Classifications and comparisons of multilocus recombination distributions

Various classifications and representations of multilocus recombination structures are delineated based on generalized notions of linkage values and recombination rates. An important class of recombination distributions (called the count-location chiasma process) is parameterized by a distribution of the number of crossover events and, for each such crossover count, by a conditional distribution of crossover locations. A number of properties of this recombination structure are developed. A multilocus definition of a "natural" recombination range is set forth. Orderings among recombination distributions in the multilocus setting are also discussed. Comparisons are made in terms of complete linkage, free assortment and noninterference schemes serving as standards.

XDH gene mutation is the underlying cause of classical xanthinuria: a second report

BACKGROUND

Classical xanthinuria is a rare autosomal recessive disorder characterized by excessive excretion of xanthine in urine. Type I disease results from the isolated deficiency of xanthine dehydrogenase (XDH), and type II results from dual deficiency of XDH and aldehyde oxidase. The XDH gene has been cloned and localized to chromosome 2p22-23. The aim of this study was to characterize the molecular basis of classical xanthinuria in an Iranian-Jewish family.

METHODS

The apparently unrelated parents originated from a community in which consanguineous marriages are common. Subtyping xanthinuria was attempted by homozygosity mapping using microsatellite markers D2S352, D2S367, and D2S2374 in the vicinity of the XDH gene. Mutation detection was accomplished by PCR-SSCP screening of all 36 exons and exon-intron junctions of the XDH gene, followed by direct sequencing and confirmation of sequence alteration by restriction analysis.

RESULTS

The index case was homozygous for all three microsatellite markers analyzed. The expected frequency of this genotype in a control population was 0. 0002. These results suggested that xanthinuria in the patient is linked to the XDH gene. Consequently, a 1658insC mutation in exon 16 of the XDH gene was identified. The 1658insC mutation was not detected in 65 control DNA samples.

CONCLUSION

A molecular approach to the diagnosis of classical xanthinuria type I in a female patient with profound hypouricemia is described. Linkage of xanthinuria to the XDH locus was demonstrated by homozygosity mapping, and a 1658insC mutation, predicting a truncated inactive XDH protein, was identified. These results reinforce the notion that mutations in the XDH gene are the underlying cause of classical xanthinuria type I.

Discrete chiasma formation models and their associated high order interference

We introduce some special chiasma formation processes. First a family of discrete chiasma formation processes is introduced and we determine the nature of higher order interference associated with those processes. Secondly we consider a two-stage chiasma formation process, where the associated recombination frequency between two markers depends not only on their map distance but also on their location along the chromosomes. We characterise under this process, in some cases, the nature of interference between two segments.

Quality of life as outcome in the treatment of osteoporosis: the development of a questionnaire for quality of life by the European Foundation for Osteoporosis

The morbidity of osteoporosis is caused by fractures. Vertebral fractures lead to pain and disability and a decrease in quality of life. A Working Party of the European Foundation for Osteoporosis has developed a specific questionnaire for patients with established vertebral osteoporosis. This questionnaire is intended for use in clinical trials. The questionnaire consists of questions and visual analogue scales in the following domains: pain, activities of daily living, jobs around the house, mobility, leisure and social activities, general health perception and mood. The questionnaire has been translated from English into French, German, Italian, Hebrew, Swedish and Dutch. The questionnaire is currently being validated in a multicentre study involving patients with stable osteoporosis and control subjects. Preliminary results indicate that the reproducibility is sufficient and that the questionnaire is able to discriminate between patients with vertebral osteoporosis and control subjects.

On the modification of recombination with sex-dependent fitnesses and linkage

According to the Reduction Principle, when a recombination-reducing allele is introduced near an equilibrium that depends on recombination, that allele will increase in frequency. If the allele increases the recombination rate, it will be expelled from the population. There are known cases where this principle fails. In this respect, an interesting question is what kind of two-sex viability regimes support a general Reduction Principle. In this paper, we construct a model of viabilities, due to two autosomal linked genes, which differ between the sexes, such that recombination is different in the sexes. A complete analysis is provided for the case where recombination is absent in one sex. It is proved that the Reduction Principle is still valid for recombination in the other sex.

