High-Order Iterative Scheme for a Viscoelastic Wave Equation and Numerical Results

In this paper, we consider a Robin problem for a viscoelastic wave equation. First, by the high-order iterative method coupled with the Galerkin method, the existence of a recurrent sequence via an N -order iterative scheme is established, and then the N -order convergent rate of the obtained sequence to the unique weak solution of the proposed model is also proved. Next, with N = 2 , a numerical algorithm given by the finite-difference method is constructed to approximate the solution via the 2-order iterative scheme. Moreover, the same algorithm for the single-iterative scheme generated by the 2-order iterative scheme is also considered. Finally, comparison with errors of the numerical solutions obtained by the single-iterative scheme and the 2-order iterative scheme shows that the convergent rate of the 2-order iterative scheme is faster than that of the single-iterative scheme.

Quadric SFDI for Laplacian Discretisation in Lagrangian Meshless Methods

In the Lagrangian meshless (particle) methods, such as the smoothed particle hydrodynamics (SPH), moving particle semi-implicit (MPS) method and meshless local Petrov-Galerkin method based on Rankine source solution (MLPG_R), the Laplacian discretisation is often required in order to solve the governing equations and/or estimate physical quantities (such as the viscous stresses). In some meshless applications, the Laplacians are also needed as stabilisation operators to enhance the pressure calculation. The particles in the Lagrangian methods move following the material velocity, yielding a disordered (random) particle distribution even though they may be distributed uniformly in the initial state. Different schemes have been developed for a direct estimation of second derivatives using finite difference, kernel integrations and weighted/moving least square method. Some of the schemes suffer from a poor convergent rate. Some have a better convergent rate but require inversions of high order matrices, yielding high computational costs. This paper presents a quadric semi-analytical finite-difference interpolation (QSFDI) scheme, which can achieve the same degree of the convergent rate as the best schemes available to date but requires the inversion of significant lower-order matrices, i.e. 3 × 3 for 3D cases, compared with 6 × 6 or 10 × 10 in the schemes with the best convergent rate. Systematic patch tests have been carried out for either estimating the Laplacian of given functions or solving Poisson’s equations. The convergence, accuracy and robustness of the present schemes are compared with the existing schemes. It will show that the present scheme requires considerably less computational time to achieve the same accuracy as the best schemes available in literatures, particularly for estimating the Laplacian of given functions.

RERconverge: an R package for associating evolutionary rates with convergent traits

Motivation When different lineages of organisms independently adapt to similar environments, selection often acts repeatedly upon the same genes, leading to signatures of convergent evolutionary rate shifts at these genes. With the increasing availability of genome sequences for organisms displaying a variety of convergent traits, the ability to identify genes with such convergent rate signatures would enable new insights into the molecular basis of these traits. Results Here we present the R package RERconverge, which tests for association between relative evolutionary rates of genes and the evolution of traits across a phylogeny. RERconverge can perform associations with binary and continuous traits, and it contains tools for visualization and enrichment analyses of association results. Availability and implementation RERconverge source code, documentation and a detailed usage walk-through are freely available at https://github.com/nclark-lab/RERconverge. Datasets for mammals, Drosophila and yeast are available at https://bit.ly/2J2QBnj. Supplementary information Supplementary data are available at Bioinformatics online.

Rate variation in the evolution of non-coding DNA associated with social evolution in bees

The evolutionary origins of eusociality represent increases in complexity from individual to caste-based, group reproduction. These behavioural transitions have been hypothesized to go hand in hand with an increased ability to regulate when and where genes are expressed. Bees have convergently evolved eusociality up to five times, providing a framework to test this hypothesis. To examine potential links between putative gene regulatory elements and social evolution, we compare alignable, non-coding sequences in 11 diverse bee species, encompassing three independent origins of reproductive division of labour and two elaborations of eusocial complexity. We find that rates of evolution in a number of non-coding sequences correlate with key social transitions in bees. Interestingly, while we find little evidence for convergent rate changes associated with independent origins of social behaviour, a number of molecular pathways exhibit convergent rate changes in conjunction with subsequent elaborations of social organization. We also present evidence that many novel non-coding regions may have been recruited alongside the origin of sociality in corbiculate bees; these loci could represent gene regulatory elements associated with division of labour within this group. Thus, our findings are consistent with the hypothesis that gene regulatory innovations are associated with the evolution of eusociality and illustrate how a thorough examination of both coding and non-coding sequence can provide a more complete understanding of the molecular mechanisms underlying behavioural evolution. This article is part of the theme issue ‘Convergent evolution in the genomics era: new insights and directions’.

