4226 | A619 | 4 |
| 4722 | A619 | 4 |
| 33-16 | A619 | 4 |
designs <- tibble(
design = c("factorial 3 x 2", "half_diallel 4", "diallel 4",
"diallel 4 with selfs", "nested 2 fathers x 2 mothers", "random 5"),
rows = c(
nrow(mating_design(pool_a[1:3], pool_b[1:2], design = "factorial")),
nrow(mating_design(pool_a[1:4], design = "half_diallel")),
nrow(mating_design(pool_a[1:4], design = "diallel")),
nrow(mating_design(pool_a[1:4], design = "diallel", allow_self = TRUE)),
nrow(mating_design(pool_a[1:4], pool_b[1:2], design = "nested",
mothers_per_father = 2)),
nrow(mating_design(pool_a, pool_b, design = "random", n_crosses = 5,
seed = 1))
),
parameter = c(3 * 2, choose(4, 2), 4 * 3, 4 * 3 + 4, 2 * 2, 5)
)
designs| design | rows | parameter |
|---|---|---|
| factorial 3 x 2 | 6 | 6 |
| half_diallel 4 | 6 | 6 |
| diallel 4 | 12 | 12 |
| diallel 4 with selfs | 16 | 16 |
| nested 2 fathers x 2 mothers | 4 | 4 |
| random 5 | 5 | 5 |
The parameter column is the number of pairs each design defines: every mother with every father for the factorial (North Carolina design II), every unordered pair once for the half diallel, every ordered pair for the full diallel, plus one self per parent when selfs are allowed, each father with its own set of mothers for the nested design (North Carolina design I), and the requested number of random pairs. ?mating_design names the Griffing (1956) methods that the diallel variants correspond to.
12.1.3 Execute the plan across pools
When parents come from several populations, the plan needs mother_pool and father_pool columns naming the arguments of mate(). All rows run in one pass, and the returned Population carries the plan with each row’s progeny identifiers.
plan <- mating_design(pool_a[1:3], pool_b[1:2], design = "factorial",
progeny_per_cross = 4) |>
mutate(mother_pool = "A", father_pool = "B")
hybrids <- mate(plan, A = pool_a, B = pool_b, seed = 1, prefix = "H")
hybrids#> <Population>
#> Individuals: 24 Markers: 10650 Chromosomes: 10
#> Genetic map: 1497 cM total span (109-219 cM per chromosome)
#> Origin: mate(6 rows)
#> Pedigree: 29 recorded individuals; generation 1
#> IDs: H_1, H_2, H_3, H_4, H_5, H_6, ... (18 more)attr(hybrids, "plan") |>
select(mother, father, n, progeny) |>
head(3)| mother | father | n | progeny |
|---|---|---|---|
| 4226 | A6 | 4 | H_1, H_2, H_3, H_4 |
| 4722 | A6 | 4 | H_5, H_6, H_7, H_8 |
| 33-16 | A6 | 4 | H_9 , H_10, H_11, H_12 |
table(families(hybrids, "full_sib"))#>
#> 4226 x A6 4722 x A6 33-16 x A6 4226 x A619 4722 x A619 33-16 x A619
#> 4 4 4 4 4 4Each row produced a full-sib family of four, and the family labels name the two parents. Progeny identifiers follow the prefix.
12.1.4 Expected ancestry
breed_composition() gives each individual’s expected fraction of genes from each founder pool, traced through the pedigree: a founder is one for its own pool, and every progeny is the mean of its parents. We build a backcross series by hand with cross() to see the familiar halving of the donor fraction.
f1 <- hybrids[1]
bc1 <- cross(f1, pool_a[1], n = 1, seed = 2)
bc2 <- cross(bc1, pool_a[1], n = 1, seed = 3)
bc3 <- cross(bc2, pool_a[1], n = 1, seed = 4)
series <- c(f1, bc1, bc2, bc3)
breed_composition(series) |>
as_tibble(rownames = "id") |>
mutate(generation = c("F1", "BC1", "BC2", "BC3"),
parameter_A = c(1 / 2, 3 / 4, 7 / 8, 15 / 16)) |>
select(generation, id, A, B, parameter_A)| generation | id | A | B | parameter_A |
|---|---|---|---|---|
| F1 | H_1 | 0.5000 | 0.5000 | 0.5000 |
| BC1 | prog_1 | 0.7500 | 0.2500 | 0.7500 |
| BC2 | prog_1_1 | 0.8750 | 0.1250 | 0.8750 |
| BC3 | prog_1_2 | 0.9375 | 0.0625 | 0.9375 |
These are expected fractions. The realized share of an individual’s genome that came from pool A varies around them, and the Population does not record per-locus founder origin, so the realized share is not available from the pedigree. The marker-assisted breeding chapter estimates it from markers instead.
