Analysis of the Unbalanced Two-Period Cross-Over Design with Negligible Residual Effects

The analysis of variance proposed by Grizzle (1965) for the unbalanced two-period crossover design when residual effects are omitted is shown to be appropriate only when the numbers of subjects in both treatment sequences, A : B and B : A, are equal. Following a general linear model approach, a valid analysis of variance is developed for the general unbalanced case. Explicit general expressions, both in algebraic and matrix form, are presented and a numerical example is given to illustrate the details of the analysis.     

Genetic variance in fitness and its cross-sex covariance predict adaptation during experimental evolution

The additive genetic variation (V_A) of fitness in a population is of particular importance to quantify its adaptive potential and predict its response to rapid environmental change. Recent statistical advances in quantitative genetics and the use of new molecular tools have fostered great interest in estimating fitness V_A in wild populations. However, the value of V_A for fitness in predicting evolutionary changes over several generations remains mostly unknown. In our study, we addressed this question by combining classical quantitative genetics with experimental evolution in the model organism Tribolium castaneum (red flour beetle) in three new environmental conditions (Dry, Hot, Hot-Dry). We tested for potential constraints that might limit adaptation, including environmental and sex genetic antagonisms captured by negative genetic covariance between environments and female and male fitness, respectively. Observed fitness changes after 20 generations mainly matched our predictions. Given that body size is commonly used as a proxy for fitness, we also tested how this trait and its genetic variance (including nonadditive genetic variance) were impacted by environmental stress. In both traits, genetic variances were sex and condition dependent, but they differed in their variance composition, cross-sex and cross-environment genetic covariances, as well as in the environmental impact on V_A.     

A Bayesian A-optimal and model robust design criterion

Summary . Suppose that the true model underlying a set of data is one of a finite set of candidate models, and that parameter estimation for this model is of primary interest. With this goal, optimal design must depend on a loss function across all possible models. A common method that accounts for model uncertainty is to average the loss over all models; this is the basis of what is known as Läuter's criterion. We generalize Läuter's criterion and show that it can be placed in a Bayesian decision theoretic framework, by extending the definition of Bayesian A‐optimality. We use this generalized A‐optimality to find optimal design points in an environmental safety setting. In estimating the smallest detectable trace limit in a water contamination problem, we obtain optimal designs that are quite different from those suggested by standard A‐optimality.     

Marker-based adjustment of the additive relationship matrix for estimation of genetic parameters��an example using Eucalyptus cladocalyx

The effects of adjusting additive (numerator) relationship matrices (A) for inbreeding estimates taken from molecular markers were investigated using a small, model population of Eucalyptus cladocalyx. A number of individual-tree, mixed-models were compared, incorporating estimates of population- and family-level selfing and ancestral inbreeding applied either as average values to the entire population or as variable estimates for subpopulation and family groups. The consequences of ignoring inbreeding were inflated additive genetic variance estimates and underestimation of residual variance, with resulting inflation of heritability. We found models that correct for differential inbreeding at the subpopulation level give similar results to more complex ones including family-level estimates. Our analysis indicates that the commonly applied coefficient of relationship for first-generation eucalypt progeny of ρ = 1/2.5 appears to be quite suitable for correcting variance component and heritability estimates. However, if inbreeding is not specifically corrected for by adjustment of A, some minor rank changes of individual breeding values can occur, especially where levels of inbreeding vary among families, and some suboptimal selections and loss of genetic gain may ensue.     

Selection,structure and the heritability of behaviour

Characters which are closely linked to fitness often have low heritabilities (V_A/V_P). Low heritabilities could be because of low additive genetic variation (V_A), that had been depleted by directional selection. Alternatively, low heritabilities may be caused by large residual variation (V_R=V_P – V_A) compounded at a disproportionately higher rate than V_A across integrated characters. Both hypotheses assume that each component of quantitative variation has an independent effect on heritability. However, V_A and V_R may also covary, in which case differences in heritability cannot be fully explained by the independent effects of elimination‐selection or compounded residual variation. We compared the central tendency of published behavioural heritabilities (mean=0.31, median=0.23) with morphological and life history data collected by 26 ). Average behavioural heritability was not significantly different from average life history heritability, but both were smaller than average morphological heritability. We cross‐classified behavioural traits to test whether variation in heritability was related to selection (dominance, domestic/wild) or variance compounding (integration level). There was a significant three‐way interaction between indices of selection and variance compounding, related to the absence of either effect at the highest integration level. At lower integration levels, high dominance variance indicated effects of selection. It was also indicated by the low CV_A of domestic species. At the same time CV_R increased disproportionately faster than CV_A across integration levels, demonstrating variance compounding. However, neither CV_R nor CV_A had a predominant effect on heritability. The partial regression coefficients of CV_R and CV_A on heritability were similar and a path analysis indicated that their (positive) correlation was also necessary to explain variation in heritability. These results suggest that relationships between additive genetic and residual components of quantitative genetic variation can constrain their independent direct effects on behavioural heritability.     