Oral alendronate induces progressive increases in bone mass of the spine, hip, and total body over 3 years in postmenopausal women with osteoporosis

To determine the effects of long-term daily oral alendronate sodium (ALN) on bone mass in postmenopausal women with osteoporosis, 19 centers enrolled 516 postmenopausal women aged 45-80 years with spine bone mineral density (BMD) at least 2.5 SD below the mean for young premenopausal women in a 3-year, double-blind, placebo-controlled study. Subjects were randomly allocated to one of four treatment groups: placebo; alendronate, 5 or 10 mg/day for 3 years; or alendronate, 20 mg/day for 2 years followed by 5 mg/day for the 3rd year. All patients received 500 mg/day of supplemental calcium to ensure adequate calcium intake. BMD was measured by dual-energy X-ray absorptiometry at several skeletal sites. Nonsignificant mean decreases in BMD of the spine, femoral neck, and trochanter of 0.6, 0.7, and 0.4%, respectively, occurred in the placebo group at 3 years. Relative to placebo-treated patients, spine BMD increased by 5.4%, 7.4%, and 8.4% in the 5, 10, and 20/5 mg ALN groups, respectively. Increases at the femoral neck were 3.5%, 5.5%, and 4.3%, and those at the trochanter were 5.1%, 7.2%, and 7.2%, respectively. Thus, efficacy of 10 and 20/5 mg ALN was similar, whereas the 5 mg dose was less effective. BMD continued to increase over the entire 3-year study duration in the ALN-treated groups and, compared with the other dosage groups, 10 mg ALN produced the largest gains in BMD during the 3rd year. Changes in biochemical markers of bone turnover and mineral homeostasis confirmed the effect of ALN to decrease bone turnover to a new steady-state level. The safety and tolerability of ALN were comparable with those of placebo. In summary, 10 mg daily oral ALN given for 3 years significantly and progressively increases bone mass and is a generally well-tolerated treatment for osteoporosis in postmenopausal women.

The development of a European questionnaire for quality of life in patients with vertebral osteoporosis

The complaints of vertebral osteoporosis usually result from wedge or crush fractures and biconcave deformities. These are caused by a decrease of bone mass and deterioration of bone structure leading to loss of strength. Treatment of osteoporosis should result in an increase of bone mass, and the incidence of new vertebral fractures should diminish. However, new vertebral fractures are not always accompanied by pain, and disability does not well correlate with the number of vertebral fractures. Patients with osteoporosis often have other problems e.g. with taking a shower, preparing meals, gardening, walking stairs, visiting friends and attending social activities. In addition, pain and disability may influence mood and lead to depression. The assessment of quality of life should be a primary endpoint in clinical trials in patients with osteoporosis and in individual patients care. Recently, the European Foundation for Osteoporosis (EFFO) has decided a develop a questionnaire for patients with vertebral osteoporosis, i.e. patients with vertebral deformities. The questionnaire is meant for use in clinical trials. A questionnaire was made including 48 questions and 6 visual analogue scales. The questions concern the following domains: pain, activities of daily living, jobs around the house, moving, leisure and social activities, general health perception and mood. The questionnaire ("Qualeffo") has now entered the validation phase. The first study in 8 centres concerns the within-subject reproducibility, the internal coherence, and the specificity by comparing osteoporotic patients with a control group not suffering from osteoporosis or backpain.

Mutations in the SLC3A1 transporter gene in cystinuria

Cystinuria is an autosomal recessive disease characterized by the development of kidney stones. Guided by the identification of the SLC3A1 amino acid-transport gene on chromosome 2, we recently established genetic linkage of cystinuria to chromosome 2p in 17 families, without evidence for locus heterogeneity. Other authors have independently identified missense mutations in SLC3A1 in cystinuria patients. In this report we describe four additional cystinuria-associated mutations in this gene: a frameshift, a deletion, a transversion inducing a critical amino acid change, and a nonsense mutation. The latter stop codon was found in all of eight Ashkenazi Jewish carrier chromosomes examined. This report brings the number of disease-associated mutations in this gene to 10. We also assess the frequency of these mutations in our 17 cystinuria families.

Theoretical recombination processes incorporating interference effects

With the acquisition of genetic, physical, and sequence maps, linkage relationships among genes (markers) may be more accurately approached in terms of global models for the distribution of recombination events that take into account interference. There are two principal analytical methods used for ascertaining linkage relationships. The first method, the Haldane-Kosambi differential equation approach, has the limitation that all of its calculations rest on consideration of only three gene markers, where recombination depends only on the physical distance between markers. In this formulation the resulting map function is in general not feasible for use with multiple markers. The second method starts with a model of the crossover process from which recombination values are determined. The best studied global recombination processes are based on sequential (renewal) crossover formation processes, the count-location crossover structure, and crossovers evolving by a cascade mechanism. This paper, containing both review and new results, concentrates on two aspects of recombination structures: (i) classifications and characterizations of multimarker crossover distributions; and (ii) analysis of regular and higher order crossover interference forms. In eucaryotic species, the general impression is that positive interference prevails, while in procaryotic and viral organisms, there may be circumstances of negative interference. We would propose in estimating the crossover formation process a binomial count distribution or any other count distribution satisfying property (a) of Theorem 9.1 and a location distribution fitted by the data. It is also reasonable to try one or more obligate crossover points superimposed on independent Poisson processes determining other crossover points. This latter model also generates a situation of positive interference (Theorem 3.1).

