Protected polymorphism in the two-locus haploid model with unpredictable fitnesses

 In an unpredictable environment, the distributions of alleles from which polymorphism can be maintained forever belong to a certain set, the C-viability kernel. Such a set is calculated in the two-locus haploid model, as well as the corresponding fitnesses at any time which make this maintenance possible. The dependence of the C-viability kernel on the set U of admissible fitnesses and on the recombination rate r is studied. Notably, the C-viability kernel varies rapidly in the neighborhood of equal fitness of AB and ab; it becomes empty when ab has a fitness below a certain function, which is delineated, of the recombination rate. The properties of the two-locus model under constraints, out of equilibrium and with unpredictable selection are thus presented. Received: 20 May 1999     

Linkage disequilibrium in a finite population that is partially selfing

The linkage disequilibrium expected in a finite, partially selfing population is analyzed, assuming the infinite allele model. Formulas for the expected sum of squares of the linkage disequilibria and the squared standard linkage disequilibrium are derived from the equilibrium values of sixteen inbreeding coefficients required to describe the behavior of the system. These formulas are identical to those obtained with random mating if the effective population size N_e = (1-½S)N and the effective recombination value r_e = (1-S)r/(1-½S), where S is the proportion of selfing, are substituted for the population size and the recombination value. Therefore, the effect of partial selfing at equilibrium is to reduce the population size by a factor 1-½S and the recombination value by a factor (1-S)/(1-½S).     

Repeatability and heritability of divergent recombination frequencies in the Iowa Stiff Stalk Synthetic (Zea mays L.)

Variability in recombination frequency was reported in the Iowa Stiff Stalk Synthetic. The objectives of the present research were to verify the differences in recombination frequency among individuals in the Iowa Stiff Stalk Synthetic maize population and to determine if the recombination frequency differences persisted among the S₁ progeny. Testcrosses to measure male recombination frequency on three chromosomes (4, su1-c2; 5, a2-bt1-pr1; 9, sh1-bz1-wx1) were repeated for eight S₀ individuals. Recombination frequencies were repeatably divergent among those individuals which were selected based on high or low recombination frequencies on specific chromosomes. Individuals which had been selected for long and short total map distances across the three chromosome regions produced repeatably divergent recombination frequencies only at the su1-c2 region. The recombination frequencies of the S₁ lines, derived from the S₀ individuals which had the most divergent recombination frequencies on a single chromosome, were significantly different. The broadsense heritability estimates derived from the regression of six S₁ lines on six S₀ individuals ranged from 0.69 to 0.20 for the five chromosome regions. We conclude that genetic differences for recombination frequency exist in this population and that modification by selection should be possible.     

Circularly polarized luminescence of Eu(III) complexes with chiral 1,1′-bi-2-naphtol-derived bisphosphate ligands

The interest for lanthanide circularly polarized luminescence (CPL) has been quickly growing for 10 years. However, very few of these studies have involved correlation between the dissymmetry factor (g_lum) and the chemical modifications in a series of chiral ligands. Four polymeric compounds of Eu(III) were prepared by using a series of binaphtyl derivatives for which the size of the π system as well as the number of stereogenic elements (i.e., the binaphtyl moiety) are modulated. The resulting {[Eu(hfac)₃((S)/(R)-L^x)]}_n (x = 1 and 3) and {[Eu(hfac)₃((S,S,S)/(R,R,R)-L^x)]}_n (x = 2 and 4) have been characterized by powder X-ray diffraction by comparison with the X-ray structures on single crystal of the Dy(III) analogs. In solution, the structure of the complexes is deeply modified and becomes monomeric. The nature of the ligand induces change in the shape of the CPL spectra in CH₂Cl₂ solution. Furthermore, a large |g_lum| = 0.12 of the magnetic-dipole transition for the [Eu(hfac)₃((S,S,S)/(R,R,R)-L²)] complex involving the ligand with three stereogenic elements and an extended ?? system has been measured. This report also shows CPL measurements in solid state for the series of {[Eu(hfac)₃((S)/(R)-L^x)]}_n (x = 1 and 3) and {[Eu(hfac)₃((S,S,S)/(R,R,R)-L^x)]}_n (x = 2 and 4) polymers.     

