The number of alleles in multigene families

The probability distribution and moments of the number of alleles present in a sample of homologous chromosomes are studied. It is assumed that there are multiple copies of the gene on each chromosome. When there are only two copies per chromosome or when there are only two or three chromosomes, it is possible to use analytic methods to tackle the problem. Otherwise, a simulation method is suggested.     

The Hill-Robertson effect and the evolution of recombination

In finite populations, genetic drift generates interference between selected loci, causing advantageous alleles to be found more often on different chromosomes than on the same chromosome, which reduces the rate of adaptation. This "Hill-Robertson effect" generates indirect selection to increase recombination rates. We present a new method to quantify the strength of this selection. Our model represents a new beneficial allele (A) entering a population as a single copy, while another beneficial allele (B) is sweeping at another locus. A third locus affects the recombination rate between selected loci. Using a branching process model, we calculate the probability distribution of the number of copies of A on the different genetic backgrounds, after it is established but while it is still rare. Then, we use a deterministic model to express the change in frequency of the recombination modifier, due to hitchhiking, as A goes to fixation. We show that this method can give good estimates of selection for recombination. Moreover, it shows that recombination is selected through two different effects: it increases the fixation probability of new alleles, and it accelerates selective sweeps. The relative importance of these two effects depends on the relative times of occurrence of the beneficial alleles.     

Frequency Distribution of Esterase-5 Alleles in Two Populations of DROSOPHILA PSEUDOOBSCURA

Statistical tests comparing allele frequencies in natural populations with those predicted by various theories of genic variation depend critically on the accurate enumeration of alleles. This study used a series of five sequential electrophoretic conditions to characterize the allele frequency distributions of esterase-5 in two large population samples of Drosophila pseudoobscura from California. In Standard chromosome lines 12 electromorphs were discriminated using a single electrophoretic condition. When four additional criteria were used, the number of electromorphs increased to 41, 33 in one population and 22 in the other. Both populations had the same two alleles in high frequency, with other alleles present in frequencies of 6% or less. Although each population had a number of unique alleles, a χ² contingency test demonstrated no significant genetic divergence between them. A statistical comparison of allele frequencies in both populations with that predicted by neutral models suggests that the individual and combined distributions deviate from neutrality in the direction of purifying selection.—Sex-Ratio chromosomes differed markedly from Standard chromosomes in both allelic content and diversity. In 32 Sex-Ratio chromosomes from one population only three alleles were found, all of which were detected under the initial "standard" electrophoretic conditions. Moreover, none of these alleles was found in the Standard chromosome lines.     

Genetic identity between populations when mutation rates vary within and across loci

A formula is derived for the probability that two genes taken at random from the same locus in two populations isolated at time t ago are of the same allelic type. The model assumed is a neutral one where there are possibly different mutation rates between different alleles. Inequalities are derived for this probability. A particular result is that for a fixed overall mutation rate, the probability is least for the infinite alleles model. Inequalities and approximations are found for Nei's genetic identity at one locus when mutation rates vary, and also for the identity across loci when the overall mutation rates per locus vary. Genetic identity at the molecular level is considered and a probability generating function found for the number of segregating sites between two randomly chosen gametes from two divergent populations, under various models.     

Two chromosomes with multigene families

Watterson's formulae for the distribution, mean, and variance of the number of alleles in common on two chromosomes with multigene families are derived as simpler forms, and extended to chromosomes with an infinite number of genes, each evolving as in an infinitely many alleles model.     

New founder haplotypes at the myotonic dystrophy locus in southern Africa.