Localization of a gene causing cystinuria to chromosome 2p

Cystinuria is an autosomal recessive disorder of amino acid transport. It is a common hereditary cause of kidney stones worldwide, and is associated with significant morbidity. In 17 affected families, we found linkage between cystinuria and three chromosome 2p markers. Maximal two-point lod scores between cystinuria and D2S119, D2S391 and D2S288 were 8.23 (theta = 0.07), 3.73 (theta = 0.15) and 3.03 (theta = 0.12), respectively. Analysis of recombinants and multipoint linkage data indicated that the most likely order is cen-D2S391-D2S119-cystinuria-D2S177-tel. We also observed high rates of homozygosity for markers in this chromosomal region among 11 affected offspring of consanguineous marriages. Based on its map position and function, the recently cloned SLC3A1 amino acid transporter gene is a primary candidate gene for this disease.

Multilocus autosomal sex determination

The two basic one locus sex determination models, diploid individual sex determination and parental sex determination, are generalized to the multilocus framework. As in the single locus case, it is established that there are two classes of polymorphic equilibria, equilibria with even sex ratio and equilibria with equal allele frequencies in the two sexes. The condition for external stability of this second class equilibria to invasion by a "new" mutant allele is that a "new" appropriately averaged sex ratio near the equilibrium be moved closer to the even sex ratio. However, stable polymorphisms with noneven sex ratio are not those that have a sex ratio as close as possible to 1/2, in contrast to the single locus case.

On the relation between the instability of ESS in discrete dynamics and segregation distortion

There is a simple correspondence between discrete dynamical systems associated with evolutionary game dynamics and general locus multiallele selection models with non-Mendelian segregation. When interpreted properly the payoff matrix has two components, a fitness matrix component and a segregation matrix component. The presence of segregation distortion which corresponds to a non-symmetric payoff matrix, is a source of instability. With non-symmetric payoff an ESS does not usually correspond to a stable equilibrium. It is always externally stable but does not necessarily have an internally stable equilibrium.

Global convergence properties in multilocus viability selection models: the additive model and the Hardy-Weinberg law

A natural coordinate system is introduced for the analysis of the global stability of the Hardy-Weinberg (HW) polymorphism under the general multilocus additive viability model. A global convergence criterion is developed and used to prove that the HW polymorphism is globally stable when each of the loci is diallelic, provided the loci are overdominant and the multilocus recombination is positive. As a corollary the multilocus Hardy-Weinberg law for neutral selection is derived.

Two-locus autosomal sex determination: on the evolutionary genetic stability of the even sex ratio

In two-locus models of sex determination, there are two kinds of interior (polymorphic) equilibria. One class has the even sex ratio, and the other has equal allele frequencies in the two sexes. Equilibria of the second class may exhibit linkage disequilibrium. The condition for external stability of these second-class equilibria to invasion by a new allele is that the appropriately averaged sex ratio near the equilibrium be moved closer to the even sex ratio than the average among the resident genotypes. However, invasion by a new chromosome depends on the recombination fraction in a way that appears to preclude general results about the evolutionary genetic stability of the even sex ratio in this situation.

Modifiers of mutation rate: a general reduction principle

A deterministic two-locus population genetic model with random mating is studied. The first locus, with two alleles, is subject to mutation and arbitrary viability selection. The second locus, with an arbitrary number of alleles, controls the mutation at the first locus. A class of viability-analogous Hardy-Weinberg equilibria is analyzed in which the selected gene and the modifier locus are in linkage equilibrium. It is shown that at these equilibria a reduction principle for the success of new mutation-modifying alleles is valid. A new allele at the modifier locus succeeds if its marginal average mutation rate is less than the mean mutation rate of the resident modifier allele evaluated at the equilibrium. Internal stability properties of these equilibria are also described.

An evolutionary reduction principle for genetic modifiers

The joint evolution of major genes under viability selection and a modifier locus that controls recombination between the major genes, mutation at the major gene, or migration between two demes is studied. The modifying locus is selectively neutral and may have an arbitrary number of alleles. For each case a class of polymorphic equilibria exists in which the frequencies of the modifying alleles are those computed by assuming that the recombination, mutation, or migration rates were viabilities and in which the major and modifier loci are not statistically associated. These are called viability-analogous Hardy-Weinberg (VAHW) equilibria. A new allele introduced near these equilibria will enter the population if its marginal average rate of recombination, mutation, or migration (whichever applies) is less than the population average prior to its introduction. Stability properties of these VAHW equilibria are also reported.

