file_gencomp

Exploring genetic and spatial competition with

Saulo Chaves

Assistant Professor

Data Science for Plant Breeding Lab

Objective

At the end of this short-course, you will be able understand the genetic competition effects on the selection process in forest tree breeding. Additionally, you will be capable of running spatial and genetic competition models, and correctly interpret their outputs.

Plants compete for above- and below-ground resources

Potencial Florestal

Competition can mislead selection

Hardy Eucalyptus

Competitive plants can impair sensitive neighbours to express their full genetic potential
- Challenging environments
Selection of aggressive individuals can decrease the correspondence between trials and commercial farms
- Clonal mixture
Weak correlation between growth and quality traits

Competition models are not trivial

Quantitative genetics base of the variance-component model (Griffing, 1967):

\[ \begin{align} u_{A_1a_1} = & \; \alpha_{A_1} + \alpha_{a_1} + \delta_{A_1a_1} + \alpha_{A_2} + \alpha_{a_2} + \delta_{A_2a_2} + \\ & \; \left(\alpha \alpha \right)_{A_1A_2} + \left(\alpha \alpha \right)_{A_1a_2} + \left(\alpha \alpha \right)_{a_1A_2} + \left(\alpha \alpha \right)_{a_1a_2} +\\ & \; \left(\alpha \delta \right)_{A_1A_2a_2} + \left(\alpha \delta \right)_{a_1A_2a_2} + \left(\delta \alpha \right)_{A_1 a_1 A_2} + \left( \delta \alpha\right)_{A_1a_1a_2} + \\ & \; \left(\delta \delta \right)_{A_1a_1A_2a_2} \end{align} \]

\[ P_v = DGE_v + \sum_{k=1}^n IGE_k + e_v \]

Fear not, presenting `gencomp`!

# install.packages("devtools")
devtools::install_github("Kaio-Olimpio/gencomp", build_vignettes = TRUE)

library(gencomp)
library(ggplot2)

Example dataset

str(cpt)

'data.frame':   3000 obs. of  9 variables:
 $ prog       : Factor w/ 60 levels "F01xF02","F01xF03",..: 50 35 55 33 51 60 7 35 17 42 ...
 $ id         : Factor w/ 600 levels "F01xF02_13","F01xF02_19",..: 500 345 542 325 505 597 64 347 169 420 ...
 $ col        : Factor w/ 75 levels "1","2","3","4",..: 1 2 3 4 5 6 7 8 9 10 ...
 $ row        : Factor w/ 40 levels "1","2","3","4",..: 1 1 1 1 1 1 1 1 1 1 ...
 $ block      : Factor w/ 5 levels "1","2","3","4",..: 1 1 1 1 1 1 1 1 1 1 ...
 $ direct     : num  0.406 -8.96 7.913 -5.263 -0.219 ...
 $ competition: num  0.578 2.277 -2.677 2.348 2.139 ...
 $ e.Trait1   : num  1.37 9.97 4.81 3.07 -2.48 ...
 $ pheno      : num  29 24.4 39.5 21.8 22.8 ...

Simulated data
Partial diallel with 20 founders
Clonal progeny trials
600 clones from 60 families
RCBD, single-tree plots, five replicates

1^st step: Competition matrix

For crops, a within-row or within-column incidence matrix (Stringer et al., 2011)

\[ {{\boldsymbol{Z}}}_{{c}_{1}}=\left[\begin{array}{cccccc}0&1&0&0&0&0\\ 1&1&0&0&0&0\\ 0&1&0&1&0&0\\ 0&1&0&0&0&0\\ 0&0&0&0&0&1\\ 0&0&1&0&1&0\\ 0&0&1&0&0&1\\ 0&0&0&1&0&0\\ 0&0&0&0&0&1\\ 1&0&0&0&1&0\\ 0&0&1&0&0&1\\ 1&0&0&0&0&0\end{array}\right]\quad {{\boldsymbol{Z}}}_{{c}_{2}}=\left[\begin{array}{cccccc}0&0&1&0&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&0&1\\ 0&0&0&0&1&0\\ 0&0&1&1&0&0\\ 1&1&0&0&0&0\\ 0&1&0&0&0&1\\ 1&0&0&0&1&0\\ 0&0&1&0&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&0&1\\ 0&0&0&0&1&0\end{array}\right] \]

