*4.1. Plant Material, Field Trials, Phenotyping, and Statistical Analyses*

A total of 200 *L. usitatissimum* accessions from the Canadian flax core collection [67] were selected for this study based on their geographic distribution and genetic diversity (Table S3). The 200 genotypes were planted in 2014 and 2015 at the Agriaquaculture Nutritional Genomic Center (CGNA) experimental stations located in Vilcún (CAR2014) and Huichahue (HU2015), La Araucania region, Chile, using a completely randomized design (CRD) with three biological replicates. Genotypes were arranged in rows and columns in order to take into account spatial heterogeneity.

The seed mucilage content (MC) was determined in three biological replicates following the procedure described by Kaewmanee et al. [4] with minor modifications. A total of 2 g of seeds were incubated in 20 mL of water at 100 ◦C for 15 min in 50 mL Falcon tubes. Next, the tubes were shaken for 30 min at 250 rpm. The soluble extract was recovered by centrifugation at 6132 relative centrifugal force (RCF) for 30 min, and the mucilage fraction was precipitated by incubating in 30 mL of ethanol (95%) overnight at 4 ◦C. The seeds were recovered, and the extraction procedure was carried out twice more to maximize mucilage recovery. The mucilage pellet was weighed and expressed as milligrams of mucilage per gram of seed (mg g−<sup>1</sup> ).

HC was determined in three biological replicates by separating the hull from the embryos using a dissecting needle and tweezers from 50 seeds after imbibition in water for 24 h. Both fractions were dried at 90 ◦C for 4 h before their dry weights were measured. HC was expressed as (hull dry weight/(hull dry weight + embryo dry weight)) × 100, averaged from 50 seeds.

Variation of phenotypic data was analyzed individually for each environment using a restricted maximum likelihood (REML) analysis. Spatial correction in row and column directions was used with different variance–covariance structures. Spatial models were compared with Akaike information criterion (AIC) and Bayesian information criterion (BIC), and the most appropriate model in each environment was used to obtain a best linear unbiased estimate (BLUEs) for mucilage and hull contents in GenStat v.16 [68]. Descriptive statistics and Shapiro–Wilk normality test were conducted in the R package MVN [69]. Narrow sense heritability (*h* 2 ) was estimated using variance components from TASSEL v.5.2.31 [70]. Trait *h* 2 estimates were computed using the equation: *h* <sup>2</sup> = σ 2 <sup>a</sup>/σ 2 <sup>a</sup> + σ 2 <sup>e</sup>, where σ 2 <sup>a</sup> is the additive genetic variance and σ 2 <sup>e</sup> is the residual error variance [70].
