P. Narain

The survival of recessive lethals in finite populations

The behaviour of recessive lethals under recurrent mutation in small populations has been investigated using matrix methods. Equilibrium gene frequencies are then far from their infinite population values and are comparatively insensitive to selection on the heterozygote. Particular emphasis was paid to the allelism, both within and between generations. The life statistics of individual lethals, underlying the similarity between successive generations, was investigated and expressions given for the average age and subsequent expectation of life of lethals present in the population at any time. At equilibrium, the two are equal and the values found from the computer using matrix methods were surprisingly high. For a population size of 50, the average age of lethals, neutral in the heterozygote, present at any time is 10.2 generations. This produces a high correlation between observations in successive generations which leads to great difficulty in obtaining accurate estimates of parameters in any population. For instance, with N = 50 and an overall mutation rate of 0.005, the mean lethal frequency estimated from an analysis of 80 successive generations had a sampling variance slightly larger than the binomial sampling variance from a sample of 100 chromosomes from a single generation. As a consequence, results from single populations are of little value. The effect of linkage was studied using simulation methods. It was found that, for lethals neutral in the heterozygote, linkage had no detectable effect at equilibrium on the proportion of chromosomes containing a lethal or on the allelism between them. Individual lethals appeared to have a slightly higher expectation of life in the absence of crossing over.

A note on the diffusion approximation for the variance of the number of generations until fixation of a neutral mutant gene.

A general expression is derived for the variance of time to fixation of a neutral gene in a finite population using a diffusion approximation. The results are compared with exact values derived by matrix methods for a population size of 8.

Quantitative genetics: past and present

Most characters of economic importance in plants and animals, and complex diseases in humans, exhibit quantitative variation, the genetics of which has been a fascinating subject of study since Mendel’s discovery of the laws of inheritance. The classical genetic basis of continuous variation based on the infinitesimal model of Fisher and mostly using statistical methods has since undergone major modifications. The advent of molecular markers and their extensive mapping in several species has enabled detection of genes of metric characters known as quantitative trait loci (QTL). Modeling the high-resolution mapping of QTL by association analysis at the population level as well as at the family level has indicated that incorporation of a haplotype of a pair of single-nucleotide polymorphisms (SNPs) in the model is statistically more powerful than a single marker approach. High-throughput genotyping technology coupled with micro-arrays has allowed expression of thousand of genes with known positions in the genome and has provided an intermediate step with mRNA abundance as a sub-phenotype in the mapping of genotype onto phenotype for quantitative traits. Such gene expression profiling has been combined with linkage analysis in what is known as eQTL mapping. The first study of this kind was on budding yeast. The associated genetic basis of protein abundance using mass spectrometry has also been attempted in the same population of yeast. A comparative picture of transcript vs. protein abundance levels indicates that functionally important changes in the levels of the former are not necessarily reflected in changes in the levels of the latter. Genes and proteins must therefore be considered simultaneously to unravel the complex molecular circuitry that operates within a cell. One has to take a global perspective on life processes instead of individual components of the system. The network approach connecting data on genes, transcripts, proteins, metabolites etc. indicates the emergence of a systems quantitative genetics. It seems that the interplay of the genotype-phenotype relationship for quantitative variation is not only complex but also requires a dialectical approach for its understanding in which ‘parts’ and ‘whole’ evolve as a consequence of their relationship and the relationship itself evolves.

Sample size for collecting germplasms: a polyploid model with mixed mating system

The present paper discusses a general expression for determining the minimum sample size (plants) for a given number of seeds or vice versa for capturing multiple allelic diversity. The model considers sampling from a large 2 k-ploid population under a broad range of mating systems. Numerous expressions/results developed for germplasm collection/regeneration for diploid populations by earlier workers can be directly deduced from our general expression by assigning appropriate values of the corresponding parameters. A seed factor which influences the plant sample size has also been isolated to aid the collectors in selecting the appropriate combination of number of plants and seeds per plant. When genotypic multiplicity of seeds is taken into consideration, a sample size of even less than 172 plants can conserve diversity of 20 alleles from 50,000 polymorphic loci with a very large probability of conservation (0.9999) in most of the cases.

Homozygosity in a selfed population with an arbitrary number of linked loci

Summary1.A generalised ‘coefficient of relationship’ between two individualsX andY has been defined with any number of linked loci.2.Recurrence relations for ϕ-, π- and ξ-functions (Schnell, 1961) in the case of two and three loci have been obtained.3.Solutions for recurrence relations have been given for ϕ-function upto the case of any number of linked loci.4.It has been found that the effect of linkage on the homozygosity of a selfed population is more with a greater number of linked loci and is maximum after one generation of selling. With three linked loci, the pairs of values ofp12, p23when taken as (.1, .1) exert a greater effect than (.3, 0) or (.5, 0) but this is true only upto two generations of selfing.5.With more than one locus, the rate of inbreeding is not constant with further generations of selfing. It depends on the number of generations of selling and the recombination values.

