The transitivity of the Hardy-Weinberg law

The reduction of multi-allelic polymorphisms to variants with fewer alleles, two in the limit, is addressed. The Hardy-Weinberg law is shown to be transitive in the sense that a multi-allelic polymorphism that is in equilibrium will retain its equilibrium status if any allele together with its corresponding genotypes is deleted from the population. Similarly, the transitivity principle also applies if alleles are joined, which leads to the summation of allele frequencies and their corresponding genotype frequencies. These basic polymorphism properties are intuitive, but they have apparently not been formalized or investigated. This article provides a straightforward proof of the transitivity principle, and its usefulness in practical genetic data analysis with multi-allelic markers is explored. In general, results of statistical tests for Hardy-Weinberg equilibrium obtained with polymorphisms that are reduced by deletion or joining of alleles are seen to be consistent with the formulated transitivity principle. We also show how the transitivity principle allows one to identify equilibrium-offending alleles, and how it can provide clues to genotyping problems and evolutionary changes. For microsatellites, which are widely used in forensics, the transitivity principle implies one expects similar results for statistical tests that use length-based and sequence-based alleles. High-quality autosomal microsatellite databases of the US National Institute of Standards and Technology are used to illustrate the use of the transitivity principle in testing both length-based and sequence-based microsatellites for Hardy-Weinberg proportions. Test results for Hardy-Weinberg proportions for the two types of microsatellites are seen to be largely consistent and can detect allele imbalance.

Nano-ZrO2/TiO2 Impregnated Orange Wood Sawdust and Peach Stone Shell Adsorbents for Cr (VI) Removal

Background: In this work, Cr (VI) adsorption on nano-ZrO2๏TiO2 impregnated orange wood sawdust (Zr๏Ti/OWS) and nano-ZrO2๏TiO2 impregnated peach stone shell (Zr๏Ti/PSS) was investigated by applying different adsorption parameters such as Cr (VI) concentrations, contact time, adsorbent dose, and pH for all adsorbents. Methods: The adsorbents were characterized by SEM and FT-IR. The equilibrium status was achieved after 120 min of contact time and optimum pH value around 2 were determined for Cr (VI) adsorption. Adsorption data in the equilibrium is well-assembled by the Langmuir model during the adsorption process. Results: Langmuir isotherm model showed a maximum adsorption value of OWS: 21.65 mg/g and Zr๏Ti/OWS: 27.25 mg/g. The same isotherm displayed a maximum adsorption value of PSS: 17.64 mg/g, and Zr๏Ti/PSS: 31.15 mg/g. Pseudo-second-order kinetic models (R2=0.99) were found to be the best models for describing the Cr (VI) adsorption reactions. Conclusıon: Thermodynamic parameters such as changes in ΔG°, ΔH°, and ΔS° have been estimated, and the process was found to be spontaneous.

Statistical measures of genetic differentiation of populations: Rationales, history and current states

Population differentiation is a fundamental process of evolution, and many evolutionary studies, such as population genetics, phylogeography and conservation biology, all require the inference of population differentiation. Recently, there has been a lot of debate over the validity of FST (and its analogue GST) as a measure for population genetic differentiation, notably since the proposal of the new index D in 2008. Although several papers reviewed or explored specific features of these statistical measures, a succinct account of this bewildering issue with an overall update appears to be desirable. This is the purpose of the present review. The available statistics generally fall into two categories, represented by FST and D, respectively. None of them is perfect in measuring population genetic differentiation. Nevertheless, they each have advantages and are valuable for current research. In practice, both indices should be calculated and a comparison of them can generate useful insights into the evolutionary processes that influence population differentiation. FST (GST) has some unique irreplaceable characteristics assuring its standing as the default measure for the foreseeable near future. Also, it will continue to serve as the standard for any alternative measures to contrast with. Instead of being anxious about making choice between these indices, one should pay due attention to the equilibrium status and the level of diversity (especially HS) of the populations, since they largely sway the power of a given statistic to address a specific question. We provide a multi-faceted comparative summary of the various statistics, which can serve as a basic reference for readers to guide their applications.