The Burau estimate for the entropy of a braid

The topological entropy of a braid is the infimum of the entropies of all homeomorphisms of the disc which have a finite invariant set represented by the braid. When the isotopy class represented by the braid is pseudo-Anosov or is reducible with a pseudo-Anosov component, this entropy is positive. Fried and Kolev proved that the entropy is bounded below by the logarithm of the spectral radius of the braid's Burau matrix, $B(t)$, after substituting a complex number of modulus~1 in place of $t$. In this paper we show that for a pseudo-Anosov braid the estimate is sharp for the substitution of a root of unity if and only if it is sharp for $t=-1$. Further, this happens if and only if the invariant foliations of the pseudo-Anosov map have odd order singularities at the strings of the braid and all interior singularities have even order. An analogous theorem for reducible braids is also proved.Comment: 28 pages, 8 figure

Bayesian meta-analysis across genome-wide association studies of diverse phenotypes

Genome-wide association studies (GWAS) are a powerful tool for understanding the genetic basis of diseases and traits, but most studies have been conducted in isolation, with a focus on either a single or a set of closely related phenotypes. We describe MetABF, a simple Bayesian framework for performing integrative meta-analysis across multiple GWAS using summary statistics. The approach is applicable across a wide range of study designs and can increase the power by 50% compared with standard frequentist tests when only a subset of studies have a true effect. We demonstrate its utility in a meta-analysis of 20 diverse GWAS which were part of the Wellcome Trust Case Control Consortium 2. The novelty of the approach is its ability to explore, and assess the evidence for a range of possible true patterns of association across studies in a computationally efficient framework.Peer reviewe

Human candidate gene polymorphisms and risk of severe malaria in children in Kilifi, Kenya: a case-control association study

Background: Human genetic factors are important determinants of malaria risk. We investigated associations between multiple candidate polymorphisms—many related to the structure or function of red blood cells—and risk for severe Plasmodium falciparum malaria and its specific phenotypes, including cerebral malaria, severe malaria anaemia, and respiratory distress. Methods: We did a case-control study in Kilifi County, Kenya. We recruited as cases children presenting with severe malaria to the high-dependency ward of Kilifi County Hospital. We included as controls infants born in the local community between Aug 1, 2006, and Sept 30, 2010, who were part of a genetics study. We tested for associations between a range of candidate malaria-protective genes and risk for severe malaria and its specific phenotypes. We used a permutation approach to account for multiple comparisons between polymorphisms and severe malaria. We judged p values less than 0·005 significant for the primary analysis of the association between candidate genes and severe malaria. Findings: Between June 11, 1995, and June 12, 2008, 2244 children with severe malaria were recruited to the study, and 3949 infants were included as controls. Overall, 263 (12%) of 2244 children with severe malaria died in hospital, including 196 (16%) of 1233 with cerebral malaria. We investigated 121 polymorphisms in 70 candidate severe malaria-associated genes. We found significant associations between risk for severe malaria overall and polymorphisms in 15 genes or locations, of which most were related to red blood cells: ABO, ATP2B4, ARL14, CD40LG, FREM3, INPP4B, G6PD, HBA (both HBA1 and HBA2), HBB, IL10, LPHN2 (also known as ADGRL2), LOC727982, RPS6KL1, CAND1, and GNAS. Combined, these genetic associations accounted for 5·2% of the variance in risk for developing severe malaria among individuals in the general population. We confirmed established associations between severe malaria and sickle-cell trait (odds ratio [OR] 0·15, 95% CI 0·11–0·20; p=2·61 × 10−58), blood group O (0·74, 0·66–0·82; p=6·26 × 10−8), and –α3·7-thalassaemia (0·83, 0·76–0·90; p=2·06 × 10−6). We also found strong associations between overall risk of severe malaria and polymorphisms in both ATP2B4 (OR 0·76, 95% CI 0·63–0·92; p=0·001) and FREM3 (0·64, 0·53–0·79; p=3·18 × 10−14). The association with FREM3 could be accounted for by linkage disequilibrium with a complex structural mutation within the glycophorin gene region (comprising GYPA, GYPB, and GYPE) that encodes for the rare Dantu blood group antigen. Heterozygosity for Dantu was associated with risk for severe malaria (OR 0·57, 95% CI 0·49–0·68; p=3·22 × 10−11), as was homozygosity (0·26, 0·11–0·62; p=0·002). Interpretation: Both ATP2B4 and the Dantu blood group antigen are associated with the structure and function of red blood cells. ATP2B4 codes for plasma membrane calcium-transporting ATPase 4 (the major calcium pump on red blood cells) and the glycophorins are ligands for parasites to invade red blood cells. Future work should aim at uncovering the mechanisms by which these polymorphisms can result in severe malaria protection and investigate the implications of these associations for wider health. Funding: Wellcome Trust, UK Medical Research Council, European Union, and Foundation for the National Institutes of Health as part of the Bill & Melinda Gates Grand Challenges in Global Health Initiative

The correlation between reading and mathematics ability at age twelve has a substantial genetic component

Dissecting how genetic and environmental influences impact on learning is helpful for maximizing numeracy and literacy. Here we show, using twin and genome-wide analysis, that there is a substantial genetic component to children’s ability in reading and mathematics, and estimate that around one half of the observed correlation in these traits is due to shared genetic effects (so-called Generalist Genes). Thus, our results highlight the potential role of the learning environment in contributing to differences in a child’s cognitive abilities at age twelve

Evidence of the interplay of genetics and culture in Ethiopia.

Funder: RCUK | Natural Environment Research Council (NERC); doi: https://doi.org/10.13039/501100000270The rich linguistic, ethnic and cultural diversity of Ethiopia provides an unprecedented opportunity to understand the level to which cultural factors correlate with-and shape-genetic structure in human populations. Using primarily new genetic variation data covering 1,214 Ethiopians representing 68 different ethnic groups, together with information on individuals' birthplaces, linguistic/religious practices and 31 cultural practices, we disentangle the effects of geographic distance, elevation, and social factors on the genetic structure of Ethiopians today. We provide evidence of associations between social behaviours and genetic differences among present-day peoples. We show that genetic similarity is broadly associated with linguistic affiliation, but also identify pronounced genetic similarity among groups from disparate language classifications that may in part be attributable to recent intermixing. We also illustrate how groups reporting the same culture traits are more genetically similar on average and show evidence of recent intermixing, suggesting that shared cultural traits may promote admixture. In addition to providing insights into the genetic structure and history of Ethiopia, we identify the most important cultural and geographic predictors of genetic differentiation and provide a resource for designing sampling protocols for future genetic studies involving Ethiopians

Two complement receptor one alleles have opposing associations with cerebral malaria and interact with α+thalassaemia

Malaria has been a major driving force in the evolution of the human genome. In sub-Saharan African populations, two neighbouring polymorphisms in the Complement Receptor One (CR1) gene, named Sl2 and McCb, occur at high frequencies, suggesting selection by malaria. Previous studies have been inconclusive. Using a large case-control study of severe malaria in Kenyan children and statistical models adjusted for confounders, we demonstrate that Sl2 and McCb have opposing malaria associations. The Sl2 polymorphism is associated with markedly reduced odds of cerebral malaria and death, while the McCb polymorphism is associated with increased odds of cerebral malaria. We also identified an unexpected interaction between Sl2 and α+thalassaemia, revealing that the protective association of Sl2 was greatest in children with normal α-globin. The complex relationship between these three mutations may explain previous conflicting findings, and the data highlight the importance of considering genetic interactions in disease-association studies