2D association and integrative omics analysis in rice provides systems biology view in trait analysis.

The interactions among genes and between genes and environment contribute significantly to the phenotypic variation of complex traits and may be possible explanations for missing heritability. However, to our knowledge no existing tool can address the two kinds of interactions. Here we propose a novel linear mixed model that considers not only the additive effects of biological markers but also the interaction effects of marker pairs. Interaction effect is demonstrated as a 2D association. Based on this linear mixed model, we developed a pipeline, namely PATOWAS. PATOWAS can be used to study transcriptome-wide and metabolome-wide associations in addition to genome-wide associations. Our case analysis with real rice recombinant inbred lines (RILs) at three omics levels demonstrates that 2D association mapping and integrative omics are able to provide a systems biology view into the analyzed traits, leading toward an answer about how genes, transcripts, proteins, and metabolites work together to produce an observable phenotype

Uniform convergence of optimal order under a balanced norm of a local discontinuous Galerkin method on a Shishkin mesh

For singularly perturbed reaction-diffusion problems in 1D and 2D, we study a local discontinuous Galerkin (LDG) method on a Shishkin mesh. In these cases, the standard energy norm is too weak to capture adequately the behavior of the boundary layers that appear in the solutions. To deal with this deficiency, we introduce a balanced norm stronger than the energy norm. In order to achieve optimal convergence under the balanced norm in one-dimensional case, we design novel numerical fluxes and propose a special interpolation that consists of a Gauss-Radau projection and a local $L^2$ projection. Moreover, we generalize the numerical fluxes and interpolation, and extend convergence analysis of optimal order from 1D to 2D. Finally, numerical experiments are presented to confirm the theoretical results.Comment: 22pages, two table