Limiting relationships between selection and recombination

It is difficult to directly observe processes like natural selection at the genetic level, but relatively easy to estimate genetic frequencies in populations. As a result, genetic frequency data are widely used to make inferences about the underlying evolutionary processes. However, multiple processes can generate the same patterns of frequency data, making such inferences weak. By studying the limits to the underlying processes, one can make inferences from frequency data by asking how strong selection (or some other process of interest) would have to be to generate the observed pattern. Here we present results of a study of the limits to the relationship between selection and recombination in two-locus, two-allele systems in which we found the limiting relationships for over 30 000 sets of parameters, effectively covering the range of two-locus, two-allele problems. Our analysis relates T _min—the minimum time for a population to evolve from the initial to the final conditions—to the strengths of selection and recombination, the amount of linkage disequilibrium, and the Nei distance between the initial and final conditions. T _min can be large with either large disequilibrium and small Nei distance, or the reverse. The behavior of T _min provides information about the limiting relationships between selection and recombination. Our methods allow evolutionary inferences from frequency data when deterministic processes like selection and recombination are operating; in this sense they complement methods based entirely on drift.     

The effects of recombination,mutation and selection on the evolution of the Rp1 resistance genes in grasses

Plant immune genes, or resistance genes, are involved in a co‐evolutionary arms race with a diverse range of pathogens. In agronomically important grasses, such R genes have been extensively studied because of their role in pathogen resistance and in the breeding of resistant cultivars. In this study, we evaluate the importance of recombination, mutation and selection on the evolution of the R gene complex Rp1 of Sorghum, Triticum, Brachypodium, Oryza and Zea. Analyses show that recombination is widespread, and we detected 73 independent instances of sequence exchange, involving on average 1567 of 4692 nucleotides analysed (33.4%). We were able to date 24 interspecific recombination events and found that four occurred postspeciation, which suggests that genetic introgression took place between different grass species. Other interspecific events seemed to have been maintained over long evolutionary time, suggesting the presence of balancing selection. Significant positive selection (i.e. a relative excess of nonsynonymous substitutions (d_N/d_S>1)) was detected in 17–95 codons (0.42–2.02%). Recombination was significantly associated with areas with high levels of polymorphism but not with an elevated d_N/d_S ratio. Finally, phylogenetic analyses show that recombination results in a general overestimation of the divergence time (mean = 14.3%) and an alteration of the gene tree topology if the tree is not calibrated. Given that the statistical power to detect recombination is determined by the level of polymorphism of the amplicon as well as the number of sequences analysed, it is likely that many studies have underestimated the importance of recombination relative to the mutation rate.     

CYCLICAL ENVIRONMENTAL CHANGES AS A FACTOR MAINTAINING GENETIC POLYMORPHISM. 2. DIPLOID SELECTION FOR AN ADDITIVE TRAIT

The subject of this paper is polymorphism maintenance due to stabilizing selection with a moving optimum. It was shown that in case of two-locus additive control of the selected trait, global polymorphism is possible only when the geometric mean fitnesses of double homozygotes averaged over the period are lower than that of the single heterozygotes and of the double heterozygote (with a multiplier [1 – r]^p, which depends on recombination rate r and period length p). But local stability of polymorphism cannot be excluded even if geometric mean fitnesses of all double homozygotes are higher than that of all heterozygotes. We proved, that for logarithmically convex fitness functions, cyclical changes of the optimum cannot help in polymorphism maintenance in case of additive control of the selected trait by two equal loci. However, within the same class of fitness functions, nonequal gene action and/or dominance effect for one or both loci may lead to local polymorphism stability with large enough polymorphism attracting domain. The higher the intensity of selection and closer the linkage between selected loci the larger is this domain. Note that even simple cyclical selection could result in two forms of polymorphic limiting behavior: (a) usually expected forced cycle with a period equal to that of environmental changes; and (b) “supercycles,” nondumping auto-oscillations with a period comprising of hundreds of forced oscillation periods.     

Comments on the Role of H2S in the Chemistry of Earth's Early Atmosphere and in Prebiotic Synthesis

Free energy calculations and experimental measurements have been used to show that H₂S/CO₂ mixtures outgassing from a prebiotic Earth's crust would have produced a reducing gas mixture containing CO, H₂, H₂O, and S _x as principal components. Due to rapid recombination of H₂, CO, and S _x to H₂S and CO₂ on cooling from a high temperature to ambient conditions, reducing components would have been retained only if efficient quenching of the reduced gas mixture had been possible. Consequently, subsea vents or vents with efficient infusion of water would have been ideal sites for retention of reduced species and for prebiotic organic synthesis. It is suggested that C/H/O/S ratios are important factors in controlling the degree of prebiotic organic synthesis and, hence, the emergence of life, since if oxygen is abundant, CO₂ and SO₂ would have been dominant species. Received: 5 March 1997 / Accepted: 15 December 1997     