The association between normal alleles at the CTG repeat and two nearby polymorphisms in the myotonin protein kinase gene, the Alu insertion/deletion polymorphism and the myotonic dystrophy kinase (DMK)(G/T) intron 9/HinfI polymorphism, has been analyzed in South African Negroids, a population in which myotonic dystrophy (DM) has not been described. South African Negroids have a CTG allelic distribution that is significantly different from that in Caucasoids and Japanese: the CTG repeat lengths of > or = 19 are very rare. The striking linkage disequilibrium between specific alleles at the Alu polymorphism (Alu(ins) and Alu(del)), the HinfI polymorphism (HinfI-1 and HinfI-2), and the CTG repeat polymorphism seen in Caucasoid (Europeans and Canadians) populations was also found in the South African Negroid population. Numerous haplotypes, not previously described in Europeans, were, however, found. It thus seems likely that only a small number of these "African" chromosomes were present in the progenitors of all non-African peoples. These data provide support for the "out of Africa" model for the origin of modern humans and suggest that the rare ancestral DM mutation event may have occurred after the migration from Africa, hence the absence of DM in sub-Saharan Negroid peoples.     

Human chromosome variation: the discriminatory power of Q-band heteromorphism (variant) analysis in distinguishing between individuals, with specific application to cases of questionable paternity.

The chromosomes from 57 persons were analyzed by means of quinacrine fluorescent staining in order to assess the amount of variation and the discriminatory power of Q-band heteromorphism analysis. Chromosomes 3, 4, 13, 14, 15, 21, 22, and Y of each person were visually compared to those of 56 others, for a total of 1,596 comparisons. No two persons were found to have the same set of variants. The number of differences between chromosomes for each comparison ranged from 2 to 12 out of a possible total of 14 for females and 15 for males. Relatives were also distinguishable, and differences ranged from two to seven. We used the frequency with which each chromosome was useful for telling two people apart, and estimated the probability of finding two persons with the same set of quinacrine variants as .0003. Distinctly different heteromorphisms were found in the 39 unrelated persons for each of the chromosomes examined. In this small population, the number of different sets of variants observed for chromosomes 3, 4, 13, 14, 15, 21, 22, and Y were six, seven, 27, 16, 20, 15, 24, and five, respectively, for a total number of possible combinations of 1.14 X 10(15). As a test of the usefulness of chromosome heteromorphisms in paternity cases, 12 father-mother-child trios of virtually certain paternity, owing to the father-child segregation of a rare structural rearrangement, were coded and recombined at random to produce 120 cases of uncertain paternity. When the code was broken, 108 "alleged fathers" had been excluded correctly and the 12 biological fathers had been included correctly.     

Paths and cycles in breakpoint graph of random multichromosomal genomes.

We study the probability distribution of the distance d = n + chi - kappa - psi between two genomes with n markers distributed on chi chromosomes and with breakpoint graphs containing kappa cycles and psi "good" paths, under the hypothesis of random gene order. We interpret the random order assumption in terms of a stochastic method for constructing the bicolored breakpoint graph. We show that the limiting expectation of E[d] = n - 1/2chi - 1/2 log n+chi/2chi. We also calculate the variance, the effect of different numbers of chromosomes in the two genomes, and the number of plasmids, or circular chromosomes, generated by the random breakpoint graph construction. A more realistic model allows intra- and interchromosomal operations to have different probabilities, and simulations show that for a fixed number of rearrangements, kappa and d depend on the relative proportions of the two kinds of operation.     

A vectorized method of importance sampling with applications to models of mutation and migration

An importance-sampling method is presented for computing the likelihood of the configuration of population genetic data under general assumptions about population history and transitions among states. The configuration of the data is the number of chromosomes sampled that are in each of a finite set of states. Transitions among states are governed by a Markov chain with transition probabilities dependent on one or more parameters. The method assumes that the joint distribution of coalescence times of the underlying gene genealogy is independent of the genetic state of each lineage. Given a set of coalescence times, the probability that a pair of lineages is chosen to coalesce in each replicate is proportional to the contribution that the coalescence event makes to the probability of the data. This method can be applied to gene genealogies generated by the neutral coalescent process and to genealogies generated by other processes, such as a linear birth-death process which provides a good approximation to the dynamics of low-frequency alleles. Two applications are described. In the first, the fit of allele frequencies at two microsatellite loci sampled in a Sardinian population to the one-step mutation model is tested. The one-step model is rejected for one locus but not for the other. The second application is to low-frequency alleles in a geographically subdivided population. The geographic location is the allelic state, and the alleles are assumed to be sufficiently rare that their dynamics can be approximated by a linear birth-death process in which the birth and death rates are independent of geographic location. The analysis of eight low-frequency allozyme alleles found in the glaucous-winged gull, Larus glaucescens, illustrates how geographically restricted dispersal can be detected.     

