Effect of Selection on the Failure Rate Function

Effect of selection procedures such as truncation and genotypic selection, are studied using selection differentials. These studies are mainly restricted to the study of mean or percentiles of the characteristics of interest. However, in situations where improvement on longetivity is itself of interest, in addition to studying mean or percentiles, one may study the reliability characteristics such as Failure Rate, Mean Residual Life etc., of the lifetimes of individuals. In this paper, we study the effect of genotypic selection when the distribution function of the lifetime is governed by a single locus with 2 alleles. We compute the difference in the failure rate function from one generation to the next generation for different values of n, the number of generations. We also obtain the limiting values of the failure rate function as n and compute the number of generations required for the failure rate to reach a value close enough to the limiting failure rate function. We also consider an example in a two-loci case and study the effect of selection on the failure rate function.     

The Point Evaluation Method for Approximating Function Means,Variances and Covariances

A perennial problem in statistics is the determination of biases, variances and covariances for functions of random variables X₁, X₂, …, X_n which themselves have a known distribution. A common approach is through equations based upon Taylor series approximations but a “point evaluation” method may sometimes be a useful alternative. This involves approximating the multivariate distribution of the X variables by the 2ⁿ points given by X₁=μ₁±₁, X₂ = μ₂ ±₂, …, X_n = = μ_n μ_n, where μ_i is the mean and σ_i the standard deviation of Xi, with appropriate point weights. An advantage over the Taylor series approach is that function derivatives do not have to be explicitely calculated. The point evaluation method is particularly useful in cases where the X variables are uncorrelated. Then the evaluation of the 2ⁿ points can be replaced by the evaluation of 2n points. The point evaluation method is illustrated with powers of a normally distributed variable, and with estimation of gene frequencies from ABO blood group frequencies.     

A survey of unequal crossover systems and their mathematical properties

We present a model of gene duplication by means of unequal crossover (UCO) where the probability of any given pairing between homologous sequences scales as a penalty factor p ^z ≤ 1, with z the number of mismatches due to asymmetric sequence alignment. From this general representation, we derive several limiting case models of UCO, some of which have been treated elsewhere in the literature. One limiting case is random unequal crossover (RUCO), obtained by setting p = 1 (corresponding to equiprobable pairings at each site). Another limiting case scenario (the ‘Krueger-Vogel’ model) proposes an optimal ‘endpoint’ alignment which strongly penalizes both overhang and deviations from endpoint matching positions. For both of these scenarios, we make use of the symmetry properties of the transition operator (together with the more general UCO properties of copy number conservation and equal parent-offspring mean copy number) to derive the stationary distribution of gene copy number generated by UCO. For RUCO, the stationary distribution of genotypes is shown to be a negative binomial, or alternatively, a convolution of geometric distributions on ‘haplotype’ frequencies. A different type of model derived from the general representation only allows recombination without overhang (internal UCO or IntUCO). This process has the special property of converging to a single copy length or a distribution on a pair of copy lengths in the absence of any other evolutionary forces. For UCO systems in general, we also show that selection can readily act on gene copy number in all of the UCO systems we investigate due to the perfect heritability (h ² = 1) imposed by conservation of copy number. Finally, some preliminary work is presented which suggests that the more general models based on misalignment probabilities seem to also converge to stationary distributions, which are most likely functions of parameter value p. An erratum to this article is available at .     

Efficient genetic transformation of red raspberry,Rubus ideaus L.

We have developed an efficient transformation system for red raspberry (Rubus ideaus L.) using Agrobacterium mediated gene transfer. Using this system we have successfully introduced a gene that encodes an enzyme, S-adenosylmethionine hydrolase (SAMase), in raspberry cultivars Meeker (MK), Chilliwack (CH) and Canby (CY). Leaf and petiole expiants were inoculated with disarmed Agrobacterium tumefaciens strain EHA 105 carrying either of two binary vectors, pAG1452 or pAG1552, encoding gene sequences for SAMase under the control of the wound and fruit specific tomato E4 promoter. Primary shoot regenerants on selection medium were chimeral containing both transformed and non-transformed cells. Non-chimeral transgenic clones were developed by iterative culture of petiole, node and leaf explants, on selection medium, from successive generations of shoots derived from the primary regenerants. Percent recovery of transformants was higher with the selection marker gene hygromycin phosphotransferase (hpt), than with neomycin phosphotransferase (nptII). Transformation frequencies of 49.6%, 0.9% and 8.1% were obtained in cultivars Meeker, Chilliwack and Canby respectively from petiole expiants using hygromycin selection. Genomic integration of transgenes was confirmed by Southern hybridization. Transgenic plants from a total of 218 independent transformation events (161 MK, 4 CH, 53 CY) have been successfully established in soil.Abbreviations ACCO amincocyclopropane-1-carboxylic acid oxidase - AS acetosyringone - BA 6-benzylaminopurine - CH cultivar Chilliwack - CY cultivar Canby - cv cultivar - hpt hygromycin phosphotransferase - IBA indolebutyric acid - MK cultivar Meeker - npt II neomycin phosphotransferase - SAMase S-adenosylmethionine hydrolase - TDZ Thidiazuron (N-phenyl-N'-l,2,3-thidiazol-5-ylurea)     

