TY - JOUR AB - Extending work of Bielawski-Dancer 3 and Konno 14, we develop a theory of toric hyperkähler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study of semi-projective toric varieties, meaning GIT quotients of affine spaces by torus actions, and specifically, of Lawrence toric varieties, meaning GIT quotients of even-dimensional affine spaces by symplectic torus actions. A toric hyperkähler variety is a complete intersection in a Lawrence toric variety. Both varieties are non-compact, and they share the same cohomology ring, namely, the Stanley-Reisner ring of a matroid modulo a linear system of parameters. Familiar applications of toric geometry to combinatorics, including the Hard Lefschetz Theorem and the volume polynomials of Khovanskii-Pukhlikov 11, are extended to the hyperkähler setting. When the matroid is graphic, our construction gives the toric quiver varieties, in the sense of Nakajima 17. AU - Hausel, Tamas AU - Sturmfels, Bernd ID - 1451 IS - 1 JF - Documenta Mathematica SN - 1431-0635 TI - Toric hyperkähler varieties VL - 7 ER - TY - JOUR AB - Transcription is a slow and expensive process: in eukaryotes, approximately 20 nucleotides can be transcribed per second at the expense of at least two ATP molecules per nucleotide. Thus, at least for highly expressed genes, transcription of long introns, which are particularly common in mammals, is costly. Using data on the expression of genes that encode proteins in Caenorhabditis elegans and Homo sapiens, we show that introns in highly expressed genes are substantially shorter than those in genes that are expressed at low levels. This difference is greater in humans, such that introns are, on average, 14 times shorter in highly expressed genes than in genes with low expression, whereas in C. Elegans the difference in intron length is only twofold. In contrast, the density of introns in a gene does not strongly depend on the level of gene expression. Thus, natural selection appears to favor short introns in highly expressed genes to minimize the cost of transcription and other molecular processes, such as splicing. AU - Castillo Davis, Cristian AU - Mekhedov, Sergei AU - Hartl, Daniel AU - Koonin, Eugene AU - Kondrashov, Fyodor ID - 897 IS - 4 JF - Nature Genetics TI - Selection for short introns in highly expressed genes VL - 31 ER - TY - JOUR AB - BACKGROUND: Gene duplications have a major role in the evolution of new biological functions. Theoretical studies often assume that a duplication per se is selectively neutral and that, following a duplication, one of the gene copies is freed from purifying (stabilizing) selection, which creates the potential for evolution of a new function. RESULTS: In search of systematic evidence of accelerated evolution after duplication, we used data from 26 bacterial, six archaeal, and seven eukaryotic genomes to compare the mode and strength of selection acting on recently duplicated genes (paralogs) and on similarly diverged, unduplicated orthologous genes in different species. We find that the ratio of nonsynonymous to synonymous substitutions (Kn/Ks) in most paralogous pairs is <<1 and that paralogs typically evolve at similar rates, without significant asymmetry, indicating that both paralogs produced by a duplication are subject to purifying selection. This selection is, however, substantially weaker than the purifying selection affecting unduplicated orthologs that have diverged to the same extent as the analyzed paralogs. Most of the recently duplicated genes appear to be involved in various forms of environmental response; in particular, many of them encode membrane and secreted proteins. CONCLUSIONS: The results of this analysis indicate that recently duplicated paralogs evolve faster than orthologs with the same level of divergence and similar functions, but apparently do not experience a phase of neutral evolution. We hypothesize that gene duplications that persist in an evolving lineage are beneficial from the time of their origin, due primarily to a protein dosage effect in response to variable environmental conditions; duplications are likely to give rise to new functions at a later phase of their evolution once a higher level of divergence is reached. AU - Kondrashov, Fyodor AU - Rogozin, Igor AU - Wolf, Yuri AU - Koonin, Eugene ID - 871 IS - 2 JF - Genome Biology SN - 1465-6906 TI - Selection in the evolution of gene duplications VL - 3 ER - TY - JOUR AB - The polymeric ubiquitin (poly-u) genes are composed of tandem 228-bp repeats with no spacer sequences between individual monomer units. Ubiquitin is one of the most conserved proteins known to date, and the individual units within a number of poly-u genes are significantly more similar to each other than would be expected if each unit evolved independently. It has been proposed that the rather striking similarity among poly-u monomers in some lineages is caused by a series of homogenization events. Here we report the sequences of the polyubiquitin-C (Ubc) genes in two mouse strains. Analysis of these sequences, as well as those of the previously reported Chinese hamster and rat poly-u genes, supports the assertion that the homogenization of the ubiquitin-C gene in rodents is due to unequal crossing-over events. The sequence divergence of noncoding DNA was used to estimate the frequency of unequal crossing-over events (6.3 x 10-5 events per generation) in the Ubc gene, as well as to provide evidence of apparent selection in the poly-u gene. AU - Perelygin, Andrey AU - Kondrashov, Fyodor AU - Rogozin, Igor AU - Brinton, Margo ID - 859 IS - 2 JF - Journal of Molecular Evolution SN - 0022-2844 TI - Evolution of the mouse polyubiquitin C gene VL - 55 ER - TY - JOUR AB - Let k⩾5 be an integer, and let x⩾1 be an arbitrary real number. We derive a bound[Formula presented] for the number of positive integers less than or equal to x which can be represented as a sum of two non-negative coprime kth powers, in essentially more than one way. AU - Browning, Timothy D ID - 204 IS - 2 JF - Journal of Number Theory SN - 0022-314X TI - Equal Sums of Two kth Powers VL - 96 ER -