1^st step: Competition matrix

For trees, distance-based weights (directional competition intensity factors):

Direction	Muir (2005)	Cappa & Cantet (2008)	Costa e Silva & Kerr (2013)
Diagonal \(\left(f_d\right)\)	\(\frac{1}{\sqrt{\mathcal{D}^2_r + \mathcal{D}^2_c}}\)	\(\frac{1}{\sqrt{2 \left(n_c + n_r\right) + n_d}}\)	\(\frac{p}{\sqrt{\left(n_r p^4 \right) + \left(n_r p^2 \right) + \left(n_c p^2 \right) + \left(n_d p^2 \right) + n_c}}\)
Row \(\left(f_r\right)\)	\(\frac{1}{\mathcal{D}_r}\)	\(\sqrt{\frac{2}{2 \left(n_c + n_r\right) + n_d}}\)	\(f_d \sqrt{1 + p^2}\)
Column \(\left(f_c\right)\)	\(\frac{1}{\mathcal{D}_c}\)	\(\sqrt{\frac{2}{2 \left(n_c + n_r\right) + n_d}}\)	\(\frac{f_d \sqrt{1 + p^2}}{p}\)

\(p = {\mathcal{D}_c}/{\mathcal{D}_r}\)

1^st step: Competition matrix

Ferreira, Chaves & Dias (2024)

1^st step: Competition matrix

Function prepfor

compmat = prepfor(
  data = cpt,
  gen = "id",
  row = "row",
  col = "col",
  trait = "pheno",
  plt = NULL,
  effs = "block",
  dist.row = 2,
  dist.col = 3,
  method = "SK",
  area = NULL,
  age = NULL,
  n.dec = 2,
  verbose = TRUE
)

1^st step: Competition matrix

Function prepfor

plot(compmat)

1^st step: Competition matrix

Function prepfor

plot(compmat, category = "boxplot") + theme(axis.text.x = element_blank(), 
                                            axis.ticks.x = element_blank())

1^st step: Competition matrix

Function prepfor

Competition matrix:

str(compmat$Z)

 num [1:3000, 1:600] 0 0 0 0 0 0 0 0 0 0 ...
 - attr(*, "dimnames")=List of 2
  ..$ : chr [1:3000] "1" "2" "3" "4" ...
  ..$ : chr [1:600] "F01xF02_13" "F01xF02_19" "F01xF02_22" "F01xF02_47" ...

Competition intensity factor \(\left(\hat{\phi}=\bar{{n}_{r}}{\bar{f}}_{r}+{\bar{n}}_{c}{\bar{f}}_{c}+{\bar{n}}_{d}{\bar{f}}_{d} \right)\)

compmat$CIF

[1] 2.675279

1^st step: Competition matrix

Function prepfor

Neighbourhood check of each tree

str(compmat$neigh_check)

'data.frame':   3000 obs. of  11 variables:
 $ gen    : Factor w/ 600 levels "F01xF02_13","F01xF02_19",..: 500 345 542 325 505 597 64 347 169 420 ...
 $ row    : num  1 1 1 1 1 1 1 1 1 1 ...
 $ col    : num  1 2 3 4 5 6 7 8 9 10 ...
 $ y_focal: num  29 24.4 39.5 21.8 22.8 ...
 $ y_row  : num  24.4 34.2 23.1 31.2 23.5 ...
 $ n_row  : num  1 2 2 2 2 2 2 2 2 2 ...
 $ y_col  : num  21.9 34.9 27.5 18.6 17.6 ...
 $ n_col  : num  1 1 1 1 1 1 1 1 1 1 ...
 $ y_diag : num  34.9 24.7 26.7 22.5 21.4 ...
 $ n_diag : num  1 2 2 2 2 2 2 2 2 2 ...
 $ y_neigh: num  27.1 30.5 25.4 25.2 21.5 ...