Simulation of X-Linked Selection in Drosophila

The change in gene frequency for two X-linked mutants, y and w, in a number of experiments was compared to that predicted from a genetic simulation program which utilized estimated differences in relative mating ability, fecundity, and viability. The simulation gave excellent predictions of gene frequency change even when experiments were started with different initial gene frequencies in the males and females or when the two loci were segregating simultaneously. The rate of elimination was slower when there were unequal initial gene frequencies than when males and females had equal initial gene frequencies. Simulation demonstrated that this was a general phenomenon when there is strong selection but that the opposite is true for weak selection. In two other experiments, the mating advantage of wild-type males was balanced by a fecundity advantage in mutant females. In all four replicates of both experiments, the mutant was maintained for several generations at the high initial frequency but then decreased quickly and was eliminated. Results obtained restarting one of these experiments with flies from a generation after the decline in gene frequency indicated that a linked gene and not frequency-dependent selection was responsible for the unpredictable gene-frequency change in the mutant. Using a least squares technique, it was found that a recessive fecundity locus 15 map units from the w locus gave the best fit for both experiments.     

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.     

Gene stacking strategies with doubled haploids derived from biparental crosses: theory and simulations assuming a finite number of loci

Recent progress in genotyping and doubled haploid (DH) techniques has created new opportunities for development of improved selection methods in numerous crops. Assuming a finite number of unlinked loci (ℓ) and a given total number (n) of individuals to be genotyped, we compared, by theory and simulations, three methods of marker-assisted selection (MAS) for gene stacking in DH lines derived from biparental crosses: (1) MAS for high values of the marker score (T, corresponding to the total number of target alleles) in the F₂ generation and subsequently among DH lines derived from the selected F₂ individual (Method 1), (2) MAS for augmented F₂ enrichment and subsequently for T among DH lines from the best carrier F₂ individual (Method 2), and (3) MAS for T among DH lines derived from the F₁ generation (Method 3). Our objectives were to (a) determine the optimum allocation of resources to the F₂ ( n₁^* \, n_{1}^{*} ) and DH generations (n - n₁^* ) (n - n_{1}^{*} ) for Methods 1 and 2 by simulations, (b) compare the efficiency of all three methods for gene stacking by simulations, and (c) develop theory to explain the general effect of selection on the segregation variance and interpret our simulation results. By theory, we proved that for smaller values of ℓ, the segregation variance of T among DH lines derived from F₂ individuals, selected for high values of T, can be much smaller than expected in the absence of selection. This explained our simulation results, showing that for Method 1, it is best to genotype more F₂ individuals than DH lines ($ n_{1}^{*} :n > 0.5 $ n_{1}^{*} :n > 0.5 ), whereas under Method 2, the optimal ratio n₁^* :n n_{1}^{*} :n was close to 0.5. However, for ratios deviating moderately from the optimum, the mean [`(X)] \overline{X} of T in the finally selected DH line ( T<sub>\textDH</sub><sup>*</sup> T_{\text{DH}}^{*} ) was hardly reduced. Method 3 had always the lowest mean [`(X)] \overline{X} of T_\textDH^* T_{\text{DH}}^{*} except for small numbers of loci (ℓ = 4) and is favorable only if a small number of loci are to be stacked in one genotype and/or saving one generation is of crucial importance in cultivar development. Method 2 is under most circumstances the superior method, because it generally showed the highest mean [`(X)] \overline{X} and lowest SD of T_\textDH^* T_{\text{DH}}^{*} for the finally selected DH.     

On a Characterization of the Exponential Distribution by Weak Homoscedasticity of Record Values

A sequence {X_n, n≤1} of independent and identically distributed random values with continuous cumulative distribution function F(x) is considered. X_j is a record value of this sequence if X_j ≤ max (X₁, X₂, …, X_j?1). We define L(n)=min.(j!j>L(n?1.), X_j<X_L(n?1)), with L(0) = 1. Let Z_n=X_L(n)? X_L(n?1), n ≤ 1. We will show that the conditional variance of Z_n given X_L(n?1)=x does not depend on × if and only if F(x) is exponential.     

