# Spatio-temporal model for the genetic resistance to ash-dieback

Facundo Munoz, Arnaud Dowkiw

► To cite this version:

Facundo Munoz, Arnaud Dowkiw. Spatio-temporal model for the genetic resistance to ash-dieback.
VIBASS València International Bayesian Analysis Summer School, Jul 2017, Valencia, Spain. hal-01569509

HAL Id: hal-01569509

https://hal.science/hal-01569509v1

Submitted on 5 Jun 2020

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY-SA 4.0 - Attribution - ShareAlike - International
License
Facundo Muñoz, Arnaud Dowkiw

## Ash Dieback

* Invasive **fungal disease** of european ash trees causing large numbers of deaths and **threatening the survival** of the species

* 2 symptoms: *Crown Dieback (CD)* and *Collar Lesion (CL)*

<figure><img src="image_2.png" /></figure>

## Scientific goals

* Estimate genetic parameters for the resistance

* Assess genetic covariation across symptoms

* Account for differences in exposure to pathogen across space and time

* Quantify the effect of potential covariates

## Data

* N = 777 trees from 23 OP progenies.

* CD assesed visually in a 6-level scale. CL measured as a girdling index.

* Other variables: bud flush precocity (BF) provenance (3-levels), and basal circumference

## Crown Dieback progress and model

<figure><img src="image_3.png" /><figcaption>Figure: Spatio-temporal progression of CD. Each pixel is a tree.</figcaption></figure>

After a normalizing transformation and a variable selection procedure, we model the response in year *i*, location *j* for the individual *k* as:

$$ 
\begin{align} 
y_{ijk} &= \text{Year}_i + \text{BF}_k + \eta_{ij} + a_k + \varepsilon_{ijk} \nonumber \\ 
\eta &\sim \mathcal{N}(\mathbf{0}, \tau_{\eta}^2 \mathbf{Q}(\rho_t, \rho_s)) \nonumber \\ 
a &\sim \mathcal{N}(\mathbf{0}, \tau_a^2 \mathbf{A}^{-1}) \nonumber \\ 
\varepsilon &\sim \mathcal{N}(\mathbf{0}, \mathbf{I}), \tag{1} 
\end{align} 
$$

which accounts for the year-wise progression of the disease, its spatio-temporal variability, the effect of the individual bud flush precocity, the individual additive genetic effect (a.k.a. *Breeding Value*) and the residual effects.

* The parametric structure matrix $\mathbf{Q}(\rho_t, \rho_s)$ is the tensor product of a Matérn spatial process and a exchangeable temporal structure

* The relationship matrix $\mathbf{A}$ is determined from the family kinkship

* Prior spatial range constrained within the field dimensions. Other priors are *reasonably* vague. See paper for details.

## Results for CD

<figure><img src="image_4.png" /><figcaption>Figure: Posterior means and 95% credible intervals of BF and family effects.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure: Posterior mean spatio-temporal effect for BF = 3</figcaption></figure>

## Collar Lesion progress and model

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>Figure: Marginal and spatio-temporal progression of CL.</figcaption></figure>

* No normalizing transformation possible here: too many zeros we model the response in year *i*, location *j* for the individual *k* as:
  For a measurement of CL $y_{ijk}$ taken in year *i*, at location *j* for individual *k*, we assume that

$$ 
\begin{align} 
\Pr[y_{ijk} = 0] &= p_{ijk}, \tag{2} \\ 
\pi(y_{ijk} | y_{ijk} > 0) &= \text{Ga}(a_{ijk}, b_{ijk}), \quad a_{ijk}, b_{ijk} > 0. \nonumber 
\end{align} 
$$

We define a hierarchical model for the parameters *p*, *a* and *b* using appropriate link functions of the expected values of the respective distributions. Specifically, calling $\mu = E(y|y > 0) = a/b$, we define two linear predictors

$$ 
\begin{align} 
\operatorname{logit}(p_{ijk}) &= \text{Year}_i^{(1)} + \eta_{ij}^{(1)} + a_k^{(1)} \nonumber \\ 
\operatorname{log}(\mu_{ijk}) &= \text{Year}_i^{(2)} + \eta_{ij}^{(2)} + a_k^{(2)} \tag{3} 
\end{align} 
$$

for the binary and continuous components, respectively. The elements in the latent linear predictors are defined as for Eq. 1

## Results for CL

<figure><img src="image_8.png" /><figcaption>Figure: Posterior mean spatio-temporal effect and empirical genetic correlations</figcaption></figure>

## Conclusions

* Narrow-sense heritabilities estimated at .42, .49 and .42 for CD, CL prevalence (Binomial component) and CL severity (Gamma component) respectively.

* Genetic correlations of up to .4 between the severities of crown and collar symptoms

* No evidence for differences between provenances. More genetic variability within than between families.

* Early flushing correlates with healthier crown.

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>