2^nd step: Fitting the model

\[ {\boldsymbol{y}}={\boldsymbol{X}}{\boldsymbol{\beta }}+{{\boldsymbol{Z}}}_{1}{\boldsymbol{u}}+{{\boldsymbol{Z}}}_{2}{\boldsymbol{p}}+{\boldsymbol{\varepsilon}} \]

\[ \begin{align} {{\boldsymbol{u}}}^{{\prime} }=\begin{bmatrix}{\underbrace{\boldsymbol g^\prime}_{\mbox{DGE}}} & \underbrace{\boldsymbol c^\prime}_{\mbox{IGE}}\end{bmatrix} && \boldsymbol Z_1 = \begin{bmatrix} \underbrace{\boldsymbol Z_g}_{\mbox{DGE}} & \underbrace{\boldsymbol Z_c}_{\mbox{IGE}} \end{bmatrix} \end{align} \]

\[ \boldsymbol{u} \sim \mathcal{N} \left(\underbrace{\begin{bmatrix} \boldsymbol 0 \\ \boldsymbol 0 \end{bmatrix}}_{\mu}, \underbrace{\begin{bmatrix} \sigma^2_g & \sigma_{gc} \\ \sigma_{gc} & \sigma^2_c\end{bmatrix} \otimes \boldsymbol K}_{\operatorname{Var}\left(\boldsymbol u\right)}\right) \]

\[ \boldsymbol \varepsilon \sim \left[\boldsymbol 0, \sigma^2_\varepsilon \left(\boldsymbol{AR1}_R \otimes \boldsymbol{AR1}_C \right)\right] \]

2^nd step: Fitting the model

Function asr: No relationship matrix

mod1 = asr(
  prep.out = compmat,
  fixed = pheno ~ block,
  random = ~ 1,
  spatial = TRUE,
  cor = TRUE,
  lrtest = TRUE
)

Function asr: With relationship ⚠️ Major Feature Release ⚠️

ped = unique(as.data.frame(cbind(
  do.call(rbind, strsplit(as.character(cpt[, 1]), split = "x")), as.character(cpt[, 2])
)))
ped = ped[, c(3, 1, 2)]
ainv = asreml::ainverse(ped)

mod2 = asr(
  prep.out = compmat,
  fixed = pheno ~ block,
  random = ~ 1,
  spatial = TRUE,
  cor = TRUE,
  lrtest = TRUE,
  K = ainv
)

2^nd step: Fitting the model

About the relationship matrix

A sparse inverse variance matrix held in three-column coordinate form in row-major order. For backwards compatibility, a three-column data frame is also accepted. In either case, the source must have a rowNames attribute. This object must have class ginv from a call to asreml::ainverse(), or have attribute INVERSE set to TRUE. If the attribute INVERSE is not set then FALSE is assumed. We recommend using the ASRgenomics package to set the genomic relationship matrix (see ASRgenomics::G.matrix() and ASRgenomics::G.inverse()). If there is no relationship information, K = NULL (default)

str(ainv)

 'matrix' num [1:1880, 1:3] 1 2 3 3 4 4 5 5 5 6 ...
 - attr(*, "inbreeding")= Named num [1:620] 0 0 0 0 0 0 0 0 0 0 ...
  ..- attr(*, "names")= chr [1:620] "F17" "F12" "F19" "F11" ...
 - attr(*, "logdet")= num -416
 - attr(*, "dimnames")=List of 2
  ..$ : NULL
  ..$ : chr [1:3] "Row" "Column" "Ainverse"
 - attr(*, "rowNames")= chr [1:620] "F17" "F12" "F19" "F11" ...
 - attr(*, "geneticGroups")= num [1:2] 0 0

3^rd step: Extracting the outputs

Function resp

outs = resp(
  prep.out = compmat,
  model = mod2,
  weight.tgv = TRUE,
  sd.class = 1
)

outs$lrt

  effect LR-statistic    Pr(Chisq)
1    DGE     541.2104 0.000000e+00
2    IGE      40.3012 1.088375e-10

outs$varcomp

                  component  std.error   z.ratio bound %ch
cor(IGE_DGE)    -0.73740806 0.07773369 -9.486338     U 0.1
DGE             21.52088403 2.11141011 10.192659     P 0.1
IGE              4.01501998 0.77251608  5.197329     P 0.0
R               37.72756142 1.16113954 32.491841     P 0.0
R=autocor(row): -0.14501420 0.02162016 -6.707362     U 0.1
R=autocor(col): -0.04126034 0.02236380 -1.844961     U 0.2

3^rd step: Extracting the outputs

If competition is significant, selection should be based on the Total Genetic Value (TGV):