Rate variation in the evolution of non-coding DNA associated with social evolution in bees

The evolutionary origins of eusociality represent increases in complexity from individual to caste-based, group reproduction. These behavioral transitions have been hypothesized to go hand-in-hand with an increased ability to regulate when and where genes are expressed. Bees have convergently evolved eusociality up to five times, providing a framework to test this hypothesis. To examine potential links between putative gene regulatory elements and social evolution, we compare alignable, non-coding sequences in eleven diverse bee species, encompassing three independent origins of reproductive division of labor and two elaborations of eusocial complexity. We find that rates of evolution in a number of non-coding sequences correlate with key social transitions in bees. Interestingly, while we find little evidence for convergent rate changes associated with independent origins of social behavior, a number of molecular pathways exhibit convergent rate changes in conjunction with subsequent elaborations of social organization. We also present evidence that many novel non-coding regions may have been recruited alongside the origin of sociality in corbiculate bees; these loci could represent gene regulatory elements associated with division of labor within this group. Thus, our findings are consistent with the hypothesis that gene regulatory innovations are associated with the evolution of eusociality and illustrate how a thorough examination of both coding and non-coding sequence can provide a more complete understanding of the molecular mechanisms underlying behavioral evolution.

RERconverge: an R package for associating evolutionary rates with convergent traits

Motivation: When different lineages of organisms independently adapt to similar environments, selection often acts repeatedly upon the same genes, leading to signatures of convergent evolutionary rate shifts at these genes. With the increasing availability of genome sequences for organisms displaying a variety of convergent traits, the ability to identify genes with such convergent rate signatures would enable new insights into the molecular basis of these traits.Results: Here we present the R package RERconverge, which tests for association between relative evolutionary rates of genes and the evolution of traits across a phylogeny. RERconverge can perform associations with binary and continuous traits, and it contains tools for visualization and enrichment analyses of association results.Availability: RERconverge source code, documentation, and a detailed usage walk-through are freely available at https://github.com/nclark-lab/RERconverge. Datasets for mammals, Drosophila, and yeast are available at https://bit.ly/2J2QBnj.Contact:[email protected] information: Supplementary information, containing detailed vignettes for usage of RERconverge, are available at Bioinformatics online.

Bayesian detection of convergent rate changes of conserved noncoding elements on phylogenetic trees

Conservation of DNA sequence over evolutionary time is a strong indicator of function, and gain or loss of sequence conservation can be used to infer changes in function across a phylogeny. Changes in evolutionary rates on particular lineages in a phylogeny can indicate shared functional shifts, and thus can be used to detect genomic correlates of phenotypic convergence. However, existing methods do not allow easy detection of patterns of rate variation, which causes challenges for detecting convergent rate shifts or other complex evolutionary scenarios. Here we introduce PhyloAcc, a new Bayesian method to model substitution rate changes in conserved elements across a phylogeny. The method assumes several categories of substitution rate for each branch on the phylogenetic tree, estimates substitution rates per category, and detects changes of substitution rate as the posterior probability of a category switch. Simulations show that PhyloAcc can detect genomic regions with rate shifts in multiple target species better than previous methods and has a higher accuracy of reconstructing complex patterns of substitution rate changes than prevalent Bayesian relaxed clock models. We demonstrate the utility of PhyloAcc in two classic examples of convergent phenotypes: loss of flight in birds and the transition to marine life in mammals. In each case, our approach reveals numerous examples of conserved non-exonic elements with accelerations specific to the phenotypically convergent lineages. Our method is widely applicable to any set of conserved elements where multiple rate changes are expected on a phylogeny.