A symmetric two-locus fertility model

A model in which selection is mediated by differential fertilities among the genotypes at two diallelic loci is proposed. Fertility depends only on the number of heterozygous loci participating in the mating. Classes analogous to symmetric equilibria in symmetric viability models are determined explicitly and shown to exhibit stability behavior very different from the viability results. Linkage equilibrium is shown to occur in a relatively asymmetric fashion and to overlap in stability with linkage disequilibrium. In many cases single-locus or two-locus polymorphism is shown to be stable simultaneously with chromosome fixation even under very tight linkage. It is suggested that historical effects may be of great significance in the evolution of systems in which fertility is the primary agent of natural selection.

On chiasma formation point processes having the count location property

The count-location (C-L) chiasma formation schemes introduced by Karlin and Liberman (1979b) encompass a broad class of map functions involving positive, negative or no chiasma interference. The C-L schemes do not explicitly assume a specific mechanism of crossover formation, but rather a statistical property of the process. If viewed as a stochastic point process along the chromosome, it is shown that a crossing over mechanism having the C-L property is actually a rescaled mixture of Poisson processes. Surprisingly it turns out that these C-L point processes involve negative interference throughout the entire genome.

ON SOME MODELS OF FERTILITY SELECTION

Additive, multiplicative and symmetric models of fertility controlled by one diallelic gene are studied. For the completely symmetric fertility system a complete equilibrium and local stability analysis is possible. Contrary to previous conjectures, asymmetric equilibria can be stable. Conditions are derived under which a multiplicative model can be regarded as equivalent to a symmetric fertility system.

Representation of Nonepistatic selection models and analysis of multilocus Hardy-Weinberg Equilibrium configurations

The paper develops conditions for the existence and the stability of central equilibria emanating from selection recombination interaction with generalized nonepistatic selection forms operating in multilocus multiallele systems. The selection structure admits a natural representation as simple sums of Kronecker products based on a common set of marginal selection components. A flexible parametrization of the recombination process is introduced leading to a canonical derivation of the transformation equations connecting gamete frequency states over successive generations. Conditions for the existence and stability of multilocus Hardy-Weinberg (H.W.) type equilibria are elaborated for the classical nonepistatic models (multiplicative and additive viability effects across loci) as well as for generalized nonepistatic selection expressions. It is established that the range of recombination distributions maintaining a stable H.W. polymorphic equilibrium is confined to loose linkage in the pure multiplicative case, but is not restricted in the additive model. In the bisexual case we ascertain for the generalized nonepistatic model the stability conditions of a common H.W polymorphism.

CENTRAL EQUILIBRIA IN MULTILOCUS SYSTEMS. I. GENERALIZED NONEPISTATIC SELECTION REGIMES

The generalized nonepistatic selection regime encompasses combinations of multiplicative and neutral viability effects distributed across a set of loci. These subsume, in particular, mixtures of the classical modes of multiplicative and additive fitness evaluations for multilocus traits. Exact analytic conditions for existence and stability of a multilocus Hardy-Weinberg (H-W) polymorphic equilibrium cmfiguration are ascertained. It is established that the central H-W polymorphism is stable only if the component loci are "overdominant" and sufficient recombination is in force. The H-W central equilibrium is never stable for tight linkage whenever some multiplicative selection effects are contributed by at least two of the loci involved. In the case of additive selection expression and individual overdominant loci, the H-W polymorphism is stable independently of the level of recombination. In the context of "natural" recombination schemes, "more recombination" enhances the stability 3f the H-W polymorphic equilibrium.

CENTRAL EQUILIBRIA IN MULTILOCUS SYSTEMS. II. BISEXUAL GENERALIZED NONEPISTATIC SELECTION MODELS

This paper is a continuation of the paper "Central Equilibria in Multilocus Systems I," concentrating on existence and stability properties accruing to central H-W type equilibria in multilocus bisexual systems acted on by generalized nonepistatic selection forces coupled to recombination events. The stability conditions are discussed and interpreted in three perspectives, and the influence of sexual differences in linkage relationships together with sex-dependent selection is appraised. In this case we deduce that the stability conditions of the H-W polymorphism in the bisexual model coincide exactly with the conditions for the corresponding monoecious model, provided that the recombination distribution imposed is that of the arithmetic mean of the male and female recombination distributions. A second concern has the same recombination distribution for both sexes, but contrasting selection regimes between sexes. It is then established that, with respect to discerning the relevance of the H-W equilibrium, there is an equivalent monoecious selection regime which is an appropriate "weighted combination" of the male and female selection forms. Finally, in the case where the selection and recombination structures are both sex dependent, a hierarchy of comparisons is elaborated, seeking to unravel the nature of selection-recombination interaction for monoecious versus diocecious systems.