A new method for the partition of allelic diversity within and between subpopulations

A method is proposed for the analysis of allelic diversity in the context of subdivided populations. The definition of an allelic distance between subpopulations allows for the partition of total allelic diversity into within- and between-subpopulation components, in a way analogous to the classical partition of gene diversity. A new definition of allelic differentiation, A _ST , between subpopulations results from this partition, and is contrasted with the concept of allelic richness differentiation. The partition of allelic diversity makes it possible to establish the relative contribution of each subpopulation to within and between-subpopulation components of diversity with implications in priorisation for conservation. A comparison between this partition and that corresponding to allelic richness is illustrated with an example. Computer simulations are used to investigate the behaviour of the new statistic A _ST in comparison with F _ST for a finite island model under a range of mutation and migration rates. A _ST has less dependence on migration rate than F _ST for large values of migration rate, but the opposite occurs for low migration rates. In addition, the variance in the estimates of A _ST is higher than that of F _ST for low mutation rates, but the opposite for high mutation rates.     

The effect of trait type and strength of selection on heritability and evolvability in an island bird population

The heritability (h²) of fitness traits is often low. Although this has been attributed to directional selection having eroded genetic variation in direct proportion to the strength of selection, heritability does not necessarily reflect a trait's additive genetic variance and evolutionary potential (“evolvability”). Recent studies suggest that the low h² of fitness traits in wild populations is caused not by a paucity of additive genetic variance (V_A) but by greater environmental or nonadditive genetic variance (V_R). We examined the relationship between h² and variance‐standardized selection intensities (i or β_σ), and between evolvability (I_A:V_A divided by squared phenotypic trait mean) and mean‐standardized selection gradients (β_μ). Using 24 years of data from an island population of Savannah sparrows, we show that, across diverse traits, h² declines with the strength of selection, whereas I_A and I_R (V_R divided by squared trait mean) are independent of the strength of selection. Within trait types (morphological, reproductive, life‐history), h², I_A, and I_R are all independent of the strength of selection. This indicates that certain traits have low heritability because of increased residual variance due to the age at which they are expressed or the multiple factors influencing their expression, rather than their association with fitness.     

The Effect of Design Imbalance on the Power of the F‐test of a Variance Component in the One‐Way Random Model

The ANOVA‐based F‐test used for testing the significance of the random effect variance component is a valid test for an unbalanced one‐way random model. However, it does not have an uniform optimum property. For example, this test is not uniformly most powerful invariant (UMPI). In fact, there is no UMPI test in the unbalanced case (see Khuri , Mathew , and Sinha , 1998). The power of the F‐test depends not only on the design used, but also on the true values of the variance components. As Khuri (1996) noted, we can gain a better insight into the effect of data imbalance on the power of the F‐test using a method for modelling the power in terms of the design parameters and the variance components. In this study, generalized linear modelling (GLM) techniques are used for this purpose. It is shown that GLM, in combination with a method of generating designs with a specified degree of imbalance, is an effective way of studying the behavior of the power of the F‐test in a one‐way random model.     

Exposure to an enriched CO2 atmosphere alters carbon assimilation and allocation in a pine forest ecosystem

