Hardy-Weinberg Model Research Articles

Testing for goodness rather than lack of fit of an X-chromosomal SNP to the Hardy-Weinberg model.

The problem of checking the genotype distribution obtained for some diallelic marker for compatibility with the Hardy-Weinberg equilibrium (HWE) condition arises also for loci on the X chromosome. The possible genotypes depend on the sex of the individual in this case: for females, the genotype distribution is trinomial, as in the case of an autosomal locus, whereas a binomial proportion is observed for males. Like in genetic association studies with autosomal SNPs, interest is typically in establishing approximate compatibility of the observed genotype frequencies with HWE. This requires to replace traditional methods tailored for detecting lack of fit to the model with an equivalence testing procedure to be derived by treating approximate compatibility with the model as the alternative hypothesis. The test constructed here is based on an upper confidence bound and a simple to interpret combined measure of distance between true and HWE conforming genotype distributions in female and male subjects. A particular focus of the paper is on the derivation of the asymptotic distribution of the test statistic under null alternatives which is not of the usual Gaussian form. A closed sample size formula is also provided and shown to behave satisfactorily in terms of the approximation error.

Mathematical modelling of vector-borne diseases and insecticide resistance evolution

BackgroundVector-borne diseases are important public health issues and, consequently, in silico models that simulate them can be useful. The susceptible-infected-recovered (SIR) model simulates the population dynamics of an epidemic and can be easily adapted to vector-borne diseases, whereas the Hardy-Weinberg model simulates allele frequencies and can be used to study insecticide resistance evolution. The aim of the present study is to develop a coupled system that unifies both models, therefore enabling the analysis of the effects of vector population genetics on the population dynamics of an epidemic.MethodsOur model consists of an ordinary differential equation system. We considered the populations of susceptible, infected and recovered humans, as well as susceptible and infected vectors. Concerning these vectors, we considered a pair of alleles, with complete dominance interaction that determined the rate of mortality induced by insecticides. Thus, we were able to separate the vectors according to the genotype. We performed three numerical simulations of the model. In simulation one, both alleles conferred the same mortality rate values, therefore there was no resistant strain. In simulations two and three, the recessive and dominant alleles, respectively, conferred a lower mortality.ResultsOur numerical results show that the genetic composition of the vector population affects the dynamics of human diseases. We found that the absolute number of vectors and the proportion of infected vectors are smaller when there is no resistant strain, whilst the ratio of infected people is larger in the presence of insecticide-resistant vectors. The dynamics observed for infected humans in all simulations has a very similar shape to real epidemiological data.ConclusionThe population genetics of vectors can affect epidemiological dynamics, and the presence of insecticide-resistant strains can increase the number of infected people. Based on the present results, the model is a basis for development of other models and for investigating population dynamics.

Population genetics of immune-related multilocus copy number variation in Native Americans.

While multiallelic copy number variation (mCNV) loci are a major component of genomic variation, quantifying the individual copy number of a locus and defining genotypes is challenging. Few methods exist to study how mCNV genetic diversity is apportioned within and between populations (i.e. to define the population genetic structure of mCNV). These inferences are critical in populations with a small effective size, such as Amerindians, that may not fit the Hardy-Weinberg model due to inbreeding, assortative mating, population subdivision, natural selection or a combination of these evolutionary factors. We propose a likelihood-based method that simultaneously infers mCNV allele frequencies and the population structure parameter f, which quantifies the departure of homozygosity from the Hardy-Weinberg expectation. This method is implemented in the freely available software CNVice, which also infers individual genotypes using information from both the population and from trios, if available. We studied the population genetics of five immune-related mCNV loci associated with complex diseases (beta-defensins, CCL3L1/CCL4L1, FCGR3A, FCGR3B and FCGR2C) in 12 traditional Native American populations and found that the population structure parameters inferred for these mCNVs are comparable to but lower than those for single nucleotide polymorphisms studied in the same populations.

The use of Hardy-Weinberg Equilibrium in clonal plant systems.