Unweighted (weight.tgv = FALSE):

\[ TGV_v = \hat{g}_v + \hat{\phi} \times \hat{c}_v \]

Weighted by the reliability (weight.tgv = TRUE):

\[ {wTGV}_{v}={\hat{g}}_{v}\times {r}_{{g}_{v}}^{2}+\hat{\phi }\times {\hat{c}}_{v}\times {r}_{{c}_{V}}^{2} \]

3^rd step: Extracting the outputs

Competition classes

\[ \,{\text{Classes}}\,=\left\{\begin{array}{l}\bar{{\rm{c}}}-\tau \; {\rm{sd}}({\rm{c}}) \,<\,{\hat{{\rm{c}}}}_{{\rm{v}}} \,<\, \bar{{\rm{c}}}+\tau \; {\rm{sd}}({\rm{c}})\to \,{\text{Homeostatic}}\,\quad \\ {\hat{{\rm{c}}}}_{{\rm{v}}} \,<\,\bar{{\rm{c}}}-\tau \; {\rm{sd}}({\rm{c}})\to \,{\text{Aggressive}}\,\quad \\ {\hat{{\rm{c}}}}_{{\rm{v}}} \,>\, \bar{{\rm{c}}}+\tau \; {\rm{sd}}({\rm{c}})\to \,{\text{Sensitive}}\,\quad \end{array}\right. \]

\(\tau \to\) weight defining the classifier sensitivity

Function resp

head(outs$blups$main[,-c(3,6)])

             id        DGE   rel.DGE       IGE   rel.IGE       class      TGV
1           F01 -3.2758934 0.8973572 3.2176001 0.8295542   Sensitive 4.201138
497         F17 -2.3111288 0.8973443 2.6525694 0.8303412   Sensitive 3.818525
155  F05xF08_93  2.2160810 0.7484024 1.1821025 0.6314346 Homeostatic 3.655404
22  F01xF04_100  3.7465442 0.7495402 0.4777074 0.6272911 Homeostatic 3.609864
218         F08  0.5368863 0.8978368 1.4116096 0.8274290 Homeostatic 3.606780
27   F01xF04_56  2.7817749 0.7493080 0.8514380 0.6304831 Homeostatic 3.520542

3^rd step: Extracting the outputs

Broad- or narrow-sense heritability

\[ H^2_g = \frac{\sigma^2_g}{\sigma^2_y} \]

Total heritable variation (Bijma et al., 2007)

\[ H^2_t = \frac{\sigma^2_g + 2 \phi \sigma_{gc} + \phi^2 \sigma^2_c}{\sigma^2_y} \]

outs$heritability

       Heritability
direct    0.3401787
total     0.2146729

3^rd step: Extracting the outputs

Function resp

plot(outs, category = "DGE.IGE") + theme(axis.text.x = element_blank(), 
                                         axis.ticks.x = element_blank())

3^rd step: Extracting the outputs

Function resp

plot(outs, category = "TGV") + theme(axis.text.x = element_blank(), 
                                         axis.ticks.x = element_blank())

3^rd step: Extracting the outputs

Function resp

plot(outs, category = "class")

3^rd step: Extracting the outputs

Function resp

plot(outs, category = "DGEvIGE")

3^rd step: Extracting the outputs

Function resp

plot(outs, category = "nneigh")  + theme(axis.text.x = element_blank(), 
                                         axis.ticks.x = element_blank())

3^rd step: Extracting the outputs

Function resp

ggpubr::ggarrange(
  plot(outs, category = "grid.res"),
  plot(outs, category = "grid.dge"),
  plot(outs, category = "grid.ige"),
  plot(outs, category = "grid.class")
)

Concluding remarks

gencomp is a user-friendly tool that facilitates the fitting and utilization of (spatial-) genetic competition models. It is worth mentioning that gencomp is a work in progress and will continue to evolve as we introduce additional functionalities

Acknowledgements

Recommended literature

Bijma, P. (2010). Multilevel Selection 4: Modeling the relationship of indirect genetic effects and group size. Genetics, 186(3), 1029–1031. https://doi.org/10.1534/genetics.110.120485

Bijma, P. (2014). The quantitative genetics of indirect genetic effects: A selective review of modelling issues. Heredity, 112(1), 61–69. https://doi.org/10.1038/hdy.2013.15