We linked a leaf-level CO₂ assimilation model with a model that accounts for light attenuation in the canopy and measurements of sap-flux-based canopy conductance into a new canopy conductance-constrained carbon assimilation (4C-A) model. We estimated canopy CO₂ uptake (A_nC) at the Duke Forest free-air CO₂ enrichment (FACE) study. Rates of A_nC estimated from the 4C-A model agreed well with leaf gas exchange measurements (A_net) in both CO₂ treatments. Under ambient conditions, monthly sums of net CO₂ uptake by the canopy (A_nC) were 13% higher than estimates based on eddy-covariance and chamber measurements. Annual estimates of A_nC were only 3% higher than carbon (C) accumulations and losses estimated from ground-based measurements for the entire stand. The C budget for the Pinus taeda component was well constrained (within 1% of ground-based measurements). Although the closure of the C budget for the broadleaf species was poorer (within 20%), these species are a minor component of the forest. Under elevated CO₂, the C used annually for growth, turnover, and respiration balanced only 80% of the A_nC. Of the extra 700 g C m⁻² a⁻¹ (1999 and 2000 average), 86% is attributable to surface soil CO₂ efflux. This suggests that the production and turnover of fine roots was underestimated or that mycorrhizae and rhizodeposition became an increasingly important component of the C balance. Under elevated CO₂, net ecosystem production increased by 272 g C m⁻² a⁻¹: 44% greater than under ambient CO₂. The majority (87%) of this C was sequestered in a moderately long-term C pool in wood, with the remainder in the forest floor–soil subsystem.     

Complex Segregation Analysis on Multi-Modal Distributions

By using sib-pairs and parent-pairs data, Tai and Gross (1988) proposed a method to test the existence of a 2-allele major locus for a quantitative trait. This method is extended to include three alleles at a major locus in this paper. One may perform this extension in a feasible and practical manner. To show the advantage of this extension, a simulated data set of nuclear families from a 3-allele ordered dominant (i.e., A > B, A > W and B > W in our notation) major locus model for a quantitative trait is analyzed by a 2-allele codominant major locus mixed model and a 3-allele ordered dominant major locus mixed model. The result shows that our method can distinguish between ambiguous trimodal distributions.     

Genotype x environment interaction for male attractiveness in an acoustic moth: evidence for plasticity and canalization

The lek paradox arises when choosy females deplete the genetic variance for male display traits from a population, yet substantial additive genetic variation (V_A) in male traits persists. Thus, the lek paradox can be more generally stated as one of the most fundamental evolutionary questions: What maintains genetic variation in natural populations? One solution to this problem may be found in the condition‐dependent nature of many sexually selected traits. Genotype × environment (G × E) interactions can maintain V_A under conditions of environmental heterogeneity provided certain restrictions are met, although antagonistic pleiotropy has also been proposed as a mechanism. Here, we provide evidence for G × E interactions and against the role of antagonistic pleiotropy in the maintenance of V_A for sexually selected traits. Using inbred lines of the lesser waxmoth Achroia grisella, we measured V_A for song attractiveness, condition and development rate under different competitive environments and found that genotypes differed in their plasticity. We argue that variation persists in natural populations because G × E interactions prevent any one variant from producing the optimal phenotype across all environments.     

Natural selection on the genetical component of variance in body condition in a wild bird population

Although there is substantial evidence that skeletal measures of body size are heritable in wild animal populations, it is frequently assumed that the nonskeletal component of body weight (or ‘condition’) is determined primarily by environmental factors, in particular nutritional state. We tested this assumption by quantifying the genetic and environmental components of variance in fledgling body condition index (=relative body weight) in a natural population of collared flycatchers (Ficedula albicollis), and compared the strength of natural selection on individual breeding values with that on phenotypic values. A mixed model analysis of the components of variance, based on an ‘animal model’ and using 18 years of data on 17 717 nestlings, revealed a significant additive genetic component of variance in body condition, which corresponded to a narrow sense heritability (h²) of 0.30 (SE=0.03). Nongenetic contributions to variation in body condition were large, but there was no evidence of dominance variance nor of contributions from early maternal or common environment effects (pre‐manipulation environment) in condition at fledging. Comparison of pre‐ and post‐selection samples revealed virtually identical h² of body condition index, despite the fact that there was a significant decrease (35%) in the levels of additive genetic variance from fledging to breeding. The similar h² in the two samples occurred because the environmental component of variance was also reduced by selection, suggesting that natural selection was acting on both genotypic and environmental variation. The effects of selection on genetic variance were confirmed by calculation of the selection differentials for both phenotypic values and best linear unbiased predictor (BLUP) estimates of breeding values: there was positive directional selection on condition index both at the phenotypic and the genotypic level. The significant h² of body condition index is consistent with data from human and rodent populations showing significant additive genetic variance in relative body mass and adiposity, but contrasts with the common assumption in ecology that body condition reflects an individual’s nongenetic nutritional state. Furthermore, the substantial reduction in the additive genetic component of variance in body condition index suggests that selection on environmental deviations cannot alone explain the maintenance of additive genetic variation in heritable traits, but that other mechanisms are needed to explain the moderate to high heritabilities of traits under consistent and strong directional selection.     