The mathematical theory of group selection. I. Full solution of a nonlinear Levins E = E(x) model

The first complete overtime solution is obtained for a group selection model of Levins E = E(x) type with recolonization but no other gene flow between islands. Assuming a subdivided population at carrying capacity, the model describes selection at a biallelic locus (A, a) where a is opposed by Mendelian selection but is favored by a lower rate of extinction of demes having high a frequency. By contrast to the linear diffusion equations encountered in classical mathematical genetics, the PDE governing the dynamics is now nonlinear in the metapopulation gene frequency distribution φ(x, t); furthermore, the initial conditions now heavily influence the equilibrium distribution φ_∞(x). A fully explicit formula (20) expressing this dependence is derived. The results indicate that a fixation is never reached, but (A, a) polymorphism in the metapopulation will result if , where s 1 parametrizes the strength of Mendelian selection, E(x) is the Levins extinction operator, h (typically in the open interval (0, 1)) is the dominance of a, and B is a parameter measuring the flatness of the initial distribution f(x) in the x → 1 limit.     

Disequilibrium, Selection, and Recombination: Limits in Two-Locus, Two-Allele Models

All possible combinations of equilibria and fitnesses in two-locus, two-allele, deterministic, discrete-generation selection models are enumerated. This knowledge is used to obtain limits (which can be calculated to arbitrary precision) to the relationships among disequilibrium, selection and recombination for fixed values of allele frequencies. In all cases, the inequality|rD| < s/10 holds, where r is recombination and D is disequilibrium, and all selection coefficients lie between 1 - s and 1 + s times that of the double heterozygote. Linear programming techniques are used to calculate the minimum strength of selection needed to explain several observed nonzero values of D reported in the literature. One conclusion is that the failure to observe nonzero values of D is not surprising.     

Simulation of ‘hitch-hiking’ genealogies

An ancestral influence graph is derived, an analogue of the coalescent and a composite of Griffiths' (1991) two-locus ancestral graph and Krone and Neuhauser's (1997) ancestral selection graph. This generalizes their use of branching-coalescing random graphs so as to incorporate both selection and recombination into gene genealogies. Qualitative understanding of a ‘hitch-hiking’ effect on genealogies is pursued via diagrammatic representation of the genealogical process in a two-locus, two-allele haploid model. Extending the simulation technique of Griffiths and Tavaré (1996), computational estimation of expected times to the most recent common ancestor of samples of n genes under recombination and selection in two-locus, two-allele haploid and diploid models are presented. Such times are conditional on sample configuration. Monte Carlo simulations show that ‘hitch-hiking’ is a subtle effect that alters the conditional expected depth of the genealogy at the linked neutral locus depending on a mutation-selection-recombination balance. Received: 21 July 2000 / Published online: 5 December 2000     

Sexual recombination under the joint effects of mutation,selection, and random sampling drift

The advantage or disadvantage of sexual reproduction or recombination for the accumulation of mutant genes in a population is studied under the joint effects of recurrent mutations, selection, and random sampling drift. To obtain the rate at which mutant genes are incorporated three different methods are used; numerical integration of Kolmogorov backward equations, simulation of stochastic difference equations, and Monte Carlo experiments. The first two methods are used in a two-locus system to obtain the fixation probability of double mutants and other related quantities under five different selection models. The third one is conducted for a multiple-locus system and the rate of accumulation of mutant genes per locus is studied. Comparison of the results between sexual and asexual populations shows that the effect of recombination depends on initial linkage disequilibrium, mutation rate v, selection intensity s, and population size N_e. The mode of selection is also an important factor and the large effect of recombination is observed when mutant genes are individually deleterious but collectively favorable. Under a given model of selection, the great advantage or disadvantage of recombination is achieved when a large extent of genetic polymorphism is produced not by mutation but by recombination. Extreme values of N_es and N_ev make the effect insignificant. The results of Monte Carlo experiments also reveal the presence of interaction between selection and sampling drift even when the loci segregate independently and selection is multiplicative. Although this interaction is usually small, there are cases in which one locus theory cannot be used freely. In those cases, the effect of recombination is prominent and one locus theory gives an overestimate of the rate.     

