BackgroundRecombination rates vary at the level of the species, population and individual. Now recognized as a transient feature of the genome, recombination rates at a given locus can change markedly over time. Existing inferential methods, predominantly based on linkage disequilibrium patterns, return a long-term average estimate of past recombination rates. Such estimates can be misleading, but no analytical framework to infer recombination rates that have changed over time is currently available.ResultsWe apply coalescent modeling in conjunction with a suite of summary statistics to show that the recombination history of a locus can be reconstructed from a time series of genetic samples. More usefully, we describe a new method, based on n-tuple dataset subsampling, to infer past changes in recombination rate from DNA sequences taken at a single time point. This subsampling strategy can correctly assign simulated loci to constant, increasing and decreasing recombination models with an accuracy of 84%.ConclusionsWhile providing an important stepping-stone to determining past recombination rates, n-tuple subsampling still exhibits a moderate error rate. Theoretical limitations indicated by coalescent theory suggest that highly accurate inference of past recombination rates will remain challenging. Nevertheless, we show for the first time that reconstructing historic recombination rates is possible in principle.

We study a class of general purpose linear multisymplectic integrators for Hamiltonian wave equations based on a diamond-shaped mesh. On each diamond, the PDE is discretized by a symplectic Runge-Kutta method. The scheme advances in time by filling in each diamond locally. We demonstrate that this leads to greater efficiency and parallelization and easier treatment of boundary conditions compared to methods based on rectangular meshes. We develop a variety of initial and boundary value treatments and present numerical evidence of their performance. In all cases, the observed order of convergence is equal to or greater than the number of stages of the underlying Runge-Kutta method.1 That is, if u 1 , u 2 are solutions to the variational equation Kut + Lux = S (z)u, then ω(

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