We study a model of elastic surfaces that was first constructed by Baillie et al. for an interpolation between the models of fluid and crystalline membranes. The Hamiltonian of the model is a linear combination of the Gaussian energy and a squared scalar curvature energy. These energy terms are discretized on dynamically triangulated surfaces that are allowed to self-intersect. We confirm that the model has not only crumpled phases but also a branched polymer phase, and find that the model undergoes a first-order phase transition between the branched polymer phase and one of the crumpled phases. We find also that the model undergoes a second-͑or higher-͒ order phase transition between the branched polymer phase and another crumpled phase.
The cloning of agriculturally important genes is often complicated by haplotype variation across crop cultivars. Access to pan-genome information greatly facilitates the assessment of structural variations and rapid candidate gene identification. Here, we identified the red glume 1 (Rg-B1) gene using association genetics and haplotype analyses in ten reference grade wheat genomes. Glume color is an important trait to characterize wheat cultivars. Red glumes are frequent among Central European spelt, a dominant wheat subspecies in Europe before the 20th century. We used genotyping-by-sequencing to characterize a global diversity panel of 267 spelt accessions, which provided evidence for two independent introductions of spelt into Europe. A single region at the Rg-B1 locus on chromosome 1BS was associated with glume color in the diversity panel. Haplotype comparisons across ten high-quality wheat genomes revealed a MYB transcription factor as candidate gene. We found extensive haplotype variation across the ten cultivars, with a particular group of MYB alleles that was conserved in red glume wheat cultivars. Genetic mapping and transient infiltration experiments allowed us to validate this particular MYB transcription factor variants. Our study demonstrates the value of multiple high-quality genomes to rapidly resolve copy number and haplotype variations in regions controlling agriculturally important traits.
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