A probabilistic and algebraic treatment of regular inbreeding systems

Summary A probabilistic and algebraic treatment of regular inbreeding systems is presented. Regular inbreeding systems can be thought of as graphs which have certain natural homogeneity properties. Random walks X_n and Y_n are introduced on the nodes of the graphs; the event {X_n = Y_n} is a renewal event by the homogeneity property. We show that in such regular inbreeding systems the population becomes genetically uniform if and only if the event {X_n = Y_n} is recurrent, which happens if 1/ A_n diverges, where A_n is the number of ancestors n generations into the past. We give two counterexamples to show the converse is false in general, but we verify the converse in the case of the graphs of certain finitely presented semigroups.An expended version of this paper was submitted as a doctoral thesis to Purdue University. This thesis was directed by Professor Stanley Sawyer.     

An example of a regular inbreeding system with cubic ancestral growth that preserves some genetic variability

A specific regular inbreeding system of quadruple half-second cousin mating is considered. A regular inbreeding system can be thought of as a graph which satisfies certain natural homogeneity properties. Random walks X _n and Y _n are introduced on the nodes of the graph; the event {X _n=Y_n} is a renewal event by the homogeneity properties. In Arzberger (1985) it is shown that 1) graphs associated with left cancellative semigroups are regular, and 2) for regular systems, the population becomes genetically uniform if and only if the event {X _n=Y_n} is recurrent. In Arzberger (1986) the system of quadruple half-second cousin mating is associated with a cancellative semigroup, thus the system is regular. In this paper we show that 1) A_n is asymptotically of the form cn ³, where A _n is the number of ancestors n generations into the past, 2) {X _n=Y_n} is not recurrent (this is shown by associating (X _n, Y _n) with a random walk in Z ³, 3) P[X _3n =Y _3n ] is asymptotically of the form cn ^–3/2. Thus, in this example, genetic heterogeneity is maintained, with a cubic rate of growth for A_n, not by an exponential growth rate, as in all previous examples of regular inbreeding systems in which genetic heterogeneity is maintained.     

Evolution of eusociality in diploid species

Haplodiploid species are naturally biased by their genetic structure toward the evolution of sterile worker castes, as shown by W. D. Hamilton (1964. J. Theoret. Biol., 7, 1–16, 17–52). Diploid species do not have this intrinsic genetic bias toward eusociality. Nonetheless, true sociality has evolved in the diploid ancestors of the modern termites, and varying degrees of quasisociality are not uncommon in diploid species, including mammals. A genetic bias toward investment in relatives rather than offspring can arise in a diploid species as a result of inbreeding. The consequences of several regular incestuous breeding systems are analyzed in detail. It is shown that, under certain conditions, there is a natural bias toward an alternation of inbred and outbred generations. As this alternation proceeds, the genetic bias toward eusociality rapidly approaches an asymptotic value of 4(1 + 2f₀)/3(1 + 3f₀), where f₀ is the average coefficient of relationship for the outbreeding pairs. For f₀ close to zero, the genetic bias toward eusociality is close to 1.33, which is even larger than the genetic bias of 1.25 in haplodiploid species. Under other conditions there may be repeated incestuous matings between successive outbreeding generations. In this case the bias toward eusociality can be as large as 2.     

Variance of prediction error with mixed model equations when relationships are ignored

