90 research outputs found
Ramsey numbers of books and quasirandomness
No full textThe book graph B^((k))_n consists of n copies of K_(k+1) joined along a common K_k. The Ramsey numbers of B^((k))_n are known to have strong connections to the classical Ramsey numbers of cliques. Recently, the first author determined the asymptotic order of these Ramsey numbers for fixed k, thus answering an old question of Erdős, Faudree, Rousseau, and Schelp. In this paper, we first provide a simpler proof of this theorem. Next, answering a question of the first author, we present a different proof that avoids the use of Szemerédi's regularity lemma, thus providing much tighter control on the error term. Finally, we prove a conjecture of Nikiforov, Rousseau, and Schelp by showing that all extremal colorings for this Ramsey problem are quasirandom
Model fit and predictions for height and body mass index (BMI), based on data from [16] and [15], respectively.
No full textIn (A), we show the fit for associated loci. In (B) and (C), we show our predictions for future increases in the heritability explained and number of variants identified as genome-wide association study (GWAS) size increases. 95% CIs are based on bootstrap; see Section 7.4 in S1 Text for details.</p
The distribution of additive genetic variance among sites.
No full textIn (A), we plot the expected contribution as a function of the scaled selection coefficient. We measure genetic variance in units of vS—the expected contribution at sites under strong selection. In (B), we show the proportion of additive genetic variance that arises from sites with minor allele frequency (MAF) greater than the value on the x-axis, for different selection coefficients.</p
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Distinguishing direct interactions from global epistasis using rank statistics
No full textThe phenotypic effect of a mutation may depend on the genetic background in which it occurs, a phenomenon referred to as epistasis. One source of epistasis in proteins is direct interactions between residues in close physical proximity to one another. However, epistasis may also occur in the absence of specific interactions between amino acids if the genotype-to-phenotype map is nonlinear. Disentangling the contributions of these two phenomena—specific and global epistasis—from noisy, high-throughput mutagenesis experiments is highly nontrivial: The form of the nonlinearity is generally not known and model misspecification may lead to over- or underestimation of specific epistasis. In contrast to previous approaches, we do not attempt to model the fitness measurements directly. Rather, we begin with the observation that global epistasis, under the assumption of monotonicity, imposes strong constraints on the rank statistics of a combinatorial mutagenesis experiment. Namely, the rank-order of mutant phenotypes should be preserved across genetic backgrounds. We exploit this constraint to devise a simple semiparametric method to detect specific epistasis in the presence of global epistasis and measurement noise. We apply this method to three high-throughput mutagenesis experiments, uncovering known protein contacts with similar accuracy to existing, more complicated procedures. Our method immediately generalizes beyond proteins, providing a simple, yet powerful framework for interpreting the epistasis observed in combinatorial datasets.</p
The distribution of effect sizes corresponding to a given selection coefficient.
No full text(A) Mutations with selection coefficient, s, lie on a hypersphere in n dimensions with radius . The probability that such mutations have effect size a1 on the focal trait is proportional to the volume of the (n − 2)–dimensional cross section of the hypersphere, with projection a1 on the coordinate corresponding to the trait. (B) The distribution of effect sizes on the focal trait, conditional on the selection coefficient being s, measured in units of the distribution’s standard deviation (see Eq 11).</p
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A population genetic interpretation of GWAS findings for human quantitative traits
No full textHuman genome-wide association studies (GWASs) are revealing the genetic architecture of anthropomorphic and biomedical traits, i.e., the frequencies and effect sizes of variants that contribute to heritable variation in a trait. To interpret these findings, we need to understand how genetic architecture is shaped by basic population genetics processes—notably, by mutation, natural selection, and genetic drift. Because many quantitative traits are subject to stabilizing selection and because genetic variation that affects one trait often affects many others, we model the genetic architecture of a focal trait that arises under stabilizing selection in a multidimensional trait space. We solve the model for the phenotypic distribution and allelic dynamics at steady state and derive robust, closed-form solutions for summary statistics of the genetic architecture. Our results provide a simple interpretation for missing heritability and why it varies among traits. They predict that the distribution of variances contributed by loci identified in GWASs is well approximated by a simple functional form that depends on a single parameter: the expected contribution to genetic variance of a strongly selected site affecting the trait. We test this prediction against the results of GWASs for height and body mass index (BMI) and find that it fits the data well, allowing us to make inferences about the degree of pleiotropy and mutational target size for these traits. Our findings help to explain why the GWAS for height explains more of the heritable variance than the similarly sized GWAS for BMI and to predict the increase in explained heritability with study sample size. Considering the demographic history of European populations, in which these GWASs were performed, we further find that most of the associations they identified likely involve mutations that arose shortly before or during the Out-of-Africa bottleneck at sites with selection coefficients around s = 10−3
The combined effect of selection and changes in population size (as inferred by [89] for Europeans) on the distribution of variances among segregating sites.
No full text(A) The cumulative variance arising from sites with contributions above a threshold as a function of the threshold, for different selection coefficients. Cumulative variance is measured in units of 4u · w2/n, the equilibrium expectation for a strongly selected site, while the threshold is in units of 10−3 · w2/n. (B) The distribution of variances among loci identified in the genome-wide association study (GWAS) of height. The empirical distribution is in solid black, and our inferred fit is in dashed black. Simulation results for each selection coefficient (in color) are normalized such that the proportion of variance at the study threshold is always 1. For similar results corresponding to BMI, see Fig A20b in S1 Text, and for further details, see Section 9 in S1 Text.</p
The proportion of heritability (A) and the number of variants (B) identified in a genome-wide association study (GWAS) as a function of study size.
No full textWe assume the pleiotropic limit and a mutational target size of 1 Mb (see Sections 3.3 and 6.1 in S1 Text for derivations). For the case without pleiotropy, see Fig A22 in S1 Text.</p
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