12.1.5 Crossbreeding systems
crossbreed() runs the classical systems on named pure-breed populations: each generation mates random pairs between the pools the system prescribes, and the attribute history records the mean expected composition per generation. Sex is not modelled; “dam” and “sire” are the roles of the two pools in the plan.
breeds <- list(A = pool_a, B = pool_b)
two_way <- crossbreed(breeds, "two_way", n_progeny = 60, seed = 5)
backcross <- crossbreed(breeds, "backcross", n_progeny = 60, seed = 6)
rotation <- crossbreed(breeds, "rotational", n_progeny = 60,
generations = 5, seed = 7)
attr(backcross, "history")| generation | sire_breed | A | B |
|---|---|---|---|
| 1 | B | 0.50 | 0.50 |
| 2 | A | 0.75 | 0.25 |
attr(rotation, "history")| generation | sire_breed | A | B |
|---|---|---|---|
| 1 | B | 0.500000 | 0.500000 |
| 2 | A | 0.750000 | 0.250000 |
| 3 | B | 0.375000 | 0.625000 |
| 4 | A | 0.687500 | 0.312500 |
| 5 | B | 0.343750 | 0.656250 |
| 6 | A | 0.671875 | 0.328125 |
A two-breed rotation alternates the sire breed, and its composition converges toward two thirds of the current sire breed and one third of the other. The three-way and terminal systems add a third pool.
pool_c <- as_population(geno, individuals = ids[17:24], pool = "C")
three <- list(A = pool_a, B = pool_b, C = pool_c)
attr(crossbreed(three, "three_way", n_progeny = 60, seed = 8), "history")| generation | sire_breed | A | B | C |
|---|---|---|---|---|
| 1 | B | 0.50 | 0.50 | 0.0 |
| 2 | C | 0.25 | 0.25 | 0.5 |
attr(crossbreed(three, "terminal", n_progeny = 60, sire_breed = "C",
seed = 9), "history")| generation | sire_breed | A | B | C |
|---|---|---|---|---|
| 1 | B | 0.50 | 0.50 | 0.0 |
| 2 | C | 0.25 | 0.25 | 0.5 |
Both systems reach the same composition, one quarter from each of the first two pools and one half from the third. The difference is in use: a terminal cross is not bred further.
12.1.6 Heterosis on a dominance architecture
heterosis() scores a crossbred population on a frozen additive-plus-dominance architecture (trait values) and reports its mean minus the composition-weighted mean of the pure pools. It also returns the expected F1 heterosis of each pair of pools, computed from the pools’ own gamete frequencies. We take 30 marker loci at random as causal, give each the same additive effect and the same dominance deviation, and compare.
set.seed(1)
causal <- sample(nrow(geno), 30)
a_eff <- rep(0.5, 30)
d_dev <- rep(0.3, 30)
h_f1 <- heterosis(two_way, breeds, qtn = causal, a = a_eff, d = d_dev)
h_bc <- heterosis(backcross, breeds, qtn = causal, a = a_eff, d = d_dev)
h_additive <- heterosis(two_way, breeds, qtn = causal, a = a_eff, d = 0)
p_a <- rowMeans(dosages(pool_a)[causal, ] == 1)
p_b <- rowMeans(dosages(pool_b)[causal, ] == 1)
expected_inbred <- sum(d_dev * (p_a + p_b - 2 * p_a * p_b))
tibble(
quantity = c("Expected F1 heterosis (package)",
"Expected F1 heterosis (inbred-line formula)",
"Realized F1 heterosis", "Realized backcross heterosis",
"Realized F1 heterosis, d = 0"),
value = c(h_f1$expected_f1["A", "B"], expected_inbred,
h_f1$realized, h_bc$realized, h_additive$realized)
)| quantity | value |
|---|---|
| Expected F1 heterosis (package) | 4.443750 |
| Expected F1 heterosis (inbred-line formula) | 4.443750 |
| Realized F1 heterosis | 4.377500 |
| Realized backcross heterosis | 4.130417 |
| Realized F1 heterosis, d = 0 | -0.037500 |
h_f1$breed_means#> A B
#> -1.750 -1.125The package’s expected value and the inbred-line formula agree, because every line in both pools is homozygous at the causal loci. The realized F1 value scatters around the expectation, since 60 progeny of random pairs sample the pool allele frequencies. With d = 0 the realized heterosis is close to zero: the additive part cancels against the mid-parent except for that sampling of parents.
The backcross retains more than the classical one half of the F1 heterosis. That is not an error. The one-half rule assumes the recurrent breed is in Hardy-Weinberg proportions at the locus, and a pool of inbred lines that differ at a locus has no heterozygotes at all. The theory section gives the exact condition.