Summary Formulas are presented to illustrate the calculation of correct variances of prediction error (PEV) and the correlation between true and predicted values (r_TI) when the incorrect variance-covariance matrix for the random effects is used in mixed-model equations (MME). The example with progeny records of highly related and inbred sires showed that PEV were underestimated from the diagonals of the inverse of the coefficient matrix of the MME when sires were assumed unrelated and not inbred and were overestimated when relationships among sires were calculated with Henderson's simple rules for the inverse of the numerator relationship matrix, A^-1, which do not consider inbreeding. When Quaas' rules for A^-1, which do consider inbreeding, are used, the correct PEV are obtained. In the example, calculations of r_TI from the diagonals of the inverse of the coefficient matrix were too large when relationships and inbreeding were ignored and were obviously wrong when the approximation to the numerator relationship matrix, A, was based on the simple rules for calculating A^-1. If the correct A is used in the MME, the calculation of r_TI may be incorrect if inbreeding of the evaluated individual is not considered. If inbreeding is known, adjustment for inbreeding is easy for calculation of r_TI.Published as paper no. 9947, Journal Ser, Nebraska Agric Res Div, University of Nebraska, Lincoln, Neb.     

Effect of recurrent selfing on inbreeding depression and mating system evolution in an autopolyploid plant

Genome duplication resulting in polyploidy can have significant consequences for the evolution of mating systems. Most theory predicts that self‐fertilization will be selectively favored in polyploids; however, many autopolyploids are outcrossing or mixed‐mating. Here, we examine the hypothesis that the evolution of selfing is restricted in autopolyploids because the genetic cost of selfing (i.e., inbreeding depression) increases monotonically with successive generations of inbreeding. Using the herbaceous, autotetraploid plant Chamerion angustifolium, we generated populations with different inbreeding coefficients (F= 0, 0.17 and 0.36) through three consecutive generations of selfing and compared their magnitudes of inbreeding depression in a common environment. Mating system estimates for four natural populations confirmed that tetraploid selfing rates (s_m= 0.25, SE = 0.02) are similar to those of diploids (s_m= 0.12, SE = 0.12; F_1,2= 1.34, P= 0.37) indicating that both cytotypes are predominantly outcrossing. Compared to an outbred control line, mean inbreeding depression for seed production, survival, and height (vegetative and total) in the inbred line differed among generations (inbreeding coefficients). Across all stages, inbreeding depression (relative to control) was positively related to generation (inbreeding coefficient). Although the initial costs of inbreeding in extant and newly synthesized polyploids may be low compared to diploids, the monotonic increase in inbreeding depression with repeated inbreeding may limit the extent to which selfing variants are favored.     

Inbreeding, pedigree size, and the most recent common ancestor of humanity

How many generations ago did the common ancestor of all present-day individuals live, and how does inbreeding affect this estimate? The number of ancestors within family trees determines the timing of the most recent common ancestor of humanity. However, mating is often non-random and inbreeding is ubiquitous in natural populations. Rates of pedigree growth are found for multiple types of inbreeding. This data is then combined with models of global population structure to estimate biparental coalescence times. When pedigrees for regular systems of mating are constructed, the growth rates of inbred populations contain Fibonacci n-step constants. The timing of the most recent common ancestor depends on global population structure, the mean rate of pedigree growth, mean fitness, and current population size. Inbreeding reduces the number of ancestors in a pedigree, pushing back global common ancestry times. These results are consistent with the remarkable findings of previous studies: all humanity shares common ancestry in the recent past.     

Effects of fragmentation phenomena on the genetic structure and gene flow in Centaurea cineraria group (Asteraceae) in the Mediterranean Basin

The complex history of the Mediterranean region illustrates how ancient and recent phenomena are closely associated with species distribution and the creation of phylogeographic divisions within Mediterranean flora. A good model to explore the genetic consequences of fragmentation can be found in Centaurea cineraria and its close relatives. We applied simple sequence repeat molecular markers to a dense population sampling throughout the distribution area of all C. cineraria taxa to study how fragmentation has altered the genetic structure and distribution of C. cineraria. The average gene diversity (H_e) was 0.286, and the average allelic richness (A_r) was 3.65 and ranged from 2.15 (C. gymnocarpa) to 5.25 (C. busambarensis). The F_IS averaged a relatively high 0.223, ranging from ? 0.724 in C. aeolica subsp. aeolica to 0.589 in C. leucadea. Our results indicate that habitat fragmentation over several generations reduced heterozygosity due to random genetic drift in populations of C. cineraria. This heterozygosity erosion becomes more severe when the inbreeding coefficient is positive and the outcrossing rates show a significant increase. The results observed for outcrossing rates and inbreeding coefficient could also indirectly support the possibility of disrupted gene flow or mating pattern changes in fragmented C. cineraria populations.     