Spontaneous noise reduction in a strongly cooperative reaction model

Noise characteristics of a simple multiparticle reaction model were investigated. The model can describe strongly cooperative reactions and is defined as follows. Particles A react in clusters of size k and each reaction forms a product molecule P. The back reaction is also allowed, and each cluster can dissociate into k reactants A. To describe a situation where the reaction is part of a pathway, the system is made open by assuming that particles A are injected and that particles P decay. This is a continuation study. The model is studied by using the same technique as employed previously, the pair approach reaction noise estimator (PARNES) method. Several new issues have been addressed. (i) In the previous work closeness to the Poisson distribution was used as a noise measure. In this work a more traditional noise measure, the ratio of the variance and the mean, was used to analyze stochastic features of the problem. (ii) The dependence of the new noise measure on k has been analyzed in detail, with an emphasis of investigating reactions with large k values. (iii) The previous study focused on understanding time-dependent issues, while this study focuses on describing the equilibrium state of the system. (iv) An exact solution (published elsewhere), available for closed system only, was used to re-investigate the validity of the PARNES method for describing reactions with large k. It was found that the PARNES method cannot describe fine details of the noise characteristics of such reactions.     

Effect of migration and environmental heterogeneity on the maintenance of quantitative genetic variation: a simulation study

The paradox of high genetic variation observed in traits under stabilizing selection is a long‐standing problem in evolutionary theory, as mutation rates appear too low to explain observed levels of standing genetic variation under classic models of mutation–selection balance. Spatially or temporally heterogeneous environments can maintain more standing genetic variation within populations than homogeneous environments, but it is unclear whether such conditions can resolve the above discrepancy between theory and observation. Here, we use individual‐based simulations to explore the effect of various types of environmental heterogeneity on the maintenance of genetic variation (V_A) for a quantitative trait under stabilizing selection. We find that V_A is maximized at intermediate migration rates in spatially heterogeneous environments and that the observed patterns are robust to changes in population size. Spatial environmental heterogeneity increased variation by as much as 10‐fold over mutation–selection balance alone, whereas pure temporal environmental heterogeneity increased variance by only 45% at max. Our results show that some combinations of spatial heterogeneity and migration can maintain considerably more variation than mutation–selection balance, potentially reconciling the discrepancy between theoretical predictions and empirical observations. However, given the narrow regions of parameter space required for this effect, this is unlikely to provide a general explanation for the maintenance of variation. Nonetheless, our results suggest that habitat fragmentation may affect the maintenance of V_A and thereby reduce the adaptive capacity of populations.     

Four Factor Balanced Optimal Designs of Resolution V in Terms of a Modified Trace Criterion

Balanced optimal two level fractional factorial designs of resolution V have been reported in series of papers by SRIVASTAVA and CHOPRA (1971a, 1971b, 1974) and CHOPRA and SRIVASTAVA (1973a, 1973b). The optimality criterion considered by them is the trace optimality which implies minimisation of the average variance of the estimated effects which include the general mean μ, main effects, A_i, and first order interactions, A_ij. In the present work some new optimal designs have been obtained on the basis of a modified criterion of trace optimality. The criterion requires minimisation of the average variance of each of the main effects, two factor interactions, etc. separately instead of minimising average variance of all the estimated effects.     

Genetics of obesity in KK mouse and effects of A y allele on quantitative regulation

KK mouse is known as a polygenic model for non-insulin-dependent diabetes mellitus with moderate obesity. To identify the quantitative trait loci (QTLs) responsible for the body weight in KK, linkage analysis with 97 microsatellite markers was carried out into 192 F₂ progeny, comprising 93 mice with a/a genotype at agouti locus and 99 mice with A ^y /a genotype, of a cross between C57BL/6J female and KK-A^y (A^y congenic) male, thereby the influence of A ^y allele on the quantitative regulation of body weight was also examined. In F₂ a/a mice, we identified a QTL on Chromosome (Chr) 4, and two loci with suggestive linkage on Chrs 15 and 18. In F₂ A ^y /a mice, a QTL was identified on Chr 6, and two loci with suggestive linkage were identified on Chrs 4 and 16. That the QTL on Chr 4 was held in common between F₂ a/a and F₂ A ^y /a progenies implies that this locus may be a primary component regulating body weight in KK and KK-A^y. These results suggest that the body weight in KK is controlled by multiple genes, and the different combination of loci is involved in the presence of A ^y allele. The QTL on Chr 6 seemed to determine the body weight by controlling fat deposition, because the linkage was identified on body weight and adiposity, and is suggested to be a component involved in the metabolic pathway in obesity caused by the A ^y allele. Received: 16 December 1997 / Accepted: 16 March 1998     