Variation in karyotype in Hemerocallis.

Fifty two taxa have been studied, out of which forty five, including the species; H. citrina, H. dumortierii, H. multiflora, H. forrestii and H. thunbergil are diploid with 22 chromosomes in the somatic complement. H. fulva var. Europa, H. fulva var. cypriana, H. kwanso "Flore Pleno" and H. disticha "Flore Pleno" are triploid with thirty three chromosomes in the somatic complement. One cultivar "Mrs. David Hall" is tetraploid with forty four chromosomes. Besides this two aneuploid clones "Garden Lady" and cv. "29" with 2n=23 and 2n=29 chromosomes respectively were also encountered. Essentially four basic karyotypes are recognizable; A(IV + 8L + 1J + 1I), B(IV + 7L + 2J + 1I), C(9L+ 1J + 1I) and d(8L + 2J + 1I), where V stands for median, L for submedian, J for subterminal and I for terminal centromere. Thirty two taxa could be relegated to these four types and 17 can be resolved as combinations of these four types. These fall under AB, AC, AD or BC, BD and CD classes. The number of nucleolar chromosomes does not show any correlation with the grade of ploidy, whereas 2, 3 or telocentric chromosomes were consistantly found in diploid, triploid and tetraploid taxa respectively. The possible origin of telocentric chromosomes and aneuploid taxa has been discussed.     

Effect of trinucleotide repeat length and parental sex on phenotypic variation in spinocerebellar ataxia I.

Trinucleotide repeat expansion has been found in 64 subjects from 19 families: 57 patients with SCA1 and 7 subjects predicted, by haplotype analysis, to carry the mutation. Comparison with a large set of normal chromosomes shows two distinct distributions, with a much wider variation among expanded chromosomes. The sex of transmitting parent plays a major role in the size distribution of expanded alleles, those with > 54 repeats being transmitted by affected fathers exclusively. Our data suggest that alleles with > 54 repeats have a reduced chance of survival; these appear to be replaced in each generation by further expansion of alleles in the low- to medium-expanded repeat range, preferentially in male transmissions. Detailed clinical follow-up of a subset of our patients demonstrates significant relationships between increasing repeat number on expanded chromosomes and earlier age at onset, faster progression of the disease, and earlier age at death.     

An exact test of the Hardy-Weinberg law.

An exact distribution of a finite sample drawn from an infinite population in Hardy-Weinberg Equilibrium is described for k-alleles. Accordingly, an exact test of the law is presented and compared with two x2-tests for two and three alleles. For two alleles, it is shown that the "classical" c2-test is very adequate for sample sizes as small as ten. For three alleles, it is shown that a simpler formulation based on Leven's distribution approximates the exact test of this paper rather closely. However, it is recommended that researchers continue to employ the standard x2-test for all sample sizes and abide by it if the corresponding probability value is not "too close" to the critical level; otherwise, an exact test should be used.     

Difficulties in parentage analysis: the probability that an offspring and parent have the same heterozygous genotype.

Parentage studies often estimate the number of parents contributing to half-sib progeny arrays by counting the number of alleles attributed to unshared parents. This approach is compromised when an offspring has the same heterozygous genotype as the shared parent, for then the contribution of the unshared parent cannot be unambiguously deduced. To determine how often such cases occur, formulae for co-dominant markers with n alleles are derived here for Ph, the probability that a given heterozygous parent has an offspring with the same heterozygous genotype, and Pa, the probability that a randomly chosen offspring has the same heterozygous genotype as the shared parent. These formulae have been derived assuming Mendelian segregation with either (1) an arbitrary mating system, (2) random mating or (3) mixed mating. The maximum value of Pa under random mating is 0.25 and occurs with any two alleles each at a frequency of 0.5. The behaviour with partial selfing (where reproduction is by selfing with probability s, and random mating otherwise) is more complex. For n < or = 3 alleles, the maximum value of Pa occurs with any two alleles each at a frequency of 0.5 if s < 0.25, and with three equally frequent alleles otherwise. Numerically, the maximum value of Pa for n > or = 4 alleles occurs with n* < or = n alleles at equal frequencies, where the maximizing number of alleles n* is an increasing function of the selfing rate. Analytically, the maximum occurs with all n alleles present and equally frequent if s > or = 2/3. In addition, the potential applicability of these formulae for evolutionary studies is briefly discussed.     