Demographic factors and genetic variation influence population persistence under environmental change

Population persistence has been studied in a conservation context to predict the fate of small or declining populations. Persistence models have explored effects on extinction of random demographic and environmental fluctuations, but in the face of directional environmental change they should also integrate factors affecting whether a population can adapt. Here, we examine the population‐size dependence of demographic and genetic factors and their likely contributions to extinction time under scenarios of environmental change. Parameter estimates were derived from experimental populations of the rainforest species, Drosophila birchii, held in the lab for 10 generations at census sizes of 20, 100 and 1000, and later exposed to five generations of heat‐knockdown selection. Under a model of directional change in the thermal environment, rapid extinction of populations of size 20 was caused by a combination of low growth rate (r) and high stochasticity in r. Populations of 100 had significantly higher reproductive output, lower stochasticity in r and more additive genetic variance (V_A) than populations of 20, but they were predicted to persist less well than the largest size class. Even populations of 1000 persisted only a few hundred generations under realistic estimates of environmental change because of low V_A for heat‐knockdown resistance. The experimental results document population‐size dependence of demographic and adaptability factors. The simulations illustrate a threshold influence of demographic factors on population persistence, while genetic variance has a more elastic impact on persistence under environmental change.     

Inbreeding and true seed in tetrasomic potato. II. Selfing and sib-mating in heterogeneous hybrid populations of Solanum tuberosum

 Inbreeding depression may affect the performance of consecutive generations of potatoes propagated by true potato seed (TPS). The effect of inbreeding was established using selfed and sib-mated generations of five TPS families. Correlation coefficients were calculated between the level of inbreeding and different traits. Inbreeding depression was expressed mainly by pollen viability, as measured by its stainability (r=−0.912, P<0.01), and tuber yield (r=−0.837, 0.01<P<0.05). The results also indicated that without unavoidable selection inbreeding depression is expected to be more evident. Furthermore, the TPS families responded quite differently to inbreeding depression. They did not show the same amount of depression for yield as they did for the characters concerning fertility. The high tuber-yielding families displayed a greater inbreeding depression for tuber yield than the lower-yielding families. Received: 10 November 1997 / Accepted: 3 March 1998     

A genetic perspective on mating systems and sex ratios of parasitoid wasps

Parasitoid sex ratios are influenced by mating systems, whether complete inbreeding, partial inbreeding, complete inbreeding avoidance, or production of all-male broods by unmated females. Population genetic theory demonstrates that inbreeding is possible in haplodiploids because the purging of deleterious and lethal mutations through haploid males reduces inbreeding depression. However, this purging does not act quickly for deleterious mutations or female-limited traits (e.g., fecundity, host searching, sex ratio). The relationship between sex ratio, inbreeding, and inbreeding depression has not been explored in depth in parasitoids. The gregarious egg parasitoid, Trichogramma pretiosum Riley, collected from Riverside, CA (USA) produced a female-biased sex ratio of 0.24 (proportion of males). Six generations of sibling mating in the laboratory uncovered considerable inbreeding depression (∼ 20%) in fecundity and sex ratio. A population genetic study (based upon allozymes) showed the population was inbred (F _it = 0.246), which corresponds to 56.6% sib-mating. However, average relatedness among females emerging from the same host egg was only 0.646, which is less than expected (0.75) if ovipositing females mate randomly. This lower relatedness could arise from inbreeding avoidance, multiple mating by females, or superparasitism. A review of the literature in general shows relatively low inbreeding depression in haplodiploid species, but indicates that inbreeding depression can be as high as that found in Drosophila. Finally, mating systems and inbreeding depression are thought to evolve in concert (in plants), but similar dynamic models of the joint evolution of sex ratio, mating systems, and inbreeding depression have not been developed for parasitoid wasps. Received: November 13, 1998 /Accepted: January 8, 1999     

