Approximating Physiological Nonlinear Genotype-Phenotype Models by Linear-Bilinear Models
Paulo C. Rodrigues1,2, Fred A. van Eeuwijk2
1Department of Mathematics, Universidade Nova de Lisboa, Portugal; 2Biometris, Wageningen University, Netherlands

Genotype by environment interactions (GEI) is frequent in multi-location trials, and represents differential responses of genotypes across environments. With the development of molecular markers and mapping techniques, researchers can go one step further and analyse the whole genome to detect specific locations of genes which influence a quantitative trait. These specific locations are called quantitative trait loci (QTL), and when these QTLs have different expression across environments we talk about QTL by environment interactions (QEI), which is the base of GEI. Simulation tools such as genotype to phenotype models have proved to be useful to better understand these GEI and QEI (van Eeuwijk et al., 2010).

In this paper, a genotype to phenotype model (i.e. a non-linear function of purely genotypic components and environmental inputs which integrates over time) with 7 physiological parameters, is used to simulate two-way tables with yield. Statistical techniques such as Factorial Regression models (Denis, 1998) are used to approximate the outcome of the non-linear genotype to phenotype model as a linear-bilinear model, by using genotypic inputs (e.g. physiological parameters, marker information, etc.) and environmental covariables. This will led to a better understand the GEI and QEI of the two-way tables, and allows to make predictions for new environments.

To evaluate the behaviour of yield along the growing season, we have simulated the two-way table for five different time points equally spaced. This results in a three-way data table with 200 genotypes, 36 environments and 5 time points. Three-way procrustean models (Borg and Groenen, 2005) and canonical polyadic decomposition (also known as PARAFAC; Kroonenberg, 1983) were used to understand the behaviour along the growing season.

The crop growth model was calibrated for sweet pepper (Capsicum annuum L.), which represents the case study of this paper.


Borg, I. and Groenen P.J.F. (2005). Modern Multidimensional Scaling. Springer, New York: 449-472.

Denis J.B. (1988) Two way analysis using covarites. Statistics 19, 123–132.

Kroonenberg P.M. (1983). Three-mode principal component analysis: theory and applications. DSWO Press, The Netherlands.

van Eeuwijk F.A., Bink, M.C.A.M., Chenu, K. and Chapman, S.C. (2010). Detection and use of QTL for complex traits in multiple environments. Current Opinion in Plant Biology 13, 193-205.

Keywords: Genotype by environment interactions; Factorial regression; Procrustean models; Canonical polyadic decomposition

Biography: I'm PhD Candidate at Universidade Nova de Lisboa, Portugal, and Wageningen University, The Netherlands. My main research is connected with statistical applications to plant breeding and genetics, with particular interest in exploring and understanding genotype by environment interactions and QTL by environment interactions, and their relation with genotype to phenotype crop growth models.

I'm also working actively to promote the activities of Young Statisticians, being Founding Chair of the y-BIS, the Young People's group in the International Society for Business and Industrial Statistics (ISBIS), ISBIS Vice President, and Co-Founder of the jSPE, the Section of Young Statisticians in the Portuguese Statistical Society. More details in