Populational interactions among underdominant chromosome rearrangements help them to persist in small demes

Populational interactions among unlinked chromosomal rearrangements with partial heterozygote sterility and multiplicative fitness were studied to verify whether they help such rearrangements to persist in small populations, in spite of a considerable migration rate with a large population. A deterministic island-continent model was studied by exact recurrence relations connecting gametic frequencies in successive generations. The increase in the number of chromosomal rearrangements with the same sterility rate (s) causes an increase in the critical value of the migration rate (m_c) over which the chromosomal rearrangements are eliminated from the small population, mainly for medium s values (0.2 ≤ s ≤ 0.5) (“reciprocal stabilization” of the chromosomal rearrangements). A comparison was made between the velocities of the fixation of a few rearrangements with high heterozygote sterility (low n/high s) and that of many rearrangements with low heterozygote sterility (high n/low s), to reach a given level of populational stability of the rearranged karyotype measured by m_c. The former process was shown to be more rapid in small demes, the latter in large ones. However, the increase in the number of unlinked chromosomal rearrangements is more efficient in reducing the neutral gene flow than in increasing the populational stability of the rearranged karyotype.     

Transformation of barley (Hordeum vulgare L.) by Agrobacterium tumefaciens infection of in vitro cultured ovules

We report on a novel transformation procedure for barley by Agrobacterium infection of in vitro cultured ovules. Ovules of the cultivar Golden Promise were isolated a few hours after pollination and infected with the Agrobacterium tumefaciens strain AGL0 carrying the binary vector pVec8-GFP. The vector harboured a hygromycin resistance gene and the green fluorescence protein (GFP) gene. GFP-expressing embryos were isolated from the ovules, regenerated to plants and investigated by Southern blot analysis. Transformation frequencies amounted to 3.1% with hygromycin selection and 0.8% without selection. Mendelian inheritance and stable expression of the GFP gene was confirmed in 18 independent lines over two generations. We conclude that the described technique allows for the rapid and direct generation of high quality transgenic plants.Communicated by W. Harwood     

The mean and variance of the number of segregating sites since the last hitchhiking event

 Tight linkage may cause a reduction of nucleotide diversity in a chromosomal region if an advantageous mutation appears in that region which is driven to fixation by directional selection. This process is usually called genetic hitchhiking. If selection is strong, the entire process takes place during a time period of length 2s ln (2N) that is very short relative to 2N generations [s is the selection coefficient of the advantageous mutation and N the effective diploid population size]. On the time scale of 2N generations, which is characteristic for neutral evolution, we may therefore call this process a hitchhiking event. Using coalescent methods, we analyzed a model in which a hitchhiking event occurred in a chromosomal region of zero-recombination in the past at time x. Such a hitchhiking “catastrophe” wipes out completely genetic variation that existed in a population before that time. Standing variation observed at present must therefore be due to mutations that have arisen since time point x. Assuming that all newly arising mutations are neutral, we derived expressions for the expectation, variance and also for the higher moments of the number of nucleotide sites segregating in a sample of n genes as a function of x. The result for the first moment is then used to estimate the time back to the last hitchhiking event based on DNA polymorphism data from Drosophila. Assuming that directional selection is the sole determinant of the level of genetic variation in the gene regions surveyed, we obtained estimates of x that were typically in the order of 0.1N generations. Received 14 May 1996; received in revised form 26 August 1996     

Characterization of the WEIBULL Distribution by Properties of Order Statistics

Let X_1:n, X_2:n, X_3:n…, X_n:n be the order statistics of n independent random variables with the common (absolutely continuous strictly increasing) distribution function F. The main results given in this article are:

• 1 For any fixed r and two distinct numbers s₁ and s₂ (1<r<s₁<s₂≦n) the distributions of V_i and W_i (defined in (1.11) and (1.12) are identical for i = 1,2 iff F(x) is WEIBULL (1.2).
• 2 The statistics D₁ and D₂ (as in (1.8) and (1.9)) are independent iff F(x) is WEIBULL (1.2).
• 3 The statistics U_i (1≦j≦n?1) and X_i:n (i ≦ j) are independent iff F(x) is WEIBULL (1.2).
• 4 Let X, X₁, X₂, …, X_k be random variables such that

These conditions are necessary and sufficient for F(x) to be WEIBULL .     

Efficiency gain of marker-assisted backcrossing by sequentially increasing marker densities over generations

Expenses for marker assays are the major costs in marker-assisted backcrossing programs for the transfer of target genes from a donor into the genetic background of a recipient genotype. Our objectives were to (1) investigate the effect of employing sequentially increasing marker densities over backcross generations on the recurrent parent genome (RPG) recovery and the number of marker data points (MDP) required, and (2) determine optimum designs for attaining RPG thresholds of 93–98% with a minimum number of MDP. We simulated the introgression of one dominant target gene for genome models of sugar beet (Beta vulgaris L.) and maize (Zea mays L.) with varying marker distances of 5–80 cM and population sizes of 30–250 plants across BC₁ to BC₃ generations. Employing less dense maps in early backcross generations resulted in savings of over 50% in the number of required MDP compared with using a constant set of markers and was accompanied only by small reductions in the attained RPG values. The optimum designs were characterized by increasing marker densities and increasing population sizes in advanced generations for both genome models. We conclude that increasing simultaneously the marker density and the population size from early to advanced backcross generations results in gene introgression with a minimum number of required MDP.     