Homoeolog Expression Is Modulated Differently by Different Subgenomes in <Emphasis Type="Italic">Brassica napus</Emphasis> Hybrids and Allotetraploids

Synthetic and natural allotetraploid Brassica napus (2n?=?38, AACC) have been widely used as a model to study the genetic changes associated with allopolyploidization; however, there has been little research on the homoeolog expression patterns and the roles of cis and trans regulation. Herein, homoeolog expression patterns were assessed by using RNA-seq for two interspecific hybrids (A_nC_o with the extracted A subgenome from natural B. napus, and A_rC_o with the A subgenome from extant B. rapa), synthetic and natural allopolyploids (C_oC_oA_rA_r and A_nA_nC_nC_n), and the diploid parents. The ranges of homoeolog expression bias decreased after hybridization, whereas the extents of homoeolog expression bias and non-conserved expression, especially transgressive expression, increased over evolutionary time. Despite sharing the same C subgenome parent, these two hybrids showed different homolog expression patterns in many respects. In A_nC_o, the trans-regulatory factors from C_o subgenome tended to cause downregulation of A_n subgenome homoeologs, but trans-regulatory factors from the A_n subgenome acted as both activators and repressors, and such asymmetric effects of trans-regulatory factors might explain why the homoeolog expression was biased toward the C subgenome after genome merger. No significant asymmetric effects of trans-regulatory factors were found in A_rC_o, which was consistent with the overall balanced expression of homoeologs. These results suggested that A subgenomes with different regulatory systems might act differently in modulating homoeolog expression after merger with the C subgenome, resulting in either balanced or unbalanced homoeolog expression biases.     

Insights into ET<Subscript>A</Subscript> subtype selectivity of benzodiazepine endothelin receptor antagonists by 3D-QSAR approaches

ET_A subtype selective antagonists constitute a novel and potentially important class of agents for the treatment of pulmonary hypertension, heart failure, and other pathological conditions. In this paper, 60 benzodiazepine derivatives displaying potent activities against ET_A and ET_B subtypes of endothelin receptor were selected to establish the 3D-QSAR models using CoMFA and CoMSIA approaches. These models show excellent internal predictability and consistency, external validation using test-set 19 compounds yields a good predictive power for antagonistic potency. Statistical parameters of models were obtained with CoMFA-ET_A (q ² = 0.787, r ² = 0.935, r ² _pred  = 0.901), CoMFA-ET_B (q ² = 0.842, r ² = 0.984, r ² _pred  = 0.941), CoMSIA-ET_A (q ² = 0.762, r ² = 0.971, r ² _pred  = 0.958) and CoMSIA-ET_B (q ² = 0.771, r ² = 0.974, r ² _pred  = 0.953) respectively. Field contour maps (CoMFA and CoMSIA) corresponding to the ET_A and ET_B subtypes reflects the characteristic similarities and differences between these types. The results of this paper provide valuable information to facilitate structural modifications of the title compounds to increase the inhibitory potency and subtype selectivity of endothelin receptor.     