12.2 Assumptions and theory
12.2.1 Expected composition
Let \(F_{i,b}\) be the expected fraction of individual \(i\)’s genes from pool \(b\). A founder has \(F_{i,b} = 1\) for its own pool and \(0\) otherwise, and a progeny of mother \(m\) and father \(f\) has
\[ F_{i,b} = \tfrac{1}{2}\left(F_{m,b} + F_{f,b}\right), \tag{12.1}\]
with a self or doubled haploid equal to its parent (package definition, breed_composition()). This is the expectation of the share of alleles inherited from each pool and says nothing about which loci came from where.
12.2.2 Realized and expected heterosis
The realized heterosis of a crossbred population is its mean total genotypic value minus the pool means weighted by its mean expected composition,
\[ H = \bar G_{\text{cross}} - \sum_b \bar F_b\,\bar G_b, \tag{12.2}\]
the mid-parent value for an F1 (package definition, heterosis(); the equation map cites Falconer and Mackay 1996, chapter 14, page unverified, for mid-parent heterosis). For the expected F1 heterosis between pools A and B, write \(p_A\) and \(p_B\) for the gamete frequencies of the allele coded \(+a\) at a locus and \(h_A\), \(h_B\) for the pools’ own heterozygote frequencies. The F1’s allele frequency is the average of the two, so the additive part cancels against the mid-parent and what remains is
\[ H_{F1} = \sum_j d_j\left[h_{AB,j} - \tfrac{1}{2}\left(h_{A,j} + h_{B,j}\right)\right], \qquad h_{AB} = p_A(1 - p_B) + (1 - p_A)\,p_B . \tag{12.3}\]
The package documents this as its own derivation from gamete frequencies, which is why linkage does not enter. Two special cases follow algebraically. When each pool is in Hardy-Weinberg proportions, \(h = 2p(1-p)\) and the expression reduces to \(\sum_j d_j (p_{Aj} - p_{Bj})^2\), the textbook \(H_{F1} = \sum d\,y^2\) with \(y\) the allele-frequency difference (Falconer and Mackay 1996, page unverified, as cited in ?heterosis). When each pool consists of fully inbred lines, \(h = 0\) and it reduces to \(\sum_j d_j\,(p_{Aj} + p_{Bj} - 2p_{Aj}p_{Bj})\), the formula evaluated in the worked example. A purely additive architecture gives zero in every case.
12.2.3 How much heterosis later generations retain
The classical fractions, one half in an F2 or backcross and two thirds in a two-breed rotation at equilibrium, come from the probability that an individual’s two alleles at a locus originate from different pools under a dominance model. The package’s documentation states the exact conditions: at a locus with non-zero F1 heterosis, a backcross to pool A retains one half if and only if \(h_A = 2p_A(1 - p_A)\), that is, only the recurrent pool needs to be in Hardy-Weinberg proportions; an F2 retains one half if and only if the two pools’ deviations \(h - 2p(1-p)\) sum to zero. The condition for the two-thirds rotation figure is not derived in the package and is given as the classical value for pools in Hardy-Weinberg proportions. Our inbred-line pools violate the backcross condition at every segregating locus, with \(h_A =
0 < 2p_A(1-p_A)\), which is why the backcross retained more than one half. heterosis() computes the realized value from the actual genotypes and does not assume any retention fraction.
12.2.4 Mating designs
The designs are bookkeeping, not genetics: they enumerate pairs. Individuals are compared by pedigree identity rather than display name, so two pools that reuse names stay distinct provided they were imported under their own pool labels, and a parent that appears twice under different names enters a design once. The random design draws a mother uniformly and then one of her admissible fathers uniformly, which is uniform over pairs only when every mother has the same number of admissible fathers.
12.3 Implementation and practical limits
- Pool labels are the only record of origin. Import each breed or heterotic group with its own
pool, or two founders with the same name and genotype are the same individual, and mating them is a self. - All populations in one plan must share one marker map and the same allele coded
1; see the crossing chapter on allele orientation. crossbreed()mates random dam-sire pairs with one progeny each; the number of progeny per generation isn_progeny, and no sexes are modelled.breed_composition()and thehistoryattribute are expectations from the pedigree. Realized genome shares require marker-based estimation, as inrecurrent_parent_recovery().heterosis()requires every founder that the crossbred pedigree traces to be present in the supplied pools, and scores the frozen architecture per se; epistasis is outside the function.- The pools here are an arbitrary split of one panel. For a study of heterotic groups, import populations with documented provenance and let the observed allele-frequency differences, not a label, carry the biology.