Approximate matching of regular expressions

Given a sequenceA and regular expressionR, theapproximate regular expression matching problem is to find a sequence matchingR whose optimal alignment withA is the highest scoring of all such sequences. This paper develops an algorithm to solve the problem in timeO(MN), whereM andN are the lengths ofA andR. Thus, the time requirement is asymptotically no worse than for the simpler problem of aligning two fixed sequences. Our method is superior to an earlier algorithm by Wagner and Seiferas in several ways. First, it treats real-valued costs, in addition to integer costs, with no loss of asymptotic efficiency. Second, it requires onlyO(N) space to deliver just the score of the best alignment. Finally, its structure permits implementation techniques that make it extremely fast in practice. We extend the method to accommodate gap penalties, as required for typical applications in molecular biology, and further refine it to search for substrings ofA that strongly align with a sequence inR, as required for typical data base searches. We also show how to deliver an optimal alignment betweenA andR in onlyO(N+logM) space usingO(MN logM) time. Finally, anO(MN(M+N)+N ²logN) time algorithm is presented for alignment scoring schemes where the cost of a gap is an arbitrary increasing function of its length.     

Additive and testcross genetic variances in crosses among recombinant inbreds

 Breeders desire populations with a high mean performance and a large genetic variance. Theory and methods are lacking for predicting additive variance (V _A ) and testcross variance (V _T ) in biparental populations. Breeders have unsuccessfully attempted to predict V _A based on the coefficient of coancestry ( f ) or molecular-marker similarity between parents. In this paper, we derive the expected values of V _A and V _T in biparental populations, examine the variability of V _A among biparental crosses, and discuss how V _A and V _T may be predicted in applied breeding programs. Suppose i is a recombinant inbred derived from the cross between inbreds P ₁ and P ₂, and inbred j is not a direct descendant of i. Let V _A(i,j) be the additive variance in the F₂ of the (i×j) biparental cross. Let V _T(i, j) be the variance among testcrosses of F₂ individuals with a specific unrelated inbred or population. Assuming linkage equilibrium and the absence of epistasis, V _A(i, j) =λV _A(P1, j) +(1−λ) V _A(P2, j) , where λ= parental contribution of P ₁ to i. Similarly, V _T(i, j) = λV _T(P1, j) +(1−λ) V _T(P2, j) . Additive variance in crosses between recombinant inbreds cannot be modelled as a function of  f if, as indicated in the literature, V _A differs among crosses of founder inbreds. If molecular-marker similarity between parents is used as an estimate of f, then a strong linear relationship is likewise not expected between V _A and marker similarity. Differences between the actual and expected λ led to variation in V _A . In applied breeding programs, modelling V _A or V _T in biparental crosses may be feasible with estimates of V _A or V _T in prior crosses and information on λ obtained from molecular-marker data. Received: 23 September 1997 / Accepted: 30 December 1997     

Blue light-regulation of cell division in Adiantum protonemata: kinetic properties of the photosystem

Abstract Blue-light-induced cell division in single-celled Adiantum protonemata was studied by using two or three pulses (pulse duration: 30 s) separated by various periods and by using relatively long irradiation (e.g. 30 min). The results showed: (1) that the response is saturated by a single pulse, (2) that after the application of a saturating pulse, the protonemata gradually become responsive to another pulse, showing time-dependent saturation to the second pulse, and (3) that although reciprocity holds in the pulse-induced response, it becomes invalid as the exposure duration extends in the range of minutes. These results were analysed in view of a reaction model in which a molecular component is considered to exist in two forms A and B. The response to a single pulse is considered to result from a light-dependent conversion of the component (A→B), and the restoration response measured by two pulses, from its dark reversion (B→A). The analyses yielded (1) the value of the constant which relates the fluence rate to the rate constant of the light-induced reaction, and (2) the rate constant of the dark reaction. The model was extended to formulate the responses to long irradiations as a function of the integrated concentration of B over time. The responses predicted by the formula by using the parameter values estimated from the pulse responses were able to explain the responses measured for long irradiation.