EFFECTIVE POPULATION SIZE IN A CONTINUOUSLY DISTRIBUTED POPULATION

An individual-based simulation model was created to examine genetic variability, time until fixation and spatial genetic structure in a continuously distributed population. Previous mathematical models for continuously distributed populations have the difficulty that the assumption of independent reproduction and independent dispersal of offspring cause clumped spatial distribution and thus violate an assumption of random spatial distribution. In this study, this problem is avoided by considering the dispersal behavior of offspring. The simulation results showed that the inbreeding effective population size estimated by the rate of decrease of heterozygosity during the first 15 generations corresponds to the neighborhood size calculated by the standard deviation of the dispersal distance (σ_T). This inbreeding effective population size does not greatly change with the area of simulation when the densities and σ_T are the same. However, the inbreeding effective population size estimated by heterozygosity using the first 500 generations is larger than the neighborhood size calculated by the dispersal distance and increases with the area of simulation with the same densities. The variance effective population size, estimated by time until fixation of alleles, increases with dispersal distance (σ_T) and with the area of simulation given the same densities. The inbreeding effective population size and variance effective population size were smaller than the actual population size unless σ_T is sufficiently large (2 σ_T > approximate L/2, where L is a side of the simulation square).     

Growth performance of hybrid families by crossing selfed lines of Betula pendula Roth

Hybrid breeding is an effective approach in many agricultural crops. In allogamous tree species severe inbreeding depression and long reproductive cycles generally prohibit its use. However, three generations of selfing in silver birch (Betula pendula Roth) were obtained by forcing trees to flowering under greenhouse conditions. Hybrids were produced by crossing first-, second and third-generation selfed lines. The effects of different levels of parental inbreeding on the growth performance of hybrid families were observed in a 9-year-old field progeny test. Also, provenance crosses were carried out between selfed lines from different parts of Finland and several other European countries. Observations of growth performance of the provenance hybrids were made in the same trial. The results indicated that the mean stem volumes were significantly different between classes of parental in breeding coefficients (F_P) (P<0.0001), and were positively correlated with F_P (r=0.9106, P<0.05). Within-family variation of the hybrid families decreased with an increase of F_P. The performance of the provenance crosses between parents at a relatively close distance did not depart significantly from the standard controls. However, when the cross distance was extended far to the south, hybrids grew faster, indicating either higher heterozygosity or an extended growth period.     

CORRELATED INBREEDING AMONG RELATIVES: OCCURRENCE,MAGNITUDE, AND IMPLICATIONS

Understanding the magnitude and causes of genetic and phenotypic resemblance among relatives is key to understanding evolutionary processes. Contrary to basic expectation, individual coefficients of inbreeding ( f) were recently hypothesized to be intrinsically correlated across parents and offspring in structured populations, potentially creating an additional source of phenotypic resemblance in traits that show inbreeding depression. To test this hypothesis, we used individual‐based simulations to quantify the parent–offspring correlations in f arising under random mating in populations of different size, immigration rate, and mating system. Parent–offspring correlations in f were typically positive (median r≈ 0.2–0.4) in relatively small and isolated populations. Relatively inbred parents therefore produced relatively inbred offspring on average, although the magnitude of this effect varied considerably among replicate populations. Correlations were higher given more generations of random mating, greater variance in reproductive success, polygynous rather than monogamous mating, and for midparent–offspring rather than parent–offspring relationships. Furthermore, f was also positively correlated across half‐siblings, and closer relatives had more similar inbreeding coefficients across entire generations. Such intrinsic resemblance in f among relatives could provide an additional genetic benefit of mate choice and bias quantitative genetic analyses that do not account for correlated inbreeding depression.     

INBREEDING AND VARIANCE EFFECTIVE SIZES FOR NONRANDOM MATING POPULATIONS

Following an inbreeding approach and assuming discrete generations and autosomal inheritance involving genes that do not affect viability or reproductive ability, I have derived expressions for the inbreeding effective size, N_eI, for a finite diploid population with variable census sizes for three cases: monoecious populations with partial selfing; dioecious populations of equal numbers of males and females with partial sib mating; and unequal numbers of males and females with random mating. For the first two cases, recurrence equations for the inbreeding coefficient are also obtained, which allow inbreeding coefficients to be predicted exactly in both early and late generations. Following the variance of change in gene frequency approach, a general expression for variance effective size, N_eV, is obtained for a population with unequal numbers of male and female individuals, arbitrary family size distribution, and nonrandom mating. All the parameters involved are allowed to change over generations. For some special cases, the equation reduces to the simple expressions approximately as derived by previous authors. Comparisons are made between equations derived by the present study and those obtained by previous authors. Some of the published equations for N_eI and N_eV are shown to be incomplete or incorrect. Stochastic simulations are run to check the results where disagreements with others are involved.     