Correlation between testcross performance of lines at early and late selfing generations

Summary In hybrid breeding programs, testcross evaluation of lines can be done during the early stages of selfing (early testing) or delayed until the lines are near-homozygous. To evaluate the usefulness of early testing, the expected genetic and phenotypic correlations between testcross performance at different selfing generations were examined. The genetic correlation (r _GnGn ) between testcross performance of S _n and S _n , (n>n) individuals or lines is equal to the square root of the ratio of their testcross genetic variances, and it is a function of the inbreeding coefficients (F) at the two selfing generations, i.e., r _GnGn=[(1+F _n )/(1+F _n )]^0.5. The genetic correlation between testcross performance of lines and their directly descended homozygous (n=) lines is 0.71 for S₁; 0.87 for S₂, 0.93 for S₃, 0.97 for S₄, 0.98 for S₅, and 0.99 for S₅ lines. The effectiveness of early testing is limited mainly by nongenetic effects. The square root of testcross heritability at generation n sets the upper limit on the correlation between phenotypic value at generation n and genotypic value at homozygosity. The probabilities of correctly retaining S _n individuals or lines that have superior testcross performance at homozygosity (n=) indicate that early testing should be effective in identifying lines with above- and below-average combining ability. However, the risk of losing lines with superior combining ability is high if strong (best 10%) selection pressure is applied during early testing. If only a small proportion of lines is retained based on testcross performance and/or if the heritability of the trait is low, selfing for two or three generations prior to testcrossing may be desirable to increase the likelihood of retaining lines that perform well at homozygosity. The theoretical results in this study support the testcross evaluation procedures for grain yield used by most maize (Zea mays L.) breeders.A contribution from Limagrain Genetics, a Groupe Limagrain company     

A fast algorithm for computing multilocus recombination

A fast algorithm for computing recombination is developed for model organisms with selection on haploids. Haplotype frequencies are transformed to marginal frequencies; random mating and recombination are computed; marginal frequencies are transformed back to haplotype frequencies. With L diallelic loci, this algorithm is theoretically a factor of a constant times (3/8)^L faster than standard computations with selection on diploids, and up to 16 recombining loci have been computed. This algorithm is then applied to model the opposing evolutionary forces of multilocus epistatic selection and recombination. Selection is assumed to favor haplotypes with specific alleles either all present or all absent. When the number of linked loci exceeds a critical value, a jump bifurcation occurs in the two-dimensional parameter space of the selection coefficient s and the recombination frequency r. The equilibrium solution jumps from high to low mean fitness with increasing r or decreasing s. These numerical results display an unexpected and dramatic nonlinear effect occurring in linkage models with a large number of loci.     

Multisubstrate kinetics of PAH mixture biodegradation: analysis in the double-logarithmic plot

The proposed method of kinetic analysis of aqueous-phase biodegradation of polycyclic aromatic hydrocarbons (PAH) mixture presupposes representation of kinetic curves for each pair of mixture components, S _x and S _y , in double-logarithmic coordinates (ln S _x ; ln S _y ). If PAH mixture conversion corresponds to the multisubstrate model with a common active site, then the graphs in double-logarithmic coordinates are straight lines with the angular coefficients equal to the ratio of respective first-order rate constants k_x^y = \fracV_y K_x K_y V_x k_{x}^{y} = {\frac{{V_{y} K_{x} }}{{K_{y} V_{x} }}} , where K _x and K _y are half-saturation constants, V _x and V _y are the maximum conversion rates for substrates S _x and S _y ; the graph slope does not depend on any concentrations and remains constant during the change of reaction rates as a result of inhibition, induction/inactivation of enzymes or biomass growth. The formulated method has been used to analyze PAH mixture conversion by the culture of Sphingomonas sp. VKM B-2434. It has been shown that this process does not satisfy the multisubstrate model with a single active site. The results suggest that the strain VKM B-2434 contains at least two dioxygenases of different substrate specificity: one enzyme converts phenanthrene and fluoranthene and the other converts acenaphthene and acenaphthylene. The ratios of first-order rate constants have been obtained for these pairs of substrates.     