Traditionally population genetics precludes the use of the same genetic individual more than once in Hardy-Weinberg (HW) based calculations due to the model's explicit assumptions. However, when applied to clonal plant populations this can be difficult to do, and in some circumstances, it may be ecologically informative to use the ramet as the data unit. In fact, ecologists have varied the definition of the individual from a strict adherence to a single data point per genotype to a more inclusive approach of one data point per ramet. With the advent of molecular tools, the list of facultatively clonal plants and the recognition of their ecological relevance grows. There is an important risk of misinterpretation when HW calculations are applied to a clonal plant not recognized as clonal, as well as when the definition of the individual for those calculations is not clearly stated in a known clonal species. Focusing on heterozygosity values, we investigate cases that demonstrate the extreme range of potential modeling outcomes and describe the different contexts where a particular definition could better meet ecological modeling goals. We emphasize that the HW model can be ecologically relevant when applied to clonal plants, but caution is necessary in how it is used, reported, and interpreted. We propose that in known clonal plants, both genotype (GHet) and ramet (RHet) based calculations are reported to define the full range of potential values and better facilitate cross-study comparisons.

Decomposition of Kullback–Leibler risk and unbiasedness for parameter-free estimators

The bias and variance of traditional parameter estimators are parameter-dependent quantities. The maximum likelihood estimate (MLE) can be defined directly on a family of distributions P and so is parameter-free. The parameter-invariance property of the MLE can be described by the fact that the MLE for the original parameter and the MLE for any reparametrization name the same distribution. We define parameter-free estimators to be P-valued random variables rather than parameter-valued random variables. The Kullback–Leibler (KL) risk is decomposed into two parameter-free quantities that describe the variance and squared bias of the estimator. We show that for exponential families the P-valued MLE is unbiased. We define the KL mean K of a P-valued random variable and show how K describes the long-run properties of this random distribution. For most families P, the KL mean of any P-valued random variable will not lie in P so that we define another mean M, called the distribution mean that is related to K and is an element of P. By allowing the distribution estimator to take values outside of P, the KL mean can be made to lie in P. We compare the MLE to non-P-valued estimators that have been suggested for the Hardy–Weinberg model. Results for the dual KL risk are also given.

The Hardy-Weinberg Principle

This module was initially developed for a course in applications of mathematics in biology. The objective of this lesson is to investigate how the allele and genotypic frequencies associated with a particular gene might evolve over successive generations. The lesson will discuss how the Hardy-Weinberg model provides a basis for comparison when analyzing the evolution of a population. [Supplementary materials are available for this article. Go to the publisher's online edition of PRIMUS for the following free supplemental resource: Sample Solution.]

The Tangled Bank: an introduction to evolution

The Tangled Bank: an introduction to evolution

Characterization of Ten Microsatellite Loci in the Blue MusselMytilus edulis

Mussels of the genus Mytilus are ecologically and commercially important worldwide, and they form hybrid complexes where their distributions overlap. Allozyme and nuclear markers have been used to investigate their genetics over many years, but successful development of reliable highly valuable microsatellite markers has lagged behind other shellfish species. We have developed and characterized ten novel microsatellite loci that amplify reliably for the blue mussel Mytilus edulis. The number of alleles among 30 individuals from a wild population (Menai Strait, North Wales, UK) ranged between 9 and 29 and the observed heterozygosity between 0.300 and 0.954. Significant heterozygote deficiencies against the Hardy-Weinberg model were observed at 6 out of 10 loci. Analyses using MICRO-CHECKER suggested the presence of null alleles at 8 out of 10 microsatellites with estimated null allele frequencies ranging from 0.105–0.305. The 10 newly developed microsatellites will have value to discriminate between Mytilus species, to support studies of introgression and hybridization and to strengthen and improve the available genetic linkage map.

Mytilus species under rope culture in Scotland: implications for management

Mussel (Mytilus spp.) production is one of the most economically important aspects of global aquaculture and, in Scotland, production has increased from 262 t in 1986 to 4,219 t in 2006. Until recently, mussel production in Scotland was considered to be based exclusively on the blue mussel Mytilus edulis, the native species. In Europe, production is known to consist of M. edulis and M. galloprovincialis, while a third less profitable species M. trossulus occurs in the Baltic Sea, where it is unexploited. In Scotland, the sympatric occurrence of M. edulis, M. trossulus, M. galloprovincialis and their hybrids in cultivation in Loch Etive has recently been reported, with significant losses in production attributed to the presence of M. trossulus. Samples of mussels were taken at three depths from 10 rope-farm sites in Loch Etive. The distribution of the Mytilus species and their hybrids in Loch Etive was investigated using the Me 15/16 nuclear DNA locus for species and hybrid identification. All three species and their hybrids were detected and the data were in agreement with the Hardy–Weinberg model suggesting panmixia. Frequencies of M. galloprovincialis and its hybrids were very low. Overall, M. trossulus (37%) was found to be more common than M. edulis (30%) and 23% of the sampled mussels were M. trossulus × M. edulis hybrids. Species distribution did not correlate with year of settlement nor with salinities taken at the time of sampling. There were significant differences in species distribution due to site location and depth, with M. edulis being more frequent at 8 m depth and M. trossulus more common at shallower depths. These differences might be exploitable in management strategies for continuing production, for example to decrease the prevalence of M. trossulus at shellfish farms in favour of the more desirable M. edulis.

