Sign patterns that require eventual exponential nonnegativity

The matrix exponential function can be used to solve systems of linear differential equations. For certain applications, it is of interest whether or not the matrix exponential function of a given matrix becomes and remains entrywise nonnegative after some time. Such matrices are called eventually exponentially nonnegative. Often the exact numerical entries in the matrix are not known (for example due to uncertainty in experimental measurements), but the qualitative information is usually known. In this dissertation we discuss what structure on the signs of the entries of a matrix guarantees that the matrix is eventually exponentially nonnegative

Parallel developmental genetic features underlie stickleback gill raker evolution.

BackgroundConvergent evolution, the repeated evolution of similar phenotypes in independent lineages, provides natural replicates to study mechanisms of evolution. Cases of convergent evolution might have the same underlying developmental and genetic bases, implying that some evolutionary trajectories might be predictable. In a classic example of convergent evolution, most freshwater populations of threespine stickleback fish have independently evolved a reduction of gill raker number to adapt to novel diets. Gill rakers are a segmentally reiterated set of dermal bones important for fish feeding. A previous large quantitative trait locus (QTL) mapping study using a marine × freshwater F2 cross identified QTL on chromosomes 4 and 20 with large effects on evolved gill raker reduction.ResultsBy examining skeletal morphology in adult and developing sticklebacks, we find heritable marine/freshwater differences in gill raker number and spacing that are specified early in development. Using the expression of the Ectodysplasin receptor (Edar) gene as a marker of raker primordia, we find that the differences are present before the budding of gill rakers occurs, suggesting an early change to a lateral inhibition process controlling raker primordia spacing. Through linkage mapping in F2 fish from crosses with three independently derived freshwater populations, we find in all three crosses QTL overlapping both previously identified QTL on chromosomes 4 and 20 that control raker number. These two QTL affect the early spacing of gill raker buds.ConclusionsCollectively, these data demonstrate that parallel developmental genetic features underlie the convergent evolution of gill raker reduction in freshwater sticklebacks, suggesting that even highly polygenic adaptive traits can have a predictable developmental genetic basis

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