A nilpotent IP polynomial multiple recurrence theorem

We generalize the IP-polynomial Szemer\'edi theorem due to Bergelson and McCutcheon and the nilpotent Szemer\'edi theorem due to Leibman. Important tools in our proof include a generalization of Leibman's result that polynomial mappings into a nilpotent group form a group and a multiparameter version of the nilpotent Hales-Jewett theorem due to Bergelson and Leibman.Comment: v4: switch to TeXlive 2016 and biblate

Population genetics of the highly polymorphic RPP8 gene family

Plant nucleotide-binding domain and leucine-rich repeat containing (NLR) genes provide some of the most extreme examples of polymorphism in eukaryotic genomes, rivalling even the vertebrate major histocompatibility complex. Surprisingly, this is also true in Arabidopsis thaliana, a predominantly selfing species with low heterozygosity. Here, we investigate how gene duplication and intergenic exchange contribute to this extraordinary variation. RPP8 is a three-locus system that is configured chromosomally as either a direct-repeat tandem duplication or as a single copy locus, plus a locus 2 Mb distant. We sequenced 48 RPP8 alleles from 37 accessions of A. thaliana and 12 RPP8 alleles from Arabidopsis lyrata to investigate the patterns of interlocus shared variation. The tandem duplicates display fixed differences and share less variation with each other than either shares with the distant paralog. A high level of shared polymorphism among alleles at one of the tandem duplicates, the single-copy locus and the distal locus, must involve both classical crossing over and intergenic gene conversion. Despite these polymorphism-enhancing mechanisms, the observed nucleotide diversity could not be replicated under neutral forward-in-time simulations. Only by adding balancing selection to the simulations do they approach the level of polymorphism observed at RPP8. In this NLR gene triad, genetic architecture, gene function and selection all combine to generate diversity

Genetic Architecture of Flowering-Time Variation in Arabidopsis thaliana

The onset of flowering is an important adaptive trait in plants. The small ephemeral species Arabidopsis thaliana grows under a wide range of temperature and day-length conditions across much of the Northern hemisphere, and a number of flowering-time loci that vary between different accessions have been identified before. However, only few studies have addressed the species-wide genetic architecture of flowering-time control. We have taken advantage of a set of 18 distinct accessions that present much of the common genetic diversity of A. thaliana and mapped quantitative trait loci (QTL) for flowering time in 17 F2 populations derived from these parents. We found that the majority of flowering-time QTL cluster in as few as five genomic regions, which include the locations of the entire FLC/MAF clade of transcription factor genes. By comparing effects across shared parents, we conclude that in several cases there might be an allelic series caused by rare alleles. While this finding parallels results obtained for maize, in contrast to maize much of the variation in flowering time in A. thaliana appears to be due to large-effect alleles

Uniformity in the Wiener-Wintner theorem for nilsequences

We prove a uniform extension of the Wiener-Wintner theorem for nilsequences due to Host and Kra and a nilsequence extension of the topological Wiener-Wintner theorem due to Assani. Our argument is based on (vertical) Fourier analysis and a Sobolev embedding theorem.Comment: v3: 18 p., proof that the cube construction produces compact homogeneous spaces added, measurability issues in the proof of Theorem 1.5 addressed. We thank the anonymous referees for pointing out these gaps in v

A Coastal Cline in Sodium Accumulation in Arabidopsis thaliana Is Driven by Natural Variation of the Sodium Transporter AtHKT1;1

The genetic model plant Arabidopsis thaliana, like many plant species, experiences a range of edaphic conditions across its natural habitat. Such heterogeneity may drive local adaptation, though the molecular genetic basis remains elusive. Here, we describe a study in which we used genome-wide association mapping, genetic complementation, and gene expression studies to identify cis-regulatory expression level polymorphisms at the AtHKT1;1 locus, encoding a known sodium (Na+) transporter, as being a major factor controlling natural variation in leaf Na+ accumulation capacity across the global A. thaliana population. A weak allele of AtHKT1;1 that drives elevated leaf Na+ in this population has been previously linked to elevated salinity tolerance. Inspection of the geographical distribution of this allele revealed its significant enrichment in populations associated with the coast and saline soils in Europe. The fixation of this weak AtHKT1;1 allele in these populations is genetic evidence supporting local adaptation to these potentially saline impacted environments

Effects of rapid prey evolution on predator-prey cycles

We study the qualitative properties of population cycles in a predator-prey system where genetic variability allows contemporary rapid evolution of the prey. Previous numerical studies have found that prey evolution in response to changing predation risk can have major quantitative and qualitative effects on predator-prey cycles, including: (i) large increases in cycle period, (ii) changes in phase relations (so that predator and prey are cycling exactly out of phase, rather than the classical quarter-period phase lag), and (iii) "cryptic" cycles in which total prey density remains nearly constant while predator density and prey traits cycle. Here we focus on a chemostat model motivated by our experimental system [Fussmann et al. 2000,Yoshida et al. 2003] with algae (prey) and rotifers (predators), in which the prey exhibit rapid evolution in their level of defense against predation. We show that the effects of rapid prey evolution are robust and general, and furthermore that they occur in a specific but biologically relevant region of parameter space: when traits that greatly reduce predation risk are relatively cheap (in terms of reductions in other fitness components), when there is coexistence between the two prey types and the predator, and when the interaction between predators and undefended prey alone would produce cycles. Because defense has been shown to be inexpensive, even cost-free, in a number of systems [Andersson and Levin 1999, Gagneux et al. 2006,Yoshida et al. 2004], our discoveries may well be reproduced in other model systems, and in nature. Finally, some of our key results are extended to a general model in which functional forms for the predation rate and prey birth rate are not specified.Comment: 35 pages, 8 figure