Bijma, P., Muir, W. M., & Van Arendonk, J. A. M. (2007). Multilevel Selection 1: Quantitative genetics of inheritance and response to selection. Genetics, 175(1), 277–288. https://doi.org/10.1534/genetics.106.062711

Cappa, E. P., & Cantet, R. J. C. (2008). Direct and competition additive effects in tree breeding: Bayesian estimation from an individual tree mixed model. Silvae Genetica, 57(2), 45–56. https://doi.org/10.1515/sg-2008-0008

Chaves, S. F. S., Ferreira, F. M., Ferreira, G. C., Gezan, S. A., & Dias, K. O. G. (2025). Incorporating spatial and genetic competition into breeding pipelines with the R package gencomp. Heredity, 134, 129–141. https://doi.org/10.1038/s41437-024-00743-9

Costa e Silva, J., & Kerr, R. J. (2013). Accounting for competition in genetic analysis, with particular emphasis on forest genetic trials. Tree Genetics & Genomes, 9(1), 1–17. https://doi.org/10.1007/s11295-012-0521-8

Ferreira, F. M., Chaves, S. F. S., Bhering, L. L., Alves, R. S., Takahashi, E. K., Sousa, J. E., Resende, M. D. V., Leite, F. P., Gezan, S. A., Viana, J. M. S., Fernandes, S. B., & Dias, K. O. G. (2023). A novel strategy to predict clonal composites by jointly modeling spatial variation and genetic competition. Forest Ecology and Management, 548, 121393. https://doi.org/10.1016/j.foreco.2023.121393

Ferreira, F. M., Chaves, S. F. S., Santos, O. P., Nunes, A. C. P., Tambarussi, E. V., Pereira, G. S., Santos, G. A., Bhering, L. L., & Dias, K. O. G. (2024). Competition effects can mislead selection in eucalypt breeding trials. Forest Ecology and Management, 561, 121892. https://doi.org/10.1016/j.foreco.2024.121892

Griffing, B. (1967). Selection in Reference to Biological Groups I. Individual and Group Selection Applied to Populations of Unordered Groups. Australian Journal of Biological Sciences, 20(1), 127–140. https://doi.org/10.1071/bi9670127

Muir, W. M. (2005). Incorporation of competitive effects in forest tree or animal breeding programs. Genetics, 170(3), 1247–1259. https://doi.org/10.1534/genetics.104.035956

Stringer, J. K., Cullis, B. R., & Thompson, R. (2011). Joint modeling of spatial variability and within-row interplot competition to increase the efficiency of plant improvement. Journal of Agricultural, Biological, and Environmental Statistics, 16(2), 269–281. https://doi.org/10.1007/s13253-010-0051-5

Objective

Plants compete for above- and below-ground resources

Competition can mislead selection

Competition models are not trivial

Fear not, presenting gencomp!

Example dataset

1st step: Competition matrix

1st step: Competition matrix

1st step: Competition matrix

1st step: Competition matrix

1st step: Competition matrix

1st step: Competition matrix

1st step: Competition matrix

1st step: Competition matrix

2nd step: Fitting the model

2nd step: Fitting the model

2nd step: Fitting the model

3rd step: Extracting the outputs

3rd step: Extracting the outputs

3rd step: Extracting the outputs

3rd step: Extracting the outputs

3rd step: Extracting the outputs

3rd step: Extracting the outputs

3rd step: Extracting the outputs

3rd step: Extracting the outputs

3rd step: Extracting the outputs

3rd step: Extracting the outputs

Concluding remarks

Acknowledgements

Recommended literature

Fear not, presenting `gencomp`!

1^st step: Competition matrix

1^st step: Competition matrix

1^st step: Competition matrix

1^st step: Competition matrix

1^st step: Competition matrix

1^st step: Competition matrix

1^st step: Competition matrix

1^st step: Competition matrix

2^nd step: Fitting the model

2^nd step: Fitting the model

2^nd step: Fitting the model

3^rd step: Extracting the outputs

3^rd step: Extracting the outputs

3^rd step: Extracting the outputs

3^rd step: Extracting the outputs

3^rd step: Extracting the outputs

3^rd step: Extracting the outputs

3^rd step: Extracting the outputs

3^rd step: Extracting the outputs

3^rd step: Extracting the outputs

3^rd step: Extracting the outputs