Molecular models for intrastrand DNA G-quadruplexes

Background  

Independent surveys of human gene promoter regions have demonstrated an overrepresentation of G₃X _n1G3X _n2G₃X _n3G₃ motifs which are known to be capable of forming intrastrand quadruple helix structures. In spite of the widely recognized importance of G-quadruplex structures in gene regulation and growing interest around this unusual DNA structure, there are at present only few such structures available in the Nucleic Acid Database. In the present work we generate by molecular modeling feasible G-quadruplex structures which may be useful for interpretation of experimental data.     

Variance of Gene Frequencies from Recurrent Selection in Finite Populations

An equation was derived for estimation of the variance of gene frequencies due to drift in the presence of recurrent selection. The equation assumes knowledge of gene frequencies in successive generations. These can be approximated in at least three ways. Simulation data demonstrated that when satisfactory approximations are used, the variances supplied by the derived formula agree well with observed variances.     

Developing a rapid and highly efficient cowpea regeneration,transformation and genome editing system using embryonic axis explants

Cowpea (Vigna unguiculata (L.) Walp.) is one of the most important legume crops planted worldwide, but despite decades of effort, cowpea transformation is still challenging due to inefficient Agrobacterium-mediated transfer DNA delivery, transgenic selection and in vitro shoot regeneration. Here, we report a highly efficient transformation system using embryonic axis explants isolated from imbibed mature seeds. We found that removal of the shoot apical meristem from the explants stimulated direct multiple shoot organogenesis from the cotyledonary node tissue. The application of a previously reported ternary transformation vector system provided efficient Agrobacterium-mediated gene delivery, while the utilization of spcN as selectable marker enabled more robust transgenic selection, plant recovery and transgenic plant generation without escapes and chimera formation. Transgenic cowpea plantlets developed exclusively from the cotyledonary nodes at frequencies of 4% to 37% across a wide range of cowpea genotypes. CRISPR/Cas-mediated gene editing was successfully demonstrated. The transformation principles established here could also be applied to other legumes to increase transformation efficiencies.     

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.     

Sexual selection,redundancy and survival of the most beautiful

A model is described of a highly redundant complex organism that has overlapping banks of genes such that each vital function is specified by several different genetic systems. This generates a synergistic profile linking probability of survival to the number of deleterious mutations in the genome. Computer models show that there is a dynamic interaction between the mean number of new deleterious mutations per generation (X), the mean number of deleterious mutations in the genome of the population (Y) and percentage zygote survival (Zs). IncreasedX leads to increasedY and a fall in Zs but it takes several generations before a new equilibrium is reached. If sexual attraction is influenced by the number of deleterious mutations in the genome of individuals thenY is reduced and Zs increased for any given value ofX. This fall inY and rise in Zs is more marked in polygamous than monogamous mating systems. The model is specified such that deleterious mutations can occur without any observable or measurable effect on function. Thus sexual selection, in this organism, for low levels of deleterious mutations cannot be based on assessment of performance. Instead it is based on a simple symmetrical surface pattern that is flawlessly reproduced by organisms with no deleterious mutations, but is less than perfect, and therefore less attractive, if genetic systems have been deleted. A complex vital task requires a system with a high level of redundancy that acts so that the loss of one component has no observable effect and therefore cannot be used for sexual selection. The reproduction of a beautiful surface pattern also requires a low error, high redundancy genetic system; however, in this case there is advantage if a single deleterious mutation produces a recognisable change. This leads to the conclusion that sexual selection and sexual attraction should be based on beauty rather than utility, and explains the common observation in nature that it is the most beautiful that survive.     

Relationships between blood groups,isozymes and halothane reaction in pigs from a selection experiment1

German Landrace pigs (n= 1500) were halothane-tested and blood samples were taken for the determination of A-O and H blood types as well as for the determination of PHI and 6-PGD isozymes. The pigs originated from two generations (7th and 8th) of a selection experiment ‘selection for activity of NADPH-generating enzymes in backfat of pigs’. The selection lines are E^-, E⁺ (selection for low and high enzyme activity), U^- (selection for low ultrasonic backfat thickness) and K (control). Preliminary results show an average proportion of halothane-susceptible animals of 49 %. The frequencies of halothane-positive pigs amount to 60 %, 46 %, 70 % and 30 % in lines E^-, E⁺, U- and K, respectively. The investigation shows a non-random combination of the marker genes caused by linkage disequilibrium, especially in line E^-. Recombination frequencies between the loci vary from 0 % to 18 %.