Molecular marker analysis of 24- and 25-chromosome plants obtained from Solanum tuberosum L. subsp. andigena (2n = 4x = 48) pollinated with a Solanum phureja haploid inducer.

Clones with 24 or 25 chromosomes were obtained by pollinating an Andean cultivated tetraploid potato (Solanum tuberosum subsp. andigena clone 94H94, 2n = 4x = 48) with the Solanum phureja haploid-inducer clone 1.22. Their genetic composition was analyzed in an RAPD assay using 135 decamer primers and in an RFLP assay using 45 single-copy DNA probes. In total, 22 RAPD and 20 RFLP markers were found to be specific to S. phureja. None of these markers were found in the 24- and 25-chromosome clones. RFLP genotypes for the 45 RFLP loci were further determined for each clone. Genotypes of the 24-chromosome clones were characterized using two alleles randomly selected from four alleles of the parental tetraploid clone for almost all RFLP loci. Five 25-chromosome clones had extra alleles for all of the RFLP loci of chromosomes 4, 8, 10, 11, and 12, respectively, suggesting primary trisomy for one of these chromosomes. Clones with genotypes showing double reduction were also identified. Therefore, the obtained clones likely originated from random samples of female gametes, and hence are euhaploids or aneuhaploids of S. tuberosum subsp. andigena, strongly supporting parthenogenesis to be a primary mechanism for haploid induction in potato.     

The number of heterozygous loci between two randomly chosen completely linked sequences of loci in two subdivided population models

The stationary probability distribution of the number of heterozygous loci in two randomly chosen sequences of completely linked infinite alleles loci, with mutation at each locus, is found in the island model for within and between islands. Results for an infinite site model are found as a limit. A single charge state locus is also studied in the island model and distributions found for the charge difference between two genes. Similar results are derived for a stepping stone model.     

Acrocentric chromosome associations in man.

Heterogeneity among chromosomes was found to be a highly significant source of variation for association proportions, while culture, slide, and observer were negligible sources of variation for association proportions although important for numbers of associations. The consequences of these results for tests of group differences are discussed. It seems evident that each pair of acrocentric chromosomes has its own characteristic probability of entering into association. This is presumably a combination of the probability for each individual member of the pair, a proposition easily tested utilizing acrocentric chromosomes carrying polymorphisms which allow each member of the pair to be individually recognized. A mathematical theory for pairwise satellite association was developed and shown to fit observations on banded chromosomes. While we found very significant heterogeneity among individuals in the frequency with which different chromosomes entered into associations, there was no significant evidence for preferential association between any particular chromosomes, either heterologous or homologous. This finding in our material of apparently random associations between different chromosomes is contrary to claims made by other investigators and should be tested on other material. No correlation was found between the phenotype of the chromosome, as judged by cytogenetic polymorphisms, and its probability of association.     

Restriction fragment length polymorphism analysis of somatic hybrids between Lycopersicon esculentum and irradiated L. peruvianum: evidence for limited donor genome elimination and extensive chromosome rearrangements

Summary The genome composition of asymmetric somatic hybrids, obtained by fusion of leaf protoplasts fromLycopersicon esculentum and gamma-irradiated leaf protoplasts fromL. peruvianum, was characterised by Southern blot analysis using 29 restriction fragment length polymorphism markers. Eight low dose hybrids and seven high dose hybrids (irradiation dose 50 Gray and 300 Gray, respectively) were analysed. By densitometric scanning of the autoradiographs, the number of alleles for each locus of the component species was established. In general, elimination of alleles from the irradiatedL. peruvianum donor genome was limited and ranged from 17%–69%. ThreeL. peruvianum loci, located on chromosomes 2, 4 and 7, were present in all asymmetric hybrids, suggesting linkage to the regeneration capacity trait which was used in selecting them. The loss of donor genome was dose-dependent. Low dose hybrids contained more alleles, loci and complete chromosomes fromL. peruvianum than high dose hybrids, whereas the high dose hybrids contained more incomplete chromosomes. In most hybrids someL. esculentum alleles were lost. The possible implications of these results for the use of asymmetric hybrids in plant breeding are discussed.     