Effect of CD14 −260C>T polymorphism on the mortality of critically ill patients

The CD14 receptor seems to be an important part of the innate immune system. A mutant CD14 can produce a reduced signal in response to infection, as a result of which an adequate inflammatory innate response is not induced, leading to a systemic infection. Defects in the innate immunity increase patient susceptibility to systemic infections and can produce a deregulated inflammatory response causing sepsis, organ failure or death in critically ill patients. We evaluated the CD14 -260C>T polymorphism genotyping as a genetic tool for risk evaluation of critically ill patients admitted to an intensive care unit (ICU) in Southern Brazil. We monitored the patients daily during their entire ICU and post-ICU (hospital) stay (measured from the ICU admission day to a maximum of 224 days). A total of 85 patients, aged 19-95 years (mean = 56 years, median = 58 years), were included in this study. Patient mortality was 58.8%. The genotypic (TT = 0.27, TC = 0.41, CC = 0.32) and allelic (T = 0.48, C = 0.52) frequencies did not differ from the values expected by the Hardy-Weinberg model and genotype distribution was random for all clinical characteristics at ICU admission. We found a statistically significant difference favouring the survival of patients with TT genotype (P = 0.042), suggesting that this CD14 gene polymorphism could be a candidate for further study in the search for a complementary prognostic tool for patient risk evaluation. Our study describes, for the first time, the effect of the CD14 gene polymorphism in critically ill Brazilian patients. Our data suggest that patients carrying the TT genotype have a better survival outcome.

Using spreadsheets to teach aspects of biology involving mathematical models

Some aspects of biology, for example the Hardy-Weinberg simulation of population genetics or modelling heat flow in lizards, have an undeniable mathematical basis. Students can find the level of mathematical skill required to deal with such concepts to be an insurmountable hurdle to understanding. If not used effectively, spreadsheet models designed to help to overcome such mathematical difficulties present a further hurdle in terms of IT skills. This paper describes examples of how spreadsheet simulations can be designed to reduce the problem of the IT interface obscuring real subject teaching. Two sets of questions designed to help students to meet learning outcomes associated with the Hardy-Weinberg model and with thermoregulation in lizards are presented together with suggested answers. This approach has been successfully used on an introductory genetics course and a control systems course within a Biosciences degree at Canterbury Christ Church University College.

Demographics of the UK cystic fibrosis population: implications for neonatal screening.

The objective was to determine the composition of the Cystic Fibrosis (CF) Population attending specialist UK CF centres in terms of age, gender, age at diagnosis, genotype and ethnicity. With the planned introduction of the national CF screening programme in the UK, cystic fibrosis transmembrane regulator (CFTR) mutations were compared between different ethnic groups enabling a UK-specific frequency of mutations to be defined. Data were analysed from the patient biographies held in the UK CF Database (see www.cystic-fibrosis.org.uk). The currently registered population of 5,274 CF patients is 96.3% Caucasian with a male preponderance that significantly increases with age. The majority of the 196 non-Caucasian CF patients are from the Indian Subcontinent (ISC), of which one in 84 UK CF patients are of Pakistani origin. The commonest CFTR mutation, deltaF508, is found in 74.1% of all CF chromosomes. In the Caucasian CF population, 57.5% are deltaF508 homozygotes but the UK ISC CF population with only 24.7%, has significantly fewer deltaF508 homozygotes patients (95% confidence interval (CI) 0.2-0.4). The distribution of Caucasian patients with deltaF508/deltaF508, deltaF508/Other and Other/Other does not fit the expected distribution with a Hardy-Weinberg model unless those patients without a detected mutation are excluded (P<0.001). The UK CF Database has shown the UK CF population to have distinct characteristics separate from the North American and European CF Registries. The ISC group contains many mutations not recognised by current genetic analysis, and one in four ISC patients have no CFTR mutations identified. The CFTR analysis proposed for the screening programme would detect 96% of patients registered in the database, but is unlikely to achieve the desired >80% detection rates in the ethnic minority groups. Screen-positive, non-Caucasian infants without an identifiable CFTR mutation should be referred for a sweat test and genetic counselling when serum trypsinogen concentrations remain elevated after birth.