Group living and inbreeding depression in a subsocial spider

Social spiders are unusual among social organisms in being highly inbred-males and females mature within their natal nest and mate with each other to produce successive generations. Several lines of evidence suggest that in spiders inbred social species originated from outbred subsocial ancestors, a transition expected to have been hindered by inbreeding depression. As a window into this transition, we examined the fitness consequences of artificially imposed inbreeding in the naturally outbred subsocial spider Anelosimus cf. jucundus. Subsocial spiders alternate periods of solitary and social living and are thought to resemble the ancestral system from which the inbred social species originated. We found that inbreeding depression in this subsocial spider only becomes evident in spiders raised individually following the end of their social phase and that ecological and demographic factors such as eclosion date, number of siblings in the group and mother's persistence are more powerful determinants of fitness during the social phase. A potential explanation for this pattern is that maternal care and group living provide a buffer against inbreeding depression, a possibility that may help explain the repeated origin of inbred social systems in spiders and shed light on the origin of other systems involving regular inbreeding.     

Hierarchical Classification I: Single-Linkage Method

This is the first part of a survey of hierarchical clustering algorithms using joining methods: the Single-Linkage algorithm. Complete-Linkage and general algorithms defined by d(A_i, B) = = α,d(A_i, A_r)±α_sd(A_i, A_s)±βd(A_r, A_s) will be discussed in two subsequent papers.     

An extension of the probability approach to genetic relationships: One locus

A path-counting method has been presented for calculating the nine phylotype pair probabilities (k-coefficients of this paper) for a diploid autosomal locus. Cockerham (1971) has provided a somewhat more elegant method for arriving at these probabilities. The procedure explained herein is more akin to the familiar calculation of Wright's inbreeding coefficient and is easier to apply to specific complex pedigrees such as arise in human genetics problems. Cockerham's more general formulation is superior for the investigation of regular systems of inbreeding.     

A comparison of pedigree‐ and DNA‐based measures for identifying inbreeding depression in the critically endangered Attwater's Prairie‐chicken

The primary goal of captive breeding programmes for endangered species is to prevent extinction, a component of which includes the preservation of genetic diversity and avoidance of inbreeding. This is typically accomplished by minimizing mean kinship in the population, thereby maintaining equal representation of the genetic founders used to initiate the captive population. If errors in the pedigree do exist, such an approach becomes less effective for minimizing inbreeding depression. In this study, both pedigree‐ and DNA‐based methods were used to assess whether inbreeding depression existed in the captive population of the critically endangered Attwater's Prairie‐chicken (Tympanuchus cupido attwateri), a subspecies of prairie grouse that has experienced a significant decline in abundance and concurrent reduction in neutral genetic diversity. When examining the captive population for signs of inbreeding, variation in pedigree‐based inbreeding coefficients (f_pedigree) was less than that obtained from DNA‐based methods (f_DNA). Mortality of chicks and adults in captivity were also positively correlated with parental relatedness (r_DNA) and f_DNA, respectively, while no correlation was observed with pedigree‐based measures when controlling for additional variables such as age, breeding facility, gender and captive/release status. Further, individual homozygosity by loci (HL) and parental r_DNA values were positively correlated with adult mortality in captivity and the occurrence of a lethal congenital defect in chicks, respectively, suggesting that inbreeding may be a contributing factor increasing the frequency of this condition among Attwater's Prairie‐chickens. This study highlights the importance of using DNA‐based methods to better inform management decisions when pedigrees are incomplete or errors may exist due to uncertainty in pairings.