On the two-locus sampling distribution

Two methods are discussed for evaluating the distribution of the configuration of unlabeled gametic types in a random sample of size n from the two-locus infinitely-many-neutral-alleles diffusion model at stationarity. Both involve finding systems of linear equations satisfied by the desired probabilities. The first approach, which is due to Golding, is to include additional probabilities in the system that allow some members of the sample to be specified at only one locus. The second approach, which is new, considers the joint distribution of the sample configuration and the number of recombination events since the time of the most recent common ancestor. The first approach is used for numerical computation, whereas the second approach is used to derive a two-locus version of Hoppe's urn model. The latter permits efficient simulation of the two-locus sampling distribution, provided the recombination parameter is not too large.Supported in part by NSF grants DMS-8704369 and DMS-8902991     

Analysis of S-RNase alleles of almond (Prunus dulcis): characterization of new sequences, resolution of synonyms and evidence of intragenic recombination

Cross-compatibility relationships in almond are controlled by a gametophytically expressed incompatibility system partly mediated by stylar RNases, of which 29 have been reported. To resolve possible synonyms and to provide data for phylogenetic analysis, 21 almond S-RNase alleles were cloned and sequenced from SP (signal peptide region) or C1 (first conserved region) to C5, except for the S ₂₉ allele, which could be cloned only from SP to C1. Nineteen sequences (S ₄ , S ₆ , S ₁₁ –S ₂₂ , S ₂₅ –S ₂₉ ) were potentially new whereas S ₁₀ and S ₂₄ had previously been published but with different labels. The sequences for S ₁₆ and S ₁₇ were identical to that for S ₁ , published previously; likewise, S ₁₅ was identical to S ₅ . In addition, S ₄ and S ₂₀ were identical, as were S ₁₃ and S ₁₉ . A revised version of the standard table of almond incompatibility genotypes is presented. Several alleles had AT or GA tandem repeats in their introns. Sequences of the 23 distinct newly cloned or already published alleles were aligned. Sliding windows analysis of Ka/Ks identified regions where positive selection may operate; in contrast to the Maloideae, most of the region from the beginning of C3 to the beginning of RC4 appeared not to be under positive selection. Phylogenetic analysis indicated four pairs of alleles had ‘bootstrap’ support > 80%: S ₅ /S ₁₀ , S ₄ /S _8, S ₁₁ /S ₂₄ , and S ₃ /S ₆ . Various motifs up to 19 residues long occurred in at least two alleles, and their distributions were consistent with intragenic recombination, as were separate phylogenetic analyses of the 5′ and 3′ sections. Sequence comparison of phylogenetically related alleles indicated the significance of the region between RC4 and C5 in defining specificity.An erratum to this article can be found at     

Sequence variability and gene structure at the self-incompatibility locus of Solanum tuberosum

Summary Allelic complexity is a key feature of self-incompatibility (S) loci in gametophytic plants. We describe in this report the allelic diversity and gene structure of the S locus in Solanum tuberosum revealed by the isolation and characterization of genomic and cDNA clones encoding S-associated major pistil proteins from three alleles (S ₁, S _r1, S ₂). Genomic clones encoding the S₁ and S₂ proteins provide evidence for a simple gene structure: Two exons are separated by a small intron of 113 (S ₁) and 117 by (S ₂). Protein sequences deduced from cDNA clones encoding S₁ and S_r1 proteins show 95% homology. 15 of the 25 residues that differ between these S ₁and S _r1alleles are clustered in a short hypervariable protein segment (amino acid positions 44–68), which corresponds in the genomic clones to DNA sequences flanking the single intron. In contrast, these alleles are only 66% homologous to the S ₂allele, with the residues that differ between the alleles being scattered throughout the sequence. DNA crosshybridization experiments identify a minimum of three classes of potato S alleles: one class contains the alleles S ₁, S _r1and S ₃, the second class S ₂and an allele of the cultivar Roxy, and the third class contains at present only S ₄. It is proposed that these classes reflect the origin of the S alleles from a few ancestral S sequence types.     

The mechanism of sex determination in Rumex acetosella

Summary Cytogenetic studies were made with particular emphasis on the sex-determining mechanism in Rumex acetosella (6 x = 42) and its hybrids (F ₁, F ₂, BC ₁ and BC ₂) with R. hastatulus (synthetic 4 x = 16 = 4 A +4 X = and 4 x = 18 = 4 A + 2 (X Y ₁ Y ₂) = ). Rumex acetosella was almost strictly dioecious with 5050 male and female. Breeding tests revealed that the males were heterogametic. The longest chromosomes (S), usually two, are the sex chromosomes of this hexaploid species. The S chromosomes are homomorphic in both male and female. The sex chromosome: autosome ratios, and the strong epistatic male effect of the S ^M chromosome in the polyploid dioecious species and in the hybrids, are evidence of an X/Y Melandrium type sex-determining mechanism controlled by a single pair of homomorphic sex chromosomes. Thus, the sex chromosome formula of the males was S ^F S ^M and that of females was S ^F S ^F. The present approach is a new method for resolving the sex-determining mechanism in a dioecious species.