Characterization of eight VNTR loci by agarose gel electrophoresis

Allelic frequencies and their confidence intervals were obtained for eight independent VNTR loci from a sample of more than 75 Utah Caucasians. Using high-resolution agarose gel electrophoresis, we were able to resolve alleles at the D17S5 locus that differed by only one repeating unit; it was therefore possible to name the alleles according to the number of repeating units each contained. Two a priori probabilities were calculated for each VNTR locus separately and for all eight loci jointly: (i) the "power of exclusion" for an alleged father/mother/child trio and for an alleged parent/child duo, and (ii) the "probability of matching" when two unrelated individuals or two siblings are genotyped.     

Molecular evidences of single mutational events followed by recurrent crossing-overs in the common delta-globin alleles in the Mediterranean area

The human delta-globin gene (HBD) is one of the beta-like globin genes expressed in adults. In the Mediterranean countries the carriers of delta-thalassemia defects or Hb A2-variants are >1% and about 40/70 known alleles have been found in families with this ethnic origin. The scope of this study was to investigate the variability of the gene and of the chromosomal background in order to highlight the origin and spreading of the delta-globin gene alleles in the Mediterranean area. We carried out the characterization of the delta-globin gene alleles and of RFLP-haplotypes, SNPs and one microsatellite associated with them in 231 carriers originating principally from East Sicily. Seventeen alleles were identified, of which five were new. The chromosomes associated with mutated alleles from unrelated carriers were 158; the allele Hb A2-Yialousa accounted for about 75% of relative frequency, Hb A2-Mitsero for about 8%. The alleles were associated with RFLP 5'-haplotypes "- - - -" or "+ - + +", prevalent in the Mediterranean area, except Hb A2-Mitsero associated with the 5'-haplotype "Benin" "- - - +" and the Hb A2' associated with "+ - - +", both of African origin. Each allele showed linkage with one haplotype with these exceptions. The Hb A2-Yialousa showed heterogeneity of the 5'-haplotype in 2/58 chromosomes; the Hb A2-Mitsero showed SNPs and (A)gamma-microsatellite typical of a "Benin" haplotype found associated with the Hb C and Hb S chromosomes; the Hb A2-Yialousa (14/58 chromosomes), Hb A2-Mitsero, Hb A2-Pylos, Hb A2-Fitzroy showed heterogeneity in the 3'-haplotypes and beta-globin gene SNPs. The Hb A2-Coburg was found associated with the haplotype "+ - + +/+ +" different from that already reported "- - - -/+ -". With the exception of this last allele, the linkage of each mutation with a core of RFLPs or SNPs around or inside the delta-globin locus suggested the unicentric origin of the mutations followed by recurrent recombination events causing the chromosomal background heterogeneity.     

The spatial distribution of transient alleles in a subdivided population: a simulation study

The spatial distributions of newly introducted alleles in a subdivided population are generated using a computer program to model the processes of selection, gene flow and genetic drift. Advantageous, neutral and deleterious alleles are considered, and certain aspects of the patterns generated by new alleles that are ultimately fixed and ultimately lost are examined. To characterize the spatial pattern of rare alleles, the distribution, P(i), the probability that the new allele is found in exactly i local populations before it is lost, is defined and estimated from the simulations. The shape of the P(i) distribution is surprisingly similar for selected and neutral alleles. For advantageous alleles going to fixation, the "wave of advance" is set up quickly, but stochastic effects reduce the wave speed from Fisher's (1937) value. Gene flow is much more effective in dispersing alleles in a two-dimensional array than in one dimension. Long distance gene flow has a much smaller effect in two dimensions than in one dimension.