Larval growth, juvenile size and heterozygosity in laboratory reared mussels, Mytilus edulis.

Studies with marine bivalve juveniles have shown a positive correlation between growth and allozyme multi-locus heterozygosity (MLH), and, in some cases, between larval growth and juvenile growth, but there has been little research on the relationship between allozyme heterozygosity and larval growth. Larvae of M. edulis from different mating systems (half-sib families with a single female, or a single male parent, a reciprocal cross of two malesxtwo females and two mass matings of 13x13 and 8x17 females and males, respectively) were reared in the laboratory and selected into fast and slow growing groups when about 10-30% were undergoing metamorphosis. Offspring were reared to the juvenile stage (>3.00 mm) and both groups of each mating were electrophoresed and genotyped at up to 12 allozyme loci. There was generally good agreement with Mendelian inheritance (half-sibs and reciprocal cross) or the Hardy-Weinberg model (mass matings). Null alleles were detected at the Odh and Lap loci but there was no evidence that null allele heterozygotes grew slower than other genotypes. Over all cohorts, juveniles from the fast growing larval group were not significantly larger, or smaller, than juveniles from the slow growing group which suggests that larval growth rate may be independent of juvenile growth rate. This observation agrees with some, but not all, earlier studies and has commercial relevance. Tests of heterozygosity and juvenile shell length indicated no association between average heterozygosity across all allozyme loci and the size of juveniles in any cohort regardless of the mating system used or their larval growth rate. The association between MLH and juvenile growth in bivalves is seldom detected in cohorts from a limited genetic background. The lack of an association between heterozygosity and size might therefore be expected in the half-sib and reciprocal cross cohorts, but not in the mass matings. The results argue against any significant association between heterozygosity and larval size in mussels.

The Hardy-Weinberg Equilibrium—Some Helpful Suggestions

IN college General Biology curricula, the topic of classical genetics is typically followed by consideration of population and evolutionary genetics. After fundamental genetic principles are studied, students are introduced to the Hardy-Weinberg equilibrium as a continuum of Mendelian Genetics in a population context. Population genetics provides a logical segue to the study of evolution. Populations adhering to the Hardy-Weinberg equilibrium provide a reference for studying evolutionary changes (Journet 1986). With this as a starting point, Darwin’s observations can be justified and quantified, as well as expanded upon, to include additional aspects of population and evolutionary genetics. The Hardy-Weinberg equilibrium is an algebraic mathematical tool for predicting allele frequencies, phenotypes and genotypes in populations (Cummings 1997; Lewis 1997; Mader 1996; Postlethwait & Hopson 1995). Mertens (1992) proposed strategies for introducing the Hardy-Weinberg equilibrium to make it meaningful and more useful to both students and teachers. It has been our experience that students have difficulty with Hardy-Weinberg problem solving. Since students often possess a genuine fear of mathematics, reinforcement of basic mathematical principles when working through Hardy-Weinberg problems will provide a less threatening experience, thereby enabling students to successfully complete problems and concentrate on understanding genetic concepts as they relate to population dynamics (Flannery 1995; Journet 1986). Various methods, some quite detailed, for presenting the Hardy-Weinberg equilibrium have appeared in journals and textbooks (Cummings 1997; Journet 1986; Lewis 1997; Mader 1996; Postlethwait & Hopson 1995). The Hardy-Weinberg model is presented in a variety of ways in General Biology textbooks. In this paper we offer suggestions for instructors to assist with their presentation of this topic. Regardless of the approach, our suggestions can assist instructors by providing a unified method for explaining the

Effect of inbreeding avoidance on Hardy-Weinberg expectations: examples of neutral and selected loci.

