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Consider the tight binding model of graphene, sharply terminated along an edgelparallel to a direction of translational symmetry of the underlying period lattice. We classify such edgeslinto those of “zigzag type” and those of “armchair type”, generalizing the classical zigzag and armchair edges. We prove that zero energy / flat band edge states arise for edges of zigzag type, but never for those of armchair type. We exhibit explicit formulas for flat band edge states when they exist. We produce strong evidence for the existence of dispersive (non flat) edge state curves of nonzero energy for mostl.

 

Animal traits develop through the expression and action of numerous regulatory and realizator genes that comprise a gene regulatory network (GRN). For each GRN, its underlying patterns of gene expression are controlled by cis -regulatory elements (CREs) that bind activating and repressing transcription factors. These interactions drive cell-type and developmental stage-specific transcriptional activation or repression. Most GRNs remain incompletely mapped, and a major barrier to this daunting task is CRE identification. Here, we used an in silico method to identify predicted CREs (pCREs) that comprise the GRN which governs sex-specific pigmentation of Drosophila melanogaster . Through in vivo assays, we demonstrate that many pCREs activate expression in the correct cell-type and developmental stage. We employed genome editing to demonstrate that two CREs control the pupal abdomen expression of trithorax , whose function is required for the dimorphic phenotype. Surprisingly, trithorax had no detectable effect on this GRN’s key trans -regulators, but shapes the sex-specific expression of two realizator genes. Comparison of sequences orthologous to these CREs supports an evolutionary scenario where these trithorax CREs predated the origin of the dimorphic trait. Collectively, this study demonstrates how in silico approaches can shed novel insights on the GRN basis for a trait’s development and evolution.