A general model for sample size determination for collecting germplasm

The paper develops a general model for determining the minimum sample size for collecting germplasm for genetic conservation with an overall objective of retaining at least one copy of each allele with preassigned probability. It considers sampling from a large heterogeneous 2k-ploid population under a broad range of mating systems leading to a general formula applicable to a fairly large number of populations. It is found that the sample size decreases as ploidy levels increase, but increases with the increase in inbreeding. Under exclusive selfing the sample size is the same, irrespective of the ploidy level, when other parameters are held constant. Minimum sample sizes obtained for diploids by this general formula agree with those already reported by earlier workers. The model confirms the conservative characteristics of genetic variability of polysomic inheritance under chromosomal segregation.

Polymorphism and evolution of the Rh blood groups

SummaryWith the aim of understanding the mechanism of maintenance of the Rh polymorphism in man, the probability and the first arrival time of an incompatibility mutant allele (recessive allele r) to reach a high frequency by genetic drift in a finite population and the allele frequency distribution under mutation pressure are studied. The deterministic changes in allele frequency in subdivided populations are also studied. The results obtained are as follows: (1) If the effective population size is 500–1,000, the probability of a single mutant allele to reach a frequency of 0.3 or 0.5 is quite small, and without recurrent mutation it is unlikely that the mutant allele becomes polymorphic. However, if the mutant allele happens to increase in frequency by genetic drift, the increase occurs quite rapidly. (2) In an infinitely large population the backward (u) and forward mutations (v) produce two stable equilibria, one of which has a frequency of 0.065 for h=0.05 and a frequency of 0.16 for h=0.01 when u=v=10−4, where h is the fitness reduction for the offspring from mating rr×RR. These frequencies are substantially higher than 0 but still lower than the frequencies in the European populations (0.3–0.6). In relatively small populations, however, the probability of the allele frequency being 0.3–0.6 becomes quite high if h=0.01. (3) If a population is subdivided into subpopulations among which small migration occurs, stable equilibria may be developed. However, the equilibrium gene frequencies do not conform to the frequencies observed in the European populations. When the migration rate becomes higher, the stable equilibria disappear, but the gene frequency change in subdivided populations is generally much slower than that in a single random mating population, so that the Rh polymorphism may be maintained for a long time even if there are no stable equilibria. (4) If we consider all these factors together, it is possible to explain the Rh polymorphism in terms of the mutation-drift hypothesis without recourse to reproductive compensation. It seems that the Rh polymorphism is transient rather than stable.

Efficiency of selective breeding based on a phenotypic index

SummaryThis article discusses the efficiency of selective breeding based on ‘phenotypic index’ which is defined as the deviation of the phenotypic value of the trait from its expected value predicted with the help of one or more auxiliary traits. The conditions under which the efficiency of such a procedure is greater than one have been theoretically studied. The practical relevance of this technique has also been demonstrated by applying it to breeding data on cattle.

A theoretical treatment of interval mapping of a disease gene using transmission disequilibrium tests

The genetic basis of the transmission disequilibrium test (TDT) for two-marker loci is explored from first principles. In this case, parents doubly heterozygous for a given haplotype at the pair of marker loci that are each in linkage disequilibrium with the disease gene with the further possibility of a second-order linkage disequilibrium are considered. The number of times such parents transmit the given haplotype to their affected offspring is counted and compared with the frequencies of haplotypes that are not transmitted. This is done separately for the coupling and repulsion phases of doubly heterozygous genotypes. Expectations of the counts for each of the sixteen cells possible with four-marker gametic types (transmitted vs not transmitted) are derived. Based on a test of symmetry in a square 4 × 4 contingency table, chi-square tests are proposed for the null hypothesis of no linkage between the markers and the disease gene. The power of the tests is discussed in terms of the corresponding non-centrality parameters for the alternative hypothesis that both the markers are linked with the disease locus. The results indicate that the power increases with the decrease in recombination probability and that it is higher for a lower frequency of the disease gene. Taking a pair of markers in an interval for exploring the linkage with the disease gene seems to be more informative than the single-marker case since the values of the non-centrality parameters tend to be consistently higher than their counterparts in the single-marker case. Limitations of the proposed test are also discussed.

Genetic differentiation of quantitative characters between populations or species II: optimal selection in infinite populations

Using a new and more general genetic model called the discrete-allelic state model and assuming discrete-time process, the evolutionary changes of genetic variation of quantitative characters, controlled by a few loci, within and between populations during the process of genetic differentiation of populations or species, are studied under the effects of mutation and centripetal selection in infinitely large populations. While in a finite population and ignoring selection, the rate of change of additive genetic variance depends on mutation and effective population size, traits under optimal selection in infinitely large populations go through the dynamics of a rather complicated form depending on the relative intensities of selection and mutation. When a population, which has reached steady-state by mutation-selection balance, splits into two, in one of which the same optimum genotype holds but in the other the optimum shifts a few standard deviations away from the original optimum, the corresponding daughter population starts differentiating from its sister population by favouring certain class of mutant alleles and discarding others which were originally favoured. During this process of turn over of genes, both the intra- and inter-populational variances undergo a complicated change, and the ratio of the former to the latter is a non-linear function of time of divergence. This pattern is qualitatively very different from the case when selection is absent. The intra-population distribution of genotypic values, during this transition, is shown to deviate considerably from normality. The presence of linkage seems to retard the accumulation of intra-populational genetic variance. The implications of these results are discussed in comparison with the earlier findings of evolutionary models of quantitative traits.