Although all human populations avoid close inbreeding, the effect of inbreeding avoidance on genotype proportions has not been formally considered. This paper examines the expected proportions of genotype frequencies after adjusting the Hardy-Weinberg model for close inbreeding avoidance in an inbred population. This corrected model was used to evaluate genotype distributions for HLA haplotypes and haplotypes created from three microsatellite loci on human chromosome 13 in a sample of 315 married Hutterites. In this sample, there were five individuals who were homozygous for an HLA haplotype and 15 individuals who were homozygous for a chromosome 13 haplotype. The expected numbers of homozygotes were 21 for HLA and 26 for chromosome 13 haplotypes using a model adjusted for overall inbreeding (P = 0.00031 for HLA and P = 0.019 for chromosome 13) and 18 for HLA and 22 chromosome 13 haplotypes using a model adjusted for both overall inbreeding and close inbreeding avoidance (P = 0. 0016 for HLA and P = 0.099 for chromosome 13). The model that takes into account both overall inbreeding and inbreeding avoidance provided a better fit to the observed genotype distributions for chromosome 13 haplotypes (P = 0.099), suggesting that nonrandom aspects of Hutterite mating structure, other than inbreeding and inbreeding avoidance, have small effects on genotype distributions in the population. The fact that significant deficiencies of HLA homozygotes persist after taking into account inbreeding avoidance further suggests that factors related to the HLA haplotype per se influence the genotype proportions at HLA loci in the Hutterites.

Multilocus heterozygosity and size: a test of hypotheses using triploid Mytilus edulis

Cohorts of meiosis I and meiosis II induced triploid Mytilus edulis were produced from mass matings in the laboratory and reared alongside normal diploid cohorts during 1990 and 1991. Diploid cohorts generally exhibited a significant positive correlation between multilocus heterozygosity and size. This correlation was absent or much more weakly expressed in cohorts of triploid mussels and this supports the ‘associative overdominance’ hypothesis rather than the ‘direct involvement’ hypothesis as the major explanation for the correlation. For both diploids and triploids, no correlation between heterozygosity and the coefficient of variation of size was evident and heterozygotes at single loci were not larger than homozygotes. Triallelic triploid heterozygotes were no larger than diallelic triploids. The diploid cohorts were generally in agreement with the Hardy-Weinberg model and there was no trend towards heterozygote deficiency.

Genetics of indigenous and transplanted populations of Pecten maximus: no evidence for the existence of separate stocks

Genetic variation at eight polymorphic enzyme loci was investigated in 13 populations of the scallop Pecten maximus (L.) from a number of sites in Scotland, UK and Brittany, France. Two of the populations had initially been transplanted, as spat, to Brittany from Scottish and Irish collection sites. In common with other studies of marine bivalves, an overall deficiency of heterozygotes was observed, but genotype frequencies of transplanted populations were generally in agreement with the Hardy-Weinberg model indicating little post-transplant selection. There was high genetic similarity among all the populations surveyed and no overall differences between the Scottish and Brittany samples. The results are discussed in relation to contrasting evidence, derived from studies on the reproductive cycle of P. maximus and the genetics of Aequipecten (Chlamys) opercularis (L.), which suggests that the Scottish and Brittany scallop populations are, nevertheless, genetically distinct.

A note on the Hardy–Weinberg model in generalized ABO systems

An alternative formulation for the Hardy–Weinberg model in generalized ABO-like systems is introduced. It is indicated how such an approach may be employed to construct explicit Wald's statistics for goodness-of-fit tests.

The Equivalence of Two Tests and Models for HLA Data with No Observed Double Blanks

Gart and Nam (1984, Biometrics 40, 887-894) consider the analysis of an ABO-like model based on the Hardy-Weinberg law in the commonly occurring case of human leukocyte antigen (HLA) data where there are no double blanks, that is, no recessive homozygotes. They derive a score test, based on the truncated likelihood, of the hypothesis that the true recessive gene (or allele) frequency is zero. Yasuda (1968, Biometrics 24, 915-935) considers a similar codominant system wherein the true recessive gene frequency is assumed zero, but the Hardy-Weinberg law does not hold. In particular, he considers the possibility of a nonzero inbreeding coefficient. We show that the two models are equivalent; each likelihood can be shown to be a reparameterization of the other. Furthermore, the score test of the zero gene frequency in Gart and Nam is identical to the score test for a zero inbreeding coefficient given by Yasuda. The results are applied to an example wherein it appears that the Hardy-Weinberg model is appropriate. Thus, it is not possible in this population to identify homozygous individuals without error from phenotypic data alone.

On multinomial parametric estimation

When the multinomial probabilities are functions of a vector 8 it is shown that when bias correction is applied, the maximum likelihood estimator of 0 is least deficient in a class of estimators which includes the minimum chi-square and the minimum modified chi-square estimators. For the Hardy-Weinberg model Monte Carlo simulation is made for sample sizes ranging from ten to hundred which indicates the superiority of the maximum likeli¬hood estimator even for moderate sample sizes.